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Energy-efficient packet transmissions with delay constraints for wireless communications

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Energy-efficient packet transmissions with delay constraints for wireless communications

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Content ENERGY-EFFICIENT PACKET TRANSMISSIONS WITH DELAY CONSTRAINTS FOR WIRELESS COMMUNICATIONS by Wanshi Chen A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) May 2007 Copyright 2007 Wanshi Chen Dedication To my family. ii Acknowledgments I express my most profound gratitude to my advisor Professor Urbashi Mitra for her guidance and support during my Ph.D. program at the University of Southern California (USC). Without her constant trust and encouragement, this dissertation would not have been possible. I would also like to sincerely thank Professor Michael J. Neely, who positioned as a co-advisor and constantly provided stimulating discussions and valuable contributions. I owe thanks to Professor Ramesh Govindan, Professor Giuseppe Caire, and Professor Ashutosh Sabharwal (at Rice University) for their encouragement and for showing so much interest in my research. I am thankful to the Communication Sciences Institute staff, in particular, Diane Demetras and Mayumi Trasher for their help. I also appreciate the support and educational assistance that I have received during my Ph.D. study from Ericsson Inc., and Qualcomm Inc., San Diego. Lastly, I would like to thank my wife, my parents and my elder sister for their endless love and patience. Wanshi Chen Los Angeles, January 2007 iii Table of Contents Dedication ii Acknowledgments iii List Of Figures viii Abstract xii Chapter 1: Introduction 1 1.1 Future Wireless Networks and Increasing System Design Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Transmission Energy and Packet Delay Tradeoff . . . . . . . . . . . . . . 3 1.3 Modeling of Packet Delay Constraints and Energy-Efficient Transmission . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.1 Continuous-Time Static Channels . . . . . . . . . . . . . . . . . . 7 1.3.2 Time Slotted Fading Channels . . . . . . . . . . . . . . . . . . . . 10 1.4 Packet Transmission over a Multihop Link . . . . . . . . . . . . . . . . . . 15 1.5 Proactive Packet Dropping . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.6 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Chapter 2: Energy-Efficient Transmissions with Individual Delay Constraints Over Static Channels 21 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 Optimal Offline Schedule. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3.1 Summary of the Single Transmission Deadline Scheduler . . . . . . 24 2.3.2 Optimal Offline Scheduler for the Individual Delay Constraint Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3.2.1 Optimal Transmission Duration of the First Packet . . . 28 2.3.2.2 Optimal Transmission Durations of Subsequent Packets . 31 2.3.2.3 The Recursive Optimal Scheduling Algorithm . . . . . . 31 2.4 Properties of the Optimal Offline Scheduling . . . . . . . . . . . . . . . . 38 2.4.1 The Symmetry Property . . . . . . . . . . . . . . . . . . . . . . . . 38 2.4.2 A Trend of E{~ τ} . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.5 Packet Delay Performance Analysis . . . . . . . . . . . . . . . . . . . . . . 46 iv 2.5.1 Packet Delay Performance for the Individual Delay Constraint Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.5.2 Packet Delay Performance for the Single Transmission Deadline Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.6 Online Schedulers and Their Properties . . . . . . . . . . . . . . . . . . . 54 2.6.1 Optimal Buffer Flushing Based Online Scheduling . . . . . . . . . 55 2.6.2 The IMET-like Online Scheduler . . . . . . . . . . . . . . . . . . . 58 2.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.7.1 Impact of Individual Delay Constraint T . . . . . . . . . . . . . . . 62 2.7.2 Tradeoff Between Transmission Energy and Packet Delay . . . . . 62 2.7.3 Properties of Optimal Offline Transmission Durations and Packet Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.7.4 Comparison Between the Offline and Online Schedulers . . . . . . 67 2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Chapter 3: Energy-Efficient Transmissions with Individual Delay Constraints Over Fading Channels 72 3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.2 Optimal Offline Scheduling over a Fading Channel . . . . . . . . . . . . . 74 3.2.1 The Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . 76 3.2.2 A Recursive Search Algorithm . . . . . . . . . . . . . . . . . . . . 83 3.2.3 Symmetry Property and Packet Delay Performance . . . . . . . . . 87 3.2.4 Majorization Property . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.2.5 A Special Case: Static Channels . . . . . . . . . . . . . . . . . . . 92 3.3 Online Scheduling over a Fading Channel . . . . . . . . . . . . . . . . . . 93 3.3.1 Channel-Independent Online Schedulers . . . . . . . . . . . . . . . 93 3.3.2 A Channel-Dependent Online Scheduler . . . . . . . . . . . . . . . 94 3.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.4.1 Properties of Optimal Offline Scheduling . . . . . . . . . . . . . . . 98 3.4.2 Comparison Between Offline and Online Schedulers. . . . . . . . . 100 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Chapter 4: Delay-Constrained Energy-Efficient Transmissions over a Multihop Link 106 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.2.1 Traffic Model and Time-Slotted Channel . . . . . . . . . . . . . . . 108 4.2.2 Multihop Access Mode . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.3 Per Hop Optimal Offline Scheduling over Static Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.4 Multihop Optimal Offline Scheduling over Static Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.4.1 Optimal Offline Schedule vs. Delay Constraints . . . . . . . . . . . 120 4.4.2 Maximum Possible Per-Hop and Total Energy-Efficiency . . . . . . 121 4.4.3 OptimalOfflineSchedulingfortheIndividualDelayConstraintModel123 v 4.4.4 Optimal Offline Scheduling for the Single Deadline Model . . . . . 125 4.4.5 Energy Saving Upper Bound . . . . . . . . . . . . . . . . . . . . . 126 4.4.6 Average Packet Delay Performance Comparison for the Individual Delay Constraint Model . . . . . . . . . . . . . . . . . . 127 4.5 Online Scheduling Over Static Channels . . . . . . . . . . . . . . . . . . . 130 4.6 Extension to Fading Channels . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.7.1 Transmission Energy Performance . . . . . . . . . . . . . . . . . . 134 4.7.1.1 TDM Mode . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.7.1.2 FDM Mode . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.7.2 Average Packet Delay Performance . . . . . . . . . . . . . . . . . . 137 4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Chapter 5: Packet Dropping for Transmission Energy Savings 144 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.3 Optimal Packet Dropping for the Single Transmission Deadline Model . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.3.1 Optimal Packet Dropping . . . . . . . . . . . . . . . . . . . . . . . 147 5.3.2 The ‘Water-Filling’ Rule . . . . . . . . . . . . . . . . . . . . . . . . 149 5.3.3 Optimal Transmission Duration Vector De-Majorization . . . . . . 153 5.4 Optimal Packet Dropping for the Individual Delay Constraint Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Table 5.1: Summary of Packet Dropping Schemes for the Individual Delay Constraint Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.5.1 Properties of ¯ τ (K) N,1 and ¯ τ (K) N,N . . . . . . . . . . . . . . . . . . . . . . 167 5.5.2 Energy Savings due to Packet Dropping . . . . . . . . . . . . . . . 170 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Chapter 6: Conclusions and Future Work 174 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 References 178 Appendix 186 A.1 Optimality of the Offline Scheduler in (2.9) . . . . . . . . . . . . . . . . . 186 A.1.1 Alternative Proof 1: Via Majorization Theory . . . . . . . . . . . . 186 A.1.2 Alternative Proof 2: Via Detailed Comparisons . . . . . . . . . . . 188 A.2 Optimal Offline Scheduling for Unequal Packet Sizes and Unequal Individual Delay Constraints . . . . . . . . . . . . . . . . . . 190 A.3 Statistical Trend of the Optimal Offline Transmission Durations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 A.4 Non-decreasing Queuing Delay Property . . . . . . . . . . . . . . . . . . . 194 vi A.5 Average Transmission Duration for the Single Deadline Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 A.6 Asymptotic Packet Delay for the Single Deadline Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 A.7 Uniqueness of Optimal Offline Schedule . . . . . . . . . . . . . . . . . . . 198 A.8 Additional Complementary Slackness Conditions . . . . . . . . . . . . . . 198 A.9 Proof of Lemmas 3.2, 3.3, 3.4, and 3.5 . . . . . . . . . . . . . . . . . . . . 199 A.10Packet Delay Lower and Upper Bounds in Fading Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 A.11Optimal Offline Scheduling Under Static Time-Slotted Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 A.12Delay Bounds for Multihopping . . . . . . . . . . . . . . . . . . . . . . . . 203 A.13Derivation of ¯ τ (K) N,N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 A.14Derivation of ¯ τ (K) N,1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 vii List Of Figures 1.1 An illustration of the strict convex and monotonically decreasing property of the transmission energy function. . . . . . . . . . . . . . . . . . . . . . 5 1.2 The single transmission deadline model. . . . . . . . . . . . . . . . . . . . 8 1.3 The individual delay constraint model. . . . . . . . . . . . . . . . . . . . . 10 1.4 The slotted single transmission deadline model. . . . . . . . . . . . . . . . 12 1.5 The slotted individual delay constraint model. . . . . . . . . . . . . . . . . 13 1.6 The L-hop transmission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1 Illustration of groups, delay-critical packets and subgroups. . . . . . . . . 34 2.2 The possible group/subgroup associations of the first packet . . . . . . . . 37 2.3 Illustration of the forward system and the reversed system. The packet departure curve in the forward system, due to the optimal transmission duration vector ~ τ, is also feasible and optimal in the reversed system. . . 43 2.4 The average packet delay vs. M for the individual delay constraint model, λ=1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.5 The average transmission duration ¯ τ M,i and delay ¯ q M,i with λ = 1 and M =100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.6 The average packet delay of the optimal scheduler for the single transmis- sion deadline model as a function of M. . . . . . . . . . . . . . . . . . . . 54 2.7 An example run, T = 1 second. . . . . . . . . . . . . . . . . . . . . . . . . 63 2.8 An example run, T = 3 second. . . . . . . . . . . . . . . . . . . . . . . . . 63 viii 2.9 An example run, T = 6 second, which becomes the same as the single transmission deadline model. . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.10 The average packet transmission delay vs. T. . . . . . . . . . . . . . . . . 65 2.11 Normalized average packet transmission energy vs. T. . . . . . . . . . . . 65 2.12 The minimum required T vs. channel use such that the average transmis- sionenergyoftheindividualdelaymodelisnomorethan10%ofthesingle deadline model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.13 Average optimal transmission durations under the optimal offline schedul- ing, M =100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.14 Average packet delay under the optimal offline scheduling, M =100. . . . 68 2.15 Averagenormalizedtransmissionenergyfortheonlineschedulers,M =1000. 69 2.16 Averagepacketdelayvspacketindicesfortheofflineandonlineschedulers, M =1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.17 Average packet delay vs T for the offline and online schedulers, M =1000. 70 3.1 An example run of the optimal offline schedule for the individual delay constraint model, M =100, D =10, and B =0.1. . . . . . . . . . . . . . 100 3.2 An example run of the optimal offline schedule for the individual delay constraint model, M =100, D =10, and B =5. . . . . . . . . . . . . . . 101 3.3 Average optimal transmission rates under the optimal offline schedule, M =100, D =5, dynamic packet sizes. . . . . . . . . . . . . . . . . . . . 101 3.4 Average packet delay under the optimal offline schedule, M =100, ¯ B =1. 102 3.5 Average transmission energy for the offline and online schedulers vs. D, M =100, ¯ B =1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.6 Average transmission energy for the offline and online schedulers vs. ¯ B, M =100, D =5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.7 Average packet delay vs D for the offline and online schedulers, M =100, ¯ B =1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.1 TDM Mode for Multihop Transmission (L=3). . . . . . . . . . . . . . . . 112 ix 4.2 FDM Mode for Multihop Transmission (L=3). . . . . . . . . . . . . . . . 113 4.3 Ratios of the total transmission energy of TDM multihop transmissions over the single-hop transmission, D=9. . . . . . . . . . . . . . . . . . . . . 135 4.4 Ratios of the total transmission energy of TDM multihop transmissions over the single-hop transmission, D=45. . . . . . . . . . . . . . . . . . . . 135 4.5 Optimal number of hops, TDM mode, D=9. . . . . . . . . . . . . . . . . . 136 4.6 Optimal number of hops, TDM mode, D=45. . . . . . . . . . . . . . . . . 137 4.7 Ratios of total transmission energy of FDM multihop transmissions over the single-hop transmission, D=9.. . . . . . . . . . . . . . . . . . . . . . . 138 4.8 Ratios of total transmission energy of FDM multihop transmissions over the single-hop transmission, D=45. . . . . . . . . . . . . . . . . . . . . . . 138 4.9 Optimal number of hops, FDM mode, D=9. . . . . . . . . . . . . . . . . . 139 4.10 Average packet delay performance vs. the number of hops, TDM mode, D=9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 4.11 Average packet delay performance vs. the number of hops, TDM mode, D=45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 4.12 Average packet delay performance vs. the number of hops, FDM mode, D=9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.13 Average packet delay performance vs. the number of hops, FDM mode, D=45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.1 Illustration of the ‘water-filling’ rule for Δ~ τ . . . . . . . . . . . . . . . . . 152 5.2 An example run of the optimal transmission durations, M = 20, T= 2, and λ=1 packets/second. . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5.3 ¯ τ (K) N,N vs. K, when N is fixed. . . . . . . . . . . . . . . . . . . . . . . . . . 168 5.4 ¯ τ (K) N,N vs. N, when K is fixed. . . . . . . . . . . . . . . . . . . . . . . . . . 168 5.5 ¯ τ (K) N,1 vs. K, when N is fixed. . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.6 ¯ τ (K) N,1 vs. N, when K is fixed. . . . . . . . . . . . . . . . . . . . . . . . . . 169 x 5.7 Normalizedtransmissionenergy(in%)vs. B forthesingledeadlinemodel, M =1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5.8 Normalized transmission energy vs. T for the individual delay constraint model, M =200, K =2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 5.9 Normalized transmission energy vs. T for the individual delay constraint model, M =1000, K =10, and B =2. . . . . . . . . . . . . . . . . . . . . 172 xi Abstract There exists a fundamental trade-off between transmission energy and packet delay in wireless communications. In a static channel, a closed form solution of the optimal of- fline scheduling (vis-` a-vis total transmission energy), assuming information of all packet arrivals, for a set of packets each subject to an individual delay constraint is derived. It is shown that when packet arrivals are identically and independently distributed, the optimal packet transmission durations (or, equivalently, transmission rates) exhibit a symmetry property, which leads to a simple and exact solution of the average packet de- lay. The delay performance for the optimal offline scheduling of a set of packets subject toasingle transmission deadlineisalsoanalyzedandshowntobepotentiallyunbounded. The problem of optimal offline scheduling is then extended to fading channels. The prop- erties of the optimal offline transmission rates and the corresponding delay performance are also characterized. Heuristic online schedulers, assuming causal information only, are also studied. The properties of the optimal offline scheduling are demonstrated via simulations. Delay-constrainedenergy-efficientpackettransmissionisthenconsideredunderamul- tihop link. For static channels, given an end-to-end delay constraint for each packet, the optimal offline scheduling over a multihop link is obtained. The transmission energy and xii average packet delay performance are analyzed and characterized. Extension to fading channelsisalsoconsidered. Itisfurtherdemonstratedviasimulationsthatenergysavings via multihopping are possible, but heavily depend on factors such as multihop resource orthogonalizationmode,schedulingalgorithms,delayconstraints,SNRoperationregions, and channel variations. Packet transmission over wireless channels is subject to losses. Note that some ap- plications can tolerate a small fraction of packet losses. Therefore, we initiate the study of proactive packet dropping, while satisfying the required maximum packet loss rate, to maximize transmission energy savings. The optimal and suboptimal packet drop- ping schemes are investigated under both the single transmission deadline model and the individual delay constraint model. It is found that proactive packet dropping yields substantial transmission energy savings, as evidenced via simulation results. xiii Chapter 1 Introduction In this chapter, we will provide a brief overview of energy-efficient transmissions with delay constraints for wireless communications. We first discuss the increasing traffic de- mand and the imposedchallenges on system design, followed by the fundamental tradeoff betweentransmissionenergyand packet delay. The two delay-constrained system models considered in this thesis are then presented, after which communication over a multihop link is described. Thereafter, proactive packet dropping schemes for maximizing energy efficiency are discussed. We conclude this chapter with an outline of the thesis. 1.1 Future Wireless Networks and Increasing System Design Challenges Future wireless applications are anticipated to require high-speed data links in a limited frequency bandwidth. As a result, high spectral efficiency physical layer design is neces- sary. For instance, the latest development in the code-division multiple access (CDMA) standard, as part of the third generation (3G) communication networks, provides the ca- 1 pability of supporting up to 3.1 Mbps in the downlink (from the base station (BS) to the mobile station (MS)), and 1.8 Mbps in the uplink (from MS to BS), within a 1.25MHz bandwidth [TA04]. This is achieved via fast channel adaptation (with fast channel state information feedback), hybrid automatic repeat request (H-ARQ), flexible medium ac- cess control (MAC) techniques, and enhanced upper-layer operations [BLB + 06][RAB06]. Similar development can also be found in the 3G wideband CDMA (WCDMA) standard [PELT06]. Research on beyond 3G (B3G) and fourth-generation (4G) wireless communi- cations, targeting up to a 100Mbps peak rate, is already under way [FCM + 05][JWY05]. Multiple antenna techniques can significantly improve bandwidth efficiency, transmit di- versity or a combination thereof [FG98] [ZT02] [GJJV03] [PGNB04], some of which were already adopted in standards such as CDMA [EIA02], IEEE 802.16e (also known as WiMAX) [80206], etc. Future wireless networks shall also be flexible, adaptive, and scalable, possibly pos- sessingtheadvantagesofconventionalcellularnetworks[TA04][PELT06],ad hocnetworks [RT99][Fre04],andsensornetworks[ASaEC02]. Unliketheconventionalcellularnetworks, the topology of future wireless networks is not necessarily fixed. Instead, it may change withtime,eitherslowlyorrapidly,inresponsetoevolvingtrafficdemands,environmental changes, and/orusermovements. Wirelessnetworksmayoperateinacentralizedmanner or a distributed manner and are evolving towards an all-IP (internet protocol) network [Mob00]. Relaying and cooperative transmissions are promising techniques in providing enhanced system coverage and system throughput in a very efficient and flexible manner [CG79][SEA03a][SEA03b] [Lan02][NHH04] [PWS + 04][POD + 04]. Both multiplexing gain andtransmitdiversityenhancementscanbeachievedviacooperationsbetweengeograph- 2 ically distributed transmitters, receivers, or both [SEA03a][JMG04], in a manner similar to the co-located scenarios. Future wireless networks will have to accommodate diverse applications such as wire- less file transfer, voice over Internet protocol (VoIP), video streaming, wireless gaming, etc. These applications impose various quality-of-service (QoS) requirements, such as maximum tolerable delay or jitter, minimum throughput guarantee, etc. Applications such as VoIP have stringent delay constraints. For instance, the mouth-to-ear delay for VoIP applications has to be less than approximately 280 ms such that most users will find the voice latency is acceptable [MHC + 05][YDK + 06]. Wireless gaming demands a even lower latency for a real-time experience. The support of large number of users and the satisfaction of their desirable QoS requirements make current system design rather challenging. Notethatapplicationlatencyconsistsofnotonlyphysicallayertransmission andprocessingdelay, butalsoupper layer delaysuch as queuing delay. Therefore, system design limited to a particular layer, e.g., physical layer itself, may not yield the optimal results. Instead, cross-layer design becomes necessary and is expected to bring significant performance enhancements [BY04][BG02][TG95] [SM05][JZS05][ZTC + 06][KPS + 06]. 1.2 Transmission Energy and Packet Delay Tradeoff Itiswell-knownthatintheadditivewhiteGaussiannoise(AWGN)channels, thechannel capacity C (in bits/transmission) can be written as C = 1 2 log 2 (1+ P N 0 ), 3 where P denotes the transmit power, and N 0 denotes the noise power. Consider a packet of size B (in units of bits/packet), and suppose the transmission time for this packet is given by τ, then the number of channel uses per bit is τ/B. The transmission energy w(τ) consumed by this packet can be shown as 1 w(τ)=N 0 τ(2 2B/τ −1), (1.1) as illustrated in Figure 1.1 with N 0 = 1 and B = 1. It can be seen that the energy function is monotonically decreasing and convex in τ. In fact, these properties of energy functions hold over a wireless link for many scenarios of interest, as shown in [UBPG02]. Thus, in the sequel, the energy function is assumed to observe the following: • w(τ) is non-negative • w(τ) is monotonically decreasing in τ • w(τ) is strictly convex in τ. Obviously, by tolerating more delay, significantly less energy is required when other conditions, such as target packet transmission quality, remain unchanged. As in Fig- ure 1.1, by increasing the number of channel uses per bit (or correspondingly, per bit transmission delay) from 0.25 transmissions/bit to 2 transmissions/bit, the transmission energy per bit can be decreased from 50 to 2, a factor of 25 reduction. The delay and transmission energy efficiency tradeoff becomes particularly important for certain system 1 Equation (1.1) comes from inverting the information theoretic channel capacity formula C = 1/2log(1 + P/N), which holds in a limit of large coding block sizes. For finite packet sizes B, (1.1) is not exact but still represents a good approximation. We use (1.1) to illustrate the representative property of diminishing returns in channel throughput with increasing power. 4 Figure 1.1: An illustration of the strict convex and monotonically decreasing property of the transmission energy function. scenarios and certain applications. In sensor networks, energy efficiency maximization directly converts to network life maximization since the network life is heavily depen- dent on the life of each individual node, typically battery-powered [PUBG01][UBPG02]. In many wireless networks, e.g., CDMA [TA04], ad hoc networks [GT02], etc., reduced packet transmission energy directly translates into less generated interference observed by other packets transmissions. This, in turn, will potentially enhance the system capa- bility of supporting packet transmission at higher rates. In the latest CDMA standard development [TA04], H-ARQ was employed to maximize power efficiency at the expense of increased packet delay. In the uplink, H-ARQ helps bring down the required MS transmit power or equivalently, the required received signal-to-noise ratio at the BS, in order to satisfy the same frame quality requirement (e.g., 1% frame error rate). Subse- 5 quently, the interference caused by the MS is reduced and system resource utilization is improved. Combining with other enabling technologies, H-ARQ almost doubles uplink system throughput [TA04]. That is, there exists a fundamental tradeoff between tolerable delay and transmis- sion energy. Such a tradeoff has been extensively studied over a single link [UBPG02] [BG02] [FMT03] [GKS03] [KS04] [ZM05], in ad hoc networks [GT02] [NM05] [TG04] [LS04] [SMS06] and in cellular networks [WTCL99] [FTV00] [Tsy02]. In [BG02], it was shown that for a single queue, there exists a square-root relationship between the opti- mal tradeoff of transmission power and packet delay in the asymptotic regime. That is, a scheduling algorithm with a transmission power 1/D close to the minimum transmit power necessarily incurs an average delay no less than √ D. This relationship was later extended to a multi-user network in [Nee06b]. In addition, under special transmit power functions and scenarios, the square-root relationship can be improved to an exponential relationship, indicating a significant improvement [Nee06b]. 1.3 Modeling of Packet Delay Constraints and Energy-Efficient Transmission We will first consider the continuous-time static channels, followed by the time-slotted fading channels. 6 1.3.1 Continuous-Time Static Channels SupposethereareM packetstobetransmitted,withpacketarrivaltimest i ,i=1,··· ,M, throughanAWGNchannel. Withoutlossofgenerality, thearrivaltimeofthefirstpacket is assumed to be 0, i.e., t 1 =0. The packet arrivals are assumed to be random, following a certain distribution function. The packet size is assumed to have B i bits, i=1,··· ,M, and in the case of equal packet size, we have B =B 1 =···=B M . In certain scenarios, it is often required that these M packets be delivered by a dead- line. That is, there exists a single transmission deadline [UBPG02], at or before which all packets have to be successfully transmitted, as illustrated in Figure 1.2. The packet inter-arrival times, denoted by d i = t i+1 −t i ,i = 1,··· ,M 2 , is also a random variable. In the case of Poisson arrivals, the inter-arrival time follows an exponential distribution. Each packet will be deliveredover the channel with the transmission duration denoted by τ i ,i∈ [1,··· ,M]. The transmission durations of the M packets are denoted by a vector ~ τ = [τ 1 ,τ 2 ,··· ,τ M ]. The goal is to find the optimal transmission vector ~ τ such that the total transmission energy of the M packets, or equivalently, the average transmis- sion energy, is minimized subject to the satisfaction of the single transmission deadline. The scheduling algorithm for ~ τ is assumed to have knowledge of the inter-arrival time d i (hence packet arrival times), and thus is termed as an offline scheduling algorithm. The optimal offline scheduling algorithm for the single transmission deadline model wasinvestigatedin[UBPG02]. Themainideaoftheoptimalscheduleristotrytoequalize transmission durations as much as possible under the feasible constraints (details later). 2 Although there is no so-called inter-arrival time for the last packet, for presentation convenience, we still call dM the inter-arrival time of packet M, defined as from the time packet M arrives to the final deadline. 7 Figure 1.2: The single transmission deadline model. Infact, aswillbeshowninthisthesis, theoptimalofflineschedulingyieldsatransmission duration vector ~ τ which is always majorized by (or more mixed than) any other feasi- ble vectors. The offline scheduling algorithm, although not feasible for implementation in practice, provides an upper bound for transmission energy efficiency and important insights for the design of online scheduling algorithms, which assume information up to the scheduling time such as queue backlog and a maximum packet arrival rate. An online packet scheduling algorithm was also developed in [UBPG02]. Online scheduling for the single deadline model was also treated in [ZM05] in which a stochastic optimal control algorithm was developed. In [NSR06], energy-efficient scheduling was studied by considering the need of idling periods for batteries to recover energy. Whiletheoptimalalgorithmin[UBPG02]providestheminimumtransmissionenergy solution, it assumed a single transmission deadline for all the packets without consider- ation of individual packet delay. In fact, the scheduling algorithm based on the optimal algorithm in [UBPG02] may result in large per packet delays, especially when the total number of packets to be transmitted is very large, as will be shown Chapter 2. Such po- tentially significant individual packet delay is not desirable especially for delay-sensitive applications and for practical implementations. 8 In this thesis, we consider energy-efficient transmissions with individual packet delay constraints, as illustrated in Figure 1.3. Compared with Figure 1.2, each packet has its own deadline T i ,i ∈ [1,··· ,M], at or before which the packet has to be successfully delivered. Note that the individual delay constraint T i is not necessarily the same for all packets. Some packets may require a less strict delay constraint than others. However, often the individual delay constraints for all the packets are the same, in which case the delay constraint is denoted by T = T 1 = ··· = T M . We will mainly focus on equal individual delay constraints in this thesis. The end time of all packet transmissions is denoted by t E . In this particular model, t E = t M +T M , i.e., the deadline of the last packet. Note that this new system model can be viewed as a generalized version of the single transmission deadline model in Figure 1.2. The deadline t E for all packets can be mapped to individual packet delay constraints given by T i =t E −t i , i=1,··· ,M. That is, the system model in Figure 1.2 can be viewed as a particular case of the system model in Figure 1.3. The system model in Figure 1.3 guarantees a bounded packet delay and provides a flexible tradeoff between transmission energy and packet delay. One of the main goals of this thesis is to investigate the optimal offline scheduling algorithm under the individual delay constraint model. A similar model was independently studied in [RSA04], [KS04], and [ZM05]. Reference [RSA04] considered online scheduling for dynamic channels. It was proven in [KS04] that all online scheduling can be expressed as a time-varying low- pass linear filter. The lower bound and upper bound on the power-minimizing online scheduling were also presented [KS04]. In [ZM05], the properties of the optimal offline schedulingalgorithmwithgeneralpacketarrivalsandQoSconstraintswerecharacterized. 9 Figure 1.3: The individual delay constraint model. A recursive algorithm to obtain the optimal transmission policy was also provided. The transmission policy was proved feasible and optimal by showing that it satisfies the opti- malitycriteria, andtheoptimaltransmissionpolicyisunique. Inthisthesis, weexplicitly derive the optimal offline scheduling algorithm for the individual delay constraint model under dynamic arrivals and static channels. The optimal transmission policy is obtained following a different recursive approach, and its optimality is proved using an alternative technique. Moreover,wewillanalyzethepropertiesoftheoptimaltransmissiondurations andtheassociatedpacketdelayperformance, andcomparedthemwiththoseofthesingle transmission deadline model. This will be detailed in Chapter 2. 1.3.2 Time Slotted Fading Channels Wireless channels are subject to shadowing, fading, and user mobility, and thus may be time-varying. Aslow-varyingflatfadingchannelisoftenmodeledasadiscrete-timeblock- fading additive white Gaussian noise (BF-AWGN) channel [OSW94]. In the BF-AWGN model, the channel gain is fixed during a whole block and varies independently from 10 block to block, where the block duration represents the channel’s coherence time. This is illustrated in Figure 1.4 and Figure 1.5, for the single transmission deadline model and the individual delay constraint model, respectively. To make a fair comparison, we have assumed that the two models have the same scheduling duration of a total M +D−1 slots, and the same packet arrivals. The channel gains are represented by g i ,i∈ [1,··· ,M +D−1]. In case of static channels, the channel gains are fixed during the entire scheduling duration, i.e., g i =c,∀i, where c is a constant. Packets arrive only at slot boundaries, and can be served immediately (i.e., the min- imum possible queuing delay is 0). Each packet is subject to a delay constraint. For the individual delay constraint model, the delay constraint is assumed to be an integer D > 1 3 (in units of slots) and the same for all packets, although it can be extended to distinct individual delay constraints per packet. The total number of slots is fixed to be M +D−1, and the slot duration is denoted as τ s . For the single deadline model, all packets observe the same transmission deadline of (M +D− 1)τ s . Effectively, the individual delay constraint in the single deadline model for packets arrived during slot m can be expressed as (M +D−1)τ s −t arrival,m , where t arrival,m =(m−1)τ s is the arrival time. Without loss of generality, a single packet is assumed to arrive at each slot with random packet sizes B i >0,i∈[1,··· ,M], and B i =0 for i∈[M +1,M +D−1]. This one-packet-per-slot assumption facilitates analysis, but is not restrictive, as the solution when some slots i, i ∈ [1,··· ,M], do not receive any packets can be obtained as a limiting case when the B i for those slots are made arbitrarily small. Likewise, the case 3 The case when D = 1 is trivial and hence not considered. 11 Figure 1.4: The slotted single transmission deadline model. when multiple packets arrive during the same slot can be treated by viewing them as a single bulk packet with a size given by the sum of the individual sizes. A fluid packet departure model is assumed. That is, a transmitted packet is not necessarily an integer number of arrived packets, but may be assembled using fragmented arrival packets up to an arbitrary precision. Different from the continuous-time case, the goal of the optimal offline schedule in the time-slot model, which assumes perfect knowledge of packet sizes and channel states for the entire duration [0,··· ,M +D−1] before scheduling, is to choose the number of transmitted bits x i , or equivalently, the optimal transmission rate r i , for each slot such that the total transmission energy of these M packets is minimized while the underlying delay constraints are satisfied. Again, the energy-rate function f(r,g) is assumed to be strictly convex and monotonically increasing in transmission rate r for each channel state g. For instance, from Shannon capacity, the energy-rate function is given by f(r,g) = N 0 (2 2r −1)/g. Maximizing channel capacity with various average power constraints has been well studied. With an infinite coding delay (i.e., delay-unconstrained), assuming perfect chan- nel state information (CSI) at both the transmitter and the receiver, the optimal power control policy for the single-user long-term ergodic capacity (also called throughput ca- 12 Figure 1.5: The slotted individual delay constraint model. pacity in [TH98]) follows the ‘water-filling’ rule [GV97]. The resulting channel capacity can be achieved by a single Gaussian codebook with variable powers [CS99], or multiple codebooks with different rates and average powers [GV97]. In another extreme case of one block coding delay, channel inversion is the optimal power control for the single-user delay-limited capacity[CTB99]. Delay-limitedcapacityformultiaccesschannelswasstud- ied in [HT98b]. For a finite coding delay with causal channel feedback, it was shown that in the limit of high signal-to-noise ratio (SNR), the optimal power control converges to a constant power transmission [LG02], while in the low SNR region, threshold-based power control appears to be optimal [NC02]. Note that no traffic variations were assumed in the above optimal power control policies. Energy efficient scheduling utilizing both queue state and fading channel state has been a popular topic recently. Transmission policies that minimize the average transmit power under an average delay constraint and a peak power constraint was studied in [CC99] using a dynamic programming formulation approach. Packets arrive randomly and are transmitted over a Gilbert-Elliott two-state Markov chain fading channel. The optimal average power vs. average delay tradeoff for a single queue was characterized in [BG02] and later extended to a multi-user network in [Nee06b]. In [UBG04], the optimalenergy-efficientofflineschedulingunderasingledeadlinewasextendedtoatime- 13 varying channel with dynamic packet arrivals. A heuristic online scheduling algorithm named look-ahead water-filling, which exploits both backlog information and channelfad- ingstate, wasstudiedandshowntobemoreenergyefficientthanthewater-fillingscheme purely based on channel states. The energy minimization problem was also considered in [FMT06] for a fixed amount of data subject to a single deadline. The optimal on- line scheduling assuming causal channel feedback was also considered. In particular, closed-form optimal online scheduling was derived for the special case of a piece-wise lin- ear energy-throughput relationship. Power optimal schedulers for transmission of bursty traffic under average delay constraints was considered in [RSA04]. An individual packet delay constraint model was also briefly discussed. Dynamic programming can be adopted to solve the optimization problem but the complexity grows exponentially with the in- dividual delay constraint and quickly becomes intractable. Dynamic programming was used in [BS06] in deriving optimal transmission rates to minimize the average transmis- sion delay under various combinations of fixed/dynamic arrival rates and fixed/dynamic transmission rates. In Chapter 3, we will consider energy-efficient packet transmission with individual packet delay constraints over a fading channel. the problem of optimal offline scheduling is formulated as a convex optimization problem with linear constraints. The optimality conditionsareanalyzed, bywhicharecursivealgorithmtosearchfortheoptimalsolution isobtained. Thepropertiesoftheoptimalofflinetransmissionratesandthecorresponding delay performance are also characterized. Heuristic online schedulers, assuming causal information only, are also studied. The properties of the optimal offline scheduling and 14 the impact of packet sizes, individual delay constraints, and channel variations will be demonstrated via simulations. 1.4 Packet Transmission over a Multihop Link Relayassistedwirelesstransmission,proposedasearlyasintheseventies[vdM77][CG79], has recently emerged as a promising technique for cellular, ad-hoc, wireless local area network, and hybrid networks. Relaying helps improve overage, capacity, robustness against channel variations, flexibility and adaptability to dynamic deployment scenarios and traffic needs, and power efficiency [GK00][LW02][LW03][SEA03a][SEA03b][LTW04]. The potential strategies employed by the relay nodes in assisting the source nodes in- clude amplify-and-forward (AF) [LTW04], decode and forward (DF) [LTW04][MY04], and compress-and-forward [KGG05]. In addition to fixed relaying schemes, relaying can also be adaptive, in response to channel measurements and/or acknowledgment feed- backs from the destination node [LTW04]. It often becomes necessary for the source and the relaying nodes to transmit with orthogonal resources, as it is difficult to im- plement simultaneous transmission and reception using the same resource in a given node. The orthogonal resources can be realized via time division multiplexing (TDM), frequency division multiplexing (FDM), code division multiplexing (CDM), or a combi- nation thereof. Therefore, due to the excessive resource requirements for relaying, relay assisted transmission is not guaranteed to be beneficial. Instead, the optimal number of hops depends on the relay strategies, the path loss exponent, the signal-to-noise (SNR) 15 operating region, locations of nodes, etc., and typically up to 4 hops is sufficient for optimality [LWZ + 03][SLH + 04][FY05]. Relaying helps shorten transmission distances between nodes. Combining with effi- cient transmission protocols (e.g., spatial diversity realized by geographically distributed nodes), relaying helps reduce transmission power/energy consumptions in the system for fixed-rate transmissions [LW02]. Optimal power allocation for different relaying schemes has been considered in [MY04][BFY04][HA04a][HA04b][HMZ05]. The bounds for min- imum required energy-per-bit for reliable communication over additive white Gaussian noise (AWGN) relay channels were discussed in [GZ03]. Using the tools developed in [Ver02], energy efficiency of several relay strategies in the low-power regime was investi- gated in [YCG05]. Note that all the above relaying results assume immediate availability of data for transmission, and the impact of traffic variations has not been considered. In Chapter 4, we will consider energy-efficient packet transmission subject to the above discussed two delay constraint models over a multihop link, as an extension of the work over the single-hop link. A time-slot model is assumed. A set of packets are transmitted through a multihop channel, instead of the conventional single-hop channel, as shown in Figure 1.6. It consists of a source node S, a destination node D, and in case of multihop, L−1 relaying nodes (R l ), or equivalently, L hops. Two types of orthogonal division multiplexing schemes, namely, TDM and FDM, will be considered. Without loss of generality, the distance between the source and the destination is assumed to be one unit. In case of equi-distant linear network, the distance between any two neighboring nodes in the multi-hop case is 1/L unit. When the transmit powers from the source and the relaying nodes are assumed to be the same, due to the reduced distance between two 16 Figure 1.6: The L-hop transmission. neighboring nodes in the multihop case, the received SNR at each hop is, on average, increased by a factor of L α compared with the single-hop transmission, where α is the path loss exponent, and without loss of generality, is assumed to be the same for all hops. We will primarily focus on static channels, in which the channel gains are fixed at each hop. However, the channel gains may be different for different hops, depending on the distance between the transmitter and the receiver, the path loss exponent, etc. We denote the fixed channel gain at hop l by g (l) , l∈ [0,··· ,L], where l = 0 represents the direct transmission, and l =[1,··· ,L] is the l-th hop in a L-hop transmission. The opti- mal offline scheduling is re-derived by incorporating the potential existence of restricted transmission durations. Given an end-to-end delay constraint for each packet, the opti- mal delay budget allocation over hops for the optimal offline scheduling is obtained. The total transmission energy and average packet delay are analyzed and characterized. We will also briefly extend our results to the fading channel case. The channel gains are represented by g (l) i , where l∈ [0,··· ,L], and i∈ [1,··· ,M +D−1]. It is demonstrated via simulations that energy savings via multihopping are possible, but depend heavily on factors such as multihop resource orthogonalization mode, scheduling algorithms, delay constraints, SNR operating regimes, and channel variations. 17 1.5 Proactive Packet Dropping Packet transmissions over wireless networks are subject to packet losses. These losses in general consist of two parts: the losses due to physical layer transmission errors over- the-air and the losses due to upper layer operations such as buffer overflows, individual delay constraints violations, etc. Packets transmissions over wireless channels suffer from channel impairments including path loss, shadowing, fading, etc., and as a result, it is often very difficult and more importantly, often inefficient, to maintain a zero or very low packet error rate (e.g., < 10 −4 ) after one round of packet transmission. Instead, the packet quality after one transmission is often targeted at a reasonably low packet error rate, say, 1% [TA04]. Packet re-transmissions, either at the physical layer (e.g., H-ARQ, quick repeat, etc.) or upper layers (e.g., controlled by the Radio Link Protocol or RLP in [TA04]), may be implemented to reduce the residual error to a desirable level. This of course comes at the expense of additional packet delays. Ontheotherhand,upperlayerconstraintsmayresultinadditionalpacketlosses. The allocated buffer for a given application is often limited in size, and buffer overflow may occur especially for bursty packet arrivals and/or time-varying channel conditions. It is possible to provision the buffer size such that the buffer overflow probability is arbitrarily low and the resulting packet losses is negligible. In addition, as mentioned earlier, some applications, in particular voice and video streaming, have stringent delay constraints. When the total packet delay for some packets, including the queuing delay and the trans- missiondelay,isexpectedtoexceedthespecifieddelayconstraint,thesepacketsnolonger have to be transmitted and should be dropped instead. The corresponding scheduling 18 dropping rate depends on the packet scheduling algorithm, the delay constraints, the packet arrival model, and channel conditions, etc. and can be controlled via careful scheduler design and/or admission/congestion control. We underscore that so far these upper layer packet losses are unintentional. We propose to consider intentional packet dropping to improve energy efficiency. Note that delay-sensitive applications such as voice and video streaming typically can tolerate a small fraction of packet losses. The QoS needs of these applications can still be satisfied when the total packet loss rate, including transmission errors and schedul- ing droppings, is maintained within a pre-defined level. This motivates the following questions: should we proactively drop packets? How much transmission energy savings can be achieved via proactive packet dropping? In Chapter 5, we initiate the study of these questions. A related work appears in [Nee06a] which discussed asymptotic trans- mission energy and packet delay tradeoff under intelligent packet dropping. Proactive packet dropping can also be seen as connected with -outage capacity for fading channels [HT98a][CTB99][LG01][LJG05]. Theeffectsofsourcefidelityoncommunicationnetworks were investigated in [Tse95]. 1.6 Organization of the Thesis Theremainderofthisthesisisorganizedasfollows: InChapter2,energy-efficientschedul- ingalgorithmsfortheindividualdelayconstraintmodelarestudiedundercontinuous-time static channels. The optimal offline transmission durations and the associated average packet delay performance are analyzed, and compared with those of the single transmis- 19 sion deadline model. The same scheduling issue is further considered under time-slotted fading channels in Chapter 3. Delay-constrained energy-efficient scheduling over a multi- hoplinkisinvestigatedinChapter4. InChapter5, proactivepacketdroppingalgorithms forboththesingletransmissiondeadlinemodelandtheindividualdelayconstraintmodel are studied. Finally, future work is discussed in Chapter 6. Detailed derivations of some results are provided in the Appendix. 20 Chapter 2 Energy-Efficient Transmissions with Individual Delay Constraints Over Static Channels 2.1 Introduction Inthischapter, wewillconsiderenergy-efficienttransmissionwithindividualpacketdelay constraints in a continuous-time static channel. Given inter-arrival times of a total of M packets, we derive explicit expressions of the optimal transmission durations for any packets m ∈ [1,··· ,M] under the optimal offline scheduling. The properties of the optimal offline transmission durations are then analyzed. It is shown that when packet inter-arrival times are identically and independently (i.i.d.) distributed, the optimal transmissiondurationsofpacketmandpacketM−m+1,m∈[1,··· ,M],M ≥1arealso identically distributed. In other words, the optimal transmission duration vector exhibits a symmetric property. This symmetric property can be interpreted via a time-reversed process. In addition, the optimal transmission durations are on average monotonically non-increasing first until around M/2, and then monotonically non-decreasing onwards. Thepacketqueuingdelay,ontheotherhand,isonaveragemonotonicallynon-decreasing, 21 and upper-bounded by the individual delay constraint. The symmetry property of the optimal transmission durations makes it possible to obtain a simple and exact solution of the average packet delay (including queuing and transmission delays) for any i.i.d. inter-arrival times under the optimal scheduling. In fact, when M is large, the expression for the average packet delay (including queuing and transmission delays) converges to (T +E[min{d,T}])/2, where T is the delay constraint and d is a random variable with the same distribution as the packet inter-arrival time. We will also analyze packet delay performance for the single transmission deadline model [UBPG02]. Under a Poisson packet arrival model, we show that the average de- lay associated with the scheme in [UBPG02] grows monotonically and at a rate close to √ M, where M is the total number of data packets. On the other hand, the transmission scheme discussed in this thesis provides a bounded delay and a flexible tradeoff between transmission energy and packet delay. Two heuristic online scheduling algorithms, which assume no future arrival information, are also studied. While both online schedulers are inherently inferior, one online scheduler is shown to achieve a comparable energy perfor- mance to the optimal offline scheduler in a wide range of scenarios. Numerical results are provided to illustrate the tradeoff under various individual packet delay constraints and bandwidth efficiencies. This chapter is organized as follows. In Section 2.2, the system model is described. The optimal offline scheduling algorithm for the individual delay constraint model is de- rivedanditsoptimalityisproveninSection2.3. Thepropertiesoftheoptimalscheduling algorithm are analyzed in Section 2.4. The packet delay performance for both models is presented in Section 2.5. The online schedulers and the corresponding delay performance 22 are investigated in Section 2.6. Numerical results are given in Section 2.7. Finally, some concluding remarks are drawn in Section 2.8. 2.2 System Model The system model was discussed in Section 1.3 and Figure 1.3. For the optimal offline scheduling algorithm, the goal is to find the optimal transmission vector ~ τ assuming the knowledge of the inter-arrival time vector ~ d (hence the packet arrival times), and the individual packet delay constraints T i ,i ∈ [1,··· ,M]. As in [UBPG02], all schedulers are assumed to follow the first-in-first-out (FIFO) service rule, the causality constraint, and the non-idling condition whenever feasible. The FIFO rule means that packets are transmitted in the same order they arrive. The causality constraint ensures that a packet cannotbetransmittedbeforeitarrives. Thenon-idlingconditioncomesfromtheassump- tion that the energy function is a decreasing function of the transmission duration τ, and subsequently, the total transmission energy w(~ τ) can always be reduced by increasing the transmission duration for one or more packets. Note that, however, different from the single transmission deadline model, idling periods may be inevitable for any feasible schedulers for the individual delay constraint model. That is, when the inter-arrival time d i > T i , i∈ [1,··· ,M], the amount of d i −T i > 0 time resource can not be utilized and an idling period becomes necessary. This issue will be discussed in more detail in the next section. 23 In the sequel, unless explicitly specified, we will focus on equal individual delay con- straints, i.e., T = T 1 = ··· = T M , and equal packet sizes, i.e., B = B 1 = ··· = B M . Extension to unequal delay constraints and unequal packet sizes will be also considered. 2.3 Optimal Offline Schedule We will start with a cursory review of the results in [UBPG02], followed by the optimal algorithm for the new system model. 2.3.1 Summary of the Single Transmission Deadline Scheduler Here we briefly summarize the optimal offline scheduler for the single transmission dead- line model [UBPG02]. The goal of minimizing w(~ τ) is achieved by a scheme which tries to equalize the transmission durations for all the M packets as much as possible, subject to the following feasibility constraints [UBPG02]:        P k i=1 τ i ≥ P k i=1 d i ,k =1,··· ,M−1,and P M i=1 τ i = P M i=1 d i =t E . (2.1) where P k i=1 τ i is the departure time of packet k, while P k i=1 d i is the arrival time of packet k + 1 (recall the assumption that the first packet arrives at time 0). The first inequality indicates that when packet k+1 arrives, packet k either is still in transmission or just finished its transmission. The expression P M i=1 τ i = P M i=1 d i means back-to-back transmissions and non-idling periods in between, as discussed in Section 2.2. 24 A necessary condition for the optimal scheduling, ~ τ, satisfies [UBPG02] τ i ≥τ i+1 , for i∈[1,··· ,M−1]. (2.2) Thus, the transmission duration of an earlier packet will never be less than that of a later packet. Otherwise, one can always increase the transmission duration of the earlier packets, without violating the feasibility constraints, to reduce the total transmission energy. The optimal scheduler [UBPG02] selects the vector of transmission duration ~ τ such that τ i =m j , if k j−1 <i≤k j , (2.3) where m 1 = max k∈[1,···,M] { 1 k P k i=1 d i } and k 1 = max{k : 1 k P k i=1 d i = m 1 }. For j ≥ 1, m j+1 = max k∈[1,···,M−k j ] { 1 k P k i=1 d i+k j }, and k j+1 =k j +max{k : 1 k P k i=1 d i+k j =m j+1 } with k 0 = 0 and k varies between 1 and M −k j . The above procedure of obtaining the pairs (m j ,k j ) is repeated until k j =M for the first time. It is worth emphasizing that the optimal solution by (2.3) does not depend on a specific w(t) function. Hence, the optimal durations τ i , ∀i ∈ [1,··· ,M], are the same for any energy function that satisfies the non-negative, monotonically decreasing, and strictly convex properties as specified in Section 1.2. 25 2.3.2 Optimal Offline Scheduler for the Individual Delay Constraint Model We now focus on the optimal offline scheduling algorithm based on the system model detailed in Section 2.2, with each packet having its own transmission deadline. As a result, the feasibility constraints in (2.1) must be modified. First,notethatifd i ≥T,thei-thpackethastobedeliveredbefore(ifd i >T)orat(if d i =T)thetimewhenthenextpacketi+1arrives. Asaresult, theschedulingofpackets with indices≤i and packets with indices≥i+1 can be decoupled. For convenience, we introduce the following definition: Definition: A scheduling separation interval is an interval during which all packets, except possibly the last one, have d i <T. That is, packets belonging to different scheduling separation intervals can be scheduled independently. Moreover, we observe the following: Proposition 2.1 For the individual delay constraint model, the optimal offline scheduler based on the original inter-arrival vector ~ d = [d 1 ,··· ,d M ] is not affected by replacing ~ d with the upper-bounded inter-arrival vector ~ ˆ d=[ ˆ d 1 ,··· , ˆ d M ], where ˆ d i =min{d i ,T}. Proof: We will first show that for any packet i∈[1,··· ,M], replacing d i with ˆ d i does affect the optimal scheduling. When d i ≤ T, d i = ˆ d i , we are done. Otherwise, when d i > T, ˆ d i = T. Packets [1,··· ,i] and packets [i + 1,··· ,M] belong to two different scheduling separation intervals, and hence the scheduling of [1,··· ,i] and the scheduling of[i+1,··· ,M]areindependent. Notethatthefirstseparationintervalisnotaffectedby 26 replacingtheinter-arrivaltimeofthelastpacket, i.e.,d i , byT = ˆ d i , duetotheindividual delay constraint T. The second separation interval is not affected either as the (i+1)-th packet is treated as if it were the first arrival anyway. Therefore, the optimal scheduling is not impacted by using ˆ d i instead of d i . Now, repeatingthesameargumentrecursively, wecanseethat ~ ˆ dand ~ dyieldthesame optimal offline scheduling algorithm for the individual delay constraint model. Therefore, in the following, we will focus on using ~ ˆ d for the optimal scheduling. Note that ~ ˆ d removes the inevitable idling periods when d i >T. Under optimal offline schedul- ing, although ~ ˆ d and ~ d may result in different absolute packet arrival times and departure times, the relative difference between the arrival and the departure times for each packet remains unchanged. In other words, ~ ˆ d can be viewed as left-shifting the packet arrival timeswhennecessaryinordertoremovethepotentiallyinevitableidlingperiodsinherent in ~ d and T. Thus, similar to (2.1), we obtain the following feasible non-idling scheduling con- straints: (i):        P k i=1 τ i ≥ P k i=1 ˆ d i ,k∈[1,··· ,M−1], and P M i=1 τ i = P M i=1 ˆ d i , (ii):q k = P k i=1 τ i − P k−1 i=1 ˆ d i ≤T,k∈[1,··· ,M], (2.4) where P k i=1 τ i isthedeparturetimeofpacketk,while P k i=1 ˆ d i isthearrivaltimeofpacket k+1 (recall the assumption that the first packet arrives at time 0). The first inequality in (i) indicates that when packet k + 1 arrives, packet k either is still in transmission or just finished its transmission. The expression P M i=1 τ i = P M i=1 ˆ d i means back-to-back transmissions with no idling in between. The variable q k in (ii) denotes the delay for 27 packet k, defined as the difference between the packet departure time (completed packet delivery) and the packet arrival time. Note that the necessary condition in [UBPG02] for optimality τ i ≥ τ i+1 for i ∈ [1,··· ,M −1], as in (2.2), is no longer necessarily true. It can be easily proved via contradiction. This is due to the introduction of the individual packet delay constraint - an increase in transmission duration for one packet will cause additional delay for later packets and hence may violate the corresponding individual delay constraints. In the sequel, we will first heuristically construct a transmission duration vector ~ τ and then prove its optimality. 2.3.2.1 Optimal Transmission Duration of the First Packet Recall that for the single transmission deadline model, the optimal transmission duration of the first packet is given by [UBPG02] τ 1 = max 1≤i≤M ( 1 i i X m=1 d m ) . (2.5) That is, τ 1 exploits future inter-arrival times as much as possible. Assume that the first maximizing index 1 in (2.5) is i 1 . Clearly, packets 1 to i 1 have the same transmission du- ration (recall that we should try to equalize transmission durations as much as possible). For the individual delay constraint model, we have to observe the delay constraints very closely. Future arrivals provide the possibility of transmitting the first packet with a 1 There may exist multiple indices giving the same maximum value. From the optimal scheduling perspective, it does not matter which index is chosen since the resulting optimal transmission duration vector will eventually be the same and unique. 28 duration as much as possible, as indicated by (2.5). On the other hand, individual delay constraintsimplythattheexploitationoffuturearrivalscannotbeunlimited. Otherwise, delay violations will occur. That is, the optimal transmission durations have to exploit the two conflicting factors, namely, future arrival information and the individual delay constraints, as much as possible. The latter condition implies that some packets may necessarily experience a delay exactly equal to the individual delay constraint. Now let us take a closer look at (2.5) 2 , which has already incorporated future arrival information, and see how we can also take the delay constraint factor into account. If the first maximum index i 1 = 1 in (2.5) and thus τ 1 = ˆ d 1 , we have q 1 = ˆ d 1 ≤ T and there will be no delay violation. Similarly, when i 1 =2, if we let τ 1 =τ 2 =( ˆ d 1 + ˆ d 2 )/2, we have q 1 =τ 1 , and q 2 = ˆ d 2 . There will be no delay violations as well. However, when i 1 =3, if τ 1 =τ 2 =τ 3 =( ˆ d 1 + ˆ d 2 + ˆ d 3 )/3, q 2 = 2( ˆ d 1 + ˆ d 2 + ˆ d 3 ) 3 − ˆ d 1 = 2 ˆ d 2 +2 ˆ d 3 − ˆ d 1 3 may be larger than T, if 2 ˆ d 2 +2 ˆ d 3 − ˆ d 1 >3T, and a delay violation occurs. Thus, τ 1 has to be reduced. The maximum value τ 1 can take without a delay violation is ( ˆ d 1 +T)/2. That is, τ 1 =τ 2 =( ˆ d 1 +T)/2 such that q 2 =T. Note that τ 3 =( ˆ d 2 + ˆ d 3 −T)≥τ 1 =τ 2 , a property to be discussed in more detail later. To summarize, when i 1 =3, τ 1 has to be such that τ 1 =min{( ˆ d 1 + ˆ d 2 + ˆ d 3 )/3,( ˆ d 1 +T)/2}. 2 Note that for the individual delay constraint model, dm in (2.5) needs to be replaced by ˆ dm based on Proposition 2.1. 29 Note that the above expression has exploited both future inter-arrival times and the individual packet delay constraint as much as possible. This idea can be easily extended to the case when i 1 ≥3, i.e. τ 1[i 1 ] =min P i 1 m=1 ˆ dm i 1 ,T, ˆ d 1 +T 2 ,··· , P i 1 −2 m=1 ˆ dm+T i 1 −1 , (2.6) where the subscript in the brackets of τ 1[i 1 ] denotes the max index in (2.5). Note that the first element on the right hand side (RHS) of (2.6) represents the exploitation of future arrivals up to packet i 1 , while the remaining entries ( P j−1 m=1 ˆ d m +T)/j, j =0,··· ,i 1 −2, represent the maximum exploitation of the individual delay constraint of packet j. That is, if ( P l−1 m=1 ˆ d m +T)/l, l ∈ [0,··· ,i 1 −2], is the smallest element on the RHS of (2.6), packet l would have a delay exactly equal to T. Therefore, combined with all potential maximum indices i 1 ∈ [1,··· ,M], the trans- mission duration of the first packet can be explicitly written as τ 1 = max 1≤i≤M τ 1[i] , (2.7) where τ 1[1] = ˆ d 1 ,τ 1[2] = ( ˆ d 1 + ˆ d 2 )/2, and τ 1[i] ,i ≥ 3 is given by (2.6). Compared with (2.5), the nested max/min structure in (2.7) makes it much more difficult to analyze the properties of τ 1 . As discussed earlier, in the particular case when ˆ d m =T for some m∈[1,··· ,M−1], packet1andpackets[m+1,··· ,M]willbelongtodifferentschedulingseparationintervals suchthatpackets[m+1,··· ,M]havenoimpactonτ 1 . Thiscanbeeasilyverifiedby(2.6) 30 and (2.7). That is, if i 1 > m, i.e., 1 i 1 P i 1 l=1 ˆ d l > 1 m P m l=1 ˆ d l = 1 m ( P m−1 l=1 ˆ d l +T), the term 1 i 1 P i 1 l=1 ˆ d l will become inactive due to the minimization operation in τ 1[i] (see (2.6)). In other words, the effect of independent scheduling separation intervals is implicitly incorporated by (2.7). 2.3.2.2 Optimal Transmission Durations of Subsequent Packets The same criterion as in (2.7) can be used to obtain the transmission durations of subse- quentpackets[m,··· ,M], wherem>1, basedontheinter-arrivaltimesoftheremaining packets and the queuing delay, denoted by ˜ q m ≥ 0, introduced by the previous m−1 packets. Note that the delay-constraint factor for packets [m,··· ,M] now has to be modified by considering the buffering delay ˜ q m . That is, instead of T, T − ˜ q m should be used in (2.6), which yields τ 1[i] =min −˜ qm+ P i j=m ˆ d j i−m+1 ,(T − ˜ q m ), (T−˜ qm)+ ˆ dm 2 ,··· , (T−˜ qm)+ P i−2 j=m ˆ d j i−m . (2.8) 2.3.2.3 The Recursive Optimal Scheduling Algorithm Now we can summarize the recursive scheduling algorithm and prove its optimality. The optimal offline schedule is achieved by the following steps: 1. Upper-bound the given set of inter-arrival times ~ d by T, i.e., ~ ˆ d=min{ ~ d,T}. 2. Initialize ˜ q 1 =0 and m=1. 31 3. Findthemaxvalueofmax m≤i≤M τ 1[i] , whereτ 1[i] isgivenby(2.8). Denotethemax index as ˜ n 3 . Note that its index in the entire set of packets is n = ˜ n−m+1. Record the pair [n,τ max,n ], where τ max,n is the max value. Let ˜ q n = P n−1 i=1 (τ i − ˆ d i ), the queuing delay of packet n. 4. Set m=n+1. Go back to Step 3 until n=M. The optimal transmission durations for packets 1 to M are stored in the pairs [n,τ max,n ] derived in Step 3. Suppose there are in total J pairs. Denote the indices as n j ,j = 1,··· ,J and let n 0 =0. The vector of transmission times ~ τ is given by τ m =τ max,n j+1 ,n j <m≤n j+1 ,j =0,··· ,J. (2.9) Beforeprovingtheoptimalityof~ τ in(2.9), wefirstintroducethefollowingdefinitions: Definition: A group is a collection of consecutive packets in a given scheduling sepa- ration interval such that, if a scheduling algorithm is applied to the entire set of packets of the scheduling separation interval, the first packet of the group would begin its trans- mission with an empty buffer, and the last packet of the group is the packet when an empty buffer first occurs after its transmission. Definition: A delay-critical packet is a packet with a packet delay equal to T. Definition: A subgroup is a collection of consecutive packets, within a group, having the same transmission duration. 3 Note that in case there exists more than one packet resulting in the same maximum transmission duration in the index finding process, from a performance perspective, it does not matter which packet is selected. However, it is better to choose the last one to expedite the search process. 32 Notethatfromthedefinition,groupsareformedbyconsecutiveanddisjointcollections of packet indices. Each packet is associated with a distinct group. While the scheduling separation intervals are determined purely by ~ d and the individual delay constraint T, groups are further determined by the particular scheduling algorithm associated with ~ d and T. Throughout the thesis, we will only consider groups resulted from the optimal offline scheduling algorithm. Note also that the group definition is applicable to the optimal scheduling algorithm for the single transmission deadline model as well, in which the scheduling separation interval consists of the entire set of M packets. Using(2.7),wecanobtainthetransmissiondurationsofpacket1toj 1 ,wherej 1 isthe packetindexatwhichτ 1 achievesitsvaluebyeither 1 j 1 P j 1 i=1 ˆ d i or 1 j 1 ( P j 1 −1 i=1 ˆ d i +T). Thus, τ 1 =τ 2 =··· =τ j 1 and packets 1 to j 1 form a subgroup. Note that a subgroup may end with either an empty buffer, i.e., q j 1 = ˆ d j 1 , or a delay-critical packet, i.e., q j 1 = T 6= ˆ d j 1 (see (2.7)). We differentiate these two scenarios by the following two definitions: Definition: A type-1 group is a group containing no delay-critical packet, except pos- sibly the last packet in the group. Definition: A type-2 group is a group containing at least one delay-critical packet, except possibly the last packet in the group. Correspondingly, we can also call a subgroup a type-1 (or type-2) subgroup if it belongs to a type-1 (or type-2) group. From the previous discussion, we can see that when a set of packets form a type- 1 group, this set of packets are only driven by the future arrival information factor. The delay-constraint factor is inactive for a type-1 group since no packet in the group experiencesadelayviolationif(2.5)isdirectlyapplied. Thus,itisnotnecessarytoreduce 33 Figure 2.1: Illustration of groups, delay-critical packets and subgroups. the transmission duration as governed by the delay-constraint factor. On the other hand, a type-2 group is created when a transmission duration reduction from (2.5) becomes necessary, in order to avoid delay violations (see (2.6)). Under (2.5),(2.7) and (2.9), all packets in a type-1 group should have the same duration, and thus, there is only one subgroupinatype-1group. Incontrast, packetsinadelay-constrainedtype-2groupmay not have the same duration, and there are two and more subgroups in a type-2 group. Due to the delay-constraint factor, each subgroup, except the last one, in a type-2 group must end with a delay-critical packet. The above definitions are illustrated in Figure 2.1 with M = 20, T = 2, and λ = 1 packets/second. Ascanbeseen,packets1to7formatype-2group,whichconsistsofthree subgroups. Packets 3 and 6 have packet delays equal to T and hence are critical packets. Similarly, packets8to12formsatype-1group, whiletheremainingpacketsformanother type-2 group. Moreover, it can be noticed that the transmission durations of subgroups 34 in a type-2 group are non-decreasing. In fact, we have the following Lemmas which characterize the transmission duration comparisons of subgroups and groups (similar observations are also found in [ZM05]): Lemma 2.1 Under the optimal offline scheduling, the transmission durations of any two adjacent groups in the same scheduling separation interval are non-increasing. The transmission duration of a type-2 group here refers to that of the last subgroup in the preceding group and that of the first subgroup in the subsequent group. Proof: By contradiction. For any group other than the last group in a given schedul- ing separation interval, there is no delay critical packet in the last subgroup. If the transmission durations between two adjacent groups are increasing, the preceding group can increase its transmission duration by exploiting additional future arrivals from the subsequent group, such that a better transmission efficiency can be achieved. Thus, the original scheduling is not optimal. Lemma 2.2 Under the optimal offline scheduling, the transmission durations of the sub- groups within a given type-2 group are monotonically non-decreasing. Proof: This is straightforward from (2.7) and (2.8). For each τ 1[i] , a type-2 subgroup is created only if the delay-constraint factor becomes active such that the transmission durations driven by the future arrival information factor has to be reduced. That is, in each τ 1[i] , the first entry (−δ + P i j=m ˆ d j )/(i−m+1) in (2.8) is no less than any other remaining entries, each of which might create a type-2 subgroup. A type-2 subgroup is always created with the minimum transmission duration in τ 1[i] and all subsequent 35 subgroups in the same group will thus have transmission durations no less than the previous subgroup. Theorem 2.1 The offline scheduling algorithm given by (2.9) is optimal in minimizing the total transmission energy. Proof: We will use the concept of majorization [Bha97] for the proof. see Ap- pendix A.1. An alternative proof can also be found in [CM06]. As shown in Appendix A.1, the optimal transmission duration vector ~ τ by (2.9) is always majorized by (or more mixed than) any other feasible vectors. This is also true for the optimal transmission duration vector in the single transmission deadline case, as proveninAppendixA.1. Thisprovidesanalternativeproofoftheoptimalofflineschedul- ing in [UBPG02]. We can also show that given the same inter-arrival time vector ~ ˆ d, the optimal transmission duration vector for the individual delay constraint case, denoted as ~ τ ind , always majorizes that of the single transmission deadline case, denoted as~ τ single , as inAppendixA.1. Hence,thetransmissionenergyoftheindividualdelayconstraintmodel is always lower-bounded by that of the single transmission deadline case. This energy comparison is obvious since the single transmission deadline model is a less constrained system. Given an inter-arrival time vector ~ ˆ d and the individual delay constraint T, the opti- mal offline scheduling algorithm is unique. This can easily verified by the majorization approach as in Appendix A.1. Similar observation was also found in [ZM05]. 36 Figure 2.2: The possible group/subgroup associations of the first packet Note that for a total of M packets, τ 1 has 2M−3 possible values (see (2.7)), namely, τ 1 =        P j 1 m=1 ˆ d m /j 1 if j 1 ∈[1,··· ,M−2,M] ( P j 1 −1 m=1 +T)/j 1 if j 1 ∈[2,··· ,M−1], (2.10) where the top entry indicates a type-1 group (M −1 possibilities), and the bottom one indicates a type-2 subgroup (M − 2 possibilities), to which packet 1 belongs. This is illustrated in Figure 2.2. It is straightforward to extend the optimal offline schedule to the case when pack- ets have different delay constraints T i ,i = 1,··· ,M. It is natural to assume that the transmission deadline is monotonically non-decreasing, i.e., t i +T i ≤ t i+1 +T i+1 , such that the packets still follow the FIFO rule. The optimal offline schedule still follows the exact procedure as discussed above, as shown in Appendix A.2. The extension of the optimal offline schedule to unequal packet sizes can be similarly done as in [UBPG02]. That is, instead of trying to equalize per packet transmission duration, we should now try to equalize per bit packet transmission duration [UBPG02]. This is also done in Appendix A.2. 37 2.4 Properties of the Optimal Offline Scheduling In Section 2.3, we have presented the optimal offline scheduling algorithm for the indi- vidual delay constraint model. Given a realization of ~ ˆ d and T, the transmission duration of the first packet is explicitly given by (2.7). It would be desirable to be able to ana- lytically show the total transmission energy resulting from the optimal offline schedule given a specific set of arrivals and an energy-rate function. However, this appears to be intractable. Thus, we will rely on numerical results to show the transmission energy performance of the optimal offline schedule. However, we are able to analytically derive the average packet delay performance. Herein, we will first analyze the properties of the optimal transmission duration vector by presenting the symmetry property. This impor- tant property not only provides an insight to the optimal scheduling algorithm, but also makes it possible to analyze the average packet delay performance into a very compact form, as will show in Section 2.5. In addition to the symmetric property, we will also present the statistical trend of the optimal transmission vector as a function of packet indices towards the end of this section. Note that in the sequel, we always assume equal delay constraints and equal packet sizes. 2.4.1 The Symmetry Property The following theorem summarizes the symmetry property. Theorem 2.2 For any M ≥ 1, when the inter-arrival times d m ,1 ≤ m ≤ M −1, are i.i.d. 4 , under the optimal offline scheduling, the optimal transmission durations τ m and 4 The result also holds for any inter-arrival times where [d1,··· ,dM] and [dM,··· ,d1] are identically distributed. However, for practical interest, we will only focus on the i.i.d. inter-arrivals in this thesis. 38 τ M−m+1 are identically distributed. In particular, E{τ m } = E{τ M−m+1 }, where E{.} denotes expectation. The proof essentially relies on a time reversal argument, where we compare a sample path trajectory of the forward running system to a corresponding time reversed system. Consider the original forward system with the individual delay constraint T and a real- ization of the inter-arrival time vector ~ d (f) =[d 1 ,··· ,d M−1 ], (2.11) where the superscript f denotes the forward system. The absolute arrival times and packet deadlines are thus given by: ~ t (f) arrival = [0,d 1 ,d 1 +d 2 ,··· ,d 1 +d 2 +...+d M−1 ], ~ t (f) deadline = ~ t (f) arrival +[T,T,··· ,T]. Now define the reversed system as a system with the inter-arrival time vector: ~ d (r) =[d M−1 ,d M−2 ,··· ,d 2 ,d 1 ], (2.12) where the superscript r denotes the reversed system. This system can be visualized by relating the arrival time of packet k in the reversed system to the deadline of packet M+1−k intheforwardsystem. Likewise,thedeadlineofpacketk inthereversedsystem 39 corresponds to the arrival time of packet M +1−k in the forward system. Specifically, the reversed system has arrival and deadline times given by: t (r) arrival,k = V −t (f) deadline,M+1−k ,∀k∈[1,··· ,M], (2.13) t (r) deadline,k = V −t (f) arrival,M+1−k ,∀k∈[1,··· ,M], (2.14) where V M = d 1 +d 2 +...+d M−1 +T is the total time duration of the scheduling algorithm in the forward running system. Thus, the first arrival of the reversed system is indeed at time 0 (corresponding to time V on the time axis of the forward system). Before proving Theorem 2.2, we introduce the following definition. Two random vectors ~ X and ~ Y are said to be identically distributed if they have the same probability distribution function, i.e., if: Pr[X 1 ≤u 1 ,X 2 ≤u 2 ,··· ,X N ≤u N ]=Pr[Y 1 ≤u 1 ,Y 2 ≤u 2 ,··· ,Y N ≤u N ] forallrealvaluedN-tuples[u 1 ,··· ,u N ]. Thefollowingbasicprobabilityresultconcerning functions of identically distributed random vectors shall be useful. Lemma 2.3 Let ~ X and ~ Y be any two random vectors with N dimensions, and let ~ Φ(~ x) be any measurable vector valued function from R N to R M . If ~ X and ~ Y are identically distributed, then ~ Φ( ~ X) and ~ Φ( ~ Y) are identically distributed. 2 Note that this lemma holds regardless of whether or not ~ X and ~ Y are independent. The optimal offline scheduling algorithm can be viewed as a function of an M −1 dimensional inter-arrival time vector ~ d = [d 1 ,d 2 ,··· ,d M−1 ] to a unique M dimensional 40 service time vector ~ τ = [τ 1 ,τ 2 ,··· ,τ M ]. Let ~ Φ( ~ d) represent this function (for a given individual deadline constraint T). Thus, ~ τ = ~ Φ( ~ d) is the unique optimal transmission duration vector for a given inter-arrival vector ~ d, where τ k = Φ k ( ~ d) for k ∈ [1,··· ,M]. Note that ~ Φ( ~ d) is indeed measurable, as the optimal algorithm in Section 2.3 involves only simple summation, multiplication, and max/min operations on the ~ d vector. Now, we are ready to prove Theorem 2.2: Proof: (Theorem 2.2) The theorem follows from the following two claims: Claim 2.1 ~ Φ( ~ d (f) )and ~ Φ( ~ d (r) )areidenticallydistributed. Consequently, if~ τ (f) = ~ Φ( ~ d (f) ) and ~ τ (r) = ~ Φ( ~ d (r) ), then τ (f) k and τ (r) k are identically distributed for any k∈[1,··· ,M]. Claim 2.2 For any particular inter-arrival time vector ~ d (f) , if ~ τ (f) =[τ 1 ,··· ,τ M ] is the unique optimal transmission time vector for the forward system, ~ τ (r) =[τ M ,··· ,τ 1 ] is the unique optimal transmission time vector for the reversed system. The proof of Claim 2.1 follows directly from Lemma 2.3. This is because the random vector ~ d (f) given in (2.11) has i.i.d. elements, it is identically distributed to the random vector ~ d (r) given in (2.12). Thus, ~ Φ( ~ d (f) ) and ~ Φ( ~ d (r) ) are identically distributed. To prove Claim 2.2, let ~ τ (f) = [τ 1 ,··· ,τ M ] be the optimal transmission duration vector for the forward system, yielding total energy expenditure: e (f) opt = M X k=1 w(τ k ). 41 Foreachpacketk∈[1,··· ,M],lett (f) arrive,k ,t (f) start,k ,t (f) end,k ,andt (f) deadline,k representthetime packet k arrives, begins its transmission, ends its transmission, and reaches its deadline, respectively, under the optimal scheduling of the forward system. Clearly, we have: t (f) arrive,k ≤t (f) start,k ≤t (f) end,k ≤t (f) deadline,k . Note that the reversed system can emulate the same transmission durations of the for- ward system, in the following sense: For each packet k in the forward running sys- tem, schedule packet M + 1− k in the backward running system according to times t (r) arrive,M+1−k ,t (r) start,M+1−k ,t (r) end,M+1−k , and t (r) deadline,M+1−k , where: t (r) arrive,M+1−k M = V −t (f) deadline,k , t (r) start,M+1−k M = V −t (f) end,k , t (r) end,M+1−k M = V −t (f) start,k , t (r) deadline,M+1−k M = V −t (f) arrive,k . (2.15) Notethatthisschedulingpolicytransmitsnomorethanonepacketatanytime, andthat it satisfies the feasibility constraints of the reversed system. Specifically, the causality constraint of the forward running system implies packet k cannot begin its service until it has arrived, so that t (f) arrive,k ≤t (f) start,k . It follows that: V −t (f) arrive,k ≥V −t (f) start,k . (2.16) 42 Figure 2.3: Illustration of the forward system and the reversed system. The packet departure curve in the forward system, due to the optimal transmission duration vector ~ τ, is also feasible and optimal in the reversed system. Applying the definitions of t (r) end,M+1−k and t (r) deadline,M+1−k in (2.15) to the inequality by (2.16) yields t (r) deadline,M+1−k ≥t (r) end,M+1−k . This ensures that packet M +1−k of the reversed system ends its service time on or before its deadline constraint. Likewise, the deadline constraint t (f) end,k ≤ t (f) deadline,k for packet k under the forward system implies the causality constraint for packet M +1−k in the reversed system. This is illustrated in Fig. 2.3. Further note that this policy schedules packet M +1−k in the reversed system for a service time exactly equal to τ k , the service time of packet k in the forward system. It follows that this emulation achieves exactly the same energy expenditure e (f) opt as the forward system. Therefore, the optimal energy expenditure of the reversed system is 43 less than or equal to that of the forward system. However, because the forward system can be viewed as a ‘reversed’ reversed system, it likewise follows that the optimal energy expenditure of the forward system is less than or equal to that of the reversed system. Hence, both the forward and reversed systems have exactly the same optimal energy expenditure, with (unique) optimal transmission duration vectors given by [τ 1 ,··· ,τ M ] and [τ M ,··· ,τ 1 ], respectively. Thetwoclaimsaboveimmediatelyimplythetheorembecause,foranyk∈[1,··· ,M], we have that τ (f) k is identically distributed to τ (r) k (Claim 2.1), and τ (r) k = τ (f) M+1−k (Claim 2.2). Thus, τ (f) k and τ (f) M+1−k are identically distributed. Whentheinter-arrivaltimesaredeterministicandsymmetric,thesymmetryproperty of ~ τ also holds. This is summarized in the following corollary: Corollary 2.1 Given a total of M ≥1 packets, if the inter-arrival time vector [ ˆ d 1 ,··· , ˆ d M−1 ] is symmetric, i.e., ˆ d m = ˆ d M−m ,∀m ∈ [1,··· ,M −1] (note that d M is fixed at T), the optimal transmission duration vector [τ 1 ,··· ,τ M ] under the optimal offline scheduling is also symmetric, i.e., τ m =τ M−m+1 ,∀∈[1,··· ,M]. Notethatforthesingletransmissiondeadlinemodel[UBPG02],theoptimaltransmis- siondurationvectorexhibitsamonotonicallynon-increasingpropertyfor each realization of inter-arrival times. This is different from the statistically symmetric property for the individual delay constraint model discussed here. In fact, we can show that for the sin- 44 gle transmission deadline model, given an inter-arrival time vector ~ ˆ d = [ ˆ d 1 ,··· , ˆ d M ], the optimal transmission duration of packet M is given by τ M = min 1≤i≤M ( 1 M−i+1 M X m=i ˆ d m ) . (2.17) This is due to the non-increasing property of optimal transmission durations (see (2.2)) suchthatpacketM canonlybeinagroupwhichhastheminimumtransmissionduration. Ontheotherhand,fromthereversalprocessinTheorem2.2,theoptimaltransmission duration of the last packet for the individual delay constraint model can be explicitly written as τ M = max 0≤i≤M−1 τ M[i] , (2.18) where τ M[i] ,i∈[0,··· ,M−1] is given by τ M [i] =min{ 1 M−i P M−1 m=i ˆ d m , 1 2 ( ˆ d M−1 +T),··· , 1 M−i−1 ( P M−1 m=i+2 ˆ d m +T)}. (2.19) Note that after simplifications, τ M[M−1] = ˆ d M−1 and τ M[M−2] =( ˆ d M−1 + ˆ d M−2 )/2. That is, while the future arrival factor is more visible at the beginning of a set of packets (see (2.7)), its effect slowly diminishes towards the end, where the delay-constraint factor gradually takes over (see (2.18)). 2.4.2 A Trend of E{~ τ} This section presents a statistical trend of ~ τ. In fact, we have the following result when the inter-arrival times are i.i.d.. 45 Lemma 2.4 For any M ≥1, when the inter-arrival times ˆ d m ,1≤m≤M−1, are i.i.d., under the optimal offline scheduling, E{τ m } is non-increasing when m ≤ bM/2c, and non-decreasing when m>bM/2c. Proof: See Appendix A.3. The following corollaries are straightforward (proofs omitted): Corollary 2.2 E{τ 1 }=E{τ M }≥E{τ m },∀m∈[2,··· ,M−1]. Corollary 2.3 When M is odd, E{τ (M+1)/2 } ≤ E{τ m }, ∀m ∈ [1,··· ,M], but m 6= (M +1)/2. Corollary 2.4 When M is even, E{τ M/2 } = E{τ M/2+1 } ≤ E{τ m }, ∀m ∈ [1,··· ,M], but m6=M/2. Itisworthnotingthatthetrendoftheoptimaltransmissiondurationvectordiscussed in Lemma 2.4 is averaged over all possible realizations of inter-arrival times. For a given realizationofinter-arrivaltimes, theoptimaltransmissiondurationvectormaynotfollow the same trend. Instead, the general properties described in Lemmas 2.1 and 2.2 will apply. 2.5 Packet Delay Performance Analysis In this section, we will analyze the packet delay performance of the optimal scheduler for both the single transmission deadline model and the individual delay constraint model. 46 2.5.1 Packet Delay Performance for the Individual Delay Constraint Model As discussed in Section 2.4, the optimal transmission duration vector ~ τ exhibits a sym- metry property under i.i.d. inter-arrivals, equal delay constraints and equal packet sizes. This property leads to a simple yet exact expression of the average packet delay perfor- mance under the optimal offline scheduler for the individual delay constraint model. Define the average packet delay as: ¯ q M = E{ 1 M M X m=1 q m } where q m is the delay experienced by packet m under the optimal offline schedule with a particular realization of the inter-arrival time vector (see (2.4)), and the expectation is taken over all realizations of packet inter-arrival times. We have Theorem 2.3 For any M ≥ 1, when the inter-arrival times d m ,1 ≤ m ≤ M −1, are i.i.d., under the optimal offline schedule, the average packet delay is given by ¯ q = ¯ ˆ d+ M +1 2M (T − ¯ ˆ d). (2.20) where ¯ ˆ d=E{ ˆ d m } and ˆ d m =min{d m ,T}. When M →∞, ¯ q→(T + ¯ ˆ d)/2. 47 Proof: First, from Proposition 2.1, we know ~ ˆ d yields the same optimal transmission durationvector~ τ as ~ d,andsubsequently,thesamepacketdelayperformance. From(2.4), the delay for packet m is q m = P m l=1 τ l − P m−1 l=1 ˆ d l . Thus, ¯ q M = 1 M P M m=1 E{q m } = 1 M P M m=1 h P m l=1 E{τ l }− P m−1 l=1 E{ ˆ d l } i (a) = 1 M P M m=1 (M−m+1)E{τ m }− 1 M P M m=1 (m−1) ¯ ˆ d (b) = 1 M P M m=1 M+1 2 E{τ m }− 1 M M(M−1) 2 ¯ ˆ d (c) = M+1 2M [(M−1) ¯ ˆ d+T]− M−1 2 ¯ ˆ d = ¯ ˆ d+ M+1 2M (T − ¯ ˆ d), where (a) holds by counting the number of occurrences of each item, the first term of (b) comes from the symmetry property E{τ m } = E{τ M−m+1 },∀m ∈ [1,··· ,M], and equivalently, there are (M +1)/2 copies of each E{τ m }, and the first term of (c) is due to the fact that P M m=1 E{τ m }=(M−1) ¯ ˆ d+T, i.e., the non-idling scheduling. Corollary 2.5 The average queuing delay (excluding the transmission time), i.e., ¯ q− 1 M P M m=1 E{τ m }, is given by ¯ ˜ q = M−1 2M (T − ¯ ˆ d). When M →∞, ¯ ˜ q→(T − ¯ ˆ d)/2. Proof: This comes directly from Theorem 2.3 and the fact that P M m=1 E{τ m } = (M−1) ¯ ˆ d+T. Theorem 2.3 indicates that under the i.i.d. assumption of ~ ˆ d, the average packet delay is roughly half of the delay constraint when M is sufficiently large and T ¯ ˆ d. This is 48 Figure 2.4: The average packet delay vs. M for the individual delay constraint model, λ=1. illustrated in Fig. 2.4 for different individual delay constraints, under a Poisson arrival model of rate λ (thus, ¯ ˆ d is equal to (1−e −λT )/λ). As can be seen, when M increases, the average packet delay decreases and converges to a fixed value. Under typical scenarios, ¯ ˆ d is roughly equal to ¯ d (without being upper-bounded by T), and hence 1/λ. From Little’s Theorem [Kle75], the average number of packets is approximately (λT +1)/2 under the optimal scheduling for the individual delay constraint model. Letusnowlookatonaveragehowpacketdelayperformanceevolveswithpacketindex m. While the optimal transmission duration vector is statistically symmetric and mono- tonically non-increasing to the middle index, followed by a monotonically non-decreasing property, the average queuing delay exhibits a different property. Denotee q m as the queu- 49 ing delay for packet m, excluding the transmission delay (i.e., e q m = q m −τ m ). Clearly, e q 1 =0, and for m∈[2,··· ,M], e q m = m−1 X i=1 (τ i − ˆ d i )= m−1 X i=1 Δe q i , (2.21) where Δe q i =τ i − ˆ d i denotes the queuing delay contributed by each packet. We have the following lemma: Lemma 2.5 For any M ≥1, when the inter-arrival times ˆ d m ,1≤m≤M−1, are i.i.d., under the optimal offline scheduling, the average queuing delay E{e q m } is monotonically non-decreasing with m. Proof: see Appendix A.4. The following corollary is directly from Lemma 2.5 and (2.21): Corollary 2.6 E{Δe q i }≥0,∀i∈[1,··· ,M]. Since Δe q i =τ i − ˆ d i , we have Corollary 2.7 E{τ i }≥ ˆ d,∀i∈[1,··· ,M], where ˆ ddenotestheaverageinter-arrivaltimeafterbeingupper-boundedbytheindividual delay constraint. In other words, under the optimal offline scheduling, the average transmission dura- tion of any packet is no less than the average inter-arrival time. This is different from the single transmission deadline model, in which the average transmission duration is monotonically non-increasing, and the last packets may have the average transmission 50 durationsbelowtheaverageinter-arrivaltime(tobeshownbelow). Itisworthemphasiz- ing that the property in Corollary 2.7 is the statistically averaged behavior. As discussed in Appendix A.1, under a specific realization of inter-arrival times, the transmission en- ergy associated with the single transmission deadline model is always no more than that of the individual delay constraint model. 2.5.2 Packet Delay Performance for the Single Transmission Deadline Model Now let us study the packet delay performance under the optimal offline scheduling for the single transmission deadline model. Similar to [UBPG02], we will focus on a Poisson arrival model such that inter-arrival times follow the exponential distribution (including d M ). It has been shown in [UBPG02] that the average transmission duration for the first packet is given by ¯ τ M,1 = 1 λ M X m=1 1 m 2 where we have added an M in the subscript to indicate a total of M packets. As M → ∞,¯ τ M,1 →π 2 /(6λ). Similarly, we can obtain the average transmission durations for any packet i ≥ 1, althoughthederivationismorecomplicated. Theresultsaresummarizedinthefollowing lemma. 51 Lemma 2.6 When the inter-arrival times exponentially distributed, the average trans- mission durations for packets i∈[2,··· ,M], M >1, under the optimal offline scheduling for the single deadline model, are: ¯ τ M,i = 1 λ M X m=i ( 1 m 2 + i−1 X l=1 l l (m−l) m−l−2 m m m l ) . (2.22) When i=M, it can be simplified to ¯ τ M,M = 1 λ M +1 2M . (2.23) Proof: see Appendix A.5. The average individual packet delay is thus given by ¯ q M,i = i X l=1 ¯ τ M,l −(i−1)/λ. (2.24) Fig. 2.5 shows the average packet transmission duration ¯ τ M,i and delay ¯ q M,i as a function of packet indices with λ = 1 and M = 100. It can be seen that the average packet transmission durations decrease as the packet index increases. The transmission durationdecreasingrateishighatthebeginningandtowardtheendofthepacketindices, andisrelativelyflatinthemiddleofthepacketindices. AtapproximatelyM/2,thepacket transmission duration is roughly equal to 1/λ. The transmission delay follows a concave curve and peaks at about packet index M/2. The transmission delay reduces to 1/λ for the last packet M. 52 Figure 2.5: The average transmission duration ¯ τ M,i and delay ¯ q M,i with λ = 1 and M =100. It is of great interest to investigate the average packet delay as a function of M for the single deadline model. We have: Lemma 2.7 When the inter-arrival times exponentially distributed, under the optimal offline scheduling for the single deadline model, the average packet delay ¯ q M for a total number of M packets can be approximated by ¯ q M ≈ 1 λ √ 2π 6 √ M +1 ! . (2.25) Proof: see Appendix A.6. Fig.2.6showstheaveragepacketdelay ¯ q M asafunctionofthetotalnumberofpackets M. Both the actual average delay (via explicit computations) and the approximate 53 Figure 2.6: The average packet delay of the optimal scheduler for the single transmission deadline model as a function of M. delay, as in (2.25), are presented. It can be seen that the average packet delay grows monotonicallywithM andthegrowthrateisroughlyproportionalto √ M (withascaling factor). In addition, the approximation given in (2.25) closely matches the actual delay. 2.6 Online Schedulers and Their Properties In this section, we will investigate two online schedulers. The online schedulers only assume information of the current scheduling backlog. The first scheduler extends the optimal static buffer flushing algorithm in [ZM05] to a system with dynamic packet arrivalsanddepartures. ThesecondschedulerisanIMET-likealgorithminwhichpackets are buffered and scheduled on a fixed duration basis. 54 2.6.1 Optimal Buffer Flushing Based Online Scheduling Theoptimalalgorithmofflushingastaticbufferwithafinitenumberofpacketsofvarious discrete individual delay constraints was investigated in [ZM05], a re-formulation of the scheduling issue in [KS04]. Assume at a time instant, there are a finite number (e.g., K ≥ 1 5 ) of discrete individual delay constraints in the buffer, i.e., T 1 ≤ T 2 ≤ ··· ≤ T K , with packet sizes of B 1 ,B 2 ,··· ,B K , respectively. It is shown in [ZM05] that the optimal transmission duration of the head packet, i.e., the packet with the smallest delay constraint, can be written as: τ 1 =B 1 min k∈[1,···,K] T k P k i=1 B i . (2.26) Here we extend this static optimal buffer-flushing algorithm to a continuous time systemwithdynamicpacketarrivalsanddepartures. However, thetransmissionduration of the head packet by (2.26) may no longer hold when a new packet arrives. In other words, the optimal transmission rate inherent in (2.26) may be kept until upon a new packet arrival, at which point a new optimal transmission rate may become necessary. Therefore, we can re-write (2.26) in terms of the optimal buffer flushing rate as r opt = max k∈[1,···,K] P k i=1 B i T k . (2.27) Itisworthnotingthatalthoughweassumeequalpacketsizesandequaldelayconstraints, at a particular time instant, the pending packets in the buffer do not necessarily have the 5 The case when K = 0 is trivial. 55 same packet sizes and delay constraints. This is because packets may arrive at different times, and the head packet in the buffer may already in the process of transmission 6 . When a new packet arrives, the new optimal flushing rate is no less than the original ratebeforethenewpacketarrival. Thisisobviousasanewtermisaddedtotherighthand side of (2.27). On the other hand, the optimal rate in (2.27) may also need to be updated upon a packet departure, as the first entry on the right hand of (2.27) disappears. If the departed packet has a delay less than T, the optimal flushing rate remains unchanged. However, if its delay is exactly equal to T, the optimal flushing rate may be decreased. This is due to the fact that the departed head packet satisfies B 1 /T 1 =r opt . To sum up, at any time, the scheduler chooses a transmission rate based on (2.27), and may • Increase the rate upon a new packet arrival, or • Decrease the rate upon a packet departure with a delay exactly equal to T, where B i and T i in (2.27) are the current residual packet size and the remaining time to the deadline for packet i. Lemma 2.8 The online scheduler by (2.27) guarantees that all packets meet their own delay constraints. Proof: This can be easily checked by noting the FIFO assumption and the fact that at any time t≥0, we have r opt ≥B 1 /T 1 . For convenience, we will denote the above online scheduler as the online flush sched- uler. It is worth emphasizing that no future arrival information, completely or partially, 6 Note that here a fluid packet departure model is assumed. That is, a transmitted packet is not necessarily an integer number of arrived packets, but may be assembled using fragmented arrival packets up to an arbitrary precision. 56 is assumed in the design of the above online scheduling algorithm. At any given time instant, the online flush scheduler yields the optimal transmission rate when there are no future packet arrivals. In a system of dynamic arrivals, the above online scheduler may no longer be optimal when additional future arrival information is anticipated and incor- porated. For instance, a stochastic optimal control algorithm, which anticipates future arrivals, was investigated in [ZM05] for packets subject to a single transmission deadline. InthespecialcaseofconstantpacketsizesB i =B,∀i, itisnotdifficulttoseethatthe transmission rate for slot m is given by r flush m =B[1−(1−1/D) m ]/τ s when 1≤m≤M, and r flush m =B[1−(1−1/D) M ]/τ s when M <m≤M +D−1, where B is the constant packet size. In this particular example, the optimal transmission rate is monotonically increasing, and very close to B when m is reasonably large. Obviously, theonlineflushschedulerresultsinatransmissionenergynolessthanthat oftheoptimalofflinescheduler. ThefollowingLemmacharacterizesitsdelayperformance: Lemma 2.9 Given M ≥1 and any particular inter-arrival time vector ~ d, for each packet m ∈ [1,··· ,M], the packet delay under the online flush scheduler by (2.27), q flush m , is no less than that of the optimal offline scheduler by (2.7), q offline m . In particular, if q offline m =T, then q flush m =T. In addition, E{q flush }≥E{q offline }. Proof: Note that due to the FIFO constraint, when both schedulers have the same queue length, they must have the same buffered packets. Now consider at any time t ≥ 0, whenever the two schedulers have the same buffered packet sizes, reflected by B 1 ,B 2 ,··· ,B K corresponding to delay constraints T 1 ≤ T 2 ≤ ··· ≤ T K , respectively. This certainly holds true at time 0 when the first packet arrives, where B 1 = B and 57 T 1 =T. At time t + , the flush scheduler chooses a transmission rate r flush (t + ) by (2.27), using only current backlog information B i and T i , i ∈ [1,··· ,K]. The optimal offline scheduler, however, in addition to current backlog information, also exploits future ar- rival information. Thus, at time t + , the optimal offline scheduler always chooses a rate r offline (t + ) no less than that of the flush scheduler, i.e., r offline (t + )≥r flush (t + ). Thus, the unfinished amount of work in the flush scheduler is never less than that of the offline scheduler. More delay performance comparisons between the optimal offline scheduler and the online flush scheduler will be investigated via simulations in Section 2.7. 2.6.2 The IMET-like Online Scheduler This online scheduler is loosely linked to the iterative minimum emptying time (IMET) algorithm in [NSM02] and thus is termed as the IMET-like scheduler. The IMET al- gorithm [NSM02] is a frame-based iterative scheduling algorithm which, given a certain traffic model and channel conditions, determines the minimum required frame duration such that all the packets buffered in the preceding frame can be fully scheduled (without packet dropping) in the current frame. This design idea leads to a simple online sched- uler for the individual delay constraint model. In this case, the goal is not to find the minimum frame duration. Instead, given the individual delay constraint T, we can just choose a frame duration of T/2, and iteratively buffer and schedule packets on a per T/2 basis. In doing this, we can guarantee that all packets will experience a delay no more than T. To be more specific, the IMET-like online scheduler is as follows: 58 1. Choose the frame duration as T f =T/2. 2. During the first frame, do nothing. 3. For each subsequent frames, ignore all new arrivals during this frame but clear the backlog due to the preceding frame such that each buffered packet is transmitted with the same duration of T f /N f , where N f ≥ 1 is the number of buffered packets during the previous frame. If N f =0, do nothing. Note that unlike the optimal offline scheduler and the online flush scheduler, the IMET-like online scheduler with inter-arrival times d i ≤T, i∈[1,...,M], may still yield idling periods, especially when d i > T/2. These potential idling periods, along with the inefficiency of potentially not fully utilizing the individual delay constraints, makes this simple scheduler inferior to both the optimal offline scheduler and the online flush scheduler, as will be demonstrated in Section 2.7. Now let us study the packet delay performance for the IMET-like scheduler. Similar to[UBPG02], we will focus on a Poisson arrival model such that inter-arrival times follow an exponential distribution. Note that when there are no packet arrivals during the previous frame, no packets will be transmitted in the current frame. When there is one or more packet arrivals, i.e., N f ≥ 1, the transmission durations of these N f ≥ 1 packets are the same and equal to T f /N f . Under the Poisson arrival model, the arrival times of these N f packets in the preceding frame follow a uniform distribution. Under the scheduling scheme, these 59 packetswilldepartinthecurrentframewithaT f /N f increment. Thus, theaveragedelay for these N f packets is T f 2 + 1 N f N f (N f +1) 2 T f N f =T f + T f 2N f ,N f ≥1, (2.28) where the first term on the left hand side is due to the average packet arrival time in the preceding frame to the boundary of the current frame, while the second term is due to the T f /N f increment for a total of N f packets. Note that when N f = 0, the average delay is zero. Now, the number of packets arrived in a frame, N f , follows a Poisson distribution, i.e., P{N f = n} = e −λT f (λT f ) n /n!, where λT f is the average number of packets arrived in a frame. Using (2.28), the average delay of these (on average) λT f packets can be obtained as ¯ q IMET = 1 λT f P ∞ n=1 ne −λT f (λT f ) n n! (T f + T f 2n ) = e −λT f /λ[ P ∞ n=1 (λT f ) n (n−1)! + 1 2 P ∞ n=1 (λT f ) n n! ] = e −λT f /λ[λT f e λT f + 1 2 (e λT f −1)] = T f + 1 2λ (1−e −λT f ) = T 2 + 1 2λ (1−e −λT/2 ) (2.29) where we have used ¯ q IMET =E{nT f +T f /2|n≥1}/(λT f ), resulting from renewal theory and the law of large numbers, where the numerator is the average sum of delays of all packets (n≥1) scheduled over a frame (from (2.28)), and the denominator is the average number of packets scheduled over a frame duration. 60 DuetothePoissonarrivalassumptioninaframe, ¯ q IMET (see(2.29))isnotafunction of the total number of packets M. However, for a given M, the number of packet arrivals in a frame is finite and upper bounded by M. Thus, the actual average delay for the IMET-like online scheduler is approximately given by (2.29), and converges to (2.29) when M approaches infinity. Notefrom(2.20)that, whenMislarge, theaveragepacketdelayoftheoptimaloffline scheduler, ¯ q offline , converges to ¯ q offline → T/2+ ¯ ˆ d/2, where under the Poisson arrival model, ¯ ˆ d = (1−e −λT )/λ. Compared ¯ q IMET in (2.29), it can be seen that when T is reasonably large, ¯ q IMET ≈ ¯ q offline , when M →∞. Thatis,theIMET-likeonlineschedulerachievesalmostthesamedelayperformanceasthe optimal offline scheduler, while its transmission energy performance may be significantly worse than that of the optimal offline scheduler, as will be shown in the next Section. 2.7 Numerical Results We assume a Poisson arrival rate of λ = 1 packet/second. The inter-arrival time d M is fixed at T for both models. The energy function is assumed to be w(τ) = τ(2 2B/τ −1) [UBPG02], where B is normalized to be the number of bits per channel use. It is worth noting the optimal transmission duration vector ~ τ and the corresponding average packet delay are not a function of B. We will first illustrate the trade-off between transmission energy and delay offered by the optimal offline schedule under the individual delay con- 61 straint model, followed by the properties of the optimal offline transmission durations and packet delays. We will then compare the offline and the online schedulers. 2.7.1 Impact of Individual Delay Constraint T Fig. 2.7, 2.8 and 2.9 show an example run of 19 packets, with different individual packet delayconstraintsof1, 3, and6seconds, respectively, fortheoptimalofflineschedule. The inter-arrival times, the optimal transmission durations, and the corresponding packet delays are shown. It can be seen that as the delay constraints become less strict, the transmission durations generally increase (which means less transmission energy), at the expense of increased average packet delays and packet delay variances. When T = 6 seconds, there are no delay constraint violations in the index-finding process, the optimal offline scheduling algorithm becomes the same optimal scheduler as in [UBPG02] for this particular case. 2.7.2 Tradeoff Between Transmission Energy and Packet Delay Fig. 2.10 shows the average packet delay resulting from the optimal offline schedule as a function of the delay constraint T, which was averaged over 100 independent runs with a total number of 1000 packets. It can be seen that under the individual delay constraint model, the average packet delay increases monotonically as the delay constraint T grows and it approaches that of the single transmission deadline model. For instance, when T = 4 seconds, the average packet delay is about 2.5 seconds, roughly 18% of the average delay (≈13.8 seconds) associated with the optimal scheduler in [UBPG02]. Fig. 2.11, on the other hand, shows the normalized average packet energy vs. T for different packet 62 Figure 2.7: An example run, T = 1 second. Figure 2.8: An example run, T = 3 second. 63 Figure 2.9: An example run, T = 6 second, which becomes the same as the single trans- mission deadline model. sizes. The normalization is done with respect to the minimum energy achieved by the optimal scheduler in [UBPG02]. Obviously, the normalized energy vs. T is a convex and monotonicallydecreasingcurve. AtT =4andapacketsizeofB =1bits/packet(normal- ized to 1 bits/transmission), the scheduler based on the new model requires about 10% more energy than that of the optimal scheduler in [UBPG02]. As a result, the individual delay constraint model provides a tradeoff between packet delay and transmission energy. Fig. 2.12 shows the minimum required individual delay constraint T as a function of the packet size B such that the average transmission energy of the new optimal scheduler is no more than 10% higher than that of the optimal scheduler in [UBPG02]. It can be seen that at a normalized packet size of 4 bits/transmission, the minimum required T is about 15 seconds. The minimum required T increases monotonically with B. 64 Figure 2.10: The average packet transmission delay vs. T. Figure 2.11: Normalized average packet transmission energy vs. T. 65 Figure2.12: TheminimumrequiredTvs. channelusesuchthattheaveragetransmission energy of the individual delay model is no more than 10% of the single deadline model. 2.7.3 Properties of Optimal Offline Transmission Durations and Packet Delays Fig. 2.13 shows the average optimal transmission durations when M = 100. The results were averaged over 10,000 independent simulations. The single transmission deadline modelis shownforcomparison. As canbe seen, the optimal transmission duration vector for the individual delay constraint model exhibits the symmetry property. The mini- mum transmission duration occurs in the middle of the packets. The single transmission deadline model, on the other hand, exhibits a different property. Its average optimal transmission durations are monotonically decreasing. Note that the average optimal transmission durations of the individual delay constraint model are lower bounded bythe average inter-arrival time ˆ d=(1−e −λT )/λ (not shown), as analyzed in Section 2.5. 66 Figure2.13: Averageoptimaltransmissiondurationsundertheoptimalofflinescheduling, M =100. Fig. 2.14 shows the average packet delay associated with the optimal transmission durations. It can be seen that the average delay for the individual delay constraint model increases with packet index m, and maximizes at T when m = M. The average packet delay, including the transmission delay, is close to (T + ˆ d)/2 when m is approximately M/2. The delay for the single transmission deadline model, however, peaks in the middle of the packets, and is much higher than that of the individual delay constraint model. 2.7.4 Comparison Between the Offline and Online Schedulers Figure 2.15 shows the normalized transmission energy performance for the two online schedulers when M = 1000. The energy normalization is performed with respect to that of the optimal offline scheduler. The results were averaged over 1,000 independent 67 Figure 2.14: Average packet delay under the optimal offline scheduling, M =100. simulations. It can be seen that the online flush scheduler achieves a comparable energy performance to the optimal offline scheduler when the normalized packet size is small and/or the individual delay constraint is large. The simple IMET-like online scheduler, however, performs significantly worse than both the optimal offline and the online flush schedulers, especially for small delay constraints and large packet sizes. Figure 2.16 and Figure 2.17 show the average packet delay performance for the offline and the online schedulers when M = 1000. The online flush scheduler, on average, achieves a larger delay performance than both the two other schedulers, which yields almost the same delay performance. The above delay performance gap grows as the individual delay constraint increases. The average delay for all the three schedulers, on average, increases rapidly at the beginning and the end of the set of M packets, and 68 Figure2.15: Averagenormalizedtransmissionenergyfortheonlineschedulers,M =1000. remains flat in the middle. The analytical delay performance results for the optimal offline scheduler in (2.20) and the IMET-like online scheduler in (2.29) are also shown in Figure 2.17 for comparison, and agree very well with the simulations. 2.8 Conclusions This chapter studies energy-efficient packet transmissions with individual packet delay constraints over a continuous-time static channel. The solution presented herein is a generalization of [UBPG02] which considered energy-efficient transmissions for a group of packets subject to a single transmission deadline. The optimal offline scheduler for the individual delay constraint model was derived. The properties of the optimal durations and the resulting packet delays were analyzed and compared with those of the single 69 Figure 2.16: Average packet delay vs packet indices for the offline and online schedulers, M =1000. Figure 2.17: Average packet delay vs T for the offline and online schedulers, M =1000. 70 transmission deadline model. While the single transmission deadline model may yield an unbounded packet delay when the number of packets approaches infinity, the individual delay constraint model provides bounded packet delay, roughly equal to half of the delay constraint on average. A heuristic online scheduler, which assumes no future arrival information, is also studied. This online scheduler is shown to achieve a comparable energy performance to the optimal offline scheduler in a wide range of scenarios. The flexible energy and delay tradeoff provided by the individual delay constraint model is further illustrated by numerical results under various individual delay constraints and bandwidth efficiencies. 71 Chapter 3 Energy-Efficient Transmissions with Individual Delay Constraints Over Fading Channels Inthischapter,wewillfocusonenergy-efficientpackettransmissionwithindividualpacket delay constraintsoverafading channel,asanextensionofChapter2foracontinuous-time arrivalmodelandstaticchannels. Theproblemofoptimalofflinescheduling(vis-` a-visto- taltransmissionenergy)overafadingchannel,assuminginformationofallpacketarrivals andchannelstatesbeforescheduling,isformulatedasaconvexoptimizationproblemwith linearconstraints. Theoptimalityconditionsareanalyzed. Fromtheanalysis, arecursive algorithmcalled‘ConstrainedFlowRight’,basedonthe‘FlowRight’algorithmforthesin- gle deadline model in [UBG04], is developed to search for the optimal offline scheduling. It is shown that the optimal scheduling tries to equalize the derivatives of the energy-rate function as much as possible subject to the causality and delay constraints, in contrast to the equalization of transmission rates in static channels. Idling periods might exist for bad channels. The optimal derivatives are non-decreasing under an active causality con- straint and non-increasing under an active delay constraint. The symmetry property of the optimal transmission rate vector (or, equivalently, the transmission duration vector) 72 still holds under the i.i.d. assumption of packet sizes and channel coefficients. Combin- ing the symmetry property with the potential idling periods, upper and lower bounds of the average packet delay (including queuing and transmission delays) are derived. The properties of the optimal offline schedule and the impact of packet sizes, individual delay constraints, and channel variations are demonstrated via simulations. Motivated by the properties of optimal offline scheduling, a heuristic online scheduling algorithm, which assumes both causal traffic and channel information, is proposed. This online scheduler tries to equalize the energy-rate derivatives for all non-idling slots by exploiting both traffic and channel information. The online scheduler achieves comparable energy and delay performance to the optimal offline scheduling, as demonstrated by simulations. This chapter is organized as follows. In Section 3.1, the system model is described. The optimal offline scheduling over a fading channel and its properties are presented in Section3.2. OnlineschedulersareinvestigatedinSection3.3. Numericalresultsaregiven in Section 3.4. Finally, some concluding remarks are drawn in Section 3.5. The closed form solution of the optimal offline scheduling algorithm over a static channel for a time slot model is presented in the Appendix. 3.1 System Model The BF-AWGN model was discussed in Section 1.3. The time slot model for the single transmission deadline model is shown in Figure 1.4, while Figure 1.5 illustrates the time- slotted individual delay constraint model. The goal of the optimal offline schedule in the time-slot model, which assumes perfect knowledge of packet sizes and channel states 73 for the entire duration [0,··· ,M +D−1] before scheduling, is to choose the number of transmitted bits x i , or equivalently, the optimal transmission rate r i , for each slot such that the total transmission energy of these M packets is minimized while the underlying delay constraints are satisfied. Online schedulers, on the other hand, only assume infor- mation of the current scheduling backlog. As in Chapter 2, all schedulers are assumed to follow the first-in-first-out (FIFO) service rule and the causality constraint. Again, the energy-ratefunctionf(r,g)isassumedtobestrictlyconvexandmonotonicallyincreasing in transmission rate r for each channel state g. 3.2 Optimal Offline Scheduling over a Fading Channel Hereweformulatetheoptimalofflinescheduleoverafadingchannelasaconvexoptimiza- tion problem. We start by analyzing the optimality conditions, and deriving a recursive algorithm to search for the optimal schedule, before presenting the symmetry property of the optimal transmission rate vector and the resulting average packet delay performance. The objective of the optimal offline scheduling is to solve the following problem min M+D−1 X m=1 f(r m ,g m ). (3.1) subject to 1) P M+D−1 i=1 r i τ s = P M+D−1 i=1 B i , 2) P m i=1 r i τ s ≤ P m i=1 B i , m=1,··· ,M, 3) P m i=1 r i τ s ≥ P m−D+1 i=1 B i ,m=D,··· ,M +D−2, 4) r m ≥0, m=1,··· ,M +D−1, 74 where conditions 1 and 2 are due to the causality constraint, i.e., the total number of delivered bits is no more than the total number of arrived bits so far, while condition 3 is due to the individual delay constraints, i.e., the total number of delivered bits so far should be no less than the total number of arrived bits accumulated up to D−1 slots earlier. The non-negative transmission rate constraint by condition 4 is natural. Note that the causality condition 2) is also implicitly valid for m = M +1,··· ,M +D−1. This can be easily checked by comparing condition 1) and condition 2) when m = M, and by realizing that B m =0, m=M +1,··· ,M +D−1. First, we have the uniqueness of optimal offline schedule: Theorem 3.1 The optimal offline schedule for the individual delay constraint model over a fading channel is unique. Proof: See Appendix A.7. In addition, the properties of the per-bit transmission duration with respect to slot boundaries can be characterized by the following Lemma (a similar observation was also obtained in [UBGP02] for a continuous-time model): Lemma 3.1 Under the optimal offline schedule, all bits transmitted over a slot have the same transmission duration. In other words, the transmission rate is constant during any slot. Proof: Since the individual delay constraint D is in units of slots, all bits to be transmitted in the same slot have transmission deadlines either at the end of the slot or beyond. Supposing there exist unequal per-bit transmission durations in a slot, one can always equalize these transmission durations without violating any individual delay 75 constraints. Due to the strict convexity of the energy-rate function, and the assumption of a fixed channel gain during a slot, equalizing transmission durations always results in less transmission energy. Therefore, unequal per-bit transmission durations in a slot are always sub-optimal. 3.2.1 The Optimality Conditions The problem in (3.1) is a convex problem with linear inequality constraints. Thus, the Karush-Kuhn-Tucker (KKT) conditions are sufficient for optimality [Ber95]. The La- grangian function is defined by L(~ r,λ r ,~ μ,~ g)= P M+D−1 m=1 f(r m ,g m )+λ r P M+D−1 i=1 r i τ s − P M+D−1 i=1 B i + P M m=1 μ 1,m h 1,m (~ r)+ P M+D−2 m=D μ 2,m h 2,m (~ r)+ P M+D−1 m=1 μ 3,m h 3,m (~ r) (3.2) where λ r , μ 1,m , μ 2,m and μ 3,m are Lagrange multipliers for conditions 1, 2, 3, and 4, respectively, and                h 1,m (~ r) M = ( P m i=1 r i τ s − P m i=1 B i )≤0 (causality), h 2,m (~ r) M = ( P m−D+1 i=1 B i − P m i=1 r i τ s )≤0 (delay), h 3,m (~ r) M = −r m ≤0 (non-negativity). Note that when the above three constraints are tight, slotm will be empty-ending, delay- critical, and idle, respectively. A slot is said to be delay-critical if, under a scheduling algorithm, it ends with an active delay constraint condition, i.e., h 2,m (~ r)=0. If the slot is non-idle, the last transmitted packet in a delay-critical slot is called a delay-critical packet. 76 Denote f 0 (r m ,g m ) as the derivative of f(r m ,g m ) with respect to r m , then there exist unique Lagrange multipliers λ ∗ r , μ ∗ 1,m ≥ 0, μ ∗ 2,m ≥ 0 and μ ∗ 3,m ≥ 0, such that the following optimality conditions hold: f 0 (r ∗ m ,g m )+λ ∗ r τ s −μ ∗ 3,m =                        τ s [− P M i=m μ ∗ 1,i + P M+D−2 i=D μ ∗ 2,i ],m=1,··· ,D τ s [− P M i=m μ ∗ 1,i + P M+D−2 i=m μ ∗ 2,i ],m=D+1,··· ,M τ s P M+D−2 i=m μ ∗ 2,i , m=M +1,··· ,M +D−2 0, m=M +D−1 (3.3) Note that the complementary slackness condition [Ber95] holds μ ∗ l,m h l,m (~ r ∗ )=0,1≤l≤3, for all feasible m (3.4) Thatis, foreachl andm, whenevertheconstrainth l,m (~ r ∗ )≤0isslack, i.e.,h l,m (~ r ∗ )<0, we must have μ ∗ l,m =0. Similarly, when μ ∗ l,m >0, we must have h l,m (~ r ∗ )=0. Under a time-invariant channel, the optimal offline scheduling observes no idling periods (see Section 2.3). This comes from the assumption that the energy function is an increasing function of the transmission rate, and subsequently, the total transmission energy can always be reduced by increasing the transmission duration (hence reducing the transmission rate) for one or more packets. However, over a time-varying channel, even though there is data in the queue, the optimal offline schedule may choose to not 77 transmit in a slot when future channel states are more energy-efficient. Hence, idle slots maybecomenecessary. Forinstance, considerasimpleexampleofschedulingtwopackets of finite sizes with D = 2, such that there are three slots. Assume the channel gains are given by [0 + ,∞ − ,∞ − ], i.e., the channel gain is arbitrarily small for the first slot, and arbitrarilylargeforthesecondandthethirdslots. Also,assumetheenergy-ratederivative function f 0 (r,g) is positive at r = 0. Obviously, it is optimal not to transmit any data over the first slot, which results in an idle slot. Besidesthestandardcomplementaryslacknessconditiongivenby(3.4),wealsonotice three additional complementary slackness conditions for this particular problem:                1):μ ∗ 1,m μ ∗ 3,m =0, m=1,··· ,M 2):μ ∗ 1,m μ ∗ 2,m =0, m=D,··· ,M 3):μ ∗ 2,m μ ∗ 3,m+1 =0, m=D,··· ,M +D−2 (3.5) That is: 1) any slot m∈ [1,··· ,M] can not be idle and end with an empty buffer at the same time (as the slot will have at least one packet, the one that arrived in that slot); 2) any slot m∈[D,··· ,M] can not be a delay-critical slot and end with an empty buffer at the same time (due to the FIFO constraint and the fact D≥ 2); 3) and if slot m+1 is idle, slot m can not be delay-critical (otherwise, a delay constraint violation will occur). The detailed proof of (3.5) can be found in Appendix A.8. It is worth emphasizing that since it is assumed that data arrives at the beginning of each slot m ∈ [1,··· ,M], an empty-ending slot means that all data is cleared at the end of that slot (while new data will arrive immediately after, i.e., at the beginning of the next slot, except for slot M.). 78 WewillnowcharacterizethepropertiesoftheseoptimalLagrangianmultipliers,which are crucial to developing a recursive algorithm to obtain the optimal offline scheduling. This is done via comparison of the derivatives of the energy-rate function over adjacent slots. As we will see, the properties of the optimal derivatives herein are similar to the properties of the optimal transmission durations or optimal transmission rates for the static channel case [ZM05][CM06]. First, consider the difference between the m-th equation and the (m+1)-th equation in (3.3), i.e., Δf 0 ∗ m,1 M = f 0 (r ∗ m ,g m )−f 0 (r ∗ m+1 ,g m+1 ). We have Δf 0 ∗ m,1 =μ ∗ 3,m −μ ∗ 3,m+1 +                −τ s μ ∗ 1,m , m=1,··· ,D−1 −τ s μ ∗ 1,m +τ s μ ∗ 2,m , m=D,··· ,M τ s μ ∗ 2,m , m=M +1,··· ,M +D−2 (3.6) Combining (3.4), (3.5), and (3.6), the following Lemma is straightforward: Lemma 3.2 When the optimal transmission rates during two adjacent slots m and m+1 are strictly positive, i.e., both slots are non-idle, we have: 1. Δf 0 ∗ m,1 =−τ s μ ∗ 1,m ≤0, if m∈[1,··· ,M−1] and slot m ends with an empty buffer 1 ; 2. Δf 0 ∗ m,1 = τ s μ ∗ 2,m ≥ 0, if m ∈ [D,··· ,M + D− 2] and slot m ends with a delay critical packet; 3. Δf 0 ∗ m,1 =0, if m=1,··· ,M+D−2, and slot m ends neither with an empty buffer nor with a delay-critical packet. 1 Note that if slot m ≥ M ends with an empty buffer, all the subsequent slots will be idle since no arrivals are assumed after slot M. 79 Proof: See Appendix A.9. Thus, this Lemma completely characterizes the Lagrangian multipliers μ ∗ 1,m and μ ∗ 2,m via Δf 0 ∗ m,1 when both slots m and m+1 are not idle. Remarks: These properties in optimal transmission rates can be interpreted as fol- lows. While an active causality constraint may result in a lower transmission rate than an otherwise more energy-efficient rate (non-increasing derivatives between two adjacent non-zero rate slots), an active individual delay constraint may require a higher transmis- sionratethananotherwisemoreenergy-efficientone(non-decreasingderivativesbetween two adjacent non-zero rate slots). When both constraints are inactive, the optimal trans- mission rate is chosen to achieve the best possible energy efficiency purely depending on the channel states (zero derivative difference, and hence a constant transmission rate in case of a time-invariant channel). Now let us consider the case when only one slot in the pair {m,m+1} is idle. The following Lemma characterizes the properties of Δf 0 ∗ m,1 in such scenarios: Lemma 3.3 When one and only one of the two slots in the pair{m,m+1} is idle (i.e., zero transmission rate), we have: 1. Δf 0 ∗ m,1 =μ ∗ 3,m +τ s μ ∗ 2,m 1 m−D ≥ 0, if m∈ [1,M +D−2] and slot m is idle but slot m+1 is not; 2. Δf 0 ∗ m,1 =−μ ∗ 3,m+1 −τ s μ ∗ 1,m 1 M−m ≤0, if m∈[1,M +D−2] and slot m+1 is idle, but slot m is not. Proof: See Appendix A.9. 80 where the indicator function 1 n is 1 when n≥ 0 and 0 otherwise. In other words, even with a zero transmission rate, idle slots still have derivatives no less than those of the non-idling neighboring slots. Incaseoftwoormoreconsecutiveidleslots,thecomparisonofthederivativesbetween two idle slots would no longer provide much information, as Δf 0 ∗ m,1 can be either non- negative or negative. However, we can still compare any idle slot with its two closest non-zero rate slots. Note that the number of consecutive idle slots has to be no more thanD−1toavoiddelayviolations. DefineΔf 0 ∗ m,l M = f 0 (r ∗ m ,g m )−f 0 (r ∗ m+l ,g m+l ),D >l≥1 and m+l <M +D. Note that Δf 0 ∗ m,l = m+l−1 X j=m Δf 0 ∗ j,1 . Using this and (3.6), we obtain Δf 0 ∗ m,l = μ ∗ 3,m −μ ∗ 3,m+l −τ s P l−1 i=0 μ ∗ 1,m+i 1 M−m−i +τ s P l−1 i=0 μ ∗ 2,m+i 1 m+i−D , (3.7) for m=1,··· ,M+D−2, where we have assumed slots m+1 to m+l−1 are idle. Now we can have results similar to Lemma 3.3: Lemma 3.4 When there are two or more consecutive idle slots (i.e., l≥2), we have: 1. Δf 0 ∗ m,l = μ ∗ 3,m +τ s P l−1 i=0 μ ∗ 2,m+i 1 m+i−D ≥ 0, if m ∈ [1,··· ,M +D−3] and slots m,m+1,··· ,m+l−1 are idle but slot m+l is not; 2. Δf 0 ∗ m,l =−μ ∗ 3,m+l −τ s P l−1 i=0 μ ∗ 1,m+i 1 M−m−i ≤0, if m∈[1,··· ,M+D−3] and slots m+1,m+2,··· ,m+l are idle, but slot m is not. 81 Proof: See Appendix A.9. That is, the comparison of an idle slot to its closest non-zero rate slots provides the same information as in the case of{m,m+1} when one of them is idle. Remarks: Lemma 3.3 and Lemma 3.4 indicate that an idle slot occurs when the channel is bad relative to its neighbors such that it is more efficient not to transmit in this slot. The occurrence of consecutive idle slots can be interpreted as a burst of deep fades, relative to the neighbors, when no delay constraints will be violated. This is demonstrated via simulations in Section 3.4. We can now generalize Lemma 3.2 to any pair of slots {m,m+l}, l ≥ 2, which are separated by one or more idle slots: Lemma 3.5 When the optimal transmission rates during two slots m and m+l, l≥ 2, m+l <M +D, are strictly positive (i.e., r ∗ m >0 and r ∗ m+l >0), but all slots in between are idle (i.e., r ∗ m+i =0 for i=1,··· ,l−1), we have: 1. Δf 0 ∗ m,l =−τ s P l−1 i=0 μ ∗ 1,m+i 1 M−m−i ≤ 0, if m∈ [1,··· ,M−1] and slot m+l−1 is not a delay-critical slot; 2. Δf 0 ∗ m,l = τ s P l−1 i=0 μ ∗ 2,m+i 1 m+i−D ≥ 0, if m ∈ [D,··· ,M +D−3] and slot m does not end with an empty buffer; 3. Δf 0 ∗ m,l =0, if m=1,··· ,M +D−3, and slot m does not end with an empty buffer and slot m+l−1 is not a delay-critical slot. Proof: See Appendix A.9. That is, it is as if all the idle slots in between can be ignored when determining the difference of the derivatives of two non-idle slots. 82 Remarks: Lemma 3.5 has a constrained ‘water-filling’ interpretation. The optimal offline scheduling tries to equalize the derivative f 0 (r m ,g m ) as much as possible (i.e., Δf 0 ∗ m,l = 0), subject to the causality and individual delay constraints. In slots where equalization is not justified from an energy-efficiency perspective, these slots should be idle instead. Thus, the derivative f 0 (r m ,g m ) can be viewed as a measure of relative cost of choosing a transmission rate r m given a channel state g m . In static channels, the energy-rate derivative function degenerates to f 0 (r m ), in which case a constant relative cost translates into a constant transmission rate. This is consistent with the efforts of equalizing transmission rates or transmission durations by the optimal offline scheduling over time-invariant channels as discussed in Section 2.3. This important observation, indeed, motivates a simple online scheduler design, as will be discussed in Section 3.3. Lemmas 3.2, 3.3, 3.4, and 3.5 provide us with a complete characterization of the optimal Lagrangian multipliers λ ∗ r , ~ μ ∗ 1 , ~ μ ∗ 2 , and ~ μ ∗ 3 in all possible scenarios. Next, we will exploit these optimality conditions to develop a recursive optimal offline scheduling search algorithm. 3.2.2 A Recursive Search Algorithm WewillfirstbrieflyreviewtheFlowRightalgorithmdevelopedin[UBG04]fortheoptimal offlineschedulingalgorithmunderasingle transmission deadline model, beforepresenting the recursive algorithm for the individual delay constraint model. The FlowRight algo- rithm in [UBG04] is based on optimizing rates over two adjacent slots {m,m+1}. The per-pair optimization is done for all pairs m=1,··· ,M +D−2, from left to right, and a pass is completed when m=M +D−2. The recursive algorithm is performed with as 83 many passes as needed until a desirable accuracy has been reached. It was proven that such an algorithm converges, and it converges to the optimal offline scheduler [UBG04]. The name of ‘FlowRight’ comes from the fact that the derivatives of two neighboring non-zero rate slots are monotonically non-decreasing, as can be seen from Lemma 3.2 and Lemma 3.5 when the impact of individual delay constraints is eliminated (i.e., no delaycriticalpacketsinanyslotm∈[1,··· ,M+D−2]). Thismeansthatintheper-pair optimization, the bits are always moved from left to right such that the transmission of earlier arrived bits can be postponed for better channel states. This information flow is limited by channel states and the causality constraint, but never from right to left. Herewedeveloparecursivealgorithmfortheoptimalofflineschedulefortheindividual delay constraint model. The algorithm is referred as a Constrained FlowRight algorithm and it differs from the original FlowRight algorithm by incorporating: • Idling slots (which was not explicitly addressed in [UBG04]), and • Individual delay constraints. To achieve this, we can re-write the delay constraint condition in (3.1) (Condition 3) as: r m ≥        0, m=1,··· ,D−1 ( P m−D+1 i=1 B i /τ s − P m−1 i=1 r i ) + , m=D,··· ,M +D−1. (3.8) where (x) +M = max{0,x}. Now the Constrained FlowRight algorithm can be developed as follows: 1. Initialize the rates for each slot as ~ r 0 = ~ B/τ s . Set k =0. 2. In increasing order, for each m∈[1,··· ,M +D−2], do the following: 84 i) Minimize f(r)+f(r total m,m+1 −r) subject to the following constraints: a) r total m,m+1 = r k m +r k m+1 ; b) r ≥ 0 (non-negativity); c) r is upper-bounded by some value that can be computed by the causality constraint (see Condition 2 in (3.1)) using the rate vector [r k 1 ,r k 2 ,··· ,r k m−1 ] 2 ; and d) r is lower-bounded by some value that can be computed by the delay constraint (3.8) using the rate vector [r k 1 ,r k 2 ,··· ,r k m−1 ]. Denote the above optimal solution as r ∗ m and let r ∗ m+1 M = r total m,m+1 − r ∗ m . Update {r k m ,r k m+1 } in ~ r k to{r ∗ m ,r ∗ m+1 }. ii) If r ∗ m+1 = 0, repeat the same per-pair rate optimization for slots {m,m+l} 3 , l≥2, which yields{r ∗ m ,r ∗ m+l }, wherer ∗ m+l M = r k m +r k m+l −r ∗ m , untilr ∗ m+l >0, or until l =D−1 (no more than D−1 consecutive idle slots), or until m+l =M +D−1 is reached. Update{r k m ,r k m+l } in ~ r k to{r ∗ m ,r ∗ m+l }. 3. After one pass (i.e., when m = M +D−2 in the above step), set ~ r k+1 = ~ r k and k =k+1. Repeat the same procedure until k =K, such that max|~ r K−1 −~ r K |<, where 1 is arbitrarily small. Theorem 3.2 The following statements hold: 1. For each step in the above Constrained FlowRight algorithm, information always flows right (i.e., P j m=1 r k m ≤ P j m=1 r k−1 m ,∀j ≥ 1) without violating the individual delay constraints; 2. The Constrained FlowRight algorithm converges to ~ r ∗ . 2 Effectively, r≤r k m , since the information always flows right as shown in [UBG04]. 3 Note that the delay constraint has to updated by replacing m by m+l− 1 in (3.8). In addition, r ∗ m+1 =··· =r ∗ m+l−1 = 0 (idle slots). 85 Proof: To prove the first statement, we need to look at the per-pair rate optimization in the above recursive algorithm. Following a procedure similar to that in [UBG04] for thesingle deadline model,wecanprovethattheinformationalwaysflowsrightintheper- pair rate optimization. The satisfaction of the delay constraints comes from the explicit rate constraint condition of (3.8). Now for the second statement, the convergence part is due to the always right-flow of information, as also shown in [UBG04] for the single deadline model. The optimality part can be easily verified by checking the properties of the optimal Lagrangian multipliers for various scenarios as characterized by Lemmas 3.2, 3.3, 3.4, and 3.5. That is, in the k-th pass, • The optimality conditions for Δf m,1 , as specified by Lemmas 3.2 and • The optimality conditions for Δf m,1 , as specified by Lemmas 3.2 and 3.3, are guar- anteed by step 2-i) in the per-pair optimization{r k m ,r k m+1 }; • The optimality conditions for Δf m,l , l >1, when slots m+1 to m+l−1 are idle, as specified by Lemmas 3.4 and 3.5, are guaranteed by step 2-ii) in the per-pair optimization{r k m ,r k m+l } with r k m+1 =···=r k m+l−1 =0; • Step 3) guarantees that after the convergence, Δf 0 ∗ m,l is achieved, l≥1. Therefore, the optimality conditions for the optimal Lagrangian multipliers discussed earlier are all satisfied. Hence ~ r ∞ =~ r ∗ . In Section 3.4, this Constrained FlowRight algorithm will be used to search for the optimal offline scheduling to investigate its performance and properties. 86 3.2.3 Symmetry Property and Packet Delay Performance Similar to the static channel case, we also have the symmetry property of the optimal transmission rates ~ r. This important property not only provides an insight to the opti- mal scheduling algorithm, but also makes it possible to analyze the average packet delay performance into a very compact closed form for continuous-time static channels (see Theorem 2.3). For time-slotted fading channels, due to the potential existence of idling slots, we can no longer obtain a closed form solution of the average packet delay perfor- mance. However, as we will show, the symmetry property leads to a lower bound and an upper bound of the average packet delay performance. Theorem 3.3 For any M ≥ 1, when the joint probability distribution for the ran- dom packet vector [B 1 ,··· ,B M ] is identically distributed to the reversed packet vector [B M ,··· ,B 1 ] (such a property clearly holds, e.g., when B i are i.i.d.), independent of the i.i.d channel gains g m ,1 ≤ m ≤ M − 1 4 , then under the optimal offline scheduling, the optimal transmission rates r m and r M−m+1 are identically distributed. In particular, E{r m }=E{r M−m+1 }. Proof: Here we provide a sketch of the proof. The detailed proof follows the same procedure as in the proof of Theorem 2.2 for the continuous-time arrival model over a static channel, compared with a time-slotted fading channel discussed herein. The proof 4 The result also holds for any channel gains when the joint probability distribution for the random packet vector [g1,··· ,gM+D−1] is identically distributed to the reversed packet vector [gM+D−1,··· ,g1]. However, for practical interest, we will only focus on the i.i.d. channel gains in this chapter. 87 essentially relies on a time reversal argument, where a sample path trajectory of the forward running system, characterized by ~ B (f) =[B 1 ,··· ,B M ], and, ~ g (f) =[g 1 ,··· ,g M+D−1 ], is compared to a corresponding time reversed system, characterized by ~ B (r) =[B M ,··· ,B 1 ], and, ~ g (r) =[g M+D−1 ,··· ,g 1 ]. Theuniqueoptimaltransmissionratevectorfortheforwardrunningsystemcanbeshown also feasible and uniquely optimal for the corresponding time reversed system. We define the average packet delay as: ¯ q(M) M = E{ 1 M M X m=1 q m } where q m is the delay (including queuing delay and transmission delay 5 ) experienced by packet m under the optimal offline schedule with a particular realization of the channel gain vector and the packet size vector, and the expectation is taken over all channel and 5 That is, the time interval from when packet Bm arrives till when its last bit’s transmission is com- pleted, which is not necessarily aligned with slot boundaries. 88 packet size realizations. By exploiting the symmetry property, the following bounds can be obtained: Theorem 3.4 For any M ≥ 1, when the joint probability distribution for the ran- dom packet vector [B 1 ,··· ,B M ] is identically distributed to the reversed packet vector [B M ,··· ,B 1 ], independent of the i.i.d. channel gains g m ,1 ≤ m ≤ M + D− 1, then under the optimal offline scheduling, the average packet delay is bounded by τ s D 2 + 1 2D ≤ ¯ q(M)≤τ s 1+ M +1 2M (D−1) (3.9) When M → ∞, τ s (D +1/D)/2 ≤ ¯ q(∞) ≤ τ s (D +1)/2. In the case of static channels where the channel gains are fixed g m = c, ∀m, where c is a constant, the average packet delay is given by ¯ q(M) static =τ s 1+ M +1 2M (D−1), (3.10) and converges to ¯ q(∞) static =τ s (D+1)/2. Proof: See Appendix A.10. Note that the upper and the lower bounds differ only by the value τ s [(1−1/D)+(D− 1)/M]/2, which converges to τ s (1−1/D)/2 as M approaches infinity. In addition, the average packet delay performance is not a function of the packet size vector. 89 3.2.4 Majorization Property Majorization is often used to characterize the mixing degrees of vectors of the same dimension [Bha97]. For vectors of the same dimension M, a vector ~ v 1 is said to be majorized by another vector ~ v 2 , denoted as ~ v 1 ≺~ v 2 , if        P m i=1 v ↓ 1,i ≤ P m i=1 v ↓ 2,i 1≤m≤M−1 P M i=1 v 1,i = P M i=1 v 2,i m=M, (3.11) where ~ v ↓ denotes a non-increasingly ordered ~ v, i.e., v ↓ i ≥ v ↓ i+1 , ∀i. When ~ v 1 ≺ ~ v 2 , ~ v 1 is more mixed. In fact, ~ v 1 ≺ ~ v 2 if and only if P M i=1 f(v 1,i ) ≤ P M i=1 f(v 2,i ), for all convex function f(.) [Nie01]. In Chapter 2, the concept of majorization was used to prove the optimality of the optimal offline schedule for the individual delay constraint model and the single deadline model in continuous-time static channels. It was shown that the optimal offline schedule resultsinatransmissionduration(orrate)vectorthatisbestmixedandmajorizedbyany otherfeasibletransmissionduration(orrate)vectors. Notethatanyfeasibletransmission duration (or rate) vector, including the optimal one, has a fixed total duration (or rate) in static channels. In time-varying channels, the optimal offline schedule chooses to optimally equal- ize the derivatives of the energy-rate function, instead of simply the transmission rates. Thus, we examine majorization theory and the energy-rate derivative vector. However, majorization theory does not necessarily hold for the derivative vector. This is due to the fact that the sum of the derivatives over all slots for all feasible schedulers may not 90 be a constant (refer to the equality condition in (3.11)), although the total transmission rate over all slots remains fixed for all feasible schedulers. This can be easily verified, for instance, if the energy-rate function is a polynomial function. Therefore, we can no longer simply claim that the derivative vector by the optimal offline scheduling is always majorized by any other feasible derivative vectors. This is not surprising, as majorization is only a partial characterization of any two vectors. However, under some specific energy-rate functions, majorization property still holds under time-varying channels. In fact, we have the following result: Theorem 3.5 When the energy-rate function f(r,g) can be written in the form of f(r+ w 1 (g))+w 2 (g), where w 1 (g) and w 2 (g) are functions of g, the generalized rate vector ~ r gM = [r g 1 ,··· ,r g M+D−1 ], where r g m M = r m +w 1 (g m ), and hence the derivative vector ~ f 0 M = [f 0 (r g 1 ),··· ,f 0 (r g M+D−1 )], resulting from the optimal offline schedule, are majorized by any other feasible generalized rate vector and derivative vector, respectively. Proof: This can be proved by contradiction (the causality constraint would be oth- erwise violated), which is very similar to the proof for the static channel case in Ap- pendix A.1 and thus is omitted. In particular, note that the sum of the modified rates ~ r g is the same for all feasible schedulers (since g is deterministic, although time-varying, 91 for all schedulers), and subsequently, the sum of the derivatives ~ f 0 is also the same for all feasible schedulers. For instance, the Shannon-type energy-rate function f(r,g) = N 0 (2 2r −1)/g can be re- written as f(r+w 1 (g))+w 2 (g)=N 0 2 2[r+log 2 (1/g)] −N 0 /g, where w 1 (g)=log 2 (1/g) and w 2 (g)=−N 0 /g. Theorem3.5indicatesthatunderspecialenergy-ratefunctions,thebest-mixtureprop- erty of the energy-rate derivative vector for the optimal offline scheduling under fading channels can be characterized by majorization theory. However, we conjecture that the best-mixturepropertyholdsforotherenergy-ratefunctionsaswell. AsinSection3.2,this property implies that an energy-efficient scheduler should try to equalize the energy-rate derivative vector as much as possible, as will be evidenced in the next Section. 3.2.5 A Special Case: Static Channels It is of interest to note that in the special case of time-invariant channels, i.e., g i =c,∀i, where c is a constant, a closed form solution of the optimal transmission rate at each slot canbederived, followingthesameapproachasinChapter2forthecontinuous-timecase. The resulting expressions are similar to those in Chapter 2, but somewhat different due to the discrete time model assumed here. In addition, similar properties as characterized by (3.5) and Lemma 3.2 for the fading channel case can also be observed for the static channel case. This is detailed in Appendix A.11. 92 3.3 Online Scheduling over a Fading Channel In this section, we will investigate the design of online schedulers. The online schedulers assume information of the current scheduling backlog and causal channel feedback (i.e., channel states g 1 ,··· ,g m are known prior to scheduling for slot m). In addition, we also assume rough knowledge of the average packet arrival rate λ, which is known a priori or can be reasonably estimated. We will start with two simple channel-independent online schedulers, followed by an online scheduler motivated by the optimality conditions for the optimal offline scheduling discussed in Section 3.2. 3.3.1 Channel-Independent Online Schedulers One simple online scheduler is to clear the buffer completely at each slot, regardless of channelconditions. ThisisequivalenttothecaseofD =1,wherethepacketdelayisfixed at one slot. We denote this scheduler as a greedy online clearing scheduler. Obviously, such a scheduler is expected to be highly energy-inefficient. In [ZM05], the optimal algorithm of flushing a static buffer with a finite number of packets of various discrete individual delay constraints over a static channel was inves- tigated. This scheduler was later generalized to the case with dynamic packet arrivals and departures in Chapter 2. This scheduler is termed as an online flush scheduler, and compared with the optimal offline scheduler via analysis and simulations in Chapter 2. For convenience, the online flush scheduler is also briefly described here. Assume at slot m, the buffer is characterized by U m,total = U m,0 +U m,1 +U m,2 +···+U m,D−1 , where 93 U m,i denotes the number of buffered bits that arrived at slot m−i and thus has a delay constraint of D−i. The transmission rate at slot m is chosen to be: r flush m = max i∈[1,···,D] P D i=1 U m,D−i iτ s . (3.12) This scheduler guarantees that all packets meet their own delay constraints, due to the FIFO assumption and the fact that r flush m ≥ U m,D−1 /τ s . Note that this online scheduler is independent of channel conditions, and tries to utilize the buffer backlogs as much as possible (without anticipation of future arrivals). 3.3.2 A Channel-Dependent Online Scheduler In Section 3.2, it is observed that the optimal offline scheduling tries to equalize the derivatives of the energy-rate function for all slots as much as possible, subject to the causalityanddelayconstraints. Whenthechannelconditionsarerelativelybad, theopti- mal offline scheduling chooses not to transmit in the corresponding slots. This motivates an online scheduler which tries to keep a constant derivative value as much as possible. We refer this scheduler as the Derivative Directed or DD online scheduler. Theconstantvalueofthederivativesthattheoptimalofflineschedulingtriestomain- tain primarily depends on two factors, namely, traffic variations and channel states (cur- rent value and its variability). In addition, the derivative observed by the optimal offline schedulingisafunctionofthedelayconstraintDaswell,asthedelayconstraintmayforce a change in the derivative values. As D grows, the delay constraint becomes increasingly inactive. 94 Traffic variations can be smoothed out by considering current backlog information and delay constraints. The optimal static buffer flushing algorithm is realized by (3.12). Obviously, the transmission rate in slot m is upper bounded by r DD m,max =U m,total /τ s , and lower bounded by U m,D−1 /τ s , where U m,D−1 the number of buffered bits arrived D−1 slots earlier. If both the channel states and packet arrivals stay unchanged, we can choose the transmission rate by (3.12). However, if one of them is random, we may want to choose a different rate. For instance, if the channel states continuously improve (or deteriorate), we choose a rate smaller (or larger) than the rate by (3.12). To account for it, we introduce a forgetting factor α≥0 and re-write (3.12) as: r DD m,min = max i∈[1,···,D] P D i=1 U m,D−i iτ s ! α i−1 . (3.13) We will lower-bound the transmission rate at slot m by (3.13). Note that when α = 0, r DD m,min reduces to U m,D−1 /τ s . When α =1, r DD m,min reduces to (3.12). Channelvariationscanbeaccommodatedbychoosingatransmissionrateasafunction of the latest estimated derivative value ˆ f 0 m−1 and current channel state g m , and updating the estimated derivative accordingly. To be more specific, let r = v(f 0 ,g) as the inverse of f 0 (r,g). The transmission rate at slot m can be chosen by: r DD m,channel =v( ˆ f 0 m−1 ,g m ). 95 For instance, for a Shannon-like energy-rate function, if f(r,g)=N 0 (2 2r −1)/g, we have f 0 = N 0 2 2r ln2/g and r = 0.5log 2 (f 0 g/(N 0 ln2)). After incorporating traffic induced rates r DD m,max and r DD m,min , we have r DD m =max{min{r DD m,channel ,r DD m,max },r DD m,min }. (3.14) The new derivative due tor DD m can be computed byf 0 (r DD m ,g m ). A filtering function can be implemented to update the estimated derivative value ˆ f 0 , e.g., by ˆ f 0 m =        β ˆ f 0 m−1 +(1−β)f 0 (r DD m ,g m ), if r DD m >0, ˆ f 0 m−1 if r DD m =0, (3.15) where 0 < β < 1 is the forgetting factor. Note that the derivative value remains un- changed in case of idling slots. This is because an idling slot is due to a bad channel state and its energy-rate derivative value may be too large. Without loss of generality, assume the average channel gain is 1. The following sum- marizes the online DD scheduling algorithm: 1. Initialize ˆ f 0 0 =f 0 (r avg ,1), wherer avg is the known or estimated average arrival rate. 2. For each slot m=1,··· ,M +D−1, do the following: i) Determine r DD m by (3.14). ii) Update ˆ f 0 m by (3.15). Lemma 3.6 The above DD online scheduler guarantees the satisfaction of the causality and the individual delay constraints. 96 Proof: The causality constraint is explicitly guaranteed by r DD m,max . The satisfaction of the individual delay constraints is guaranteed by the FIFO assumption and the fact that r DD m,min ≥ U m,D−1 in (3.13) for α ≥ 0, where U m,D−1 denotes the buffered packets that must be delivered by the end of slot m. Note that zero-rate transmissions occur when v( ˆ f 0 m−1 ,g m )≤ 0 and r DD m,min = 0. For instance, if f(r,g) = N 0 (2 2r −1)ln2/g, the channel threshold under or at which an idle slot occurs is proportional to 1/f 0 . Generally, we expect f 0 ∝ 2 2ravgτs . Therefore, when the average transmission rate is small, f 0 is close to 1. Thus the channel threshold is relatively high and we may choose to transmit only in good channel conditions. On the other hand, when high transmission rates are necessary, f 0 becomes very large, and the channel threshold is close to 0. In this case, the DD online scheduler tends to transmit over all channel states. Such a phenomenon is similar to that of the optimal offline scheduler, as will be shown via simulations. 3.4 Numerical Results Without loss of generality, we assume a slot duration is one unit (i.e., τ s = 1). The packet sizes are normalized based on frequency bandwidth and slot duration and can be interpreted as the number of bits per channel use. A large B typically corresponds to a high SNR operating point. The energy-rate function is assumed to be f(r)=(2 2r −1)/g [UBG04], resulting from Shannon capacity, where r is the transmission rate and g is the channel gain. Unless specified, the packet sizes are assumed to be random and follow an exponentialdistribution. ThechannelcoefficientsareassumedtobeRayleighdistributed. 97 We will first illustrate the properties of the optimal transmission durations and packet delays, followed by the performance comparison of the optimal offline scheduler and the online schedulers. 3.4.1 Properties of Optimal Offline Scheduling Figure 3.1 illustrates one example run of the optimal offline schedule for the individual delay constraint model. The normalized packet size is very small and fixed at B = 0.1 such that the energy-rate function approaches a linear relationship. It can be observed that the optimal schedule chooses to transmit only in good channel conditions, and stays idle in bad channels. Indeed, it is not difficult to see that, under a strict linear energy- rate function, the optimal schedule would choose to transmit packet m only in the best slot m opt ∈ [m,··· ,m+D−1] which has the largest channel gain. Such a threshold- like scheduling was also observed in other different delay/power tradeoff settings, e.g., [CC99],[FMT06]. The derivatives of the energy-rate function for the non-idling slots are also shown. The derivatives under the optimal offline scheduling exhibit a stair-case property, which tends to remain unchanged while in response to active causality and delay constraints. The idling slots have larger derivatives (not shown), such that the ‘water-filling’ rule prohibits transmissions in these slots. Incontrast,whenthenormalizedpacketsizeisverylarge(fixedatB =5),theoptimal offline scheduling tends to transmit over all slots (with rates proportional to channel gains), and approaches a constant transmission rate in the limit. This is illustrated in Figure 3.2. This is not surprising as when B increases, the energy-rate function f(r) = (2 2r −1)/g isincreasinglydominantbyr,whilethechannelgainhasandecreasingimpact. 98 Thus,theidlingconstraintbecomesmoreslack,anditismorejustifiedtotransmitpackets overallslots, withtransmissionratesproportionaltothechannelgains. Inthisparticular example, the optimal derivatives of the energy-rate function are the same for all slots, except the last one, which has a smaller value due to an active delay constraint at slot M +D−2 (the second slot to the last). Figure3.3showstheaverage(overindependentruns)optimaltransmissionrateswhen M = 100 with dynamic packet sizes. The single transmission deadline model, which as- sumes the same duration (M+D−1), packet sizes (B i >0, fori=1,··· ,M, andB i =0 for i = M + 1,··· ,M + D− 1) and channel gains as the individual delay constraint model, is shown for comparison. As can be seen, the optimal transmission rate vector for the individual delay constraint model exhibits the symmetry property. The maximum transmission rate occurs in the middle of the packets. This property is similar to the case of dynamic arrivals with fixed channel states as in Chapter 2, in which the opti- mal transmission duration vector exhibits an inversed property. The single transmission deadline model, on the other hand, exhibits a different property. Its average optimal transmission rates are monotonically increasing, similar to the decreasing trend of the optimal transmission durations for static channels [UBPG02]. This is because earlier packets can exploit more future arrivals and potentially postpone some transmissions for better channel states. However, this may lead to significant individual packet delays, as analytically shown in Chapter 2 for static channels. Figure 3.4 shows the average packet delay associated with the optimal transmission rates, when the average normalized packet sizes ¯ B is fixed at 1. The average delay for the individual delay constraint model increases with packet index m, and maximizes 99 Figure 3.1: An example run of the optimal offline schedule for the individual delay con- straint model, M =100, D =10, and B =0.1. when m = M. The average packet delay, including the transmission delay, is close to (D+1)/2 when m is approximately M/2. The delay for the single transmission deadline model, however, peaks in the middle of the packets for small Ds, and is much higher than that of the individual delay constraint model. In the case of D = 10, since the last D−1=9 slots are assumed to have no packet arrivals, the average transmission rate for the single deadline model is in general less than 1, and hence the average packet delay is an increasing function of packet indices. 3.4.2 Comparison Between Offline and Online Schedulers Figure 3.5 shows the average transmission energy performance for the offline and online schedulers when M = 100 and ¯ B = 1. The filtering coefficients α and β discussed in 100 Figure 3.2: An example run of the optimal offline schedule for the individual delay con- straint model, M =100, D =10, and B =5. Figure 3.3: Average optimal transmission rates under the optimal offline schedule, M = 100, D =5, dynamic packet sizes. 101 Figure 3.4: Average packet delay under the optimal offline schedule, M =100, ¯ B =1. Section 3.3 are chosen to be 0.98 and 0.95, respectively. The single transmission deadline model is also shown for comparison. It can be seen that the online flush scheduler and the DD online scheduler achieve comparable energy performance to the optimal offline scheduler. The DD online scheduler yields a better energy performance than the online flush scheduler via efficient utilization of channel information. Both online schedulers and the optimal offline scheduler require significantly less energy than the greedy buffer clearing scheduler. As the individual delay constraint D increases, the energy required by the schedulers for the individual delay constraint model approaches that of the single deadlinemodel. Figure3.6presentstheenergycomparisonasafunctionofthenormalized packet size. Again, the DD scheduler outperforms the online flush scheduler and has comparable energy performance to the optimal offline scheduler. 102 Figure 3.5: Average transmission energy for the offline and online schedulers vs. D, M =100, ¯ B =1 Figure 3.6: Average transmission energy for the offline and online schedulers vs. ¯ B, M =100, D =5 103 Figure 3.7: Average packet delay vs D for the offline and online schedulers, M = 100, ¯ B =1. Figure 3.7 shows the average packet delay performance vs D for the offline and the online schedulers when M =100. The lower and upper bounds, derived in Theorem 3.4, are also shown. The DD online scheduler, on average, results in a delay performance slightly less than that of the optimal offline scheduler. It is worth noting that our goal is to optimize transmission energy instead of delay. Thus, it is not surprising that the DD online scheduler consumes more energy than the optimal offline scheduler, but yields less packet delay. The online flush scheduler, however, achieves a delay performance noticeably higher. The average packet delay for the individual delay constraint model is much less than that of the single deadline model, especially for small delay constraints. 104 3.5 Conclusions This chapter focuses on energy-efficient packet transmission with individual packet de- lay constraints over a fading channel, as an extension of the work in Chapter 2 for a continuous-timearrivalmodelandstaticchannels. Theproblemofoptimalofflineschedul- ing (vis-` a-vis total transmission energy), assuming information of all packet arrivals and channel states before scheduling, is formulated as a convex optimization problem with linear constraints. The optimality conditions are analyzed, from a recursive search al- gorithm is obtained. The symmetry property of the optimal transmission rate vector (or, equivalently, the transmission duration vector) still holds under the i.i.d. assump- tion of packet sizes and channel coefficients. Combining the symmetry property with the potential idling periods, upper and lower bounds of the average packet delay (includ- ing queuing and transmission delays) are derived. The properties of the optimal offline scheduling and the impact of packet sizes, individual delay constraints, and channel vari- ations are demonstrated via simulations. A heuristic online scheduling algorithm, which assumes both causal traffic and channel information, is proposed, which achieves compa- rable energy and delay performance to the optimal offline scheduling, as demonstrated by simulations. 105 Chapter 4 Delay-Constrained Energy-Efficient Transmissions over a Multihop Link 4.1 Introduction In this chapter, we focus on energy-efficient packet transmission over a multihop link, subject to two delay constraint models, namely, the single transmission deadline model and the individual packet delay constraint model. We will consider a static channel first, followed by an extension to the fading channel case. This can be viewed as an extension of the work in [UBPG02][UBG04] for the single deadline model, and Chapters 2 and 3 for the individual delay constraint model over the single-hop link. We want to address the following questions: given a set of randomly arriving packets, how to schedule these packets for maximum energy efficiency over a multihop link subject to the underlying delay constraints? When does multihop yield better energy efficiency than the single-hop transmission? How does the delay performance of multihop transmissions compare with the single-hop transmission? For the static channel case, we begin by re-deriving the optimal offline scheduling (vis-` a-vis total transmission energy) for the single transmission 106 deadline model and for the individual delay constraint model under a time-slot TDM- relayedandFDM-relaychannel. IntheTDM-relayedchannel, anode(sourceorrelaying) is only allowed to transmit in selective time slots. Given an end-to-end delay constraint for each packet, the optimal offline schedule over multihop transmissions is derived. It turns out that the optimal offline schedule relies on a simple delay budget allocation scheme, whichallocatesthedelaybudgettothefirsthop(fromsourcetothefirstrelaying node) as much as possible. All the relaying nodes have a delay budget exactly equal to one slot and thus simply perform buffer-clearing during any transmission opportunities. This optimal allocation is independent of traffic characteristics (e.g., number of packets, packetsizes,burstiness, etc.) andchannelcharacteristics(e.g.,pathlossexponents,inter- distancebetweennodes,etc.). Thetotaltransmissionenergyandaveragepacketdelayare analyzed and characterized. The heuristic online flush scheduling algorithm, discussed in Chapter 2 for single-hop system, is investigated for multihop transmissions. Extension to the fading channel case is also considered, in which the optimal offline scheduling over multihop channels no longer permits a simple optimal delay budget allocation, but rather depends on the specific realization of packet sizes and channel states. We provide an upper bound and a lower bound for the optimal offline schedule. The upper bound is derived by exploiting the arrival/departure properties, while the lower bound is the same simple budget allocation for the static case. The energy savings with multihop transmissions for offline and online schedulers are shown via computer simulations. It is demonstrated that energy savings via multihopping are possible, but depend heavily on factors such as multihop resource orthogonalization mode, packet scheduling algorithms, delay constraints, SNR operating regimes, and channel variations. 107 Throughout this Chapter, we use the superscript (0) to denote the single-hop trans- mission, and the superscripts (l), l ∈ [1,··· ,L], to denote the l-th hop in an L-hop transmission. This chapter is organized as follows. In Section 4.2, the system model is described. For static channels, the optimal offline scheduling algorithm at each node for the TDM- relayed and FDM-relayed transmissions is described in Section 4.3, while the optimal delay budget allocation for optimal offline scheduling over a hopping channel and its properties are presented in Section 4.4. A heuristic online scheduler is investigated in Section 4.5. Extension to fading channels is considered in Section 4.6. Numerical results are given in Section 4.7. Finally, some concluding remarks are drawn in Section 4.8. 4.2 System Model 4.2.1 Traffic Model and Time-Slotted Channel The time slotted model for the single transmission deadline model is shown in Figure 1.4, while Figure 1.5 illustrates the time-slotted individual delay constraint model. In this chapter, we will mainly focus on the individual delay constraint model. The single dead- line model will also be briefly treated as a special case. A set of M packets are transmitted through an L multihop channel, as shown in Figure1.6. Wewillprimarilyfocusonstaticchannels,inwhichthechannelgainsarefixed at each hop. However, the channel gains may be different for different hops, depending on the distance between the transmitter and the receiver, the path loss exponent, etc. We denote the fixed channel gain at hop l by g (l) , l∈ [0,··· ,L], where l = 0 represents 108 the direct transmission, and l = [1,··· ,L] is the l-th hop in a L-hop transmission. We will also briefly extend our results to the fading channel case. The goal of the optimal offline schedule, which assumes perfect knowledge of packet sizesand,inthecaseoffadingchannels,channelstates,fortheentireduration[0,··· ,M+ D−1]beforescheduling,istochoosetheoptimaltransmissionrater (l) i (numberofbitsper slotduration),0≤l≤L,foreachslotiandhopl,suchthatthetotaltransmissionenergy of these M packets is minimized while the underlying delay constraints are satisfied. The case when l = 0, i.e., r (0) i , refers to the single-hop transmission case, while l = 1,··· ,L correspond to the hopped transmissions in the L-hop case. Online schedulers, on the other hand, only assume information of the current scheduling backlog. As in Chapters 2 and 3, all schedulers are assumed to follow the first-in-first-out (FIFO) service rule, and the causality constraint. In addition, for static channels, non-idling scheduling is assumed as in Chapter 2. The energy-rate functions, denoted by w (l) (r) for a certain transmission rate r over different hops l ∈ [0,··· ,L], are not necessarily the same. For any hop l ∈ [0,··· ,L], its energy-rate function w (l) (r) is assumed to be strictly convex and monotonically increasing in the transmission rate r. Without loss of generality, the processing delays at each node are ignored such that a packet received at the end of a slot m by a node is available for transmission immediately at the next slot m + 1. It is also assumed that scheduling is only performed at slot boundaries. Thus, the minimum packet delay is at least one slot for each hop, and subsequently, the minimum end-to-end delay is L slots. Therefore, to ensure feasibility, we need D≥ L. In addition, since optimal scheduling for D = L is trivial, we will focus on D >L in the sequel. Note that in an L-hop transmission, the maximum packet delay 109 (including queuing and transmission delays) at any hop l∈ [1,··· ,L], for any packet, is no more than D−L+1 slots. 4.2.2 Multihop Access Mode We consider two types of orthogonal division multiplexing schemes. The first one is TDM, in which case, at any point of time at most one node is transmitting, and there is no interference from any other nodes’ transmissions. Given a total number of M +D−1 slots, theoptimalpercentageoftimeanodeisallowedtotransmitmaydependonfactors such as the topology of the relaying system, channel variations, etc. However, here we consider a simple TDM scheme, in which all nodes (source and relaying) have equal share of partial transmissions following a round-robin pattern, as illustrated in Fig. 4.1 for a 3-hop scenario, where slots are labeled with different colors indicating whether a slot is restricted for transmission or not. In this particular example, it is assumed that mod(M+D−1,L)=0, wheremoddenotestheModulooperation, suchthatateachslot, there is one and only one hop in transmission. If mod(M+D−1,L)=d>0, we assume that the lastd slots are not used by any hops, such that the length of the scheduling span (SS) (i.e., the duration from the start of the first non-restricted slot till the end of the last non-restricted slot) is the same for any hop l ∈ [1,··· ,L]. To be more specific, the scheduling span for hop l is given by: SS (l) TDM ={l,··· ,L(b(M +D−1)/Lc−1)+l}. (4.1) 110 whereb.c denotes the floor(.) operation. The hop-independent length of the scheduling spanisL(b(M +D−1)/Lc−1)+1slots. Thenon-restrictedslotsm∈[1,··· ,M+D−1] for hop l satisfy the condition that mod(m−l,L) = 0. That is, the l-th hop is allowed to transmit during slots [1,L + 1,2L + 1,···] within its scheduling span SS (l) TDM (see Fig.4.1). Inotherwords,thetransmissionopportunitiesarehop-independentconditioned on SS (l) TDM . To characterize this conditional independence, we introduce an indicator vector ~ I TDM , of dimension L(b(M +D−1)/Lc−1)+1, to indicate whether a slot is available for transmission (1) or not (0), conditioned on SS (l) TDM , in an L-hop link, i.e., ~ I TDM =[1,0,··· ,0 | {z } L−1 ,1,0,··· ,0 | {z } L−1 ,··· ,1,0,··· ,0 | {z } L−1 ,1], (4.2) Note that there are in total b(M +D−1)/Lc transmittable slots per-hop. It is easy to verify that the indicator vector ~ I TDM is symmetric, i.e., I TDM,i =I TDM,L(b(M+D−1)/Lc−1)+2−i ,∀i∈[1,··· ,L(b(M +D−1)/Lc−1)+1]. Therefore, the combination of SS (l) TDM and ~ I TDM completely characterizes the TDM access mode for all hops l∈[1,··· ,L] in an L-hop link. The second type of multiplexing is assumed to be FDM, in which resource orthog- onality is achieved in the frequency domain. Due to resource orthogonality, each hop’s transmissionisinterference-free. Allnodesareallowedtotransmitatanytime. However, due to the minimum one-slot delay assumption between two adjacent hops, hop l can not 111 Figure 4.1: TDM Mode for Multihop Transmission (L=3). start its transmission until slot l, while it has to finish its transmission at the end of slot M +D−1−L+l such that the delay constraint is not violated, i.e., SS (l) FDM ={l,··· ,M +D−1−L+l}. (4.3) Again, the length of the scheduling span is the same for all hops l ∈ [1,··· ,L] and is given by M +D−L slots. The indicator vector in this case is an all-one vector, i.e., ~ I FDM =[1,··· ,1 | {z } M+D−L ], (4.4) which is symmetric as well. The combination{SS (l) FDM , ~ I FDM } completely characterizes the FDM access mode for all hops l∈[1,··· ,L] in an L-hop link. In the sequel, we will first consider optimal offline scheduling over static channels on a per-hop basis, taking into account the potential existence of restricted slots within each hop, followed by optimal offline scheduling over multihop transmissions. A heuristic 112 Figure 4.2: FDM Mode for Multihop Transmission (L=3). online schedule is also considered. We will then extend the results to the fading channel case. 4.3 Per Hop Optimal Offline Scheduling over Static Channels Inthissection, wewillderivetheoptimalofflineschedulingateachnode, givenafixedset of transmission opportunity slots (as indicated by ~ I as in (4.2) and (4.4)), packet sizes, and a delay constraint. The derivation is essentially based on the closed form solution of the optimal offline scheduling in a time-slotted unrestricted system as outlined in Appendix A.11 for a single-hop transmission. Note that in the single hop case, a node is allowed to transmit during any slots. The optimal transmission rate at any slot m, 1 ≤ m ≤ M +D−1, is given by (A.17). The optimal transmission rate vector~ r M = [r 1 ,··· ,r M+D−1 ] observes a symmetry property, when the vector of packet sizes are symmetric themselves, i.e., [B 1 ,··· ,B M ] has a joint probability distribution that is identical to that of its reversed vector [B M ,··· ,B 1 ]. In 113 particular, the optimal transmission rates r m and r M+D−m are identically distributed. This symmetry property leads to a simple and exact solution of the average packet delay performance given by ¯ q(M)=τ s 1+ M +1 2M (D−1) . (4.5) In the following, we will derive the optimal offline scheduling for a set of randomly arrived packets under a general indicator function ~ I and a general per-hop individual delay constraint, denoted by D I ≥2 (an integer multiple of slots) 1 . The scheduling for a specific indicator vector (e.g., ~ I TDM in (4.2) or ~ I FDM in (4.4)) and a delay constraint can thenbereadilyobtained. Notethattheper-hopschedulepresentedhereincorrespondsto asub-problemwithinanend-to-endmultihoplink. Clearly, theper-hopdelayconstraint 2 D I satisfies D I ≤D, where D is the end-to-end delay constraint for a given packet. The full multi-hop problem will treated in the next section. Given M packets and a delay constraint of D I slots, the length of ~ I is M +D I −1. Again, similar to the single-hop case, we assume that B i > 0,i ∈ [1,··· ,M] and B i = 0,i∈[M+1,··· ,M+D I −1]. Thus, thenumberofconsecutivezerosin ~ I cannotexceed D I −1. Otherwise, delay violations would occur. Note that in the non-restricted single- hop transmission case, packet m, with a delay constraint of D slots, has transmission opportunities in D slots starting from slot m. However, in the case of slot-selective transmissions, packet m, subject to a delay constraint of D I slots, has transmission opportunities only in P m+D I −1 i=m I i ≤D I slots starting from slot m. Therefore, instead of 1 The case when D I = 1 is trivial and hence not discussed here. 2 Itmayseemthatitisnecessarytoconsider unequalpacketdelayconstraintsinoptimalofflineschedul- ing for a given hop as part of the multi-hop link. As we will show in Section 4.4, it is sufficient to consider equal packet delay constraints for any hop in a multihop link. 114 trying to equalize the transmission rates as much as possible over a span of D I slots for each packet m, the equalization by the optimal offline schedule should now be performed over a number of P m+D I −1 i=m I i slots instead. Obviously, when slot m is restricted for transmission (i.e., I m =0), we have to choose a zero transmission rate. For a non-restricted slot m, assume that the buffer can be characterized by U I m =U m,1 +U m,2 +···+U m,D I −1 , where U I m is the total buffer size before slot m (i.e., excluding the packets arrived at slot m), and where U m,i denotes the number of buffered bits that arrived at slot m−i. The buffered bits U m,i have a delay constraint of D−i, 1≤ i≤ D−1. Recall that for the single-hop case (see Appendix A.11), the optimal transmission rate during slot m, 1≤m≤M +D−1, is given by r m = min m≤i≤M+D−1 r m[i] , (4.6) where r m[i] ,i∈[m,··· ,M +D−1], is given by r m[i] =max n Um+ P i l=m B l (i−m+1)τs , U m,D−1 τs , U m,D−2 +U m,D−1 2τs , ··· , P D−1 l=1 U m,l (D−1)τs , Um+Bm Dτs , Um+Bm+B m+1 (D+1)τs ,··· , Um+ P i−1 l=m B l (D+i−m−1)τs , (4.7) where the buffer before slot m is similarly characterized by U m = U m,1 +U m,2 +···+ U m,D−1 . 115 Thus, for the slot-selective transmission case, the first term on the right hand side (RHS) of (4.7) can be re-written as U I m + P i j=m B j ( P i j=m I j )τ s , where the denominator (the total number schedulable slots) has been changed from (i− m+1)τ s to( P i j=m I j )τ s , accountingforthepotentiallyrestrictedslotsfromslotmtoslot i. Similarly, for the delay constraint terms on the RHS of (4.7), we need to equalize the transmission rates over the number of non-restricted slots spanning from slot m to slot m+j−1 (upper bounded by M +D I −1), where j∈[1,··· ,i−m+D I −1] represents all the possible distinct delay constraints associated with the buffered packets and future arrivals up to slot i−1. To sum up, after incorporating ~ I and U I m , (4.7) can be re-written as r I m =        min m≤i≤M+D I −1 r I 1[i] if I m =1, 0 if I m =0, (4.8) where r I 1[i] ,i∈[m,··· ,M +D I −1] is r I 1[i] =max ( U I m + P i l=m B l ( P i j=m I j )τs , U m,D I −1 τs , U m,D I −2 +U m,D I −1 ( P (m+1) j=m I j )τs ,··· , P D I −1 l=1 U m,l ( P (m+D−2) j=m I j )τs , U I m +Bm ( P (m+D−1) j=m I j )τs , U I m +Bm+B m+1 ( P (m+D) j=m I j )τs ,··· , U I m + P i−1 l=m B l ( P (i+D−2) j=m I j )τs ) , (4.9) where x M = min{x,M +D I −1}. Therefore, we have 116 Theorem 4.1 For slot-selective transmissions as indicated by ~ I over a one-hop trans- mission, the transmission scheme denoted by (4.8) is the optimal offline scheduling for a group of packets subject to the same individual delay constraint D I in a static discrete- time slot model . Proof: This is directly derived from the optimal offline scheduling in (4.6) for the non slot-selective case. Thus, the proof is omitted. In the case of the single transmission deadline model in which all packets observe a single deadline, the optimal transmission rate during slot m would be r I single,m =        min m≤i≤M U I m + P i l=m B l ( P i j=m I j )τs if I m =1, 0 if I m =0, (4.10) an expression simpler than that of (4.8) and (4.9), due to the fact that there is only one single common deadline. Under a general indicator function ~ I, the symmetry property of the optimal trans- mission rate for the individual delay constraint model no longer holds, even when the vector of packet sizes [B 1 ,··· ,B M ] has a joint probability distribution that is identical to that of its reversed vector [B M ,··· ,B 1 ]. Instead, we consider a special case where ~ I is symmetric, i.e., I i = I M+D I −i , ∀i∈ [1,··· ,M +D I −1]. This symmetry property is validfor ~ I TDM (or ~ I FDM )fortheTDM(orFDM)accessmodediscussedherein(see(4.2) and (4.4) in Section 4.2). Under a symmetric ~ I, the symmetry property of the optimal transmission rate vector still holds: 117 Theorem 4.2 For any M ≥ 1, when the vector of packet sizes [B 1 ,··· ,B M ] has a joint probability distribution that is identical to that of its reversed vector [B M ,··· ,B 1 ], and when the indicator vector ~ I is symmetric, under the optimal offline scheduling, the optimal transmission rates r I m and r I M+D I −m are identically distributed. Thus, E{r I m }= E{r I M+D I −m }, where E{.} denotes expectation. Proof: The proof essentially relies on a time reversal argument, where a sample path trajectory of the forward running system is compared to a corresponding time reversed system. The same proof procedure in Theorem 2.2 for the continuous-time arrival model can be followed and thus is omitted here. The symmetry property of the optimal transmission rate ~ r, in combination with the potential restricted slots as indicated by ~ I, leads to an upper bound and a lower bound of the average packet delay performance: Theorem 4.3 For any M ≥1, when the vector of packet sizes [B 1 ,··· ,B M ] has a joint probability distribution that is identical to that of its reversed vector [B M ,··· ,B 1 ], and whentheindicatorvector ~ I issymmetric, undertheoptimalofflinescheduling, theaverage packet delay is bounded by τ s 1+ M +1 2M (D I −1) −δ≤ ¯ q I (M)≤τ s 1+ M +1 2M (D I −1) . (4.11) where δ M = τ s c 0 /(2M) and c 0 is the total number of restricted slots, defined as c 0 M = M+D I − 1− P M+D I −1 m=1 I m . When M →∞, τ s (D I +1)/2−δ≤ ¯ q I (∞)≤τ s (D I +1)/2. Proof: See Appendix A.12. 118 Note that when there are no restricted slots, i.e., c 0 =0, the lower bound and the upper bound are tight. 4.4 Multihop Optimal Offline Scheduling over Static Channels In the previous section, we have examined the optimal offline schedule for one hop trans- missioninanLhoplink,givenadelayconstraintandageneralindicatorvectorindicating whether slots are restricted for transmissions or not. In this section, we will address the following questions for a multihop link: • Givenanend-to-enddelayconstraint, howdoweperformoptimalofflinescheduling over multihop transmissions? • Do the relaying nodes need to know the time stamp information of each packet under the optimal offline schedule? • How much energy savings can we achieve under the optimal offline schedule for multihop transmissions vs. single hop transmissions? • Howdoestheaveragepacketdelayperformanceofamultihoptransmissioncompare with that of the single-hop transmission? We start by presenting two key properties, which lead to the optimal offline schedule over a multihop link, followed by the performance analysis. 119 4.4.1 Optimal Offline Schedule vs. Delay Constraints Although we assume the same end-to-end individual packet delay constraint for the orig- inal packet arrival process ~ B, the per-hop packet delay constraints in the L-hop link may notnecessarilybethesame. Inaddition,duetothefluidpacketdeparturemodel,asingle packet of size B m in ~ B may be fragmented during its transmission over the L hops and thus can be treated as multiple packets at any given hop l > 1. Indeed, the number of packets of distinct delay constraints may not be the same at different hops. Now consider a general packet size vector ~ ˜ B of an arbitrary length ˜ M ≥1, associated with a delay constraint vector ~ ˜ D of the same length ˜ M. The delay constraints ˜ D 1 to ˜ D ˜ M in ~ ˜ D are not necessarily the same. Denote the optimal transmission rate vector ~ r( ~ ˜ B, ~ ˜ D). Note that the optimal transmission rate vector may also depend on other factors (e.g., the indicator vector ~ I), but for notational convenience, such dependence is omitted. It is intuitive that as the delay constraints ~ ˜ D increases, the resulting optimal rate vector ~ r( ~ ˜ B, ~ ˜ D) should become no less energy-efficient. That is, Claim 4.1 Giventhesamesetsofpacketarrivals ~ ˜ B andtransmissionopportunities ~ I, the resulting optimal offline rate vector is no less energy-efficient when the delay constraints ~ ˜ D increase : e( ~ ˜ B, ~ ˜ D+Δ ~ ˜ D)≤e( ~ ˜ B, ~ ˜ D). where e( ~ ˜ B, ~ ˜ D) denotes the total transmission energy associated with the rate vector ~ r derived based on ~ ˜ B and ~ ˜ D, and Δ ~ ˜ D M = [Δ ˜ D 1 ,··· ,Δ ˜ D ˜ M ], with Δ ˜ D m ≥0, ∀m. 120 Proof: This is straightforward. Obviously, ~ r( ~ ˜ B, ~ ˜ D) is also a feasible transmission rate vector 3 for scheduling with a delay constraint of ~ ˜ D +Δ ~ ˜ D and thus it can not be more energy-efficient than the optimal rate vector ~ r( ~ ˜ B, ~ ˜ D+Δ ~ ˜ D) assuming a delay constraint of ~ ˜ D+Δ ~ ˜ D. Theaboveclaimholdsforanyhopl,regardlessofthecorrespondingpacketsizevector ~ B (l) and the delay constraint vector ~ D (l) . 4.4.2 Maximum Possible Per-Hop and Total Energy-Efficiency Recall that since scheduling is performed only at slot boundaries, each of the first L−1 hops introduces at least one slot packet delay. In addition, the packet delays introduced by the first L−1 hops are always integer number of slots. Since the transmission delays at the last hop are always positive, given an end-to-end delay constraint of D slots, any packet in any hop thus has a maximum delay constraint of D−L+1 slots, regardless of the scheduling algorithms in any other L−1 hops. In other words, ~ D (l) ≤(D−L+1) ~ 1,∀l, where ~ 1 is an all-one vector of length M (l) ≥ 1. From Claim 4.1, we have the following claim to characterize the maximum possible energy-efficiency at any hop and over all hops: 3 A transmission rate vector is feasible if it satisfies all the scheduling constraints, including FIFO, non-idling, causality, and individual delay constraints. See (2.4) in Chapter 2. 121 Claim 4.2 The total transmission energy expended by any hop l, denoted by e (l) ( ~ B (l) , ~ D (l) ), is no less than the total energy expended by scheduling using the original packet arrival process ~ B with an equal packet delay constraint of D−L+1 slots (denoted by e (l) ( ~ B,(D− L+1) ~ 1)), i.e., e (l) ( ~ B (l) , ~ D (l) )≥e (l) ( ~ B,(D−L+1) ~ 1),∀l. The total energy over all hops is thus always lower-bounded by P L l=1 e (l) ( ~ B,(D−L+1) ~ 1). Proof: Under any scheduling algorithms, at any hop l, any packet B m , m=1,··· ,M, in the original packet arrival process ~ B can be expressed as ~ B (l) m of length m (l) ≥ 1, and a corresponding delay constraint vector ~ D (l) m of the same length. Since there is no transmission loss, we have P m (l) i=1 B (l) m,i =B m . Note that packet B m arrives at slot m−1 in the first hop, and each of the preceding hops before hop l introduces at least one slot delay, theearliestarrivaltimeassociatedwith ~ B (l) is thus noless timem+l−2(inslots). Therefore, the scheduling of these ~ B (l) m packets with the delay constraints ~ D (l) m can always be emulated by scheduling a single packet B m with a delay constraint ˜ D (l) m , where ˜ D (l) m is the difference between time m+l−2 (in slots) and the maximum departure deadline associated with ~ B (l) m and ~ D (l) m . From previous discussions, obviously, ˜ D (l) m ≤ D−L+1, 122 ∀l,m. Thus, for any ~ B (l) and ~ D (l) , given the same indicator vector ~ I over all hops, we have e (l) ( ~ B (l) , ~ D (l) ) = e (l) ([ ~ B (l) 1 ,··· , ~ B (l) M ],[ ~ D (l) 1 ,··· , ~ D (l) M ]) (a) ≥ e (l) ([B 1 ,··· ,B M ],[ ˜ D (l) 1 ,··· , ˜ D (l) M ]) (b) ≥ e (l) ([B 1 ,··· ,B M ],[D−L+1,··· ,D−L+1]), where (a) is due to the above emulation argument, and (b) is due to Claim 4.1. This completes the proof. 4.4.3 Optimal Offline Scheduling for the Individual Delay Constraint Model Now consider a simple delay budget allocation scheme given by [D−L+1,1,··· ,1 | {z } L−1 ]. That is, all packets at the first hop have the same delay constraint of D−L+1 slots, while all the remaining hops have a delay constraint of exactly one slot. Scheduling is only done at the source node. All the relaying nodes simply perform a buffer-clearing of all the pending packets at each transmission opportunity. Thus, the transmission rate r (l) i during slot i at hop l is the same as r (l−1) i−1 during slot i− 1 at the previous hop. In other words, under this delay budget allocation scheme, the transmission rate vectors over all hops are the same, except the deterministic right-shifts as characterized by (4.1) 123 and (4.2) for the TDM mode, and by (4.3) and (4.4) for the FDM mode, respectively. That is, ~ r (1) (SS (1) )=~ r (2) (SS (2) )=···=~ r (L) (SS (L) ). (4.12) In fact, this simple delay budget allocation scheme is optimal, as summarized below: Theorem 4.4 In an L hop link where all hops have the same time-shifted transmission opportunities, characterized by (4.1) and (4.2) for the TDM mode, and by (4.3) and (4.4) for the FDM mode, respectively, the optimal offline schedule under the individual delay constraint model is to perform the scheduling only at the source node, assuming an equal delay constraint of D−L + 1 slots for all packets, while all the relaying nodes simply perform a buffer-clearing at each transmission opportunity. Proof: First, note that for any single queue, the optimal offline transmission rate vector is independent of the energy function w(r) (as long as it is strictly convex and monotonically increasing in r, see Chapter 2). On the other hand, from (4.12), under the simple delay budget allocation scheme, the transmission rate vectors over all hops are the same, except the deterministic time shifts. From Claim 4.2, the transmission rate vector forthefirsthop~ r (1) ( ~ B,(D−L+1) ~ 1)achievestheenergyperformancelowerboundandis thusoptimal. Asthetransmissionratesofeachsubsequenthoparethesame, theperhop energy expended is also the minimum possible. Therefore, the optimal offline schedule associated with the simple delay budget allocation achieves the energy expenditure of P L l=1 e (l) ( ~ B,(D−L+1) ~ 1), which is the minimum possible across all hops. It is worth emphasizing that, similar to the single queue case (see Chapter 2), the optimality of the simple delay budget allocation scheme holds for any arrival processes. 124 Moreover, the optimality of the above delay budget allocation for the multihop link is independent of w (l) (r),∀l, as well. The optimality holds regardless of whether the energy functionsfordifferenthopsarethesameornot. Inotherwords, thesimplebudgetalloca- tion scheme is optimal for any channel characteristics (e.g., distances between the nodes, path loss exponents, etc.) associated with the multihop link, as long as for any given hop, the channel is time-invariant and the energy-rate function has the strict convexity and monotonically increasing properties. Note that, under the optimal offline schedule, there is no need to propagate the time stamp information of any packets to any relaying nodes as no scheduling is necessary at any relaying nodes. This leads to a very simple design at the relaying nodes. 4.4.4 Optimal Offline Scheduling for the Single Deadline Model The same derivation is also applicable to the single deadline model. Note that in this model, all packets have to be delivered by the single common deadline of (M +D−1)τ s . The optimal offline schedule is summarized below: Theorem 4.5 In an L hop link where all hops have the same time-shifted transmission opportunities, characterized by (4.1) and (4.2) for the TDM mode, and by (4.3) and (4.4) for the FDM mode, respectively, the optimal offline schedule under the single transmission deadline model is to perform the scheduling only at the source node, assuming a common deadline of M +D−L slots for all packets, while all the relaying nodes simply perform a buffer-clearing at each transmission opportunity. 125 4.4.5 Energy Saving Upper Bound Here, in addition to the strict convexity and monotonically increasing properties, we assume the energy-rate function w (l) (r) for any hop l∈[0,1,··· ,L], where l =0 denotes the single-hop transmission, is inversely proportional to the received SNR, γ (l) . This is trueforatypicalwirelesslinkandinparticular,itholdsfortheShannon-typeenergy-rate function given by w(r) = τ s (2 2r −1)/γ. In addition, conditioned on the same γ (l) , we assume the energy-functions are the same for all hops as a function of r. Without loss of generality, we assume that the received SNR under the single-hop transmission is 1 and the distance between the source and the destination nodes is 1. Denote the inter-node distance for hop l is x (l) . Clearly, P L l=1 x (l) ≥ 1, where the equality holds for a linear network when all nodes are on the same line and all the relaying nodes are in between the source and the destination nodes. Under the optimal offline schedule over a multihop link, the same transmission rates are used by all hops over the corresponding non-restricted slots (see (4.12)). However, different hops may potentially consume different transmission power in proportional to 1/γ (l) , which in turn proportional to (x (l) ) α , where α is the path loss exponent. It is not difficult to see that the minimum total transmission energy over all hops is achieved by a linear network with equi-distant nodes, in which γ (1) = ··· = γ (L) = L α . In this case, the per-hop transmission in anL hop link consumesL −α of the energy by the direct transmission. On the other hand, regardless of the multihop access mode, under the optimal offline schedule for the individual delay constraint model, the optimal offline transmission rates 126 at any hop correspond to the scheduling with a delay constraint (D−L+1) slots for all packets. The single-hop transmission, however, assumes a delay constraint of D slots for allpackets. Thus,fromClaim4.1,thesingle-hoptransmissionisatleastasenergy-efficient as any per-hop transmission in a multihop link. Denote the optimal offline transmission rate vector for the single-hop transmission as ~ r (0) , we have: ~ r (0) ≺~ r (l) ,∀l≥1. Consequently, w(~ r (0) ) ≤ w(~ r (l) ). Therefore, we can obtain the following energy-saving lower bound: Theorem 4.6 Under the optimal offline scheduling, the total transmission energy con- sumed by all nodes in an L hop link is no less than L −α+1 of that by the single-hop transmission, i.e., L X l=1 M+D−1 X m=1 w (l) m ≥L −α+1 M+D−1 X m=1 w (0) m ,L>1. This lower bound holds for both the individual delay constraint model and the single transmission deadline model. 4.4.6 Average Packet Delay Performance Comparison for the Individual Delay Constraint Model In Section 4.3, lower and upper bounds of the average packet delay performance were characterizedbyTheorem4.3foranyhopwhenthevectorofpacketsizes[B 1 ,··· ,B M ]has 127 ajointprobabilitydistributionthatisidenticaltothatofitsreversedvector[B M ,··· ,B 1 ], and when the indicator vector ~ I is symmetric. In the sequel, we will apply Theorem 4.3 to compare the average end-to-end packet delay performance between the single-hop and the multihop links, under the optimal offline schedule. Under the FDM mode, over the scheduling span of slots l to (M +D−1−L+l), ~ I L,FDM is an all-one vector. Thus, from (4.5), the average delay introduced by the first hop (from source to the first relaying node) is given by ¯ q (1) (M)=τ s 1+ M +1 2M (D−L) , while the remaining hops introduce a fixed one-slot delay per hop. Therefore, the total end-to-end average packet delay for a L-hop link is: ¯ q FDM (M)=τ s 1+ M +1 2M (D−L) +τ s (L−1). (4.13) When M →∞, ¯ q FDM (∞)=τ s [1+(D−L)/2+L−1]=τ s (D+L)/2. Comparing with the single hop transmission (see (4.5)), after some derivations, we have Δ¯ q FDM (M) M = ¯ q FDM (M)− ¯ q (0) (M) = τ s (L−1)(M−1)/(2M)≥0. (4.14) In particular, Δ¯ q FDM (∞)=τ s L−1 2 . (4.15) 128 UndertheTDMmode,theaveragedelayperformanceisdifficulttocharacterizeunder a general ~ I. However, when ~ I L,TDM is symmetric (which is true as in (4.2)), the lower and upper bounds of the average delay performance can be obtained by Theorem 4.3. i.e., ¯ q TDM,LB (M) = τ s 1+ M+1 2M (D−L) +τ s (L−1)−τ s (M+D−L−1)(L−1) 2LM , (4.16) which approaches to ¯ q TDM,LB (∞)=τ s (D+L−1+1/L)/2. Similarly, we can get Δ¯ q TDM,LB (M) M = ¯ q TDM,LB (M)− ¯ q (0) (M) = τ s (L−1)(LM−M−D+1) 2LM ≥ 0, (4.17) which converges to τ s (L−1) 2 /(2L), when M →∞. On the other hand, the upper bound of the average end-to-end packet delay satisfies ¯ q TDM,UB (M)= ¯ q FDM (M). To sum up, when M →∞, the difference between the average end-to-end delays for the TDM multihop and the single-hop transmissions, denoted by Δ¯ q TDM (∞), is bounded by τ s (L−1) 2 2L ≤Δ¯ q TDM (∞)≤τ s L−1 2 . (4.18) As can be seen, both the FDM and the TDM modes result in an average end-to- end delay slightly larger than the single-hop link. Recall that ¯ q (0) (∞) for the single-hop 129 transmission is given by τ s (D+1)/2. Thus, when M is sufficiently large, the increase in the average packet delay from single-hop to multihop transmissions is roughly by a ratio of (L−1)/(D+1). 4.5 Online Scheduling Over Static Channels In this section, we will extend the heuristic online flush scheduler for multihop transmis- sions, discussed in Section 2.6, to multihop transmissions by utilizing the indicator vector ~ I. We have, r flush,I m =        max i∈[1,···,D] P D i=1 U m,D−i P min{M+D−1,m+D−1} i=m I i τs if I m =1, 0 if I m =0. (4.19) Unfortunately,duetonoanticipationoffuturearrivals,theresultingtransmissionrate vector~ r flush,I is not necessarily more energy-efficient as the delay constraint D increases. Thiscanbeverifiedbyacounterexample. ConsiderM =2withpacketsizesof[1,1],and two delay constraint cases D = 1 and 2. Suppose ~ I = [1,1,0]. Under the online flushing scheduling in (4.19), when D = 1, the resulting rate vector is [1,1,0] over a three-slot time span. When D = 2, the resulting rate vector is [0.5,1.5,0]. Clearly, in this case, [1,1,0] is more mixed and more energy-efficient. However, generally speaking, we can still expect an improved energy efficiency as D increases. Therefore, we will use the same delay budget allocation as in the offline scheduling case for the online flushing scheduler, although it is not necessarily optimal. The performance results will be shown in Section 4.7. 130 4.6 Extension to Fading Channels In this section, we will extend the study of the optimal offline schedule over multihop transmissions to the fading channel case, also as an extension of the work in [UBG04] for the single deadline model, and in Chapter 4 for the individual delay constraint model over the single-hop link. First, it is ofinterestto see whether the simple delay budget allocation scheme forthe static channel case, namely, [D−L+1,1,··· ,1 | {z } L−1 ], under the individual delay constraint model, still holds or not for the fading channels. Note that in the fading channel case, besides traffic variations, channel variations also need to be taken into account in the optimal offline schedule, such that the best-mixture of the energy-rate derivative vector ~ w 0 (r,g) is obtained (see Chapter 4). Now consider the delay budget allocation [D−L+ 1,1,··· ,1 | {z } L−1 ],suchthatthesametransmissionratevectoristransmittedoverallhopswithin the respective scheduling spans. For a particular transmission rate r m , it is transmitted underthechannelgainsg (l) m , g (2) m+1 ,···, g (L) m+L−1 inhops1, 2,··· ,L, respectively. Assume that the energy-rate function is inversely proportional to the channel gains, which is valid, e.g., for a Shannon-capacity type energy-rate function w(r) = τ s (2 2r −1)/g. The equivalent composite channel gain experienced by the transmission rate r m thus satisfies 1/g Equiv m =1/g (l) m +1/g (2) m+1 +···+1/g (L) m+L−1 . (4.20) Therefore,theoptimalofflinescheduleatthefirsthopshouldtrytoequalizethecomposite energy-rate derivative ~ w 0 (r,g Equiv ) as much as possible subject to the causality and delay constraints. 131 However, it can be easily shown by a counter example that the above scheduling with the simple delay budget allocation is not necessarily optimal. Due to dynamic channel gains as a function of both time and hops, it may become necessary to transmit channel- adaptive transmission rate vectors over different hops. This may necessitate an advanced delay budget allocation scheme, and/or require the propagation of the time stamp in- formation of some or all packets over hops for a better rate adaptation. Nevertheless, the simple delay budget allocation [D−L + 1,1,··· ,1 | {z } L−1 ] still provides a simple system design and an upper bound of the energy performance for the optimal offline schedule over multihop transmissions in fading channels. On the other hand, based on the minimum one-slot per-hop delay assumption and packet arrival/departure properties, the largest possible delay constraint for any packet in any hop is always bounded by (D−L+1) slots. Thus, the optimal offline schedule in any hop is always lower-bounded in the transmission energy performance by assuming a delay constraint of (D−L+1) slots for all packets. Subsequently, the optimal offline schedule over the multihop link always yields an energy performance lower-bounded by the sum of the above per-hop performance lower-bounds. To sum up, we have Theorem 4.7 The total transmission energy of the optimal offline schedule over L hops undertheindividualdelayconstraintmodelinfadingchannels, denotedbye L,ind opt , satisfies: L X l=1 e l,ind ≤e L,ind opt ≤Le Equiv,ind where e l,ind is the total transmission energy achieved by the optimal offline schedule as- suming a delay constraint of (D − L + 1) slots, the original packet arrivals, and the 132 channel gains during the l-th hop scheduling span, while e Equiv,ind is the transmission energy achieved by the optimal offline schedule assuming a delay constraint of (D−L+1) slots, the original packet arrivals, and the composite channel gains over L hops, e.g., as given by (4.20). Similarly, for the single deadline model, we have Theorem 4.8 The total transmission energy of the optimal offline schedule over L hops under the single deadline model in fading channels, denoted by e L,single opt , satisfies: L X l=1 e l,single ≤e L,single opt ≤Le Equiv,single where e l,single is the total transmission energy achieved by the optimal offline schedule assuming a deadline of (M +D−L) slots, the original packet arrivals, and the channel gains during the l-th hop scheduling span, while e Equiv,single is the transmission energy achieved by the optimal offline schedule assuming a deadline of (M +D−L) slots, the original packet arrivals, and the composite channel gains over L hops, e.g., as given by (4.20). 4.7 Numerical Results Without loss of generality, we assume a slot duration is one unit (i.e., τ s = 1). The packet sizes are normalized based on frequency bandwidth and slot duration and can be interpreted as the number of bits per channel use (or bandwidth efficiency). Thus, a large B typically corresponds to a high SNR operating point. The energy-rate function 133 is assumed to be w(r) = (2 2r −1)L −α , resulting from Shannon capacity, where a linear network with equi-distant nodes is assumed. The path loss exponent is assumed to be α =2. Thepacketsizesareassumedtoberandomandfollowanexponentialdistribution. The number of packets is fixed at M =1000. We will focus on static channels. 4.7.1 Transmission Energy Performance 4.7.1.1 TDM Mode Fig. 4.3 and Fig. 4.4 show the ratios of the total transmission energy between TDM multihoptransmissions(upto4hops)andthesingle-hoptransmissionasafunctionofthe averagenormalizedpacketsize,withD =9andD =45 4 ,respectively. Theoptimaloffline schedule under the individual delay constraint model is shown here. As the normalized packet size increases, it becomes preferable to use fewer hops. The regions corresponding to the optimal numbers of hops (vis-` a-vis minimum total transmission energy) are also shown. It can be seen that hopping-advantageous regions, regardless of D, are rather limited. This is due to the TDM operation, in which packets are forced to accumulate during restricted slots, which necessitates higher transmission rates during subsequent non-restricted slots. Fig. 4.5 and Fig. 4.6 illustrate the optimal number of hops for different scheduling cases. As expected, the online flush scheduler for the individual delay constraint model hasthemostlimitationsinbenefitingfrom multihopping, whilethe single deadlinemodel 4 To facilitate analysis of average packet delay performance, the delay constraints chosen here, along with M, ensure mod(M +D−1,L) = 0 for L=2, 3, and 4. See (4.1). 134 Figure 4.3: Ratios of the total transmission energy of TDM multihop transmissions over the single-hop transmission, D=9. Figure 4.4: Ratios of the total transmission energy of TDM multihop transmissions over the single-hop transmission, D=45. 135 Figure 4.5: Optimal number of hops, TDM mode, D=9. has the least limitations, although the difference between the optimal offline schedulers for both models becomes trivial when D is very large (D =45). 4.7.1.2 FDM Mode Fig. 4.7 and Fig. 4.8 show the ratios of the total transmission energy between FDM mul- tihop transmissions and the single-hop transmission. Again, the optimal offline schedule under the individual delay constraint model is shown. In this case, even with a strin- gent delay constraint (D=9), multihop transmissions still provide energy savings over the single-hop transmission in reasonably large regions. In fact, when D is large (D=45), the 4-hop transmission always yields the least transmission energy even when the normalized packet size is as large as 15 (number of bits/channel use). From Theorem 4.6, the lower bounds on the energy ratios are simply L −α+1 = 1/L, where α = 2, as can be observed. 136 (a) D=45 Figure 4.6: Optimal number of hops, TDM mode, D=45. The optimal number of hops for different scheduling cases when D = 9 is also shown in Fig.4.9, while the casewhenD =45 is not shown as the 4-hop transmissionis always op- timal for all the scheduling cases. It can be seen that for the single deadline model, 4-hop transmission is optimal even when D = 9. Comparing energy performance between the TDMandFDMmultihoptransmissions,itcanbeseenthatfromthetrafficvariationsand energy-efficient scheduling perspective, it is preferable to not orthogonalize multihopping resources in the time domain, but rather in other domains such as frequency. 4.7.2 Average Packet Delay Performance Fig.4.10andFig.4.11presenttheaveragepacketdelayperformancefordifferentschedul- ing cases under the TDM mode. The lower and upper bounds for the optimal offline 137 Figure 4.7: Ratios of total transmission energy of FDM multihop transmissions over the single-hop transmission, D=9. (a) D=45 Figure 4.8: Ratios of total transmission energy of FDM multihop transmissions over the single-hop transmission, D=45. 138 Figure 4.9: Optimal number of hops, FDM mode, D=9. schedule under the individual delay constraint model are also shown. It can be seen that the lower and upper bounds are very tight. The optimal offline schedule for the single deadline model results in an average packet delay much larger than that of the individual delay constraint model. The average packet delay for both offline schedulers increases noticeably as the number of hops increase. The online flush scheduler, on the other hand, tries to push packet transmissions as late as possible and achieves average packet delays close to the delay constraint D. The decreasing average packet delay trend for the online flush scheduler in TDM multihop transmissions is attributed to the increasing number of restricted slots as L grows and hence the forced early transmissions. Similar behaviors canalsobeobservedforFDMmultihoptransmissions,asshowninFig.4.12andFig.4.13. In this case, all schedulers have the average packet delays that monotonically grow with L. 139 Figure 4.10: Average packet delay performance vs. the number of hops, TDM mode, D=9. (a) D=45 Figure 4.11: Average packet delay performance vs. the number of hops, TDM mode, D=45. 140 Figure 4.12: Average packet delay performance vs. the number of hops, FDM mode, D=9. (a) D=45 Figure 4.13: Average packet delay performance vs. the number of hops, FDM mode, D=45. 141 4.8 Conclusions This chapter studies delay-constrained energy-efficient packet transmission over a multi- hop link, as an extension of the work in Chapters 2 and 3 for the conventional single hop transmission. Two types of multihopping modes are considered, namely, time-division- multiplexing (TDM) and frequency-division-multiplexing (FDM). We mainly focus on static channels. The optimal offline scheduling (vis-` a-vis total transmission energy), as- suming information of all packet arrivals before scheduling, is re-derived by incorporating the potential existence of restricted transmission durations, e. g., in the TDM mode. Given an end-to-end delay constraint for each packet, the optimal offline schedule over multihop transmissions is derived. It turns out that the optimal offline schedule relies on asimpledelaybudgetallocationscheme, whichallocatesthedelaybudgettothefirsthop (fromsourcetothefirstrelayingnode)asmuchaspossible. Allthe relayingnodeshavea delaybudgetexactlyequaltooneslotandthussimplyperformbuffer-clearingduringany transmission opportunities. This optimal allocation is independent of traffic characteris- tics (e.g., number of packets, packet sizes, burstiness, etc.) and channel characteristics (e.g., path loss exponents, inter-distance between nodes, etc.). The transmission energy saving due to multihopping is lower bounded by L −α+1 , where L is the number of hops and α is the path loss exponent. The average packet delay performance for multihop transmissions is slightly larger than that of the direct transmission for both the TDM and the FDM hopping modes. A heuristic online scheduling algorithm is investigated for multihop transmissions. Extension to fading channels is also discussed, in which the optimal offline scheduling no longer permits a simple optimal delay budget allocation, 142 but rather depends on the specific realizations of packet sizes and channel states. An upper bound and an lower bound for the optimal delay budget allocation over fading channels are provided. Performance of energy-efficient scheduling over multihop trans- missions are further evaluated via simulations. It is demonstrated that energy savings via multihopping are possible, but heavily depend on factors such as multihop resource orthogonalization mode, packet scheduling algorithms, delay constraints, SNR operation regions, and channel variations. 143 Chapter 5 Packet Dropping for Transmission Energy Savings 5.1 Introduction Proactive packet dropping for transmission energy savings in a static channel is investi- gated in this chapter. We start with the single transmission deadline model [UBPG02] andshowthatdroppingthelastsetofK packetsresultsinmaximumtransmissionenergy savings. Weshowthatafterpacketdropping,thetransmissiondurationsoftheremaining packets are increased and the increments are subject to the ‘water-filling’ rule [CT91]. The resulting transmission duration vector due to the optimal packet dropping is more mixedandisalwaysmajorized[Bha97]bytheoriginaloptimaltransmissiondurationvec- tor. Assuming a Poisson arrival model, we analytically derive the optimal transmission durations of the first and the last packets after dropping the last K ≥ 1 packets. Under a Poisson arrival model, it is found that on average the optimal transmission duration of the last packet can be significantly increased, even when K = 1. When M approaches infinity, optimally dropping one packet on average improves the optimal transmission du- ration of the last packet from approximately 0.5/λ to (e−2)/λ≈0.72/λ, where λ is the 144 arrival rate. Since the last packet contributes the most in the total transmission energy, a significant increase in transmission duration of the last packet directly translates into substantial energy savings. For the individual delay constraints model, optimal packet dropping is much more complicated and seems intractable. Furthermore, we see that with the individual delay constraints, an optimal dropping scheme is energy function and packet-size dependent. As a result, we study asymptotically optimal dropping schemes when the packet size is sufficiently large. We derive the asymptotically optimal scheme of dropping one packet (K = 1). When the number of dropped packets K > 1, the asymptotically optimal schemes can be derived via a brute-force search. We provide two upper bounds for the asymptotically optimal dropping schemes by exploiting packet arrival and departure propertiesandbymaintainingtheindividualdelayconstraints. Wealsoproposetwosub- optimal packet dropping schemes, which yield energy savings very close to those achieved by the asymptotically optimal dropping schemes. Significant transmission energy savings are possible via intelligent packet dropping as evidenced by the numerical results. This chapter is organized as follows. Section 5.3 presents the optimal dropping al- gorithm for the single deadline model and its impact. The intelligent packet dropping scheme for the individual delay constraint model is discussed in Section 5.4. Numerical resultsaregiveninSection5.5. Finally,someconcludingremarksaredrawninSection5.6. 145 5.2 System Model In the single transmission deadline model [UBPG02], the goal is to deliver all packets by the time deadline T while minimizing the total packet transmission energy. The optimal scheduling algorithm, without packet dropping, is explicitly specified in [UBPG02]. One feature of the optimal solution for this model is that transmission durations are non- increasing,i.e.,τ i ≥τ i+1 fori∈[1,··· ,M−1]. Theoptimalofflineschedulerendeavorsto equalizethetransmissiondurationsasmuchaspossiblesubjecttothefollowingfeasibility constraints:        P k i=1 τ i ≥ P k i=1 d i ,k =1,··· ,M−1,and P M i=1 τ i = P M i=1 d i =T (5.1) In contrast, for the individual packet delay constraint model, each packet has its delay constraint T i ,i ∈ [1,··· ,M], at or before which the packet has to be successfully delivered after its arrival. In this chapter, we will focus the case when the individual delay constraints are equal, i.e., T i =T,∀i. The optimal offline scheduling algorithm was derived in Chapter 2. Note that the property of τ i ≥ τ i+1 may no longer hold. Using ˆ d i = min{d i ,T} instead of d i ,i ∈ [1,··· ,M], the optimal offline scheduleris subject to the following constraints: (i):        P k i=1 τ i ≥ P k i=1 ˆ d i ,k∈[1,··· ,M−1], and P M i=1 τ i = P M i=1 ˆ d i , (ii):q k = P k i=1 τ i − P k−1 i=1 ˆ d i ≤T,k∈[1,··· ,M], (5.2) 146 where q k denotes the delay for packet k, defined as the difference between the packet departure time (completed packet delivery) and the packet arrival time. Inthefollowing, wewillinvestigateoptimalschemesfordroppingK ≥1packetsfrom a total of M packets for both the single transmission deadline model and the individual delayconstraintmodel. Theoptimalcriterionhereinisdefinedasminimizingthe average transmission energy of the remaining N = M −K packets. Note that once a set of K packets to drop is identified, the optimal scheduling strategies of [UBPG02][ZM05] and Chapter 2 can be employed to minimize energy for the remaining packets. The challenge is to identify the best K packets to drop. 5.3 Optimal Packet Dropping for the Single Transmission Deadline Model We assume the common transmission deadline T remains unchanged after packet drop- ping. We will start by presenting the optimal packet dropping scheme, followed by an analysis of its properties. 5.3.1 Optimal Packet Dropping We first observe that it is certainly not optimal to drop the first packet as this will shrink the total overall time resource for scheduling. We have the following proposition: Proposition 5.1 Dropping the last set of K packets from the M packets and then apply- ing the optimal scheduler of [UBPG02] to the remaining packets results in the minimum 147 transmission energy and hence is an optimal packet dropping scheme for the single trans- mission deadline model. Proof: If we drop the last K packets, the arrival times of the remaining N packets are thesameastheoriginalfirstN packets. ConsideranewschemeofdroppingK packetsin whichatleastonedroppedpacketk∈[2,··· ,N]. Thisnewsetofpacketarrivaltimescan always be emulated by the original first N packets by buffering these packets separately and releasing them to the actual queuing system exactly at the emulated arrival times. Thus,droppingthelastK packetscanemulateanyotherdroppingschemeandisoptimal. Note that given a vector of inter-arrival times and a delay constraint, the optimal scheduling algorithm is unique [ZM05]. However, the optimal packet dropping scheme may not be unique (the optimal scheduling for the remaining N packets is still unique). It is straightforward to construct examples where dropping a different set of K packets yields exactly the same energy savings as dropping the last K packets. However, we have: Proposition 5.2 Suppose we have an optimal scheme for dropping K packets from M packets; denote the indices of the dropped K packets as [m 1 ,··· ,m K ] with m i <m j ,∀i< j. Then, dropping any combination of K packets within [m 1 ,··· ,M] such that m 0 i ≥ m i ,1 ≤ i ≤ K, will result in the same minimum transmission energy and is thus also energy optimal. Proof: Consider dropping a new set of K packets with indices [m 0 1 ,··· ,m 0 K ] such that m 0 i ≥m i , 1≤i≤K. The remaining N packets after dropping packets [m 0 1 ,··· ,m 0 K ] can at least emulate the remaining N packets if packets [m 1 ,··· ,m K ] were dropped instead. 148 Thus, dropping [m 0 1 ,··· ,m 0 K ] will achieve an average transmission energy no more than that of dropping [m 1 ,··· ,m K ], which yields the minimum transmission energy. As a result, dropping packets [m 0 1 ,··· ,m 0 K ] is also optimal. In such cases where optimality is non-unique, it is desirable from a delay perspective to drop packets as early as possible. 5.3.2 The ‘Water-Filling’ Rule Note that although there might exist multiple optimal dropping schemes, the resulting optimaltransmissiondurationvectorfortheremainingN =M−K packetsisstillunique. To analyze the impact of optimal packet dropping, we focus on the case of dropping the last K packets. Given a packet inter-arrival time vector ~ d, we can obtain a unique optimal transmission duration vector ~ τ. Note that, however, different inter-arrival time vectors may result in the same vector ~ τ. In fact, recall that under the optimal offline scheduling algorithm [UBPG02], the optimal transmission durations are non-increasing, i.e., τ i ≥ τ i+1 for i∈ [1,··· ,M −1]. An inter-arrival time vector ~ d = ~ τ would yield the same optimal transmission duration vector~ τ. As a result, from the optimal transmission duration perspective, the packets with an inter-arrival vector ~ d and an optimal duration vector~ τ can be viewed as if~ τ were both the inter-arrival vector and the optimal duration vector. In other words, these packets can be seen as having a virtual inter-arrival vector ~ τ. Note that the two inter-arrival vectors ~ d and ~ τ would of course yield different packet scheduling delays under the optimal offline scheduling algorithm. This virtual-arrival concept helps us understand the impact of dropping the last K packets. After dropping the last K packets from M packets, the remaining N packets 149 are provided additional time resources (recall that T is assumed to be unchanged). As a result, the optimal transmission durations of the remaining N packets are expected to increase by Δ~ τ (K) N = [Δτ (K) N,1 ,··· ,Δτ (K) N,N ] with Δτ (K) N,n ≥ 0,1≤ n≤ N, where, for clarity, the first subscript N is introduced to denote the dimension of the vector. The following proposition characterizes Δ~ τ (K) N : Proposition 5.3 The transmission durations of the remaining N packets after dropping K packets, denoted as ~ τ (K) N , are increased from ~ τ M (1:N) 1 by Δ~ τ (K) N =[Δτ (K) N,1 ,··· ,Δτ (K) N,N ], where Δ~ τ (K) N is derived based on the ‘water-filling’ rule [CT91], with the additional time resource given by T K = P M i=N+1 τ M,i = T − P N i=1 τ M,i . That is, Δτ (K) N,i = (ν−τ M,i ) + 2 , where ν is chosen so that P (ν−τ M,i ) + =T K . Proof: Before packet dropping and under the optimal offline scheduling algorithm, the first N packets are transmitted with durations ~ τ M (1 : N), while the last K packets are transmitted with durations ~ τ M (N + 1 : M). The dropping of the last K packets means these packets are omitted for transmission. Effectively, T K = P M i=N+1 τ M,i = T − P N i=1 τ M,i , the amount of time resource that is released. From the transmission duration perspective, the remaining N packets can be viewed as having a virtual inter- arrival vector of [~ τ M (1 : N−1),τ M,N +T k ]. In other words, the last virtual packet sees an increase of time duration from its arrival to the deadline by T K . Under the optimal 1 Here ~ vM(i :j) denotes the i-th to the j-th elements of an M-dimensional vector ~ v. 2 Here (x) + denotes the positive part of x, i.e., (x) + =x if x≥ 0; 0 otherwise. 150 offline scheduling algorithm, this additional time resource can potentially be borrowed by any preceding packets. The last one or more groups under the original ~ τ M (1 : N) can thus have a transmission duration increase. If two or more groups benefit from T K , these groups would become a new group and the packets belonging to these groups would have the same new transmission duration. This is illustrated in Figure 5.1. The following two corollaries come directly from the non-increasing property of the transmission duration vector under the optimal offline scheduling (proofs omitted): Corollary 5.1 Δτ (K) N,i ≤Δτ (K) N,i+1 , ∀i∈[1,··· ,N−1]. Corollary 5.2 P N i=N−n+1 Δτ (K) N,i ≥ nT K /N,1 ≤ n ≤ N. When n = N, the equality holds. These two Corollaries indicate that the benefit of packet dropping is non-increasing withpacketindices. Wealsohavethefollowingcorollary, whichwillbecomehelpfullater: Corollary 5.3 The following holds: N X i=N−n+1 τ (K) N,i ≥ T T −T K N X i=N−n+1 τ M,i , for 1≤n≤N. When n=N, the equality holds. Proof: Since under the optimal offline scheduling, ~ τ M is non-increasing, we have P N i=N−n+1 τ M,i ≤ (T − T K )n/N. From Corollary 3.3.2, we have P N i=N−n+1 Δτ (K) N,i ≥ T K /(T − T K ) P N i=N−n+1 τ M,i . Thus, P N i=N−n+1 τ (K) N,i = P N i=N−n+1 (τ M,i + Δτ (K) N,i ) ≥ T/(T −T K ) P N i=N−n+1 τ M,i . 151 Figure 5.1: Illustration of the ‘water-filling’ rule for Δ~ τ Corollary 5.3 indicates that after packet dropping, the total time resource consumption by the last n ≥ 1 packets is always no less than that of the non-dropping case, after a necessary normalization. Since the optimal transmission duration vector is always non- increasing, this translates into a more equalized vector after packet dropping, as will be proved below. The dropping of the last K packets has two effects. It first avoids the transmission of the packets that are of the minimum transmission durations and hence demand high energy (recall that w(τ) is a decreasing function of τ). It also increases the transmission durations of packets that are of relatively high energy-demand as well, as dictated by the ‘water-filling’ rule. Thus, the transmission energy for these packets will be reduced. Therefore, under the optimal offline scheduling, the average transmission energy of the remainingN packets(withthedurationvector~ τ (K) N )isalwayslessthanthatoftheoriginal M packets (with the duration vector ~ τ M ), as will be demonstrated via simulations. 152 5.3.3 Optimal Transmission Duration Vector De-Majorization In this section, we will utilize the concept of majorization to compare the optimal trans- mission duration vector before and after packet dropping. We will also use a Poisson arrival model as an example to show the effect of mixing after packet dropping. Now let us compare the optimal transmission duration vector ~ τ (K) N (of dimension N) after packet dropping to the optimal transmission duration vector ~ τ N if only the first N packets are transmitted assuming the inter-arrival times ~ d M (1 : N). Note that ~ τ N may not be the same as ~ τ M (1 : N), which is obtained assuming the inter-arrival times ~ d M of M packets. To that end, we first compare ~ τ (K) N and ~ τ M (1:N). Clearly, we have to normalize one of the two vectors such that c P N i=1 τ (K) N,i = P N i=1 τ M,i , where c = (T −T K )/T < 1 is the normalizationfactor(recallthetotaltimedurationof~ τ (K) N islargerthan~ τ M (1:N)bythe amount T K ). Since ~ τ (K) N is formed based on ~ τ M (1:N) using the ‘water-filling’ rule, ~ τ (K) N should be more mixed than ~ τ M (1 : N). In fact, from Corollary 5.3, it is straightforward to see that c N X i=N−n+1 τ (K) N,i ≥ N X i=N−n+1 τ M,i ,∀n∈[1,··· ,N]. Whenn=N, theequalityholds. Thus, wehave~ τ (K) N ≺~ τ M (1:N)(thenormalizedfactor has been omitted for presentation convenience). Similarly, we can compare ~ τ M (1 : N) and ~ τ N . Note that ~ τ N only relies on the inter- arrival times of the first N packets. The vector ~ τ M (1 : N), however, may potentially benefit from the inter-arrival times of packets N+1 and higher. If there is any borrowed 153 amount in~ τ M (1:N), compared with~ τ N , we can follow the same procedure as before and show ~ τ M (1:N)≺~ τ N . Thus, we have: Lemma 5.1 For any inter-arrival time vector ~ d M , the following majorization relation- ship holds: ~ τ (K) N ≺~ τ M (1:N)≺~ τ N . The following corollary is obvious: Corollary 5.4 After necessary normalizations, ~ τ (1) M−1 (1:N)~ τ (2) M−2 (1:N)··· ,~ τ (K) M−K ,∀K ≥1. In other words, the normalized transmission duration vector under the optimal offline scheduling becomes more and more mixed as the number of optimally dropped packets increases. Lemma 5.1 and Corollary 5.4 suggests that after packet dropping, the average trans- missiondurationofthelastpacketshouldbeincreased(afternormalization),whiletheav- erage transmission duration of the first packet should be decreased (after normalization). We quantify these properties by considering a Poisson arrival model (with a rate of λ) to showtheamountofdurationincrementforτ (K) N,N anddecrementforτ (K) N,1 afterpacketdrop- ping. It was shown that, without packet dropping, E{τ M,1 }≡ ¯ τ M,1 = (1/λ) P M m=1 1/m 2 (approaching to π 2 /(6λ) when M is large) [UBPG02] and ¯ τ M,M = (M +1)/(2λM) (see Lemma 2.6) (approaching to 0.5/λ when M is large). Note that since the transmission deadline T remains the same after packet dropping, the average inter-arrival time of the 154 remaining N packets is increased from 1/λ to M/(Nλ). Thus, in order to make a fair comparison,itisreasonabletonormalizethetransmissiondurationsafterpacketdropping by N/M. We have the following result for the last packet, after packet dropping: Lemma 5.2 Under optimal offline scheduling, the normalized average transmission du- ration of the last packet (packet N) after dropping K packets from M packets is given by ¯ τ (K) N,N = 1 λ 1− N X i=2 c i N K+i−1 (i+K) N +K−1 K +i−1 ! , where c i = (i− 1) P i−1 j=1 (−1) i−j−1 j K+i−2 K+i−1 i−j−1 , when the inter-arrival times assume the exponential distribution. Proof: see Appendix A.13. Corollary 5.5 The average transmission duration of the last packet (packet N) after dropping one packet from N +1 packets is given by ¯ τ (1) N,N = (1+N −1 ) N −2N/(N +1) /λ, when the inter-arrival times assume the exponential distribution. When N →∞, ¯ τ (1) N,N →(e−2)/λ≈0.72/λ. Proof: see Appendix A.13. 155 Similarly, for the first packet after packet dropping, we have: Lemma 5.3 The normalized average transmission duration of the first packet after drop- ping K packets from M packets (with N remaining packets) is given by ¯ τ (K) N,1 = N λM N X n=1 1 n 2 + K X k=1 δ k ! , whentheinter-arrivaltimesassumetheexponentialdistribution, whereδ k = (1+k/N) N+k−2 N 2 − P k−1 j=1 c k,j δ j (N+k−1)! N k−j (N+j−1)! , with1≤k≤K,c 1,1 =1, andc k,j = P k l=max{j,2} c k−1,l−1 (l−1)! (l−j)! with 1≤j≤k and 2≤k≤K. Proof: see Appendix A.14. When K =1, Lemma 5.3 reduces to: Corollary 5.6 The average transmission duration of the first packet after dropping one packet from N +1 packets is given by ¯ τ (1) N,1 = N λ(N +1) N X n=1 1 n 2 + (1+1/N) N−1 N 2 ! , when the inter-arrival times assume the exponential distribution. Proof: see Appendix A.14. From Corollaries 5.5 and 5.6, we can see that when dropping a single packet from a sufficiently large number of packets, the transmission duration of the last packet is in- creased from approximately 0.5/λ to 0.72/λ, which represents a significant increase. The transmission duration of the first packet, on the other hand, is decreased from approxi- mately π 2 /(6λ) to π 2 /(6λ)−ζ, where ζ = [π 2 /6−e/N]/(λ(N +1)) > 0 when N →∞. 156 This trend is not surprising from the majorization result. The trend when K > 1 is not obviousfromLemmas5.2and5.3. Instead,wewillrelyonnumericalresultsinSection5.5 to show ¯ τ (K) N,1 and ¯ τ (K) N,N as a function of K and N. 5.4 Optimal Packet Dropping for the Individual Delay Constraint Model Note that for the single transmission deadline model, the optimal packet dropping algo- rithm is simple and its optimality is regardless of packet size. However, as we will show, the optimal packet dropping algorithm for the individual delay constraint model is much more complicated and appears intractable. Indeed, the optimal choice of packets to drop is packet size and energy function dependent. Generally, it is not optimal to drop a packet right before or right after a packet with an inter-arrival time equal to or larger than T, as it will reduce the total allowable time resources, compared with dropping any other packet. Recall the definitions of scheduling separationinterval,group,subgroup,delay-criticalpacket,type-1groupandtype-2group inSection2.3. Underoptimalscheduling,asubgroupmayendwitheitheranemptybuffer oradelay-criticalpacket. Atype-1grouponlyhasonesubgroup. Atype-2grouphastwo ormoresubgroupsandthetransmissiondurationsofthesesubgroupswithinagiventype- 2 group are monotonically increasing (see Lemma 2.2). For convenience, we introduce the following definitions: Definition: A minimum group is a group which contains the minimum transmission duration. If two or more groups have the same minimum transmission duration, the 157 Figure 5.2: An example run of the optimal transmission durations, M = 20, T= 2, and λ=1 packets/second. minimum subgroup refers to one of the type-2 groups (if it exists), or the last type-1 group in a given scheduling separation interval. Definition: Aminimumsubgroupisthesubgroupwhichhastheminimumtransmission duration in the minimum group. For a type-1 group, the minimum subgroup is the minimum group itself. For a type-2 group, the first subgroup is referred as the minimum subgroup. Correspondingly, we can also call a subgroup a type-1 (or type-2) subgroup if it belongs to a type-1 (or type-2) group. Figure 5.2 shows an example run of the optimal transmission durations of 20 packets. It can be seen that the minimum group (a type-2 group) consists of packets 1 to 7, with packets 1 to 3 belonging to subgroup 1. Packet 3 is a delay-critical packet. Packets 8 to 12 belong to a type-1 group. 158 Claim 5.1 If the minimum group is a type-1 group, it must be the last group of a schedul- ing separation interval. Proof: By contradiction. From the definition of the minimum group, and the non- increasingpropertyofthetransmissiondurationsbetweenadjacentgroups(seeLemma2.1 in Chapter 2), if a type-1 group minimum group is followed by one or more groups, its transmission duration has to be less than that of the immediate subsequent group. This causes a contradiction. Claim 5.2 When dropping a single packet, the transmission duration of the minimum subgroup can be increased only if the dropped packet is part of the minimum subgroup. Proof: Note that all preceding (sub)groups have transmission durations no less than that of the minimum subgroup and a minimum subgroup always starts with an empty buffer,theminimumsubgroupcannotbenefitfromthereleasedtimeresourcebydropping anyprecedingpackets. Ontheotherhand,sincethelastpacketoftheminimumsubgroup is a delay-critical packet, the minimum subgroup cannot benefit from dropping a packet after the minimum subgroup. Now it may seem that dropping one packet from the minimum subgroup would yield less total transmission energy compared dropping one packet from any other subgroups. Thisisnotnecessarilythecase. InFigure5.2,themultiplicity(i.e.,thenumberofpackets) oftheminimumsubgroupis3(packets1to3). Notethatthesubgroupcontainingpackets 13 to 18 has the second shortest transmission duration with multiplicity of 6. Due to the multiplicitydifferenceandminimumsubgroupswitchoverafterpacketdropping,dropping 159 packet 2 may not necessarily be advantageous over dropping packet 16 for a given packet size. To summarize: Claim 5.3 Whendroppingapacket, itisnotalwaysoptimaltodropitfromtheminimum subgroup. That is, in most cases it is impossible to design an optimal packet dropping scheme based on transmission durations alone, as the full energy function w(τ,B) is required. This is very different from the single deadline case, in which the optimal transmission durations do not depend on the packet size or energy function. Claim 5.3 is not surprising. As discussed in Section 5.3, the relationship of~ v 1 ≺~ v 2 is equivalentto P M i=1 f(v 1,i )≤ P M i=1 f(v 2,i ),foranyconvexfunctionsf(.)[Nie01]. However, majorization only gives a partial, instead of a complete, characterization of two vectors. There always exist vectors which are not comparable such that we do not have either ~ v 1 ≺ ~ v 2 or ~ v 1 ~ v 2 . Recall that for the single transmission deadline model, the ‘water- filling’ rule applies for the entire set of remaining packets. However, for the individual delay constraint model, due to the existence of scheduling separation intervals and delay- criticalpackets,packetdroppingonlyhaslocalizedeffectsandsomerelativelyhighenergy demanding packets may not benefit from the dropping at all. Thus, dropping a packet from the minimum subgroup does not necessarily yield a transmission vector majorized by that of any other dropping scheme. It is often of interest to limit the maximum transmit power, which is determined by the minimum transmission duration of the minimum subgroup. Moreover, in scenarios when the minimum subgroup is outweighed in energy contribution by another group (or 160 subgroup), the difference in contribution by these two groups typically is not significant. Thus,weintroducethefollowingdefinitionforaclassofpacket-sizeindependentdropping schemes: Definition: An asymptotically optimal dropping scheme is a packet dropping scheme which results in the minimum average transmission energy as the packet size approaches infinity. In order to have the minimum subgroup contribute the most, asymptotically, to the total transmission energy, regardless of its multiplicity, the energy function is limited to have some desirable properties. Denote τ min as the minimum transmission duration with amultiplicity ofL min andanother transmission durationτ other with a multiplicityL other . Suppose that the energy function satisfies the property that w(τ min ,B)/w(τ other ,B) is a monotonicallyincreasingfunctionofB, therealwaysexistsaB 0 , suchthatwhenB >B 0 , L min w(τ min ,B)>L other w(τ other ,B),ortheminimumsubgroupistheleadingcontributor in the total transmission energy. This property is typical for energy functions, and is satisfied, e.g., by the energy function w(τ,B)=τ(2 2B/τ −1) from [UBPG02]. Before we present the asymptotically optimal way of dropping one packet, we intro- duce the following definition: Definition: The multiplicity of a (sub)group is the number of consecutive packets around the minimum subgroup (which may include previous and/or subsequent sub- groups) having the same minimum optimal transmission duration. 161 Now, from Claim 5.2, we have: Proposition 5.4 Assume that the energy function satisfies the property as discussed above. When dropping one packet, the packet has to be dropped from the minimum sub- group in order to achieve an asymptotically optimal total transmission energy. If there are multiple minimum subgroups, choose the one with the largest multiplicity. Proof: The proof of the case when the minimum subgroup has a transmission dura- tion strictly less than any other subgroups is straightforward. Consider the case when there exist multiple (sub)groups having the same minimum transmission duration. To prove the asymptotic optimality of choosing the largest multiplicity, we just need to show that optimally dropping a packet from the minimum subgroup will not just benefit the subgroup itself, but also benefit all the preceding and/or subsequent subgroups having the same minimum transmission duration. First, it is not difficult to see that after optimally dropping one packet from the min- imum subgroup, all the remaining packets in the given minimum subgroup will have a positive increase in the transmission durations. Now, recall the non-increasing properties between adjacent groups, and the non-decreasing properties between adjacent subgroups in a type-2 group. If some preceding and/or subsequent subgroups have the same mini- mum transmission duration before packet dropping, after packet dropping and the opti- mal offline scheduling, these subgroups will have a positive increase in the transmission durations as well. This completes the proof. Now, what if we need to drop multiple packets? Consider a recursive packet dropping scheme in which we will drop one packet at a time and repeat for K packets. For each 162 step, the packet is dropped based on Proposition 5.4. For convenience, we denote this scheme as the ‘recursive min subgroup’ approach. However, is this recursive approach asymptotically optimal when dropping K >1 packets? We have the following claim: Claim 5.4 The ‘recursive min subgroup’ approach described above is not necessarily asymptotically optimal when K >1 packets are dropped. Proof: We provide a counter example of 5 packets with inter-arrival times [1,1,1,1,T] with T = 2. Suppose we need to drop two packets. An asymptotically optimal dropping scheme (also an optimal scheme in this case) is to drop packets 2 and 4, such that the new inter-arrival times are given by [2, 2, 2], the most equalized vector and hence the leastpossibletotaltransmissionenergy. Bytherecursiveapproach, wewoulddrop3first, followed by either packet 2 or 4, resulting in a suboptimal set of inter-arrival times of [3, 1, 2] or [1, 3, 2], respectively. The same result in Claim 5.4 also holds for optimal packet dropping schemes given a specific energy function and a packet size. In fact, the example in the proof of Claim 5.4 demonstrates that it is not optimal (vs. asymptotically optimal) to sequentially drop the best single packet. Thus, optimally dropping K packets does not seem to admit an algorithm with tractable complexity. Note that when two or more packets arrive very close in time, these packets are likely to become delay-constrained. Thus, dropping a packet with the minimum inter-arrival timeintheminimumsubgroupisexpectedtosignificantlyalleviatethedelay-constrained issue. If K > 1, this can be repeated K times, similar to the ‘recursive min subgroup’ approach. Wedenotethisrecursiveapproachasthe‘recursivemininter-arrival’approach. 163 One may of course further simplify the dropping scheme by just dropping the K packets with the minimum inter-arrival times among the M packets, which is denoted as ‘simple min inter-arrival’ approach. It is worth mentioning that the K dropped packets by the ‘recursive min inter-arrival’ scheme do not necessarily have the minimum inter-arrival times among the M packets. Due to the difficulty in obtaining explicit (asymptotically) optimal packet dropping schemes, we introduce two performance bounds for asymptotically optimal packet drop- ping schemes in the sequel. The extension of the performance bounds to optimal packet dropping schemes is straightforward. Consider dropping a packet from the scheduling separation interval containing the minimum subgroup. However, instead of dropping it from the minimum subgroup, we dropthesecondtolastpacketofthegivenschedulingseparationintervalandschedulethe packetsinthisschedulingseparationintervalusingthesingletransmissiondeadlinemodel. This can be repeated K times. Note that those scheduling separation intervals with no packets dropped are still scheduled based on the individual delay constraint model. We denotethis droppingandscheduling approach as the ‘recursive single deadline’ approach. We have the following proposition: Proposition 5.5 The total transmission energy resulted from the asymptotically optimal packet dropping schemes for the individual delay constraint model is asymptotically lower bounded by that achieved by the ‘recursive single deadline’ approach described above. Proof: Consider dropping of K ≥ 1 packets, and assume the number of scheduling separation intervals is S. Suppose under an asymptotically optimal dropping scheme, 164 the number of dropped packets from each scheduling separation interval is K aopt = [K 1 ,··· ,K S ], where K i ≥ 0, and P S i=1 K i = K. Similarly, denote K rsd as the set of dropped packets from each scheduling separation interval for the ‘recursive single dead- line’approach. IfK rsd =K aopt ,foreachseparationinterval,the‘recursivesingledeadline’ approach yields a transmission energy no more than that of the asymptotically optimal dropping scheme, due to less stringent delay constraints. Thus, over the entire set of packets, the ‘recursive single deadline’ provides a lower bound of performance. Now, if K rsd 6=K aopt , duetotherecursiveprocedure, the‘recursivesingledeadline’yieldsalower transmission energy with the dropping set K rsd compared with the set K aopt . Therefore, the ‘recursive single deadline’ with K rsd 6= K aopt still provides a lower bound of the asymptotically optimal scheme with K aopt . We can improve the performance bound by observing the following. Regardless of which packet is dropped in the s-th scheduling separation interval (note that the first packet and the last packet, M s , are not dropped), the resulting arrival times are al- ways lower bounded by [t 1 ,··· ,t Ms−2 ,t Ms ] and the resulting departure time constraints are always upper bounded by [t 1 ,t 3 ,··· ,t Ms−1 ,t Ms ]+T, where t j is the arrival time of packet j in the scheduling separation interval. Similarly, when K s > 1 packets need to be dropped, the energy performance is lower bounded by assuming arrival times of [t 1 ,··· ,t Ms−Ks−1 ,t Ms ] and departure time constraints of [t 1 ,t Ks+2 ,··· ,t Ms−1 ,t Ms ]+T (equivalenttodroppacketsM s −K s toM s −1, orthelastsetofK s packetsexceptpacket M s , with a new delay constraint vector [0, Ks X j=1 d j+1 , Ks X j=1 d j+2 ,··· , Ks X j=1 d j+Ms−Ks−2 ,0]+T. 165 A recursive procedure, similar to that in the ‘recursive single deadline’ approach, can be derived based on the above dropping and scheduling scheme. Such an approach is denoted as the ‘recursive arrival departure’ approach. Proposition 5.6 Thetotaltransmissionenergyresultingfromtheasymptoticallyoptimal packet dropping schemes for the individual delay constraint model is asymptotically lower bounded by that achieved by the ‘recursive arrival departure’ approach described above. Proof: The proof is similar to that of Proposition 5.5, and thus is omitted. The following corollary is straightforward: Corollary 5.7 E g {Asymptotically Optimal} ≥ E g {Recursive Arrival Departure Bound} ≥ E g {Recursive Single Deadline Bound}, where E g {.} denotes the asymptotic average transmission energy of a given scheme. It is worth emphasizing that both the ‘recursive arrival departure’ approach and the ’recursive single deadline’ approach only serve as the lower energy bounds for the asymptotically optimal dropping schemes. These two bounds may not be achievable by the asymptotically optimal dropping schemes. Table 5.1 summarizes all the packet dropping schemes, including the two bounds, discussed above. The transmission energy savings from the dropping schemes will be illustrated via numerical results in Section 5.5 under various scenarios. 166 Table 5.1: Summary of Packet Dropping Schemes for the Individual Delay Constraint Model Dropping Scheme Performance Complexity Asymptotically Optimal Optimal Very High Recursive Min Sub-group Near optimal High Recursive Min Inter-Arrival Suboptimal Medium Simple Min Inter-Arrival Fair Low Recursive Arrival Departure Bound Fairly tight bound Low-Medium Recursive Single Deadline Bound Loose bound Low-Medium 5.5 Numerical Results In this section, we will first investigate ¯ τ (K) N,1 and ¯ τ (K) N,N vs. N and K, followed by energy savings due to intelligent packet dropping. We assume a Poisson arrival rate of λ = 1 packet/second. An energy function of w(τ)=τ(2 2B/τ −1) [UBPG02] is used. 5.5.1 Properties of ¯ τ (K) N,1 and ¯ τ (K) N,N Figures 5.3 and 5.4 show ¯ τ (K) N,N in various scenarios. As can be seen, ¯ τ (K) N,N increases significantly with K, even if only one packet is dropped (K = 1). The transmission duration ¯ τ (K) N,N vs. K exhibits a concave curve. Note that the number of remaining packets after packet dropping, N, does not impact ¯ τ (K) N,N much (see Figure 5.3), when N is reasonably large. In other words, ¯ τ (K) N,N is primarily determined by K when N is sufficiently large. Figures 5.5 and 5.6 show ¯ τ (K) N,1 in similar scenarios. As can be seen, ¯ τ (K) N,1 decreaseswithK. Notethatdifferentfrom ¯ τ (K) N,N , ¯ τ (K) N,1 stillnoticeablygrowswithN, even when N is reasonably large. This is not surprising as ¯ τ (K) N,1 may always potentially benefit from additional packets (recall ¯ τ (K) N,1 = max n∈{1,···,N} P n j=1 d j /n). Note that ¯ τ (K) N,1 is lower-bounded by ¯ τ N,1 (without packet dropping), while ¯ τ (K) N,N is upper-bounded by 167 Figure 5.3: ¯ τ (K) N,N vs. K, when N is fixed. Figure 5.4: ¯ τ (K) N,N vs. N, when K is fixed. 168 Figure 5.5: ¯ τ (K) N,1 vs. K, when N is fixed. Figure 5.6: ¯ τ (K) N,1 vs. N, when K is fixed. 169 ¯ τ N,N (without packet dropping) for all K ≥ 1. This confirms the de-majorization effect of optimal packet dropping. 5.5.2 Energy Savings due to Packet Dropping Forthesingletransmissiondeadlinemodel, Figure5.7showsthenormalizedtransmission energy 3 for the no packet-dropping case, and the optimal dropping (dropping the last K packets) schemes. The transmission deadline T is set to 1/λ seconds after the last packet arrives. The results were averaged over 1000 independent runs. It can be observed that compared with no packet dropping, the optimal dropping scheduler yields significant energy savings even when K = 1 and it achieves transmission energy close to the lower bound of w(1/λ). The energy savings due to the optimal dropping increases as the normalized packet size increases. For the individual delay constraint model, Figure 5.8 compares the performance dif- ferencebetweenanasymptoticallyoptimalpacketdroppingschemeviabrute-forcesearch and the ‘recursive min subgroup’ scheme. The transmission energy is normalized by that of the no-dropping case with a reduced packet arrival rate of λ(1−K/M). The results are averaged over 10 independent runs. The two performance bounds are also shown for comparison. As can be seen, the ‘recursive min subgroup’ packet dropping scheme yields a transmission energy which is indistinguishable from that of the asymptotically optimal scheme achieved via brute-force search. The performance bound by the ‘recursive arrival departure’ scheme is fairly tight, especially when the individual delay constraint is rela- 3 We first multiply by (1− K/M) in order to make a fair comparison between the schedulers with and without packet dropping, and then normalize with respect to the ideal case when packets have fixed inter-arrival times of 1/λ. 170 Figure 5.7: Normalized transmission energy (in %) vs. B for the single deadline model, M =1000. tively small or large. The performance bound by the ‘recursive single deadline’, on the other hand, is rather loose. Figure 5.9 shows similar comparisons with M = 1000, and K = 10. As can be seen, the energy savings due to packet dropping decrease with T, which shows that the dropping scheme is more effective for more demanding delay constraints. The ‘recursive min inter-arrival’, based on the observation of the minimum inter-arrival time in the minimum subgroup in each iteration, achieves almost the same energy savings as the more complicated ‘recursive min subgroup’ scheme. The simplest scheme, ‘simple min inter-arrival’, which blindly drops the packets with minimum inter-arrival times, yields some energy savings, but noticeably less than other suboptimal dropping schemes. 171 Figure 5.8: Normalized transmission energy vs. T for the individual delay constraint model, M =200, K =2. Figure 5.9: Normalized transmission energy vs. T for the individual delay constraint model, M =1000, K =10, and B =2. 172 5.6 Conclusions This chapter investigates proactive packet dropping to achieve transmission energy sav- ings. The optimal packet dropping schemes for the single transmission deadline model is derived and its impact is analyzed. For packets subject to individual delay constraints, the optimal scheme appears to be intractable. Thus, asymptotically optimal dropping schemes are pursued, along with sub-optimal packet dropping schemes and performance bounds. Simulations demonstrate significant energy savings are possible via intelligent packet dropping schemes. 173 Chapter 6 Conclusions and Future Work 6.1 Conclusions In this thesis, we considered delay-constrained energy-efficient packet transmissions in a static channel (Chapter 2) and in a fading channel (Chapter 3) for the conventional singlehoptransmission(i.e., directtransmission), andextendedtheresultstoamultihop transmission in Chapter 4. Intelligent packet dropping was investigated in Chapter 5, which shows that significant energy savings are possible. Two delay constraint models were considered, namely, the single transmission deadline where a set of packets observe a single common transmission deadline, and the individual delay constraint model where each packet has its own transmission deadline. The optimal offline scheduling (vis-` a-vis total transmission energy), which assumes traffic and channel information a priori, for a set of packets subject to individual delay constraintswasderivedandanalyzedinstaticchannels(Chapter2)andinfadingchannels (Chapter 3), for a single hop link, and then generalized to any single hop in a multihop link, where the per-hop transmission may occur only in selective time slots due to a time 174 divisionmultiplexingoperationamongnodes(Chapter4). Optimalofflineschedulingover a multihop link was studied in Chapter 4, and the optimal delay budget allocation was derived for static channels and bounded for fading channels. Heuristic online schedulers were also investigated under various scenarios. The properties of the offline and online schedulers were analyzed and characterized, and further demonstrated via simulations under a wide range of conditions. It was shown that energy-efficient scheduling under the individual delay constraint model provides a more flexible tradeoff between transmission energy and packet delay, compared with the scheduling under the single deadline model. Moreover, energy-efficient scheduling may yield significant energy savings, especially un- der highly bursty packet arrivals and/or dynamic channel variations. Energy savings can furtherbeobtainedviaproactive packet dropping, exploiting the packetloss-tolerantfea- ture for some applications. Intelligent packet dropping schemes are studied and analyzed for both delay models in Chapter 5. 6.2 Future Work In this thesis, from the analysis of the properties of the optimal offline scheduling, we are able to obtain a simple and exact closed form solution of the average packet delay performance. We rely on numerical results to characterize the transmission energy per- formance of the optimal offline schedule. It would be desirable if the transmission energy resulting from the optimal offline schedule can also be analytically characterized under various scenarios. 175 A single user transmission is considered in this thesis. While it is relatively straight- forward to extend some of the results presented here in a multi-user setting (e.g., see [UBG04] for discussion of multi-user scheduling under the single deadline model), multi- user scheduling generally imposes a lot of design challenges. Besides energy-efficiency, other performance metrics may need to considered as well, such as fairness among users. The design becomes particularly more challenging when users have heterogeneous QoS requirements, such as delay constraints, minimum throughput requirements, etc. In multi-hop scenarios, we have derived the optimal delay budget allocation for the optimalofflinescheduleobservingthesingledeadlinemodelandtheindividualdelaycon- straint model in a static channel. The same budget allocation, however, is shown to be not necessarily optimal for the considered heuristic online schedulers, and in particular, for the case of fading channels. More work in this area is desirable. In particular, it remains to be seen whether the same or a similar delay budget allocation exists for the optimal online scheduling algorithm, typically obtained through dynamic programming (e.g., see [ZM05]). Moreover, in this thesis, we have assumed a simple and equal time resource allocation among nodes in a multihop link. Flexible time allocation over hops may be possible, especially when different hops observe distinct channel conditions and have heterogeneous energy requirements. The study of flexible time allocation and its interaction with delay budget allocation for maximum energy efficiency seem to be ap- pealing. Energy-efficientschedulingovermultihoppingmayfurtherbeinvestigatedunder a multi-user setting. Multi-user scheduling may exist at the source node, and/or at some or all relaying nodes. 176 In this thesis, we did not assume any constraint on the maximum transmit power at a node. Inreality,thereisalwaysafinitemaxtransmitpowerlevel. Amaxpowerlimitation may imply that there is no guarantee that all the buffered packets can be delivered by the specified transmission deadline(s). Thus, packet dropping may be inevitable. In addition, we have assumed that a node may transmit with any transmission rates. In practical scenarios, a node may only have a small finite set of available transmission rates. It is worth to incorporating this constraint in energy-efficient scheduling as well. The intelligent packet dropping schemes presented in Chapter 5 also assumed offline information. More intelligent online packet dropping schemes need to be investigated. In addition, intelligent packet dropping is expected to improve asymptotic energy and delay trade-off, as was shown in [Nee06a]. 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A.1.1 Alternative Proof 1: Via Majorization Theory Majorization is often used to characterize the mixing degrees of vectors of the same dimension [Bha97]. To facilitate the analysis, we repeat the definition of majorization here (see [Bha97] for more details). Definition: For vectors of the same dimension M, a vector ~ v 1 is said to be majorized by another vector~ v 2 , denoted as~ v 1 ≺~ v 2 , if P m i=1 v ↓ 1,i ≤ P m i=1 v ↓ 2,i , when 1≤m≤M−1, and P M i=1 v 1,i = P M i=1 v 2,i , where ~ v ↓ denotes a non-increasingly ordered ~ v. When ~ v 1 ≺ ~ v 2 , ~ v 1 is more mixed. In fact, ~ v 1 ≺ ~ v 2 if and only if P M i=1 f(v 1,i ) ≤ P M i=1 f(v 2,i ), for all convex function f(.) [Nie01]. The proof below mainly utilizes the properties of the optimal transmission durations specified by Lemma 2.1 for adjacent groups and Lemma 2.2 for type-2 groups. Lemma A.1 Given an inter-arrival time vector ~ d, the offline scheduling algorithm in (2.9) yields an optimal transmission duration vector ~ τ that is always majorized by any other feasible transmission duration vectors and thus is optimal in minimizing the total transmission energy. Proof: By contradiction. Consider a feasible transmission duration vector ~ τ o 6= ~ τ (see (2.4)). By the non-idling scheduling constraint, we have P M i=1 τ o,i = P M i=1 τ i . Both ~ τ and ~ τ o are not necessarily non-increasing. Therefore, each vector has to be ordered before comparison. Denote the vectors after the non-increasing ordering as ~ τ ↓ and ~ τ ↓ o , respectively, where in ordering ~ τ, we assume that if τ i = τ j and i < j, packet i appears before packet j in ~ τ ↓ . We want to show that P m i=1 τ ↓ o,i ≥ P m i=1 τ ↓ i ,∀m∈[1,··· ,M−1]. Suppose that there exists a j,∀j ∈ [1,··· ,M − 1], which is the first such that P j i=1 τ ↓ o,i < P j i=1 τ ↓ i . Under the optimal offline scheduling ~ τ, packet j must belong to either a type-1 or type-2 group. Denote j 1 and j e as the start and the end of a set of consecutive packets within the group, respectively, such that j 1 ≤j≤j e and τ j =τ l ,l∈ [j 1 ,··· ,j e ]. Using the non-increasing trend of τ ↓ o , we obtain P je i=1 τ ↓ o,i < P je i=1 τ ↓ i . Since packets in a type-1 group or in a type-2 subgroup have the same transmission duration, 186 packets [1,··· ,j e ] in ~ τ ↓ thus consist of an integer multiple of type-1 groups and/or an integer multiple of type-2 subgroups. In addition, from Lemma 2.2 (i.e., subgroups in a type-2grouparenon-decreasingintransmissiondurations), ifa type-2 groupunder~ τ has one or more packets ∈ [1,··· ,j e ] under ~ τ ↓ , its last subgroup must be entirely contained by [1,··· ,j e ] under ~ τ ↓ . These packets [1,··· ,j e ] under ~ τ ↓ correspond to a set of packets [k 1 ,··· ,k je ] in the original non-ordered vector ~ τ, where k 1 < ··· < k je . Now consider the first k je packets under ~ τ. From previous discussions, it can be seen that packets [1,··· ,k je ] under ~ τ are formed by an integer multiple of groups. That is, P k je i=1 τ i = P k je i=1 ˆ d i . If there are no type-2 groups within [1,··· ,k je ] under~ τ, from Lemma 3.1 (i.e. adjacent groups are non- increasingintransmissiondurations), theset[1,··· ,j e ]mustbethesameas[k 1 ,··· ,k je ], and hence, k je X i=1 τ o,i < k je X i=1 τ i . Since the transmission of packet k je ends with an empty buffer under~ τ, there must exist an idling period∈[0,··· , P k je i=1 ˆ d i ] under ~ τ o . Thus, ~ τ o is not feasible. Iftherearetype-2groupswithin[1,··· ,k je ]under~ τ, thetotaltransmissiondurations of packets / ∈ [k 1 ,k 2 ,··· ,k je ] can not be increased due to the existence of delay-critical packet(s). That is, for i≤k je , X i/ ∈[k 1 ,···,k je ] τ i ≥ X i/ ∈[k 1 ,···,k je ] τ o,i Combining with P i∈[k 1 ,···,k je ] τ i = P je i=1 τ ↓ i > P je i=1 τ ↓ o,i = P i∈[k 1 ,···,k je ] τ o,i , we also have P k je i=1 τ o,i < P k je i=1 τ i . Thus, ~ τ o is not feasible as well. Therefore,~ τ ≺~ τ o and the scheduling in (2.9) is optimal for any strictly convex energy functions. Similarly, we can also provide an alternative proof of the optimal offline scheduler for the single transmission deadline model as in [UBPG02], summarized by the following Lemma: Lemma A.2 For the single transmission deadline model, given an inter-arrival time vector ~ d, the optimal offline scheduling algorithm in [UBPG02] yields an optimal trans- mission duration vector ~ τ that is always majorized by any other feasible transmission duration vectors and thus is optimal in minimizing the total transmission energy. Proof: By contradiction. Consider a feasible transmission duration vector ~ τ o 6=~ τ (see (2.1)). By the non-idling scheduling constraint, we have P M i=1 τ o,i = P M i=1 τ i . Since both ~ τ and ~ τ o are non-increasing, there is no need to order the two vectors and we just need to show that P m i=1 τ o,i ≥ P m i=1 τ i ,∀m ∈ [1,··· ,M −1]. An idling period is inevitable if there exists a j,∀j ∈ [1,··· ,M −1], which is the first such that P j i=1 τ o,i < P j i=1 τ i , following the same justification as in Lemma A.1. 187 The following Lemma compares the optimal transmission duration vectors for the single transmission deadline model (denoted as ~ τ single ), and for the individual delay con- straint model (denoted as ~ τ ind ): Lemma A.3 Given an inter-arrival time vector ~ ˆ d=[ ˆ d 1 ,··· , ˆ d M ], ˆ d i ≤T,i∈[1,··· ,M], ~ τ ind ~ τ single . Proof: Under ~ τ single , there exists one or more groups. Consider any group in ~ τ single . It is straightforward to see that the packets in the group would belong to one or more groups (refer to (2.5) and (2.7)) under ~ τ ind . Since the packets in any given group under ~ τ single have the same transmission duration, the corresponding transmission duration vector for this group is always majorized by that of the counterpart under ~ τ ind . Since the majorization relationship still holds for a concatenation of vectors [Bha97], we have ~ τ ind ~ τ single . A.1.2 Alternative Proof 2: Via Detailed Comparisons Here we provide an alternative proof of Theorem 2.1. We will follow the same approach as in [UBPG02] by studying a new arbitrary scheduler which satisfies all the feasibility conditions and proving that the new scheduler yields a higher total transmission energy. For convenience, the following definitions are introduced: Considertheoptimalschedule~ τ ∗ andanyotherfeasibleschedule~ τ forthefirstschedul- ing separation interval (the proof for other scheduling separation intervals are similar). Let i be the first index where τ i 6= τ ∗ i . We will show that w(~ τ) > w(~ τ ∗ ). There are two possibilities to consider: Case 1: τ i > τ ∗ i , since P M 1 j=1 τ j = P M 1 j=1 ˆ d j (otherwise, ~ τ would idle for some time, making it sub-optimal), where M 1 is the total number of packets in the first scheduling interval, there must be at least one j > i for which τ j < τ ∗ j . Let r = min{j : i < j < M 1 ,τ j <τ ∗ j }. Claim A.1 For the optimal schedule ~ τ ∗ , there is no packets between packet j and packet r with a delay equals to T. Proof: Suppose there is one packet m, i < m < r, such that its delay under the optimal offline schedule satisfies q ∗ m = T. Since τ i > τ ∗ i , and τ l = τ ∗ l , i < l < r, we have q m >T. This causes a delay violation. As a result, the transmission durations between packets i and r follows the same non-decreasing property as in the single deadline model and we have τ ∗ i ≥τ ∗ r [UBPG02]. Thus, τ i >τ ∗ i ≥τ ∗ r >τ r . Now consider the schedule ~ σ defined as follows: σ i M = τ i −Δ,σ r M = τ r +Δ, and σ j M = τ j ,∀j6=i,r where 0<Δ≤min{(τ i −τ ∗ i ),(τ ∗ r −τ r )}. Claim A.2 The schedule ~ σ does not idle and is feasible. 188 Proof: since P M 1 j=1 τ j = P M 1 j=1 σ j = P M 1 j=1 ˆ d j , it does not idle. By the definition of the indices i and r, and the feasibility of ~ τ and ~ τ ∗ , it follows that (note that d j < T, ∀j∈[1,··· ,M 1 ], by the scheduling separation interval definition):      P k j=1 σ j = P k j=1 τ j ≥ P k j=1 ˆ d j , 1≤k≤i−1 P k j=1 σ j ≥ P k j=1 τ ∗ j ≥ P k j=1 ˆ d j , i≤k≤r−1 P k j=1 σ j ≥ P k j=1 τ j ≥ P k j=1 ˆ d j , k≥r This verifies feasibility condition (i) in (2.4). Now we check the second condition in (2.4). Note that for packets 1 ≤ k ≤ i−1 and k ≥ r, we have P k j=1 σ j ≥ P k j=1 τ j . The scheduling delays of these packets are the same as those in the schedule ~ τ. For packets i ≤ k ≤ r−1, P k j=1 σ j < P k j=1 τ j , then q l M = P l j=1 σ j − P l−1 j=1 ˆ d j ≤ T, ∀l ∈ [i,··· ,r−1]. So, condition (ii) for feasibility is also verified. Claim A.3 w(~ σ)<w(~ τ). Proof: Similar to that in [UBPG02] and thus omitted. Therefore, under case 1, any feasible schedule ~ τ can be modified to obtain a more energy efficient schedule ~ σ. As a result, schedulers which are different from ~ τ ∗ in Case 1 are sub-optimal. Case 2: τ i < τ ∗ i , such a case is not feasible in [UBPG02]. However, it may be feasible due to the introduction of individual packet delay constraints. Again, due to the non-idling condition, there must be at least one j > i for which τ j > τ ∗ j . Let r =min{j :i<j <M 1 ,τ j >τ ∗ j }. Claim A.4 If there are no type 2 groups, or if the i-th and the r-th packets do not belong to the same type 2 group, ~ τ is not feasible. Basically, thisscenariodegeneratestothesingletransmissiondeadlinecaseandtheabove claim has been proven in [UBPG02]. Claim A.5 If there is at least one type 2 group and the i-th and the r-th packets belong to the same type 2 group, the schedule ~ τ does not idle and is feasible. The proof can be done by showing by one example, which is omitted here. Note that from Lemma 2.2, τ ∗ r ≥τ ∗ i , which leads to the next claim: Claim A.6 τ r >τ ∗ r ≥τ ∗ i >τ i . Now, q l = ( P l j=1 τ j −τ l−1 j=1 ˆ d j = P l j=1 τ ∗ j −τ l−1 j=1 ˆ d j =q ∗ l , l =1,··· ,i−1 P l j=1 τ j −τ l−1 j=1 ˆ d j < P l j=1 τ ∗ j −τ l−1 j=1 ˆ d j =q ∗ l ≤T, l =i,··· ,r−1 Since q l <T, l =i,··· ,r−1, consider the schedule ~ σ defined as follows: σ i M = τ i +Δ,σ r M = τ r −Δ, and σ j M = τ j ,∀j6=i,r 189 where 0<Δ≤min{(τ ∗ i −τ i ),(τ r −τ ∗ r )}. Claim A.7 The schedule ~ σ does not idle and is feasible. Proof: The non-idling condition for ~ τ can be easily verified. From the definition of the indices i and r, and the feasibility of ~ τ and ~ τ ∗ , it follows that:      P k j=1 σ j = P k j=1 τ j ≥ P k j=1 ˆ d j , 1≤k≤i−1 P k j=1 σ j > P k j=1 τ j ≥ P k j=1 ˆ d j , i≤k≤r−1 P k j=1 σ j ≥ P k j=1 τ j ≥ P k j=1 ˆ d j , k≥r Thus, ~ σ satisfies feasibility condition (i) in (2.4). The satisfaction of feasibility condition (ii) is guaranteed by definition of the new scheduler. Claim A.8 w(~ σ)<w(~ τ). Proof: Similar to that in [UBPG02] and thus omitted. This completes of the proof of Theorem 2.1. A.2 Optimal Offline Scheduling for Unequal Packet Sizes and Unequal Individual Delay Constraints Here we extend the optimal offline schedule for the individual delay constraint model to unequalpacketsizesandunequalindividualdelayconstraints. Wefirststartwithunequal packet sizes but equal individual delay constraints. The optimal offline schedule for the single deadline model with unequal packet sizes was discussed in [UBPG02]. Similar to [UBPG02], we should now try to equalize per bit packet transmission duration instead of per packet transmission duration. However, we will adopt a derivation approach different from the one in [UBPG02]. Indeed, this approach is also applicable to the single deadline model and can be used as an alternative to derive the optimal offline scheduler under unequal packet sizes for the single deadline model [UBPG02]. Note that the optimal offline scheduler with equal packet sizes and equal individual delay constraints discussed in Section 2.3 is valid for any packet inter-arrival times. Now, in case of unequal packet sizes, we can view each bit in a packet as a virtual packet, and the inter-arrival times between all bits in the same packet are essentially zero. In other words, a packet of B i bits is treated as if there were B i equal-size (i.e., one bit) virtual packet arrivals, with virtual inter-arrival times e i,j =0,j∈[1,··· ,B i −1], and e i,B i = ˆ d i . Applying these virtual packet arrivals to (2.7) and (2.8), after some derivations, it can be shown the optimal transmission duration for any packet m, under a queuing delay of ˜ q m , is given by τ m =B m max m≤i≤M τ 1[i] , (A.1) where τ 1[i] , i∈[m,··· ,M], given by τ 1[i] =min −˜ qm+ P i l=m ˆ d l P i l=m B l , (T−˜ qm) Bm , (T−˜ qm)+ ˆ dm Bm+B m+1 ,··· , (T−˜ qm)+ P i−2 l=m ˆ d l P i−1 l=m B l . (A.2) 190 Now when packets have different delay constraints T i ,i = 1,··· ,M, it is natural to assume that the transmission deadline is monotonically non-decreasing, i.e., t i + T i ≤ t i+1 +T i+1 , such that the packets still follow the FIFO rule. Similar scheduling feasibility constraints in (2.4) still hold, with ˆ d i = min(d i ,T i ), for 1 ≤ i ≤ M. In addition, the check of packet delay constraints should be based on the different individual packet delay constraints T i , instead of the same constraint T. The optimal offline schedule still follows the exact procedure as discussed in Section 2.3. The optimal transmission duration for packet m∈[1,··· ,M] can be obtained as τ m =B m max m≤i≤M τ 1[i] , (A.3) where τ 1[i] , i∈[m,··· ,M], given by τ 1[i] =min −˜ qm+ P i l=m ˆ d l P i l=m B l , (Tm−˜ qm) Bm , (T m+1 −˜ qm)+ ˆ dm Bm+B m+1 ,··· , (T i−1 −˜ qm)+ P i−2 l=m ˆ d l P i−1 l=m B l . (A.4) where ˜ q m ≥0 again is the queuing delay of packet m. A.3 Statistical Trend of the Optimal Offline Transmission Durations This is to prove Lemma 2.4, the statistical trend of the optimal offline transmission durations. Letusstartbycomparingτ m andτ m+1 ,1≤m<M. Foragivenm,weassume for now ˆ d m <T. The case when ˆ d m =T will be considered later. For indices j6=m,j∈ [1,··· ,M], we still have the same constraint ˆ d j ≤ T. Under this assumption, packets m andm+1belongtothesameschedulingseparationinterval. DenoteΔτ m,m+1 =τ m −τ m+1 as the difference between the optimal transmission durations of packets m and m+1. Obviously, if packetsm andm+1 belong to the same subgroup, Δτ m,m+1 =0. If the two packets belong to two adjacent subgroups of the same type-2 group, Δτ m,m+1 ≤0. When the two packets belong to two different groups (regardless of type-1 or type-2 groups), we will have Δτ m,m+1 ≥0. For any m∈[1,··· ,M−1], if τ m 6=τ m+1 , there are two possibilities (see also (2.10)): packets m and m + 1 are either of different groups (hence Δτ dg(m,m+1) m,m+1 ≥ 0, where the superscript dg(m 1 ,m 2 ) denotes that packets m 1 and m 2 are of different groups), or of different subgroups of the same type-2 group (hence Δτ dsg(m,m+1) m,m+1 ≤ 0, where the superscript dsg(l 1 ,l 2 ) denotes that packets l 1 and l 2 are of different subgroups in the same type-2 group). The difference Δτ dg(m,m+1) m,m+1 can be computed as Δτ dg(m,m+1) m,m+1 =τ dg(m,m+1) m −τ dg(m,m+1) m+1 =max 1≤i≤m τ dg(m,m+1) m[i] −max 1≤i≤M−m τ dg(m,m+1) 1[i] , (A.5) where τ dg(m,m+1) m[i] is similarly defined as τ M[i] in (2.18), using the inter-arrival time vector [ ˆ d 1 ,··· , ˆ d m ], 191 and τ dg(m,m+1) 1[i] is similarly defined as τ 1[i] in (2.7), using the inter-arrival time vector [ ˆ d m+1 ,··· , ˆ d M−1 ,T]. Similarly, Δτ dsg(m,m+1) m,m+1 =τ dsg(m,m+1) m −τ dsg(m,m+1) m+1 =max 1≤i≤m τ dsg(m,m+1) m[i] −max 1≤i≤M−m τ dsg(m,m+1) 1[i] , (A.6) where τ dsg(m,m+1) m[i] is derived based on the inter-arrival time vector [ ˆ d 1 ,··· , ˆ d m−1 ,T], and τ dsg(m,m+1) 1[i] is derived based on the inter-arrival time vector [ ˆ d m + ˆ d m+1 −T, ˆ d m+2 ,··· , ˆ d M ]. Using (2.7), (2.18), (A.5), (A.6), and the i.i.d. assumption of ~ ˆ d, it is not difficult to see that τ dg(m,m+1) m+1 and τ dsg(M−m,M−m+1) M−m are identically distributed, and τ dg(m,m+1) m and τ dsg(M−m,M−m+1) M−m+1 are identically distributed. This is consistent with the symmetry property presented by Theorem 2.2. Therefore, we claim: Claim A.9 For any M ≥ 1, when the inter-arrival times ˆ d m ,1 ≤ m ≤ M − 1, are i.i.d., under the optimal offline scheduling, Δτ dg(m,m+1) m,m+1 and −Δτ dsg(M−m,M−m+1) M−m,M−m+1 are identically distributed. Similarly, we have: Claim A.10 For any M ≥ 1, when the inter-arrival times ˆ d m ,1 ≤ m ≤ M − 1, are i.i.d., under the optimal offline scheduling, Δτ dsg(m,m+1) m,m+1 and −Δτ dg(M−m,M−m+1) M−m,M−m+1 are identically distributed. In the previous discussions, when comparing the transmission durations of two adja- centpacketsmandm+1, wehaveassumedthat ˆ d m 6=T. Now, when ˆ d m =T, packetsm andm+1 would belong to different scheduling separation intervals. Thus, the scheduling ofthesetwopacketsisde-coupled. Packetmisthelastpacketofitsschedulingseparation interval, while packet m+1 is the first packet of its scheduling separation interval. As a result, we have Δτ dssi(m,m+1) m,m+1 =τ dssi(m,m+1) m −τ dssi(m,m+1) m+1 =max 1≤i≤m τ dssi(m,m+1) m[i] −max 1≤i≤M−m τ dssi(m,m+1) 1[i] , (A.7) where the superscript dssi(m 1 ,m 2 ) denotes that packets m 1 and m 2 belongs to different scheduling separation intervals, τ dssi(m,m+1) m[i] is similarly defined as τ M[i] in (2.18), using 192 the inter-arrival time vector [ ˆ d 1 ,··· , ˆ d m−1 ,T] (recall ˆ d m = T), and τ dssi(m,m+1) 1[i] is simi- larly defined as τ 1[i] in (2.7), using the inter-arrival time vector [ ˆ d m+1 ,··· , ˆ d M−1 ,T]. The following claim is thus straightforward: Claim A.11 For any M ≥ 1, when the inter-arrival times ˆ d m ,1 ≤ m ≤ M − 1, are i.i.d., under the optimal offline scheduling, Δτ dssi(m,m+1) m,m+1 and −Δτ dssi(M−m,M−m+1) M−m,M−m+1 are identically distributed. As a result, from Claims A.9, A.10 and A.11, we have Claim A.12 For any M ≥ 1, when the inter-arrival times ˆ d m ,1 ≤ m ≤ M − 1, are i.i.d., under the optimal offline scheduling, Δτ m,m+1 and−Δτ M−m,M−m+1 are identically distributed. Again, it is consistent with the symmetry property presented by Theorem 2.2. Inordertocomparethestatisticaltrendoftheoptimalofflinetransmissiondurations, weneedthefollowingtwoclaimswhichcharacterizethetrendoftheoptimaltransmission durationasafunctionofthenumberofpacketsforaspecificrealizationoftheinter-arrival times: Claim A.13 Given m packets with an inter-arrival time vector of [ ˆ d 1 ,··· , ˆ d m−1 ,T] and the individual delay constraint T, the optimal transmission duration of the first packet, denoted as τ m,1 , is non-increasing in m. Proof: From(2.7), foratotalnumberofmpackets, theoptimaltransmissionduration of the first packet is given by τ m,1 = max 1≤i≤m τ 1[i] , where τ 1[i] is defined in (2.6). Now consider a new inter-arrival vector of m+1 packets given by [ ˆ d 1 ,··· , ˆ d m−1 , ˆ d m ,T]. Since q m ≤ T, τ m+1,1 is upper-bounded by the optimal transmission duration based on the inter-arrival time vector [ ˆ d 1 ,··· , ˆ d m−1 ,T]. Therefore, τ m+1,1 ≤τ m,1 . Claim A.14 Given m packets with an inter-arrival time vector of [ ˆ d m , ˆ d m−1 ,··· , ˆ d 1 ] (note the reverse indices) and the individual delay constraint T, the optimal transmission duration of the last packet, denoted as τ m,m , is non-increasing in m. Proof: The proof is similar to that of Claim A.13. Consider a new inter-arrival time vectorofm+1packetsgivenby[ ˆ d m+1 , ˆ d m ,··· , ˆ d 1 ]. Clearly, theoptimalschedulingofthe firstpacketsatisfiesτ m+1,1 ≥ ˆ d m+1 . Thatis, theschedulingoftheremainingmpacketsis upper-bounded (in terms of the total time resource) by an inter-arrival time vector given by [ ˆ d m ,··· , ˆ d 1 ]. Thus, τ m,m ≥τ m+1,m+1 Based on Claim A.13 and Claim A.14, the following claims are straightforward (proof is omitted): Claim A.15 Under the i.i.d. assumption of ˆ d i ,i∈[1,··· ,M−1], E{τ dg(m,m+1) m+1 }, based on the inter-arrival time vector [ ˆ d m+1 ,··· , ˆ d M−1 ,T], is a non-decreasing function of m. This also hold true for E{τ dsg(m,m+1) m+1 } and E{τ dssi(m,m+1) m+1 }. 193 Claim A.16 Under the i.i.d. assumption of ˆ d i ,i∈ [1,··· ,M], E{τ dg(m,m+1) m }, based on [ ˆ d 1 ,··· , ˆ d m ], is a non-increasing function of m. This also holds true for E{τ dsg(m,m+1) m } and E{τ dssi(m,m+1) m }. Therefore, from (A.5), (A.6) and (A.7), it can be seen that E{Δτ m,m+1 } is non- increasing and non-negative when m≤bM/2c and non-decreasing and non-positive oth- erwise, where b.c denotes the floor operation. When M is even, E{Δτ M/2,M/2+1 } = 0. Therefore, the Lemma is proved. A.4 Non-decreasing Queuing Delay Property This appendix provides the proof of Lemma 2.5. Consider two adjacent packets m and m+1, m∈ [1,··· ,M −1]. Assume that for a given realization of ~ ˆ d, the queuing delay is of packet m is e q m = δ,∀δ ≥ 0. In this case, τ m = max 0≤i≤m−1 τ m[i] , where τ m[i] is similarly defined as in (2.18) based on the inter-arrival time vector [ ˆ d 1 ,··· , ˆ d m−1 , ˆ d m + δ]. Similarly, τ m+1 = max 1≤i≤M−m τ 1[i] , as in (2.7) with the inter-arrival time vector [ ˆ d m+1 −δ, ˆ d m+2 ,··· , ˆ d M−1 ,T]. Note that the relationship between τ m and τ m+1 depends on the scheduling separation interval/group associations. For instance, when δ = 0 and ˆ d m < T, packets m and m+1 must belong to two groups and thus τ m ≥ τ m+1 . When δ >0, packets m and m+1 may belong to the same subgroup (thus τ m =τ m+1 ), or two adjacent type-2 subgroups (thus τ m ≤ τ m+1 ). When ˆ d m = T, the scheduling of packets m and m+1 becomes independent. Now, consider a new arrival vector [ ˆ d M−1 , ˆ d 1 ,··· , ˆ d m−1 , ˆ d m +δ] and the transmission durationofthelastpacket(denotedasτ new m+1 ). FromClaimA.14,τ new m+1 ≤τ m . Ontheother hand, consider a new arrival vector [ ˆ d m+1 −δ, ˆ d m+2 ,··· , ˆ d M−2 ,T] (without ˆ d M−1 ), and thetransmissiondurationofthefirstpacket(denotedasτ new m+2 ). FromClaimA.13,τ new m+2 ≥ τ m+1 . Thus, if we join the two new arrival vectors together and form an M-dimensional inter-arrival time vector as ~ ˆ d new = [ ˆ d M−1 , ˆ d 1 ,··· , ˆ d M−2 ,T], under the optimal offline scheduling, potentially packet m+1 in ~ ˆ d new may benefit more from subsequent packets instead of just by the original amount of δ. Thus, the queuing delay for the (m+1)-th packet satisfies e q new m+1 ≥ e q m . Under the i.i.d. assumption of ~ d, E{e q new m+1 } = E{e q m+1 }. Thus, E{e q m+1 }≥E{e q m }. This completes the proof. A.5 Average Transmission Duration for the Single Deadline Model Herewewillderivetheaveragetransmissiondurationforthesingledeadlinemodelundera Poissonarrivalmodel. DenoteΔτ i,i+1 =τ M,i −τ M,i+1 asthedifferenceofthetransmission durations between two adjacent packets i and i+1. Clearly, Δτ i,i+1 = 0 if packet i and i + 1 belong to the same pair [UBPG02]. Otherwise, Δτ i,i+1 > 0. Thus, in order to compute Δτ i,i+1 , we can just focus on the case when τ M,i >τ M,i+1 . 194 Denote τ M,i+1 = max i+1≤m≤M {S (i+1,m) /(m−i)}, where S (j,i) = P i l=j d l , the sum of inter-arrival times from packet j to packet i. The case when τ M,i > τ M,i+1 occurs if and only if the following condition is met: τ M,i+1 <min{S (i,i) , S (i−1,i) 2 , S (i−2,i) 3 ,··· , S (1,i) i }. Now the distribution function of the above expression can be expressed as Pr{min{S (i,i) , S (i−1,i) 2 , S (i−2,i) 3 ,··· , S (1,i) i }>t}=Pr{ S (j,i) i−j+1 >t,∀j∈[1,i]} = R ∞ it R s (1,i) (i−1)t ··· R s (i−1,i) t λ i e −λs (1,i) ds (i,i) ···ds (2,i) ds (1,i) =e −λit P i j=1 j(λit) i−j i(i−j)! , where we have used the fact that f S (i,i) ,···,S (1,i) (s (i,i) ,··· ,s (1,i) ) = f D 1 ,···,D i (d 1 ,··· ,d i ) = λ i e −λs (1,i) given the assumption that the inter-arrival times are i.i.d. following the expo- nentialdistribution,and R λe −λs s n−1 /(n−1)!ds=−e −λs P n j=1 (λs) n−j /(n−j)!. Similarly, the distribution function of τ M,i+1 = max i+1≤m≤M {S (i+1,m) /(m−i)} can be expressed as (see also in [UBPG02]): f τ M,i+1 (t)=λe −λt − P M−i n=2 λ n−1 t n−2 e −λtn h n n−2 (n−2)! − λtn n−1 (n−1)! i . (A.8) Therefore, the average transmission duration of packet i + 1, conditioned on that packet i and i+1 belong to different pairs, denoted by E{τ M,i+1 |(i,i+1) in different pairs}, is given by: E{τ M,i+1 |(i,i+1) in different pairs} = R ∞ 0 tf τ M,i+1 (t)Pr{min{S (i,i) , S (i−1,i) 2 ,··· , S (1,i) i }>t}dt = 1 λ P M−i n=1 P i j=1 ji i−j n n−2 (n+i−nj)(n+i−j−1)! i(n+i) n+i−j+1 (i−1)!(n−1)! . On the other hand, the average transmission of packet i, conditioned on that packet i and packet i+1 belong to different pairs, denoted by E{τ M,i |(i,i+1) in different pairs} can be similarly derived, i.e., E{τ M,i |(i,i+1) in different pairs} = R ∞ 0 tf τ M,i (t)Pr{max{S (i+1,i+1) , S (i+1,i+2) 2 ,··· , S (i+1,M) M−i }>t}dt. It can be shown that f τ M,i (t)= i X j=1 jλ(λit) i−j−1 e −λit λit (i−j)! − 1 (i−j−1)! , and Pr{max{S (i+1,i+1) , S (i+1,i+2) 2 , S (i+1,i+3) 3 ,··· , S (i+1,M) M−i }>t} =1− P M−i j=1 (λt) j−1 e −λit j j−2 (n−1)! . 195 As a result, E{τ M,i |(i,i+1) in different pairs} can be derived as E{τ M,i |(i,i+1) in different pairs}= 1 λ h i+1 2i − P M−i n=1 P i j=1 j 2 n n−1 i i−j−1 (n+i−j−1)! (i−j)!(n−1)!(n+i) n+i−j+1 i . Now, Δ¯ τ i,i+1 >0 can be obtained as Δ¯ τ i,i+1 = 1 λ h i+1 2i − P M−i n=1 P i j=1 jn n−2 i i−j−1 (n+i−j−1)! (i−j)!(n−1)!(n+i) n+i−j i = 1 λ h i+1 2i − P M m=i+1 i i (m−i) m−i−2 m m m i i , where we have used P i j=1 jn n−2 i i−j−1 (n+i−j−1)! (i−j)!(n−1)!(n+i) n+i−j = i i n n−2 (n+i) n+i n+i i . Given that ¯ τ M,1 = 1 λ P M m=1 1 m 2 , and τ M,i = τ M,1 − P i−1 m=1 Δτ m,m+1 , it can be shown that ¯ τ M,i = 1 λ M X m=1 " 1 m 2 + i−1 X l=1 l l (m−l) m−l−2 m m m l # (A.9) The average transmission duration of the last packet (packet M) can also be derived based on the fact that τ M,M ≤ τ M,i ,∀i ∈ [1,··· ,M −1]. The distribution function of τ M,M can be obtained as Pr{τ M,M >t}=Pr{ S (i,M) M−i+1 ≥t,∀i∈[1,··· ,M]}=e −λMt P M j=1 j(λMt) M−j M(M−j)! . (A.10) Thus, ¯ τ M,M = Z ∞ 0 Pr{τ M,M ≥t}dt= M +1 2λM . (A.11) This completes the proof of Lemma 2.6. Note that by comparing (A.9) and (A.11), we can get the the following interesting results 1 1 M 2 + M−1 X l=1 l l (M−l) M−l−2 M M M l = M +1 2M ,∀M ≥1 (A.12) Also, using M+1,M i in (A.14), (A.11), (A.12), and ¯ τ M+1,M+1 + P M i=1 M+1,M i = 1/λ, after some simplifications, we have 1 M +1 + M X l=1 l l (M−l+1) M−l (M +1) M+1 M +1 l =1,∀M ≥0, or, equivalently, M X l=1 l l (M−l+1) M−l (M +1) M+1 M +1 l = M M +1 ,∀M ≥1. (A.13) 1 Note that these results ((A.12) and (A.13)) are side results and are of independent interest. 196 A.6 Asymptotic Packet Delay for the Single Deadline Model In this appendix, we will provide the proof of Lemma 2.7. Consider the impact of an additionalpacket(packetM+1)onthetransmissiondurationsofpackets1toM. Denote M+1,M i = ¯ τ M+1,i −¯ τ M,i as the average difference of the transmission durations of packet i between the case when there are M+1 packets and the case when there are M packets. From (2.22), we have λ M+1,M i = 1 (M +1) 2 + i−1 X l=1 l l (M−l+1) M−l−1 (M +1) M+1 M +1 l . (A.14) Denote the total transmission delay of packets 1 to M as Q M = P M i=1 q M,i , and let ΔQ M+1,M =Q M+1 −Q M . After some derivation, we obtain: Δ ¯ Q M+1,M = 1 λ + M X i=1 i M+1,M i , (A.15) in which we have used P M+1 i=1 ¯ τ M+1,i =(M +1)/λ. Using (A.14), (A.15), and (A.12), we can further write P M i=1 i M+1,M i as 1 λ h 1 (M+1) 2 P M m=1 m+ P M−1 l=1 l l (M−l+1) M−l−1 (M+1) M+1 M+1 l P M−l j=1 j i = 1 2λ h M M+1 + P M−1 l=1 l l (M−l+1) M−l (M+1) M+1 M+1 l (M−l) i = 1 2λ h P M l=1 l l (M−l+1) M−l (M+1) M+1 M+1 l + P M−1 l=1 l l (M−l+1) M−l (M+1) M+1 M+1 l (M−l) i = 1 2λ h P M l=1 l l (M−l+1) M−l+1 (M+1) M+1 M+1 l i = 1 2λ h P M l=1 l l l! (M−l+1) M−l+1 (M+l−1)! (M+1)! (M+1) M+1 i . Using Stirling’s approximation, i.e., n!/n n ≈ √ 2πne −n , we get M X i=1 i M+1,M i ≈ 1 2λ " r M +1 2π M X l=1 1 p l(M−l+1) # . Note that since R 1 0 1/ √ 1−x 2 dx=π/2, we obtain lim M→∞ M X l=1 1 p l(M−l+1) ≈ Z 1 0 1 p x(1−x) dx=π. Thus, M X i=1 i M+1,M i ≈ π 2λ p (M +1)/(2π)= p 2π(M +1)/(4λ), 197 and, Δ ¯ Q M+1,M = 1 λ 1+ p 2π(M +1)/4 . Given that the total packet transmission delay when M = 1 is on average 1/λ and the average packet delay ¯ q M = ¯ Q M /M, we thus get ¯ q M ≈ 1 λ 1+ √ 2π 4M M X m=2 √ m ! ≈ 1 λ 1+ √ 2π 6 √ M ! , where we have used R √ xdx= 2 3 x 3/2 +c, wherec is a constant. This completes the proof. A.7 Uniqueness of Optimal Offline Schedule The uniqueness of the optimal offline schedule can be proved by contradiction. First, a transmission rate vector ~ r is said to be feasible if it satisfies all the four constraints in (3.1). Because these constraints are all linear, it is easy to see that the set of all feasible vectors is convex. That is, a convex combination of two feasible vectors yields another feasible vector. Now suppose there are two distinct optimal rate vectors ~ r (1) = [r (1) 1 ,··· ,r (1) M+D−1 ] and ~ r (2) = [r (2) 1 ,··· ,r (2) M+D−1 ]. Define ~ r (3) = δ~ r (1) +(1−δ)~ r (2) , 0 < δ < 1. Obviously, ~ r (3) is feasible. Since f(r i ,g i ) is a strictly convex function of r i for any i ∈ [1,··· ,M +D−1], we have f(r (3) i ,g i ) ≤ δf(r (1) i ,g i )+(1−δ)f(r (2) i ,g i ), where the equality holds if and only if r (1) i = r (2) i . Since ~ r (1) 6= ~ r (2) , there exists at least one j∈[1,··· ,M+D−1]suchthatr (1) j 6=r (2) j , andconsequently, P M+D−1 i=1 f(r (3) i ,g i )<e opt , wheree opt M = P M+D−1 i=1 f(r (1) i ,g i )= P M+D−1 i=1 f(r (2) i ,g i ). Thus,~ r (3) isstrictlymoreenergy- efficient than the two optimal rate vectors ~ r (1) and ~ r (2) , which causes a contradiction. A.8 Additional Complementary Slackness Conditions Thisistoprovetheadditionalcomplementaryslacknessconditionsin(3.5)fortheoptimal offline schedule. Recall that: 1. h l,m (~ r ∗ )<0 implies that μ ∗ l,m =0; 2. μ ∗ l,m >0 implies that h l,m (~ r ∗ )=0, for any l and m, due to the conventional complementary slack condition by (3.4). The first condition, μ ∗ 1,m μ ∗ 3,m = 0,m = 1,··· ,M, is due to the fact that given B m > 0,m = 1,··· ,M, if μ ∗ 3,m > 0 (hence h 3,m (~ r ∗ ) =−r ∗ m = 0 and so slot m is an idle slot), we have h 1,m (~ r ∗ ) = P m i=1 r ∗ i τ s − P m i=1 B i = P m−1 i=1 r ∗ i τ s − P m i=1 B i < P m−1 i=1 r ∗ i τ s − P m−1 i=1 B i ≤ 0 Thus, μ ∗ 1,m = 0 or slot m does not end with an empty buffer. Similarly, the reverse case, i.e., μ ∗ 1,m > 0 implying μ ∗ 3,m = 0, can also be derived. In other words, any slot 198 m ∈ [1,··· ,M] can not be idle and end with an empty buffer at the same time. The second condition, μ ∗ 1,m μ ∗ 2,m = 0,m = D,··· ,M, is due to the fact that if μ ∗ 1,m > 0, slot m must satisfy h 1,m (~ r ∗ )=0, i.e., it must end with an empty buffer, h 2,m (~ r ∗ ) = P m−D+1 i=1 B i − P m i=1 r ∗ i τ s = − P m i=m−D+2 B i −h 1,m (~ r ∗ ) = − P m i=m−D+2 B i < 0 Hence, μ ∗ 2,m = 0. Similarly, if μ ∗ 2,m > 0, we have μ ∗ 1,m = 0. In other words, any slot m ∈ [D,··· ,M] can not be a delay-critical slot and ends with an empty buffer at the same time. For the third condition, first note that an idle slot may possibly be a delay-critical slot,i.e.,h 2,m (~ r ∗ )= P m−D+1 i=1 B i − P m i=1 r ∗ i τ s andr ∗ m =0maybesatisfiedsimultaneously, such that μ ∗ 2,m μ ∗ 3,m >0. However, if slotm+1 is idle, slotm can not be delay-critical, and vice versa, as indicated by condition 3 (μ ∗ 2,m μ ∗ 3,m+1 = 0,m = D,··· ,M + D− 2). Consider if μ ∗ 2,m > 0, we have h ∗ 2,m = 0 and hence slot m is delay-critical. That is, slot m ends with completing transmissionofpacketsarrivedatslotm−D+1. SinceB m−D+1 >0,form=D,··· ,M+ D−1, slot m+1 has to at least serve packets arrived at slot m−D+2 and thus can not be idle, or μ ∗ 3,m+1 =0. Similarly, one can also show that if μ ∗ 3,m+1 >0, we must have μ ∗ 2,m =0. A.9 Proof of Lemmas 3.2, 3.3, 3.4, and 3.5 For Lemma 3.2, since r ∗ m > 0 and r ∗ m+1 > 0, we have μ ∗ 3,m = μ ∗ 3,m+1 = 0. Case 1 is further due to an empty buffer at the end of slot m, and hence a non delay-critical slot such that μ ∗ 2,m = 0 in (3.6). Case 2 is further due to μ ∗ 1,m = 0 as slot m is delay-critical and hence non-empty ending. Case 3 is further due to μ ∗ 1,m = 0 and μ ∗ 2,m = 0 as slot m is neither empty-ending nor delay-critical. ForLemma3.3,Case1isduetoμ ∗ 3,m+1 =0(non-idlingslot)andμ ∗ 1,m =0(non-empty ending slot) in (3.6). Case 2 is due to μ ∗ 3,m = 0 (non-idling slot) and the last condition in (3.5), i.e., slot m can not be delay-critical if slot m+1 is idle. For Lemma 3.4, Case 1 is due to μ ∗ 3,m+l = 0 (non-idling slot) and μ ∗ 1,i = 0,i = m,··· ,m+l−1 (idling slots can not be empty-ending), while Case 2 is due to μ ∗ 3,m =0 (non-idling slot) and μ ∗ 2,i = 0,i = m,··· ,m+l−1 (an idling slot can not be preceded by a delay-critical slot) in (3.7). For Lemma 3.5, since r ∗ m > 0 and r ∗ m+l > 0, we have μ ∗ 3,m = μ ∗ 3,m+l = 0. Case 1 is further due to μ ∗ 2,i =0,i=m,··· ,m+l−2 (as slots i=m+1,··· ,m+l−1 are idle), and μ ∗ 2,m+l−1 = 0 (non delay-critical) in (3.7). Case 2 is further due to μ ∗ 1,m = 0 (non empty-ending), and μ ∗ 1,i = 0,i =m+1,··· ,m+l−1 (as slots i =m+1,··· ,m+l−1 are idle) in (3.7). The last case is further due to the combination of Case 1 and Case 2 (i.e., μ ∗ 1,i =0 and μ ∗ 2,i =0, i=m,··· ,m+l−1). 199 A.10 Packet Delay Lower and Upper Bounds in Fading Channels First, denote t start,m as the time when the first bit of packet m is transmitted. Specially, let t start,1 = 0 because from a delay perspective, if there are any idling periods before the first packet transmission, the first packet is effectively delayed starting from time 0. Similarly, denote t end,m as the departure time of the last bit of packet m’s transmission. Note that both t start,m and t end,m are not necessarily aligned with slot boundaries. Also, t start,m ≥ t end,m−1 , and the equality holds only if there is no idling period between the departure time of packet m− 1 and the start time of packet m’s transmission. By defining the inter-departure time of packet m, m∈ [1,··· ,M], as φ m M = t end,m −t end,m−1 , with t end,0 M = 0, the delay for packet m can thus be computed as q m = m X l=1 φ l −(m−1)τ s , where P m l=1 φ l = t end,m and (m− 1)τ s are the departure time and the arrival time of packet m, respectively. Now, define the virtual start time of packet m ∈ [1,··· ,M] as t v start,m M = (t end,m−1 + t start,m )/2,andthevirtual departure timeofpacketm∈[1,··· ,M−1]ast v end,m M = (t end,m + t start,m+1 )/2. Let t v end,M M = (M +D−1)τ s , regardless of any potential idling slots after t end,M . Subsequently, define the virtual inter-departure time of packet m as φ v m M = t v end,m −t v start,m . Note that if there are no idling slots between t end (m− 1) and t start (m), and between t end (m) and t start (m+1), φ v m corresponds to φ m . Otherwise, φ v m also incorporates half of the idling period between t end (m−1) and t start (m), and half of the idling period between t end (m) and t start (m+1), for 2≤m≤M−1. For the first packet, φ v 1 includes the entire idling period, if any, before its transmission, while for the last packet, φ v M includes the entire remaining idling period after t end (M), if any. Denote q v m = m X l=1 φ v l −(m−1)τ s . Note that since t v end,m =t v start,m+1 and t v end,m ≥t end,m , we have q v m ≥q m . Duetothesymmetrypropertyof ~ B andhence~ r ∗ , asinTheorem 3.3, itisnotdifficult to show that, using a sample path trajectory of the forward running system and the corresponding time reversed system, the same symmetry property holds for the virtual inter-departure time vector ~ φ v as well, i.e., E{φ v m }=E{φ v M+1−m },∀m. 200 Therefore, ¯ q(M) M = 1 M P M m=1 E{q m } ≤ 1 M P M m=1 E{q v m } = 1 M P M m=1 [ P m l=1 E{φ v l }−(m−1)τ s ] (a) = 1 M P M m=1 (M−m+1)E{φ v m }− τs M P M m=1 (m−1) (b) = 1 M P M m=1 M+1 2 E{φ v m }− 1 M M(M−1) 2 τ s (c) = M+1 2M (M +D−1)τ s − M−1 2 τ s = τ s + M+1 2M (D−1)τ s , (A.16) where (a) holds by counting the number of occurrences of each item, the first term of (b) comes from the symmetry property E{φ v m } = E{φ v M−m+1 },∀m ∈ [1,··· ,M], and equivalently, there are (M +1)/2 copies of each E{φ v m }, and the first term of (c) is due to the fact that P M m=1 E{φ v m }=(M +D−1)τ s . On the other hand, re-writing the average delay computation as ¯ q(M) M = 1 M P M m=1 E{q m } = E{ 1 M P M m=1 P m l=1 φ l }− 1 M P M m=1 (m−1)τ s we can see that P m l=1 φ l = t end,m is used only once for each m ∈ [1,··· ,M] before taking the average (1/M) and the expectation. Similar statement also holds true for P m l=1 φ v l = t v end,m as in (A.16). To quantify the difference between t v end,m and t end,m , denote Δt v m M = t v end,m −t end,m =(t start (m+1)−t end (m))/2≥0. First note that any idling periods between time 0 and t start (1), and in between a packet transmission have no impact on Δt v m , ∀m. For a particular m∈ [1,··· ,M−1], if there is one or more idle slot between t end (m) and t start (m+1), Δt v m >0 and it equals to half of the idling period. However, any other idling periods between t end (j) and t start (j+1), j 6= m, have no impact on Δt v m . In the special case of m = M, if there are idling slots after packet M transmission, we have Δt v M = (M +D−1)τ s −t end,M . However, due to the symmetry property, there must exist a realization of the same length of an idling period between time 0 and t start (1) (which does not impact Δt v 1 ). Effectively, the idling period after t end,M , if any, only contributes half of its duration to Δt v M . Therefore, we have: Lemma A.4 Any idling period effectively at most contributes once to the difference in the average delay computation using ~ φ v and ~ φ, and the contribution is at most half of its duration. This leaves us to count the total idling duration within M +D−1 slots. Due to the delay constraint D, there are at leastb(M +D−1)/Dc≥M/D non-idling slots between slots 1 and M +D−1. Therefore, we have, M X m=1 ( m X i=1 (φ v i −φ i ) ) ≤τ s [(M +D−1−M/D)/2] 201 Defining δ M = τ s [(M +D−1−M/D)/2]/M = τ s [(1−1/D)+(D−1)/M]/2, we finally have ¯ q(M)≥τ s 1+ M +1 2M (D−1) −δ =τ s D 2 + 1 2D . Note that δ converges to 0.5τ s (1−1/D) when M approaches infinity. In case of static channels, there will be no idling periods under the optimal offline schedule. Thus, φ m =φ v m , and the equality holds in (A.16). A.11 Optimal Offline Scheduling Under Static Time-Slotted Channels Denote the total number of buffered bits right before slot m as U m (i.e., excluding the packets arrived at slot m). Clearly, U 1 =0. Due to the causality constraint, we have U m = m−1 X i=1 (B i −x i )≥0,m>1, where x i is the number of transmitted bits during slot i. Due to the delay constraint D, the buffered bits U m may contain bits arrived at up to D−1 slots earlier. That is, we can write U m as U m =U m,1 +U m,2 +···+U m,D−1 , where U m,i denotes the number of buffer bits that arrived at slot m−i. The buffered bits U m,i have a delay constraint of D−i, 1 ≤ i ≤ D−1. Note that due to the FIFO constraint, if U m,j > 0,j∈ [1,··· ,D−1], packets that arrived during slots m−j +1 to m−1 have not started transmission yet, i.e., we have U m,i =B m−i ,∀i>j. The optimal transmission rate during slot m, 1≤m≤M +D−1, is given by r m = min m≤i≤M+D−1 r m[i] , (A.17) where r m[i] ,i∈[m,··· ,M +D−1], is given by r m[i] =max n Um+ P i l=m B l (i−m+1)τs , U m,D−1 τs , U m,D−2 +U m,D−1 2τs , ··· , P D−1 l=1 U m,l (D−1)τs , Um+Bm Dτs , Um+Bm+B m+1 (D+1)τs ,··· , Um+ P i−1 l=m B l (D+i−m−1)τs . (A.18) The optimal number of transmitted bits during slot m is simply given by x m =r m τ s . To summarize, we have Theorem A.1 Thetransmissionschemedenotedby(A.17)istheoptimalofflineschedul- ing for a group of packets subject to the same individual delay constraint D in a discrete- time slot model under the AWGN channel. 202 Proof: This can be derived from the optimal offline scheduling algorithm for the continuous-time model as in Section 2.3. Thus, the proof is omitted. Note that there are no idling periods under the optimal offline schedule for the static channel case. Under the optimal offline scheduling, the two events: 1) slot m ends with an empty buffer (i.e., U m+1 = 0, such that packet m finishes its transmission at the end of slot m), and 2) slot m ends with a delay critical packet (i.e., packet m− D + 1 finishes its transmission at the end of slot m), are mutually exclusive, except for the last slot M + D− 1. This is similar to the complementary conditions in (3.5) for the fading channel case. SimilartothepropertiesoftheoptimalderivativescharacterizedbyLemma3.2forthe fading channel case, we can also characterize the properties of the optimal transmission rate 2 , r m , as below: Lemma A.5 Under the optimal offline scheduling for the slot individual delay constraint model in the AWGN channel: 1. r m ≤r m+1 , if m=1,··· ,M +D−2, and slot m ends with an empty buffer; 2. r m ≥ r m+1 , if m = D,··· ,M +D−2, and the last departed packet in slot m is delay-critical; 3. r m =r m+1 , if m=1,··· ,M+D−2, and slot m ends neither with an empty buffer nor with a delay-critical packet. Proof: This is directly from the properties in the continuous-time model (see Lem- mas 2.1 and 2.2) and can also be verified from (A.17) and (A.18). In the case of the single transmission deadline model in which all packets observe a single deadline, the optimal transmission rate during slot m would be r single,m =min m≤i≤M Um+ P i l=m B l (i−m+1)τs , (A.19) a simplified version of (A.17) and (A.18). In addition, it is not difficult to see that the optimal number of transmitted bits per slot for the single transmission deadline model is non-decreasing, i.e., r single,i ≤ r single,j ,∀i < j, similar to the non-increasing property of the optimal transmission durations in the continuous time case [UBPG02]. A.12 Delay Bounds for Multihopping Here we prove Theorem 4.3. This is similar to the proof in Section A.10. The upper bound is due to the symmetry property of the optimal transmission rate vector, and the fact that the existence of restricted slots may force early packet transmissions and 2 Note that in the static channel case, the energy-rate derivative vector is just a scaled version of the transmission rate vector. 203 subsequently, may reduce the average packet delay, compared with the non-restricted case. Therefore, ¯ q I (M)≤τ s 1+ M +1 2M (D I −1) . On the other hand, as shown in Section A.10, the amount of the reduced average packetdelayduetorestrictedslotsisnomorethanthetotaldurationoftherestrictedslots divided by 2M. Denote the number of restricted slots as N 0 M = M+D I −1− P M+D I −1 m=1 I m and δ M = τ s N 0 /(2M), we thus have ¯ q I (M)≥τ s 1+ M +1 2M (D I −1) −δ. A.13 Derivation of ¯ τ (K) N,N AfterdroppingthelastK packetsfromM packets(withN packetsremaining), theinter- arrival times of the first N−1 packets still observe exponential distributions. However, the inter-arrival time of the last packet (packet N) follows the Gamma distribution given by: f(x)= λ K+1 x K K! e −λx . Recallthatundertheoptimalschedulingalgorithm,theoptimaltransmissiondurationsof theremainingN packetssatisfyτ (K) N,N ≤τ (K) N,i ,∀i∈[1,··· ,N−1]. DenoteS (j,i) = P i l=j d l , the sum of inter-arrival times from packets j to packet i. The distribution function of τ (K) N,N can be obtained as Pr[τ (K) N,N ≥t]=Pr h S (i,N) N−i+1 ≥t,∀i∈[1,··· ,N] i = R ∞ Nt R s (1,N) (N−1)t ··· R s (N−1,N) t f S (N,N) ,···,S (2,N) ,S (1,N) (s (N,N) , ··· ,s (2,N) ,s (1,N) )ds (N,N) ···ds (2,N) ds (1,N) = R ∞ Nt R s (1,N) (N−1)t ··· R s (N−1,N) t λ N+K e −λs (1,N) (s (N,N) ) K K! ds (N,N) ···ds (2,N) ds (1,N) , (A.20) where we have used the fact that f S (N,N) ,···,S (1,N) (s (N,N) ,··· ,s (1,N) )=f D 1 ,···,D N (d 1 ,··· ,d N ) =λ N+K e −λs (1,N) (s (N,N) ) K /K!, given the assumption that the inter-arrival is identically and independently distributed following the exponential distribution for the first N−1 packets and the Gamma distri- bution of the N-th packet. It can be shown that R s (1,N) (N−1)t ··· R s (N−1,N) t (s (N,N) ) K K! ds (N,N) ···ds (2,N) = (s (1,N) ) N+K−1 (N+K−1)! − P N i=2 c i t K+i−1 (s (1,N) ) N−i (K+i−1)!(N−i)! , (A.21) 204 wherec i =(i−1) P i−1 j=1 (−1) i−j−1 j K+i−2 K+i−1 i−j−1 . Giventhis,Pr[τ (K) N,N ≥t]canbederived as Pr[τ (K) N,N ≥t] = R ∞ Nt λ N+K e −λs (1,N) h (s (1,N) ) N+K−1 (N+K−1)! − P N i=2 c i t K+i−1 (s (1,N) ) N−i (K+i−1)!(N−i)! i ds (1,N) = h P N+K j=1 (λNt) N+K−j (N+K−j)! − P N i=2 c i (K+i−1) P N−i+1 j=1 (λt) (N+K−j) N N−i−j+1 (N−i−j+1)! i e −λNt . (A.22) Using the identity E[Y] = R ∞ 0 Pr(Y > t)dt for any positive random variable Y, we obtain from (A.22) that E[τ (K) N,N ]= R ∞ 0 Pr[τ (K) N,N ≥t]dt = R ∞ 0 h P N+K j=1 (λNt) N+K−j (N+K−j)! − P N i=2 c i (K+i−1) P N−i+1 j=1 (λt) (N+K−j) N N−i−j+1 (N−i−j+1)! i e −λNt dt = P N+K j=1 N N+K−j λ(N+K−j)! (N+K−j)! N N+K−j+1 − P N i=2 c i λ(K+i−1)! P N−i+1 j=1 N N−i−j+1 (N−i−j+1)! (N+K−j)! N N+K−j+1 = N+K λN − P N i=2 c i λN K+i P N−i+1 j=1 (N+K−j)! (K+i−1)!(N−i−j+1)! = N+K λN − P N i=2 c i λN K+i P N−i+1 j=1 N+K−j N−i−j+1 = N+K λN − P N i=2 c i λN K+i P N l=i K+l−1 l−i = N+K λN − P N i=2 c i λN K+i N+K N−i , (A.23) where we have used the fact that R ∞ 0 (λt) k e −λnt dt = k! λn k+1 , and N+K N−i = P N l=i K+l−1 l−i by the triangle equality of n k + n k+1 = n+1 k+1 . After applying the normalization factor N/M, we finally obtain: ¯ τ (K) N,N = N N+K E[τ (K) N,N ]= 1 λ − P N i=2 c i( N+K N−i ) λN K+i−1 (N+K) = 1 λ − P N i=2 c i( N+K−1 K+i−1 ) λN K+i−1 (i+K) . (A.24) This completes the proof of Lemma Lemma 5.2. When K = 1 (i.e., only one packet is dropped), c i is reduced to c i = i−1, we thus have E[τ (1) N,N ]= N+1 λN − P N i=2 (i−1)( N+1 N−i ) λN i+1 = N+1 λN +2 P N i=2 ( N+1 N−i ) λN i+1 − P N i=2 (i+1)( N+1 N−i ) λN i+1 = N+1 λN +2 P N i=2 ( N+1 N−i ) λN i+1 − N+1 λN P N i=2 ( N+1 i ) λN i = N+1 λN +2λ (1+N −1 ) N+1 −1− N+1 N − ( N+1 2 ) N 2 − N+1 λN (1+N −1 ) N −1− N N = (1+N −1 ) N+1 −2 /λ. (A.25) After normalization, we have: ¯ τ (1) N,N = N N+1 E[τ (1) N,N ]= 1 λ h (1+N −1 ) N −2 N N+1 i . (A.26) This completes the proof of Corollary 5.5. 205 A.14 Derivation of ¯ τ (K) N,1 This appendix provides the proof of Lemma 5.3 and Corollary 5.6. The proof approach is very similar to that in Appendix A.13. Recall that τ (K) N,1 ≥ τ (K) i ,∀i ∈ [2,··· ,N], the distribution function of τ (K) N,1 can be obtained as Pr[τ (K) N,1 ≤t]=Pr h S (1,i) i ≤t,∀i∈[1,··· ,N] i = R t 0 R 2t s (1,1) ··· R Nt s (1,N−1) f S (1,N) ,···,S (1,2) ,S (1,1) (s (1,N) ,··· ,s (1,2) ,s (1,1) )ds (1,N) ···ds (1,2) ds (1,1) = R t 0 R 2t s (1,1) ··· R Nt s (1,N−1) λ N+K e −λs (1,N) (s (1,N) −s (1,N−1) ) K K! ds (1,N) ···ds (1,2) ds (1,1) , (A.27) where we have used the fact that f S (1,N) ,···,S (1,1) (s (1,N) ,··· ,s (1,1) )=f D 1 ,···,D N (d 1 ,··· ,d N ) =λ N+K e −λs (1,N) (s (1,N) −s (1,N−1) ) K /K!, given the assumption that the inter-arrival is identically and independently distributed following the exponential distribution for the first N−1 packets and the Gamma distri- bution of the N-th packet. Equation (A.27) can be further derived as: Pr[τ (K) N,1 ≤t]= R t 0 R 2t s (1,1) ··· R Nt s (1,N−1) λ N−1 (e −λs (1,N−1) −e −λNt ) − P K j=1 λ N+j−1 e −λNt (Nt−s (1,N−1) ) j j! i ds (1,N−1) ···ds (1,2) ds (1,1) = R t 0 R 2t s (1,1) ··· R Nt s (1,N−1) λ N e −λs (1,N) ds (1,N−1) ···ds (1,2) ds (1,1) − P K j=1 R t 0 R 2t s (1,1) ··· R Nt s (1,N−1) λ N+j−1 e −λNt (Nt−s (1,N−1) ) j j! ds (1,N−1) ···ds (1,2) ds (1,1) . (A.28) Itcanberecognizedthatthefirstterm onthe righthandsideis equivalenttothe original no-droppingcase,whosecontributiontoE[τ (K) N,1 ]equalsto P N n=1 1/(λn 2 )[UBPG02]. Now, it is known that [UBPG02]: R t 0 R 2t s (1,1) ··· R (N+k−1)t s (1,N+k−2) ds N+k−1 1 ···ds (1,2) ds (1,1) = t N+k−1 (N+k) N+k−2 (N+k−1)! . (A.29) On the other hand, we have R t 0 R 2t s (1,1) ··· R (N+k−1)t s (1,N+k−2) ds N+k−1 1 ···ds (1,2) ds (1,1) = R t 0 R 2t s (1,1) ··· R (N−1)t s (1,N−2) P k j=1 c k,j t k−j (Nt−s (1,N−1) ) j j! ds (1,N−1) ···ds (1,2) ds (1,1) , (A.30) 206 with c 1,1 = 1, and c k,j = P k l=max{j,2} c k−1,l−1 (l− 1)!/(l− j)!, 1 ≤ j ≤ k. Note that c k,k =1,∀k≤1. Thus, R t 0 R 2t s (1,1) ··· R (N−1)t s (1,N−2) (Nt−s (1,N−1) ) k k! ds (1,N−1) ···ds (1,2) ds (1,1) = t N+k−1 (N+k) N+k−2 (N+k−1)! − R t 0 R 2t s (1,1) ··· R (N−1)t s (1,N−2) P k j=1 c k,j t k−j (Nt−s (1,N−1) ) j j! ds (1,N−1) ···ds (1,2) ds (1,1) , (A.31) with R t 0 R 2t s (1,1) ··· R (N−1)t s (1,N−2) (Nt−s (1,N−1) )ds (1,N−1) ···ds (1,2) ds (1,1) = R t 0 R 2t s (1,1) ··· R Nt s (1,N−1) ds (1,N) ···ds (1,2) ds (1,1) = t N (N+1) N−1 N! . (A.32) Using the identity E[Y] = R ∞ 0 Pr(Y > t)dt for any positive random variable Y, we have E[τ (K) N,1 ]= R ∞ 0 1−Pr[τ (K) N,1 ≤t] dt= P N n=1 1 λn 2 + P K j=1 δ k , (A.33) with δ k given by δ k = R ∞ 0 R t 0 R 2t s (1,1) ··· R (N−1)t s (1,N−2) λ N+j−1 (Nt−s (1,N−1) ) j j! ds (1,N−1) ···ds (1,2) ds (1,1) dt = R ∞ 0 (λt) N+k−1 (N+k) N+k−2 (N+k−1)! e −λNt dt− P k−1 j=1 c k,j R ∞ 0 R t 0 ··· R (N−1)t s (1,N−2) λ N+k−1 t k−j (Nt−s (1,N−1) ) j j! ds (1,N−1) ···ds (1,1) dt = (N+k) N+k−2 λ(N+k−1)! (N+k−1)! N N+k − P k−1 j=1 c k,j δ j (N+k−1)! N N+k N N+j (N+j−1)! = (1+k/N) N+k−2 λN 2 − P k−1 j=1 c k,j δ j (N+k−1)! N k−j (N+j−1)! . (A.34) When K = 1, we have δ 1 = (1+1/N) N−1 /(λN 2 ), and E[τ (1) N,1 ] = P N n=1 1/(λn 2 )+(1+ 1/N) N−1 /(λN 2 ). Thus, thenormalizedtransmissiondurationofthefirstpacketafterdroppingK pack- ets is given by ¯ τ (K) N,1 = N N +K N X n=1 1 λn 2 + K X k=1 δ k ! . (A.35) When K =1, (A.35) reduces to ¯ τ (1) N,1 = N N +1 N X n=1 1 λn 2 + (1+1/N) N−1 λN 2 ! . (A.36) This completes the proof of Lemma 5.3 and Corollary 5.6. 207

Abstract (if available)

Abstract There exists a fundamental trade-off between transmission energy and packet delay in wireless communications. In a static channel, a closed form solution of the optimal offline scheduling (vis-à-vis total transmission energy), assuming information of all packet arrivals, for a set of packets each subject to an individual delay constraint is derived. It is shown that when packet arrivals are identically and independently distributed, the optimal packet transmission durations (or, equivalently, transmission rates) exhibit a symmetry property, which leads to a simple and exact solution of the average packet delay. The delay performance for the optimal offline scheduling of a set of packets subject to a single transmission deadline is also analyzed and shown to be potentially unbounded. The problem of optimal offline scheduling is then extended to fading channels. The properties of the optimal offline transmission rates and the corresponding delay performance are also characterized. Heuristic online schedulers, assuming causal information only, are also studied. The properties of the optimal offline scheduling are demonstrated via simulations.

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Asset Metadata

Creator Chen, Wanshi (author)

Core Title Energy-efficient packet transmissions with delay constraints for wireless communications

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 02/16/2009

Defense Date 12/04/2006

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag fading channels,individual delay constraint,majorization theory,minimum energy transmission,multithop,OAI-PMH Harvest,optimal scheduler,packet dropping,single transmission deadline

Language English

Advisor Mitra, Urbashi (committee chair), Govindan, Ramesh (committee member), Neely, Michael J. (committee member)

Creator Email wanshic@qualcomm.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m265

Unique identifier UC1203650

Identifier etd-Chen-20070216 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-168069 (legacy record id),usctheses-m265 (legacy record id)

Legacy Identifier etd-Chen-20070216.pdf

Dmrecord 168069

Document Type Dissertation

Rights Chen, Wanshi

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu