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Aging analysis in large-scale wireless sensor networks
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Aging analysis in large-scale wireless sensor networks
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AGING ANALYSIS IN LARGE-SCALE WIRELESS SENSOR NETWORKS by Jae-Joon Lee 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 (COMPUTER ENGINEERING) May 2007 Copyright 2007 Jae-Joon Lee Dedication This dissertation is dedicated to my parents for their love. ii Acknowledgements IwouldliketothankProf. C.-C.JayKuoforhissupportandencouragementduring thecourseofmyPhD.IwouldalsoliketothankProf. BhaskarKrishnamacharifor his advice throughout my PhD. I would also thank Prof. Gaurav Sukhatme for his feedback on the thesis. Finally, I would like to thank group members in the Media Communication Lab and friends in USC for their help and friendship. iii Table of Contents Dedication ii Acknowledgements iii List Of Tables vii List Of Figures viii Abstract xii Chapter 1: Introduction 1 1.1 Significance of the Research . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Review of Previous Work . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Contributions of the Research . . . . . . . . . . . . . . . . . . . . . 5 1.4 Organization of the Dissertation . . . . . . . . . . . . . . . . . . . . 7 Chapter 2: Background Review 9 2.1 Node Aging Modeling. . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Device Reliability . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.2 Energy Depletion . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Network Aging Modeling . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Aging Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Aging Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4.1 Energy-efficient Schemes . . . . . . . . . . . . . . . . . . . . 20 2.4.2 Redeployment or Recharging . . . . . . . . . . . . . . . . . . 20 Chapter 3: Aging Analysis of Multi-hop Data Gathering Trees 22 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Hop-Level Aging Analysis for Sparse Node Deployment . . . . . . . 22 3.2.1 Workload at Each Hop Level . . . . . . . . . . . . . . . . . . 23 iv 3.2.2 Device Reliability . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2.3 Analysis Validation . . . . . . . . . . . . . . . . . . . . . . . 26 3.2.4 Connectivity Aging Function. . . . . . . . . . . . . . . . . . 30 3.3 Distance-Level Dynamic Aging Analysis for Dense Node Deployment 33 3.3.1 Dynamic Aging Analysis . . . . . . . . . . . . . . . . . . . . 33 3.3.2 Geometric Calculation for Dynamic Data Gathering Construction . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3.3 Understanding Network Aging via Case Study . . . . . . . . 40 3.3.4 Computer Simulation with Realistic Wireless Links . . . . . 41 3.3.5 First-hop Node Aging Analysis . . . . . . . . . . . . . . . . 45 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Chapter 4: Impact of Localized Topology Construction on Network Lifetime 54 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.2 Case Study with Real Empirical Data . . . . . . . . . . . . . . . . . 56 4.2.1 Data Gathering Topology Maps . . . . . . . . . . . . . . . . 57 4.2.2 Link Quality and Communication Load . . . . . . . . . . . . 58 4.3 Analysis of Localized Topology Construction Schemes . . . . . . . . 64 4.3.1 Energy Consumption of Localized Schemes . . . . . . . . . . 64 4.3.1.1 Lowest ETX Parent Selection . . . . . . . . . . . . 65 4.3.1.2 Random Parent Selection with MHR . . . . . . . . 66 4.3.1.3 Lowest ETX Parent Selection with MHR . . . . . . 68 4.3.1.4 Balanced Parent Selection with MHR . . . . . . . . 69 4.3.2 Comparison of Localized Topology Schemes . . . . . . . . . 70 4.3.3 Conditions for Longer Lifetime of Balanced Scheme . . . . . 71 4.4 Comparison to Global Optimum . . . . . . . . . . . . . . . . . . . . 73 4.5 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . 76 Chapter 5: Aging in Heterogeneous Deployment 78 5.1 System Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.1.1 Heterogeneous Deployment Model . . . . . . . . . . . . . . . 80 5.1.2 Coverage Evaluation Models . . . . . . . . . . . . . . . . . . 82 5.1.2.1 Total Sensing Coverage . . . . . . . . . . . . . . . 82 5.1.2.2 Information Utility . . . . . . . . . . . . . . . . . . 84 5.2 Coverage Aging Analysis with Single-hop Direct Communication . . 87 5.2.1 Coverage Area Aging and Its Initial Time . . . . . . . . . . 88 5.2.2 Lifetime Sensing Coverage and Optimum No. of Sinks. . . . 90 5.2.3 Lifetime Information Utility . . . . . . . . . . . . . . . . . . 93 5.3 Coverage Aging Analysis with Multi-hop Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.3.1 Coverage Aging with Simulation Results . . . . . . . . . . . 96 v 5.3.1.1 Total Sensing Coverage Aging (S tot ) . . . . . . . . 98 5.3.1.2 Coverage Area Loss Rate . . . . . . . . . . . . . . 99 5.3.1.3 Sensing Coverage Degree Aging . . . . . . . . . . . 100 5.3.1.4 Lifetime Sensing Coverage (S life ) . . . . . . . . . . 101 5.3.2 Lifetime Sensing Coverage and Optimum No. of Sinks. . . . 102 5.3.3 Lifetime Information Utility . . . . . . . . . . . . . . . . . . 104 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Chapter 6: Conclusion and Future Work 109 6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.2 Future Research Direction . . . . . . . . . . . . . . . . . . . . . . . 112 Reference List 114 vi List Of Tables 3.1 Summary of notation for a multi-hop communication tree. . . . . . 35 3.2 Parameters of realistic wireless link model (PRR: packet reception rate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.1 Summary of notation . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.1 Key findings (n H : no. of high cost devices, n H¡opt : the optimum n H ) 80 5.2 Summary of notation for a heterogeneous deployment model . . . . 81 vii List Of Figures 2.1 The probability density function of the Weibull distribution with ¯ from 1 to 4 when ´ =150. . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 The diagram of the relationship among aging factors (applications, deployments and operations). . . . . . . . . . . . . . . . . . . . . . 16 3.1 The illustration of a data gathering tree, where h represents the hop level from the sink and r indicates the radio range of a sensor device. 24 3.2 The average number of (a) children and (b) descendants in each hop level when the maximum hop distance L is 7. . . . . . . . . . . . . 24 3.3 Thepercentagesofnodeconnectionineachhopleveltothesink: (a) analytical and (b) simulation results. . . . . . . . . . . . . . . . . . 27 3.4 The percentages of dead nodes in each hop level over time: (a) the uniform random device failure case, (b) the energy depletion case with perfect data aggregation, and (c) the energy depletion case without data aggregation. Note that the x-axis scale in Figure (c) is finer than that in Figures (a) and (b). . . . . . . . . . . . . . . . . . 29 3.5 The percentages of live nodes and connected nodes in each hop level overtime: (a)theuniformrandomdevicefailurecase,(b)theenergy depletion case with perfect data aggregation, and (c) the energy depletion case without data aggregation. Note that the x-axis scale in Figure (c) is finer than that in Figures (a) and (b). . . . . . . . . 30 3.6 The log-log plot of the percentages of node connection to the sink in each hop level over time as node death occurs due to (a) the devicefailureeffect(80%ofthenodebecomesdisconnectedwithover 45% of the node deaths), and (b) the energy depletion effect, where perfect data aggregation is performed (80% of the node becomes disconnected with over 18% of the node deaths. The curves are obtained by the least-squares approximation. . . . . . . . . . . . . . 32 viii 3.7 Selectionofanewparentafterenergydepletionofthecurrentparent (a) Case I: Select a new parent in the live upper hop level (b) Case II: Select a new parent in the previously same hop level . . . . . . . 34 3.8 The energy consumption distribution as a function of time and the distance from the sink. . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.9 The residual energy distribution (in terms of percentages) as a func- tion of time and the distance from the sink. . . . . . . . . . . . . . 42 3.10 The maximum energy consumption (the dashed line) and the per- centage of nodes connected to the sink (the solid line) in the first hop level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.11 The residual energy distribution at time with 10% connection loss. The solid line represents the average residual energy . . . . . . . . . 44 3.12 Energy consumption of the first-hop nodes as a function of time. Each line represents the energy consumption of a node along time. . 45 3.13 The average residual energy with realistic wireless links, where ana- lytical and simulation results are indicated by dash and solid lines, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.14 (a),(b)Theseparateaverageresidualenergyofnodesin2ndand3rd hop levels with realistic wireless links. Analytical and simulation results are indicated by dash and solid lines, respectively, and (c) the distribution of 2nd and 3rd hop level nodes at each distance unit. 46 4.1 The indoor deployment location of nodes and four data gathering topology maps with localized topology construction schemes using thePRRdataobtainedbyUCLA/CENSgroup: (a) thelowestETX parentselection,(b)therandomparentselectionscheme(allpossible links) with MHR, (c) the lowest ETX parent selection scheme with MHR and (d) the balanced parent selection scheme with MHR. . . 59 4.2 The comparison of localized topology construction schemes: the av- erage ETX of data gathering paths with respect to link threshold. . 60 4.3 Thenumberofdescendantsforthefirsthoplevelnodesasafunction oftheirdistancetothesinkunderthreeparentselectionschemeswith MHR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.4 The ratio of maximum energy consumption of MHR schemes to the maximum energy consumption of the lowest ETX scheme. . . . . . 63 ix 4.5 Illustration of the linear topology. . . . . . . . . . . . . . . . . . . . 65 4.6 The lowest ETX parent selection scheme. . . . . . . . . . . . . . . . 66 4.7 The random parent selection scheme with MHR. . . . . . . . . . . . 66 4.8 The lowest ETX parent selection scheme with MHR. . . . . . . . . 69 4.9 Teh balanced parent selection scheme. . . . . . . . . . . . . . . . . 69 4.10 The maximum energy consumption as a function of the network size (i.e. the total number of nodes, N, in a network). . . . . . . . . . . 70 4.11 The maximum energy consumption as the number of nodes in a hop (r). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.12 The topology of an exemplary network. . . . . . . . . . . . . . . . . 75 4.13 The optimal flow with link thresholds equal to 0.6 and 0.9. . . . . . 75 4.14 Comparison of the lowest ETX scheme and the balanced scheme in terms of (a) the maximum energy consumption and (b) the network lifetime normalized to the optimal network lifetime. . . . . . . . . . 77 5.1 Heterogeneous deployments with single-hop direct communication: (a) the fixed area deployment and (b) the fixed coverage degree de- ployment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2 The analyzed aging process of the total sensing coverage, S tot , over time with respect to the number of sinks. . . . . . . . . . . . . . . . 90 5.3 The analyzed lifetime sensing coverage, S life , with respect to the numberofsinks, wheretwolifetimesareconsidered(timetothefirst node’s death and time to all nodes’ death). . . . . . . . . . . . . . . 92 5.4 The optimum number of sinks with respect to the initial number of low-cost devices.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.5 Theoptimumnumberofsinkswithrespecttothecostratio(C H =C L ) and the sensing range ratio (R H =R L ). . . . . . . . . . . . . . . . . . 94 5.6 The lifetime information utility with respect to the number of sinks with various correlation parameters (c=0.1, 1, 5, 10). Fixed area deployment case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.7 Illustrationofthemulti-hopnetworkmodelwith n H =4insimulation. 97 x 5.8 The total sensing coverage (S tot ) aging over time with respect to the number of sinks (n H ) for multi-hop networks. . . . . . . . . . . . . 99 5.9 The coverage area loss rate (the averaged grid point loss per round) under a given number of sinks (n H ).. . . . . . . . . . . . . . . . . . 100 5.10 Thesensingcoveragedegree(S d )agingovertimewithrespecttothe number of sinks (n H ).. . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.11 The lifetime sensing coverage (S life ) with respect to the number of sinks (n H ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.12 The optimum number of sinks with respect to non-communication energy consumption (E1) and the initial network size (n ¤ L ). . . . . . 104 5.13 The lifetime information utility with respect to the number of sinks with various correlation parameters (c=0.1, 1, 5, 10). . . . . . . . . 106 5.14 Thenetworklifetimewithrespecttothenumberofsinkswithvarious correlation parameters (c=0.1, 1, 20). . . . . . . . . . . . . . . . . . 107 xi Abstract Understanding the aging process of a sensor network, i.e. its performance degrada- tion over time, is important since this information can be used for network main- tenance and planning in long-term deployment. Prior research on the lifetime of wireless sensor networks has primarily focused on the energy depletion of the very first node. In this study, we analyze the aging process of the entire sensor network with respect to different network operations and deployment conditions. First, we examine the impact of node death on connectivity to the sink in a multi-hop data gathering tree. Then, we provide distance-level analysis for the dense deployment case by taking into account the re-construction of a data gathering tree and work load shift caused by energy depletion. Second, we provide an extensive study on localized topology construction schemes and their impact on the network lifetime. We present the precedence condition among these schemes in terms of network lifetime. Third, we analyze the aging process in heterogeneous deployments and deriveexpressionsfortheoptimalmixtureofheterogeneousdevicesthatmaximizes the lifetime coverage. We examine the spatial correlation effect on the optimal heterogeneous deployment. Our analysis explains diverse aging behaviors, which are highly dependent on the network operations and deployment conditions, and provides useful insights into long-term deployment and network maintenance and planning. xii Chapter 1 Introduction 1.1 Significance of the Research A wireless sensor network typically consists of a large number of low-cost sensor devices with limited battery energy in an unattended manner. Many applications require sensor nodes to operate against limited resources, node failure and an un- predictable dynamic environment in a distributed and self-organized way. Under the above constraints, a sensor network should provide a certain degree of net- work performance required by applications for a target period of time. The aging problemisimportantinthefieldsofbiologyandsystemreliabilityandhasbeenex- tensively studied therein. Studies in biology and system reliability have attempted to understand the causes of the aging process and predict the lifetime of organisms or systems by characterizing their aging process. Asimilaragingphenomenonoccursinwirelesssensornetworks. Sincetheaging process of a network is caused by the aging of its constituent sensor nodes, we need to examine the aging process of each individual node, including the energy consumption rate and the device reliability rate. The energy consumption rate is however not a trivial problem since a sensor network has to be regarded as 1 a whole system (or an organism) that allows dynamic interaction among sensors with respect to the evolving nature of the environment and the survival state of neighboring nodes. To analyze the aging process of a wireless sensor network, we would like to un- derstandhownodesapproachtheendoftheirlifeandhowthenetworkperformance degrades over time. Generally speaking, we have to examine this problem from a microscopic view of energy consuming in a sensor device as well as a macroscopic view of the network environment and condition needs in order to understand the overallagingprocessofanetwork. Applications,operations,theinitialdeployment, and the operating environment all affect the aging process. Basedonagoodunderstandingoftheagingprocessandanaccurateprediction, we can provide proper network maintenance planning and operations effectively. Theobjectiveofnetworkmanagementistomaintainthenetworksystemtoprovide therequiredfunctionalityforacertainperiodoftimeataminimumcost. Itincludes energy-efficient operations such as load balancing and power saving so as to delay the aging process as well as proper planning in redeployment and/or recharging time to extend network’s life. Predicting when and where nodes are approaching to death can provide timely network maintenance without interrupting network operations. 1.2 Review of Previous Work Agingisaninherentphenomenoninanorganismand/oranengineeringsystem. We review previous work on the modeling of the aging process in the fields of biology, system reliability and wireless sensor networks below. 2 ² Biology and demography The aging process of organisms has been extensively studied in order to un- derstand deterioration of species over time in biology and demography. The mortality rate, which is equivalent to the hazard rate and the failure rate, is the number of deaths in a given living number. According to the Gompertz law [21], the living adult organism’s mortality rate increases exponentially withage. Thisrelationhasbeenobservedinvariousbiologicalspeciesinclud- ing humans, fruit flies, mice [18,41]. In addition, many evidences have shown that mortality rates decelerate at older ages [6]. The aging study attempts to identify the causes of the aging pattern shown in many species and to provide a theoretical model for the aging process. Evolutionary theory was one of ways to explain aging from the view of evolutionary biology. Recently, reliability theory, which is widely used in system lifetime analysis, has also been considered to characterize the aging phenomenon in biology [19]. ² Engineering system A complex engineering system is often made up of multiple components, and its function is affected by the failure of individual components. A device is likely to lose its capability to conduct required operations as it becomes older duetoeitherinternallimitationsorexternalenvironmentaleffects. Reliability theory intends to model the failure rate of a component, a subsystem or the whole system as a function of time and the impact of interconnected components failure on the whole system. The failure rate z(t) of an item, which is also called the hazard rate, is proportional to the ratio of the failure probability density function f(t) of the item and the reliability function R(t) [42]. The failure rate is the same as the mortality rate or the mortality force 3 in the field of biology and demography. R(t) is the probability that the item does not fail till time t. Note that R(t) is also the complement of the failure distribution function F(t). Most recently, the failure of a network system has been studied. It provides an analytical model to understand how fast the failure of some component of a network diffuses across the network system, where the phase transition of the whole system due to the abrupt failure of a component or a sub-system is examined [13]. A wireless sensor network is a system consisting of tens or thousands of sensors that are interconnected via wireless links. The aging phenomenon occurs in each individual sensor device (i.e. a node) as well as the network that is affected by the death of nodes. A sensor device is equipped with limited energy supplied by batteries. In addition, they are connected with wireless links via single or multiple hops. There can be multiple paths for message passing, which provide alternate routes for robust performance, e.g. some nodes die along the previous path. Fi- nally, manyapplicationssuchasenvironmentalmonitoringofthewildlifeortarget trackinginabattlefielddemandsensornodestooperateinaself-organizedwayand adapt itself against limited resources, node failure and an unpredictable dynamic environment. These facts lead to an aging process in wireless sensor networks that is different from biological organisms and other engineering systems. Previousworkonagingconsiderationinwirelesssensornetworksmainlyfocuses on the time till the initial node death. There was some research on the network performance degradation process caused by the node death [28,44]. Shakkottai et al. [44] used a node failure rate to derive a bound on the probability for all nodes to be connected and the network to be covered in a grid deployment. Their main result shows that the network is connected and covered if the transmission radius 4 multipliedbythesquarerootofthenodesurvivalprobabilityisoforder p log(n)=n. They only examined the device reliability effect and did not consider the energy depletioneffectonthenetworkconnectivity. KunniyurandVenkatesh[28]analyzed networkdevolutioncausedbynodedeathandexaminedthetransitionalbehaviorof connectivity between live nodes. The battery lifetime of nodes is assumed to be an independent random variable, which is basically identical to the device reliability model. While the main focus of these two studies is on the radio radius in order to maintain connectivity among nodes, the communication operation, which is the most dominant energy consumption operation in sensor networks, has not been considered in [28,44]. 1.3 Contributions of the Research In this work, aging of wireless sensor networks due to energy depletion and device failure in diverse network operations and deployment conditions are extensively examined. It reveals that diverse aging phenomena depend on network conditions including communication models, topology construction schemes as well as hetero- geneous device deployment. ² In a multi-hop data gathering tree for a sparse node deployment, we present the hop-level analysis of communication load. We examine the connectivity agingperhoplevelovertimewiththreeconditionsincludingenergydepletion without data aggregation, with data aggregation and due to device failure. We observe that the existence of multiple alternate paths toward the sink leads to a power law relation, where the probability of connection to the sink decreases in proportion to the hop level with an exponent, as node death 5 occurs. From this observation, we characterize connectivity to a sink in each hop level over time after the initial node death. ² In the dense node deployment case with a multi-hop data gathering tree, we present the aging analysis that can provide the prediction of the energy consumption distribution and the residual energy distribution in a large-scale network over time after initial node death with low complexity. In addition, our first-hop node aging analysis in a dense deployment provides the expres- sion of the node density effect on the network disconnection time and the prediction of energy depletion time of the first-hop nodes. We show that the node density change with a fixed radio range and hop levels does not affect the network disconnection time due to energy depletion of all first-hop nodes regardless of workload distribution among them. ² By extensive study of four localized topology construction schemes with the empiricaldata,mathematicalanalysisandcomparisonwiththeoptimaltopol- ogy based on the global knowledge, we identify the trade-off between the link-quality-based scheme and the minimum-hop-routing-based schemes. We show how network conditions such as the network size, the node density, and the link quality for direct communication affect the network lifetime of dif- ferent localized schemes. We present the link quality threshold of the radio range that determines the longer network lifetime between the load balanced scheme and the link-quality-based scheme. ² Westudytheimpactofheterogeneousdeploymentsontheagingprocessunder the budget constraint and identify the trade-off in deploying heterogeneous devices, coverage performance and network lifetime. Specifically, we analyze 6 the lifetime coverage under various deployment conditions and communica- tion models and derive expressions for heterogeneous device deployment that maximizes the lifetime coverage. The results of this analysis help understand the effect of several parameters including the cost ratio, the sensing range ratio, the network size and the ratio of energy consumption operations. ² Weexaminetheeffectofspatialcorrelationinthenetworkfieldonthelifetime informationinconjunctionwithheterogeneousdeployment. Weobservethat, as the correlation increase, the optimal number of high-cost sink increases in the case of the single-hop communication system while the optimal number decreases in the case of the multi-hop communication system because of the effect of data aggregation. In addition, the result indicates that the optimal mixture of heterogeneity based on the low correlation case can provide a high lifetimecoverageclosetothemaximumvalueunderawiderangeofcorrelation conditions. 1.4 Organization of the Dissertation The rest of the dissertation is organized as follows. The background on the aging process in wireless sensor networks, including node aging, network aging, aging factors, and aging maintenance is reviewed in Chapter 2. Chapter 3 provides a study on the aging process in a multi-hop data gathering tree that goes beyond the initial node death in sparse and dense deployments. Chapter 4 presents a studyontheimpactoflocalizedtopologycontrolschemesonnetworklifetime. The effect of heterogeneity on the aging process and the coverage of sensor networks including the information utility based on the spatial correlation are examined in 7 Chapter5. Finally,concludingremarksandfutureresearchdirectionsarepresented in Chapter 6. 8 Chapter 2 Background Review Somebackgroundknowledgeofourcurrentresearchisreviewedinthischapter. We examine issues associated with the modeling of node and network aging processes in Sec. 2.1 and Sec. 2.2. Then, various aging factors are considered in Sec. 2.3. Finally, the problem of aging maintenace is discussed in Sec. 2.4. 2.1 Node Aging Modeling Anodeisanindividualsensordevicethathastheabilityofsensing, processingand communication. Node aging comes from two main causes: energy dissipation and thelikelihoodofdevicefailureovertime. Thenodeagingprocesscontinuesuntilthe eventualtermination of its function due to energy depletion or device failure. Node aging metrics provide indication about the length for a node to survive. These metrics include: the energy dissipation rate, the residual energy, and the device failure function parameterized by the scale and shape factors. 9 2.1.1 Device Reliability Alow-costsensordeviceisvulnerabletofailureduetoexternalorinternalproblems. Sensor devices can be deployed in diverse environments including hostile areas. The external environment such as temperature, pressure, etc. can produce device malfunctions. Thedevicecanalsoexperiencesoftwarefailure, whichpreventssome required operation. In case of device failure modeling, the Weibull distribution, which is one of the most commonly used in reliability analysis, can provide the lifetimedistributionofasensordevice[42]. TheWeibulldistributionhasbeenused to provide various types of failure models in real life including semiconductors, engines, biological organisms and others by varying parameters properly [42]. The probability density function of the Weibull distribution has the following form: f(t)= ¯ ´ ( t ´ ) ¯¡1 e ¡( t ´ ) ¯ ; where ¯ and ´ are the shape and the scale parameters, respectively. This function indicates the likelihood of failure at time t. Fig. 2.1 shows the f(t) with the shape parameter from 1 to 4 and the scale parameter 150. When ¯ =1, the f(t) is equal to the exponential distribution. The reliability function of the Weibull distribution is given by R(t)=e ¡( t ´ ) ¯ ; which is the complement of the cumulative distribution function of F(t) of the Weibull density function. The reliability function is the probability that a device is functioning at time t. The failure rate function is z(t)=f(t)=R(t), which is the probability that an item fails when the item is functioning at time t. When the 10 shape parameter ¯ = 1, the failure rate is constant, and when ¯ > 1, the failure rate increases over time. 0 50 100 150 200 250 300 350 0 0.002 0.004 0.006 0.008 0.01 0.012 Time f(t) beta=1 beta=2 beta=3 beta=4 Figure 2.1: The probability density function of the Weibull distribution with ¯ from 1 to 4 when ´ =150. 2.1.2 Energy Depletion In many battery-power supplied devices, the lifetime of the device is highly de- pendent on battery lifetime rather than component failure. Especially, in the data gathering application of wireless sensor networks, nodes near a sink, where all data are collected from other nodes, consume significantly more energy due to frequent data relay, which results in their earlier death than other nodes. In the first phase of the network where all nodes survive, energy consumption of nodes depends on deployment conditions, event distributions, the initial com- munication topology and operational models. Power saving and topology control can defer node aging. In the second phase of the network where nodes begin to die either due to energy depletion and/or device failure, since the communication topology changes over time with node’s death, energy consumption of remaining 11 nodes is affected by such network dynamics. This dynamic nature makes node aging modeling challenging. Major energy consumption activities include communications, data processing and sensing. Communications are needed for situations such as data gathering to the sink, in-network data communication within a cluster, and control message exchanges. According to power analysis in [40], power consumption in the commu- nication operation is twice as much as the processing operation with a radio sleep mode. For example, for MEDUSA-II sensor nodes, when the CPU and the sensing components are running, power consumptions for each operation are 27mW for transmission, 22mW for the receiving mode, 22mW for the radio idle mode, and 10mW for the radio sleep mode. Since the communication cost plays a dominant role in the battery energy consumption, we focus on the data gathering communi- cation of nodes for the energy consumption analysis. Energy consumption in the communication operation during a sampling round, which refers to the time unit for periodic data gathering, can be written as E(r;k t ;k r ) = E rx ¢k rx +E tx ¢k tx = E elec ¢(k tx +k rx )+E amp ¢k tx ¢r · ; (2.1) where E rx and E tx denote the amount of energy consumption per data unit for receiving and transmitting, and k rx and k tx are the amount of data receiving and transmitting per round, respectively, and r is the radio radius. Data transmission andreceptioncanbeexpressedbyanelectronicoperationandanamplifyingopera- tion, whichare accountedfor by E elec andE amp , respectively, and · is the path loss exponentinEq. (2.1). In[24],thevaluesforE elec andE amp aresetE elec =50nJ=bit and E amp =100pJ=bit=m 2 . 12 The energy consumption model and the network lifetime analysis until the first node death has been intensively studied in prior work. Heinzelman et al. [24] presentedageneralenergyconsumptionmodelforcommunicationcomponentsina sensordeviceandanalyzedthecluster-baseddatagatheringscheme. Bhardwajand Chandrakasan [5] discussed the upper bound of the network lifetime by considering thenetworktopologyandthedataaggregationscheme. Theyusedtheoptimization model to compute the maximum lifetime of a sensor network. Lotfinezhad and Liang[32]analyzedtheenergyconsumptionofdataforwardingwithadistancelevel when all nodes are alive, and showed that high energy consumption is required in the outer border of each hop level when a random parent selection scheme is used. Several operational modes of a sensor node, including sleep and active modes, were modeled in [11] as a Markov chain and the energy dissipation level was computed withstationaryprobabilitiesofoperationalmodes. Thedatacommunicationrateof anodewasderivedusinganetworkmodel,whereparameterswereobtainedthrough iterations of a closed loop of a node model, a wireless link interference model and a network model. Chen and Zhao [10] presented the average network lifetime that incorporates the average unused energy in the network and the expected energy consumption. The lifetime expression is a general form that is independent of underlying networks. Using the expression, they proposed a lifetime maximization MACprotocol. ZhangandHou[50]studiedthenodedensityandthelifetimeupper bound that maintains a certain portion of network area to be covered. None of the above work is concerned with the aging effect based on the energy consumption of survival nodes. 13 2.2 Network Aging Modeling The problem of network aging is concerned with network performance degradation overtime. Itisresultedfromtheaggregationofindividualnodedeathduetoenergy depletion or device failure. In network aging analysis, we attempt to identify how and when the performance of the wireless sensor network goes below a certain threshold, which represents the minimum requirement with respect to a specific application. Network performance metrics include: the connectivity, the coverage, and in- formation quality degradation over time. Connectivity can be measured using the probability of connectivity between arbitrary end-points or one-to-many points (data gathering), the degree of connectivity which indicates the fault-tolerant con- nection. Inmanycases,connectionofnodestoasinkisthemostimportantrequire- ment in a data gathering application. Coverage metrics include the percentages of the covered area, the coverage degree, the coverage loss rate of the worst, average and best cases. Information quality can be measured by the fidelity of in-network processed data such as the false alarm rate, the precision of target location, data accuracy against faulty samples, which are required by applications and related to coverage. Information utility in a network field can be also represented as the coverage metric. Spatial correlation among data collected by nodes is incorporated into information utility metric. Network performance degrades drastically from the time when the number of death nodes reaches a certain threshold. At this point, the network will demand redeployment or energy supply to maintain acceptable performance for long-term deployment. Connectivityandcoverageissuesinwirelesssensornetworkshavebeen 14 discussed in the conjunction with the deployment problem, but network aging is not considered beyond the initial node death in most existing work. Many studies on connectivity have focused on identifying a proper radio range to achieve the connectivity between any two nodes in an uniformly distributed sen- sorenvironmentatthedeploymenttime[4,23]. GuptaandKumar[23]showedthat anetworkwithanode’sradioradiussetto p (log(n)+c(n))=n¼ 2 isconnectedwith probability one if and only if c(n) goes to infinity. Bettstetter [4] derived the ex- pressionfortheprobabilityoftheminimumnodedegreeandak-connectednetwork of a certain radio range and node density using the Poisson point formulation and performed various simulation results. There has been also research on the network performance degradation process caused by the node death. Shakkottai et al. [44] derived a bound on the probability that all nodes are connected and the network is covered in a grid deployment when a node has a device failure rate. Kunniyur and Venkatesh [28] presented analysis of network connectivity degrading by assum- ing that the battery lifetime of nodes is an independent random variable, which is basically identical to the device reliability model. The coverage problem in wireless sensor networks has been discussed in many papers. Meguerdichian et al. [36] discussed the coverage metrics that measure the quality of service provided by a sensor network. They considered the maximal breach path for the worst case, which is the maximum distance path from the closest nodes lying in the Voronoi diagram, and the maximal support path for the best case. They presented an algorithm to calculate the worst-case and best- case coverage metrics. Wang et al. [46] discussed the relation between coverage and connectivity, and presented a protocol that can maintain the desired degree of coverage, where every location is covered by at least a certain number of nodes. In their algorithm, the K-coverage degree can be achieved dynamically inside the 15 network with the minimum number of nodes active by determining the coverage degree of intersection points among neighbor nodes and a node itself. Gupta et al. [22] presented algorithms that provide the subset of nodes that are connected and are covering the target area, which can reduce communication cost during query response. Liu et al. [31] examined the effect of mobility on coverage and showed that the sensor mobility actually enhances network coverage by covering the positions which might be undetected by stationary nodes. 2.3 Aging Factors There are many factors that affect node aging, especially energy dissipation. Fig. 2.2 shows the relationship of three aging factors: applications, deployments, and operationsinadditiontotheenvironment. Thenodeandnetworkagingphenomena lead to the aging maintenance issue such as energy-efficient schemes and mainte- nance (redeployment or recharging). Figure 2.2: The diagram of the relationship among aging factors (applications, deployments and operations). 16 ² Applications Applications determine the type of sensors, targeting event characteristics, requirements of data processing, and the sampling rate of each sensor. Ba- sically, application specific sensors are attached to nodes, including acoustic, seismic, video camera, or temperature. When an event occurs or a data gath- ering request is made by sinks, data collected by sensors are processed in nodes and forwarded to the sink. All these elements significantly affect the energy consumption of a sensor device. The pattern of data communication depends on the specific application of interest. For example, target tracking, industrial and structural monitoring, traffic monitoring and control, and in- ventory monitoring applications would require different network operations and requirements [12,29,35]. An event induces the occurrence of active sens- ing operations and, as a result, the temporal and spatial event distribution affects in-network data aggregation and processing. ² Deployments Depending on environments and applications, random deployment or deter- ministic deployment with predetermined positioning can be considered. The communication range and the sensing range of a device are among the most significant factors that affect the overall performance of network operations. The spatial node distribution and density will result in different aging pro- cesses over the network. Locations of sinks affect data communication pat- terns among nodes. Node redundancy should be considered to preserve cov- erage and connectivity against node failure in the network aging process. 17 Furthermore, we may adopt multiple device types in deployment. Heteroge- neous devices have different characteristics in energy consumption, the mem- ory size, the processing speed, device reliability, etc. They lead to different cost-performance tradeoff, which affect the deployment decision and aging processes in networks. Heterogeneous sensor deployment has been discussed in recent years. Stud- ies in [8] and [37,38] focused on minimizing the total deployment cost while guaranteeing some requirements. Chakrabarty et al. [8] presented a solution for the integer linear programming problem that minimizes the cost of het- erogeneous sensor deployment for complete coverage in a grid-based sensor deployment network. However, they did not consider the energy consump- tion of nodes and the communication in the network. Mhatre et al. [37,38] provided the optimal heterogeneous sensor deployment that minimizes the deployment cost in different communication modes. In their model, the cost of the cluster head device is determined by the amount of initial battery energy, which depends on the number of cluster members and the commu- nication model. Kumar et al. [27] presented a hierarchical network system with heterogeneous devices that have different capabilities and showed that a partition of the computation task with heterogeneous devices enhances the network performance as compared with the homogeneous node deployment. Duarte-Melo and Liu [17] examined the effect of heterogeneous deployments in the direct single-hop communication. They considered the MAC collision as one of the main factors in the determination of the optimal number of clusters. Girod et al. [20] presented the tools for simulation and analysis of heterogeneous wireless sensor network systems. Ma and Aylor [34] presented 18 the topology formation protocol that utilizes the heterogeneous resource ca- pacitiesofsensornodes. Yarvisetal.[48]examinedtheeffectofheterogeneous link capabilities and energy capacities among nodes on network lifetime with analysis as well as simulation and real deployment test. ² Operations The single-hop and multi-hop communication models lead to different ag- ing patterns in a network. For the multi-hop communication case, rout- ing schemes may significantly affect the energy consumption among nodes caused by different communication paths. For example, a geographic for- warding scheme [26] may not require as many resources as the other schemes that demand route establishment and maintenance. Depending on the ap- plications, communication patterns can be characterized by the data collec- tion/gather schemes such as event-driven data collection and periodic data gathering [15,25,45]. The one-to-one, one-to-many, and many-to-one com- munication patterns will incur different aging processes across networks. For the data gathering case, there exists a significant workload difference among nodes of different distance from the sink. Data gathering tree construc- tion schemes [51] and routing strategies can result in different aging process. Linklayerschemesthatprovidecontentionresolutionandreliablehop-by-hop transmissionalsoaffecttheamountofdatapacketshandledbyanode. Many energy-efficient schemes can prolong the aging process. ² Environments Inadditiontotheabovethreeagingfactors, theoperationalenvironmentofa sensor network affects the wireless link condition and determines the amount 19 of successful data transmission for a given number of trials, which affects the energy consumption of a node. 2.4 Aging Maintenance By monitoring and predicting the aging process of a network, proper network man- agement operations can be done to extend the network life. They include energy- efficient operations and redeployment/recharging as detailed below. 2.4.1 Energy-efficient Schemes Developing the aging model and predicting the aging function in a network on- the-fly provides useful information for network management such as load balancing communication and topology control. To avoid a large amount of communication overhead, a localized and distributed method to identify the aging of nodes is desirable. Based on localized decision, each individual node or the cluster head can providevariousload-balancingoperationssuchasclustering[1,24],topologycontrol (radiorangecontrolorsleepandactivecontrol)[30,43,47], andenergy-awareMAC protocol[33,49]. Mostoftheseenergyefficientschemesintendtoreducetheenergy usage of nodes by rotating the high energy consumption operation among nodes or adopt power on-off schemes using localized algorithms. Different energy-efficient schemes will result in a diverse aging process of nodes and the network. 2.4.2 Redeployment or Recharging Sincesensordeviceshavelimitedenergycapacity, weneedtoredeploysensornodes and/or supply new energy resources to existing devices to maintain the continuous 20 operation of the system. Issues in re-deployment include: when to redeploy sensors (the proper frequency), which and how many nodes to be recharged or replaced (depending on the proper spatial distribution, the percentages of nodes) with a minimum cost. A good understanding of the aging process would be critical to an economical yet effective operation of the wireless sensor network. Batalin and Sukhatme [3] presented an algorithm that allows a mobile robot to navigate in a field and perform deployment and maintenance for damaged nodes to achieve coverage and connectivity. Barroso et al. [2] addressed the overall mainte- nance cost for battery recharging or replacement and examined maintenance effi- cient routing protocols. They considered the access cost as the Euclidian distance from a certain point to the targeting sensors and compared different forwarding schemes. However, the access cost model may not represent the cost well in an actual deployment. 21 Chapter 3 Aging Analysis of Multi-hop Data Gathering Trees 3.1 Introduction Theagingprocessinamulti-hopdatagatheringtreewithasinkispresentedinthis chapter. First, hop-level analysis for sparse node deployment is examined. Then, finer-grained distance-level analysis for dense node deployment is discussed, where a dynamic data gathering construction that causes workload changing over time is introduced. Finally, the effect of the workload distribution and the node density on the aging process in the first-hop nodes, which determine the connection of the whole network, is examined. 3.2 Hop-Level Aging Analysis for Sparse Node Deployment The aging process in a multi-hop data gathering tree with a sink is presented. First, hop-level analysis for sparse node deployment is examined. It is assumed 22 that nodes are deployed uniformly at random. A data gathering tree is shown in Figure3.1. Itisformedsuchthatalldeployedsensornodesareconnectedtoasink, and can be constructed as follows. First, we find the set of first hop nodes that can directly connect to the sink within its radio range. Then, other nodes select the parentnodesthatarewithintheirradiorangeandhaveashorterhoplevelfromthe sink to establish the forwarding path to the sink [51]. In wireless sensor networks, major energy consumption activities include communications, data processing and sensing. The communicationplaysa dominantrole in the batteryenergy consump- tion [40] as compared to other activities, and the data gathering communication towards a sink accounts for the major portion of communication in most wireless sensor networks. Thus, the average number of children and descendants provides a good estimation of workload in each hop level. We assume that every node pe- riodically generates one data unit and sends it to the sink in each data sampling round, which is adopted as the discrete time unit t. 3.2.1 Workload at Each Hop Level As shown in Figure 3.1, the area of the h-hop level from the sink can be computed via h 2 ¼r 2 ¡(h¡1) 2 ¼r 2 =(2h¡1)¼r 2 ; h=1;2;3;¢¢¢ : By assuming that the average number of nodes at each hop level is proportional to its area, the average child number of a node in hop level h can be computed as n c h = 2h+1 2h¡1 . Then, we can obtain the average number of descendants (n d h ) of a node in hop level h as n d h = L¡1 X i=h ( i Y j=h n c j )= L 2 ¡h 2 2h¡1 23 where L is the maximum hop level from the sink. The average numbers of children and descendants in each hop level with L=7 are shown in Figure 3.2 (a) and (b), respectively. Figure 3.1: The illustration of a data gathering tree, where h represents the hop level from the sink and r indicates the radio range of a sensor device. 1 2 3 4 5 6 0 0.5 1 1.5 2 2.5 3 3.5 4 Hop Level Average Number of Children (a) The number of children 1 2 3 4 5 6 0 5 10 15 20 25 30 35 40 45 Hop Level Average Number of Descendants (b) The number of descendants Figure 3.2: The average number of (a) children and (b) descendants in each hop level when the maximum hop distance L is 7. If no data aggregation takes place during data forwarding, then the number of descendants of a node affects the communication workload. When perfect data aggregation is performed, which aggregates multiple data received from children into one data unit (e.g., MIN, MAX, SUM), the number of packets handled by a node is determined by the number of children. We observe that even in the case of perfect data aggregation the workload in the first hop nodes is al least twice as 24 much as that in the other hop level nodes. Non-uniform density deployment and a sleep scheduling scheme for balancing energy consumption among the hop levels may use this approximate analysis of average workload. We further analyze the dynamic energy consumption in finer-grained distance level in Section 3.3. 3.2.2 Device Reliability Alow-costsensordeviceisvulnerabletofailureduetoexternalorinternalproblems. Sensor devices can be deployed in diverse environments including hostile areas. The external environment such as temperature, pressure, etc. can produce device malfunctions. The device can also experience software failure, which prevent some required operation. In this work, we model the reliability of a sensor node using a classical distri- bution known as the Weibull distribution, which is described in Chapter 2 Section 2.1.1. The sensor node survival function, denoted by S i (t), characterizes the node aging process in a data gathering tree. It is defined as the probability that node i is functioning at the data sampling round t. This function is primarily dependent on the energy consumption rate and the device reliability of a node. Initially, when t = 0, S i (0) = 1. For t > 0, S i (t) is R i (t) if there exists residual energy and R i (t) is the reliability function and otherwise, S i (t)=0. Let C h i (t) be the event where node i in hop level h is connected to the sink at time t. We use p h¡1 i to refer to any candidate ascendent nodes in hop level h¡1 of node i in a dynamic data gathering tree. Then, we have Pr(C h i (t))= 8 > < > : S h i (t); h =1; Pr( S C h¡1 p h¡1 i (t))S h i (t); h¸2: (3.1) 25 As given above, the probability that a node is connected to the sink is equal to the probability that any of the candidate parents is connected and the node is alive. The lower bound of Eq. (3.1) is the case where there is only one available parent for all nodes in the forwarding path, and it can be written as Pr(C h i (t))¸( h¡1 Y k=1 S k p k i (t))S h i (t); h>1 (3.2) A more general expression for the connectivity probability at hop level h over time, Pr(C h (t)), which provides average value for each hop level, can be expressed as Pr(C h (t))= 8 > < > : S h (t) h=1; (1¡(1¡Pr(C h¡1 (t))) n p )S h (t) h¸2; (3.3) where n p is the average number of candidate parents in the upper hop level h¡1. When n p =1, Pr(C h (t)) becomes the lower bound as Eq. (3.2). 3.2.3 Analysis Validation Figure 3.3(a) shows the connection probability in a data gathering tree in each hop level according to Eq. (3.3) where n p is set to 2. In this figure, all nodes have the uniform survival function that follows the Weibull reliability function with ´ =160 and¯ =3regardlessofhoplevels. ThesimulationresultsgiveninFigure3.3(b)and theanalyticalresultsgiveninFigure3.3(a)areconsistentintheiroverallshapeand thedecreasingpatternofthedifferencesbetweenadjacenthoplevels. Itisobserved that the connection probability becomes similar as hop level increases, which is characterized in the following subsection. The simulation results are obtained as theaverageof50differentrandomnodedeploymentswhereeachsimulationrunsfor 200 rounds with 1600 nodes which are distributed uniformly at random. The radio 26 range is set to provide around 9 neighboring nodes within the range on the average and the average furthest hop level from the sink is around 16 in the complete circle network where the sink is located at the center. (a) Analytical result (b) Simulation result Figure 3.3: The percentages of node connection in each hop level to the sink: (a) analytical and (b) simulation results. In order to examine the effect of energy depletion of nodes on connectivity, we also performed two different energy consumption cases - with perfect data aggre- gation and without data aggregation with the same network simulation setup (50 differentrandomdeploymentsof1600nodes)asdevicerandomfailurecase. Energy consumptionofanodeiduringthesamplingroundattcanbecalculatedusingthe numberofcommunicateddataunitsandtheconstantenergyoperationassumption as E i (t) = E rx ¢n i (t)+E tx ¢(1+n i (t))+E other = 2c 1 ¢E elec ¢n i (t)+c 2 ¢E amp ¢(1+n i (t))¢r · +E other ; (3.4) 27 where E rx and E tx denote the amount of energy consumption per data unit for receiving and transmitting, respectively, E other is the energy consumption for non- communication operations such as sensing. n i (t) represents the amount of data packet received from the children of the node i. As explained in Section 3.2.1, n i (t) wouldbeproportionaltothenumberofchildrenincaseofperfectdataaggregation or to the number of descendants in case of without data aggregation. Please note thatadataunitgeneratedbyitselfisaddedtothenumberofdatapacketsreceived in data transmission. Furthermore, it was described in [24] that data transmission and reception can be expressed by an electronic operation and an amplifying oper- ation. They are accounted for by E elec and E amp , respectively. Finally, c 1 and c 2 are constant factors and · is the path loss exponent in Eq. (3.4). Eq. (3.4) can be re-written to the following simplified expression: E(d;t)=E 1 ¢n(d;t)+E 2 ¢(1+n(d;t))¢r · +E other ; where E 1 is the energy required for distance-independent operations, and E 2 is the energy for the amplifying portion. The parameters of the network are given below. · = 2, E 1 = 1, E 2 = 0:001 and E other = 1. Please note that E 1 and E 2 are chosen based on the work in [24], where E elec =50nJ=bit and E amp =100pJ=bit=m 2 . AsdiscussedinSection3.2.1,thedataaggregationdegreecansignificantlyaffect the workload in the nodes closer to the sink, which determine the connectivity to the sink. In a multi-hop data gathering tree topology, the energy consumption of the first hop nodes is significantly higher than the other nodes since they have more child nodes as shown in Figure 3.2. According to this workload difference, the network operation stops due to the death of the first hop nodes even though most nodes have enough residual energy. Figure 3.4 shows that a higher death rate 28 appears in the hop levels closer to the sink in the energy depletion case with a uniform device reliability function at all nodes. Tosimulatetheenergydepletioncasewithoutdataaggregation, weincreasethe initial battery capacity 20 times as large as that for the data aggregation case due to its faster energy depletion and network disconnection. We observe much faster energy depletion especially in the first-hop nodes as shown in Figure 3.4(c) with the finer scale of the x-axis. (a) Device failure effect (b) Energy depletion effect (Data aggregation) (c) Energy depletion effect (No data aggregation) Figure 3.4: The percentages of dead nodes in each hop level over time: (a) the uniform random device failure case, (b) the energy depletion case with perfect data aggregation, and(c)theenergydepletioncasewithoutdataaggregation. Notethat the x-axis scale in Figure (c) is finer than that in Figures (a) and (b). Figure 3.5 shows that, when the same ratio of nodes discontinue their func- tioning, the node death caused by energy consumption even with perfect data aggregation has a more significant impact on network performance than the device reliability effect. This figure presents the percentage of nodes connected to the sink in each hop level. A much lower degree of node death causes significant connection loss due to energy depletion. When no data aggregation is performed, the majority ofconnectionsarelostevenwithlessthana2%rateofnodedeath. Thisisbecause 29 significantly more workload is given to nodes in the first hop level as compared to the case of data aggregation. (a) Device failure effect (b) Energy depletion effect (Data aggregation) (c) Energy depletion effect (No data aggregation) Figure 3.5: The percentages of live nodes and connected nodes in each hop level over time: (a) the uniform random device failure case, (b) the energy depletion case with perfect data aggregation, and (c) the energy depletion case without data aggregation. Note that the x-axis scale in Figure (c) is finer than that in Figures (a) and (b). 3.2.4 Connectivity Aging Function A power-law relationship is observed between the connectivity and the hop level whennodedeathoccursduetodevicefailurefromthelog-logplotsinFigure3.6(a). Thefigureshowsthattheprobabilityofconnectiontoasinkdecreasesinproportion to the hop level with an exponent as node death occurs. The least-squares approx- imation using linear regression provides straight lines that fit simulation data in these two plots. The power law between connectivity and the hop level can be well conjectured from this observation. This relation is observed in the device failure case. Figure 3.6(a) shows up to 45% node death case since around 45% of node death results in the disconnection of almost 80% of nodes. This power-law rela- tion can be explained as follows. In the device uniform random failure case, the 30 samepercentageofnodedeathoccursinallhoplevels,whichareincludedindiscon- nectednodes. Inaddition,ashoplevelsincrease,thepercentageofconnectednodes decreases and its decreasing rate in the longer hop level saturates faster than the exponentiallydecayingcurve, whichisthelowerboundoftheconnectivityfunction as shown in Eq. (3.2), since multiple candidate parents exist to re-construct the data forwarding path to the sink. In case of the energy depletion effect with perfect data aggregation in Figure 3.6(b), the longer hop level nodes show higher connectivity to the sink since most node deaths occur in the closer hop levels to the sink. Since over 80% of nodes become disconnected after 18% of node death, the connectivity up to 18% of node death is shown in this figure. The reason for the higher connectivity in the longer hoplevelupto15%ofnodedeathinthefigureisthatmostenergydepletionoccurs in the hop levels closer to the sink, which decreases the connectivity in those hop levels, while most nodes in the longer hop level are alive and find alternate routes to the sink which result in higher connectivity. However, as more node deaths occur (over 17% of node death), a similar power-law relationship is observed. If no data aggregation is performed, most of initial node deaths occur in the first-hop nodes due to a drastic workload difference between the first-hop nodes and nodes in other hops. This results in drastic connectivity loss of all hop level nodes which have maintained higher connectivity except the first-hop level nodes. On the other hand, with perfect data aggregation, energy depleted nodes exist in all hop levels with much less difference than in the case without data aggregation. Definition 1. The connectivity aging function, denoted by Pr(C h (t)), is de- fined as the probability that a node in hop level h is connected to the sink at time t. 31 (a) Device failure effect (b) Energy depletion effect Figure 3.6: The log-log plot of the percentages of node connection to the sink in each hop level over time as node death occurs due to (a) the device failure effect (80% of the node becomes disconnected with over 45% of the node deaths), and (b) the energy depletion effect, where perfect data aggregation is performed (80% of the node becomes disconnected with over 18% of the node deaths. The curves are obtained by the least-squares approximation. When nodes die due to the uniform random device failure, the connectivity aging function Pr(C h (t)) at time t decreases as hop level h increases at each time instance. Their relationship can be expressed as Pr(C h (t))/h ¡H(t) ; H(t)¸0; (3.5) where H(t) is called the hop connectivity exponent and¡H(t) indicates the slope of the log-scale plot for the percentage of nodes that are connected to the sink with respect to the hop level at each time. As to the device reliability effect, H(t) also depends on the device reliability function, which can be characterized by the Weibull distribution according to the device’s characteristics and environmental conditions. Sparse node deployments, e.g., 10 or less neighboring nodes, would re- sultintheaboverelationship. Generallyspeaking, moredenselydeployednetworks 32 can maintain higher connectivity to the sink in the longer hop level nodes against node death. The multi-hop tree structure for data gathering to a sink requires a proper function of first hop nodes to maintain the connectivity from the other nodes. In the sparse node deployment case, frequent redeployment will be needed to preserve the longer-hop’s connectivity to a sink since the small fraction of node death can cause the connectivity loss in the longer hop levels. This can be characterized and predicted using the presented power-law relation in the case of random device failure. Ontheotherhand,densenodedeploymentcanmaintaintheinitialnetwork connectivity from longer hop level until most of the first-hop nodes die, which is examined in the next section. 3.3 Distance-Level Dynamic Aging Analysis for Dense Node Deployment In this section, we examine finer-grained distance level node aging in a data gath- ering tree by analyzing the dynamic energy consumption change beyond the initial node death with tree re-construction. Then we examine the effect of the workload distributionandthenodedensityontheagingprocessinthefirst-hopnodes,which determine the connection of the whole network. 3.3.1 Dynamic Aging Analysis In the following analysis, we consider the dynamic change of the number of descen- dants caused by the re-construction of the data gathering tree due to the death of 33 other nodes. This process is illustrated in Figure 3.7, where the center is the sink location and r is the radio range, and other notations are summarized in Table 3.1. (a) Case I (b) Case II Figure 3.7: Selection of a new parent after energy depletion of the current parent (a) Case I: Select a new parent in the live upper hop level (b) Case II: Select a new parent in the previously same hop level It was observed in [32] that energy consumption is the highest at the border of each hop level when the random parent selection scheme is used for minimum- hop data gathering tree construction as discussed in Section 3.2. The workload difference among nodes of the same hop level leads to gradual energy depletion of nodes from the border of each hop level and the workload change over time. The maximum distance of live nodes at hop level h over time is denoted by d h max (t). Figure 3.7 shows two cases of data gathering tree re-construction caused by energy depletion of some nodes. ² Case I: Consider a node in hop level h+1 at distance c 1 from the sink that has a parent node in area D(c 1 ;t), where all nodes deplete their energy. Thus, it has to select a new parent in area A 1 (c 1 ;t), where nodes still have energy. 34 Table 3.1: Summary of notation for a multi-hop communication tree. d distance from the sink c distance from the sink at which the children of a node at d is located n(d;t) number of descendants of a node at distance d and sampling round t n 0 (d;t) n(d;t) with the children that join before initial node death n 1 (d;t) n(d;t) with the children that newly join from the next hop area n 2 (d;t) n(d;t) with the children that newly join from the previous same hop area A(c) upper hop area within the radio range of a node at c A 1 (c;t) live upper hop area within the radio range A 2 (c;t) live same hop area within the radio range D(c;t) energy depleted upper hop area within the radio range d h max (t) maximum distance of live nodes in hop level h r radio range Δd distance unit ¸ node density h hop level of nodes at d This re-construction process increases the energy consumption of live nodes in that area. ² Case II: Consider another node at distance c 2 , which is located between d h max (t)+r andd h+1 max (t). Ifallnodesintheupperhopregionwithinitsradiorangedeplete their energy, nodes that lose their parent should select a new parent in area A 2 (c 2 ;t), which was in the same hop level previously, to forward the data towards the sink. This process splits one hop level into two. To compute the expected number of children that are connected to a node, we canfollowtheanalysisin[32],whichprovidestheaveragenumberofdescendantsof a node at distance d from the sink before any node death. This number is denoted by n 0 (d). Let Pr(A(c)) be the probability that a node at distance c from the sink 35 selects one node in the upper hop area A(c) to the sink within its radio range as Eq. (3.8) following [32], which is presented in Appendix 3.3.2. We use n 1 (d;t) to denote the newly added number of descendants of a node at d for the first case in Figure 3.7, where nodes at c 1 select a new parent in area A 1 . Pr(D(c;t)) is the probability that node c 1 selected a parent which was located in area D(c;t) in the upper hop area, which is D(c 1 ;t)=(D(c 1 ;t)+A 1 (c 1 ;t)). Then, we have n 1 (d;t)= d+r X c=hr+Δd fC(d;c)¸(1+n(c;t))Pr(D(c;t))Pr(A 1 (c;t))g; (3.6) where (h¡1)r < d < d h max (t), h is the hop level of the node at d, ¸ is the node density,C(d;c)isthelowerhopareawithadistanceofc¡Δdtocfromthesinkand within the radio range of the node at d, and n(c;t) is the number of descendants of a node at c, which can be calculated recursively. Basically, if the children of a node at d exist in the area from hr to d+r and within its radio range, we add the number of their descendants. If there is no node available in the upper hop level, a child node will select a parent among nodes in the same hop level, which is the second case given in Figure 3.7. As shown in the figure, node c 2 will select a new parent in area A 2 (c;t). Let n 2 (d;t) be the newly added number of descendants from the same hop level before initial node death. Then, we can derive the following n 2 (d;t)= d h max (t) X c=d h¡1 max (t)+r+Δd fC(d;c)¸(1+n(c;t))Pr(A 2 (c;t))g; (3.7) 36 where (h¡1)r < d < d h¡1 max (t)+r. In words, if the children of a node at d exist in the area from d h¡1 max (t)+r to d h max (t) and within its radio range, we add the number of their descendants. Areas C(d;c), D(c;t), A(c), A 1 (c;t) and A 2 (c;t) can be computed based on geometry. These calculations is presented in the following Section 3.3.2. Then, the average number of descendants of a node at d, denoted by n(d;t), is the sum of three types of descendants obtained above, i.e. n(d;t)=n 0 (d;t)+n 1 (d;t)+n 2 (d;t): EnergyconsumptionE(d;t)duringthesamplingroundattcanbecalculatedusing thenumberofcommunicateddataunitsandtheconstantenergyoperationassump- tion as Eq. (3.4) and can be re-written to the following simplified expression: E(d;t)=E 1 ¢n(d;t)+E 2 ¢(1+n(d;t))¢r · +E other ; ThevalueE(d;t)ofnodesinaliveregionisrecomputedwheneverenergydepletion of some node occurs. The energy depletion time of nodes is determined by the cumulative effect of the energy consumption dynamics among nodes over time. 37 3.3.2 Geometric Calculation for Dynamic Data Gathering Construction The probability, Pr(A(c)), that a node at distance c from the sink selects one node in the upper hop area A(c) to the sink within its radio range was derived in [32]. It is given below Pr(A(c))= 1 X k=1 1 k Pr(N(A(c))=kjN(A(c))¸1)= e ¡¸A(c) 1¡e ¡¸A(c) 1 X k=1 (¸A(c)) k k!k : (3.8) Furthermore, areas C(d;c), D(c;t), A(c), A 1 (c;t) and A 2 (c;t) can be computed using geometry. These results are summarized as follows. ² Calculation of C(d;c). C(d;c)= Z c c¡Δd 2cos ¡1 ( d 2 +l 2 ¡r 2 2ld )ldl: ² Calculation of A(c), which is the upper hop area within the radio range of a node at c distance from the sink. First, we have w 01 =cos ¡1 µ r 2 +c 2 ¡(hr) 2 2rc ¶ ; w 02 =cos ¡1 µ hr 2 +c 2 ¡r 2 2hrc ¶ ; Then, we obtain A(c)=r 2 (w 01 ¡sin(2w 01 )=2)+(hr) 2 (w 02 ¡sin(2w 02 )=2): 38 ² Calculation of A 1 (c;t), which is the area within A(c) except for the energy depletion area D(c;t) (=A(c)¡A 1 (c;t)). First, we have w 11 =cos ¡1 µ r 2 +c 2 ¡d h max (t) 2 2rc ¶ ; w 12 =cos ¡1 µ d h max (t) 2 +c 2 ¡r 2 2cd h max (t) ¶ Then, we get A 1 (c;t)=r 2 (w 11 ¡sin(2w 11 )=2)+d h max (t) 2 (w 12 ¡sin(2w 12 )=2) ² Calculation of A 2 (c;t), which is the same hop area within the radio range of a node at c. First, we have w 21 =cos ¡1 µ (d h¡1 max (t)+r) 2 +c 2 ¡r 2 2c(d h¡1 max (t)+r) ¶ ; w 22 =cos ¡1 µ r 2 +c 2 ¡(d h¡1 max (t)+r) 2 2rc ¶ : Then, we find A 2 (c;t)= (d h¡1 max (t)+r) 2 (w 21 ¡sin(2w 21 )=2)+r 2 (w 22 ¡sin(2w 22 )=2) ¡fr 2 (w 0 01 ¡sin(2w 0 01 )=2)+((h¡1)r) 2 (w 0 02 ¡sin(2w 0 02 )=2)g where w 0 01 and w 0 02 are obtained by replacing h in w 01 and w 02 with h¡1, respectively. 39 3.3.3 Understanding Network Aging via Case Study We evaluate the performance of a wireless sensor network based on our analysis conducted above by considering an example. The parameters of the network are given below. The radio range r is 25m, the network radius is 100m from the sink, a total of 500 nodes are deployed and the other parameters follow the values used in Section 3.2.3. Theenergyconsumptiondistributionasafunctionoftimeandthedistancefrom the sink is shown in Figure 3.8. As shown in the figure, the energy consumption rateissignificantlylargerfornodesinthefirstandthesecondhoplevelsofthesink. As time increases, nodes in the initial second hop area are split into two groups. One is directly connected to the first hop nodes while the other is not due to the energy depletion of nodes in the border of the first hop area. It is also worthwhile to point out the high energy consumption for nodes in the second hop close to the first hop border, which is caused by the rapid increase of newly joined descendants, n 2 (d;t), from the same hop area previously. Theresidualenergydistributionasafunctionoftimeandthedistancefromthe sink is shown in Figure 3.9. Initially, energy depletion occurs in the border of each hop level. For the region in the second hop level, the residual energy of nodes in the area near to the first hop level becomes smaller rapidly. The maximum energy consumption and the percentage of nodes connected to the sink in the first hop level are shown in Figure 3.10. We observe the sharp increase of maximum energy consumption in the first hop nodes at t=40, which is followed by the sudden loss of all network connections. Thus, by observing a rapid increase of the maximum energy consumption rate, we can predict an immediate transitiontocompletenetworkconnectionloss. Duetoalargevariationofworkload 40 Figure 3.8: The energy consumption distribution as a function of time and the distance from the sink. in a dense deployment under a random parent selection scheme, we see that there is a substantial amount of time between the initial energy depletion of a node and the complete network disconnection. 3.3.4 Computer Simulation with Realistic Wireless Links Inthissubsection,weperformcomputersimulationwith500nodesand100different random deployments by incorporating a realistic wireless link model given in [52]. Thismodelprovidesthepacketreceptionratealongthedistancebyconsideringthe pathlossexponent,theshadowingeffect,andthephysicalcommunicationschemes. The key parameters used in the simulation are listed in Table 3.2. The residual energy values of nodes at the same distance unit are averaged and plotted in Figure 3.11 for each random deployment when 10% of nodes are disconnected from the sink. The figure consists results of 100 runs. We see a large 41 Figure 3.9: The residual energy distribution (in terms of percentages) as a function of time and the distance from the sink. Table3.2: Parametersofrealisticwirelesslinkmodel(PRR:packetreceptionrate). Parameters Simulation Values Transmitting power -3dBm Shadowing effect standard deviation 2 Path loss exponent 2.7 Beginning of the transitional region (PRR=0.9) 16.4m End of the transitional region (PRR=0.1) 38.5m Modulation FSK Encoding scheme MANCHESTER variation of residual energy values at the same distance. Even though a similar trend is observed by analysis, the variation obtained by analysis is smaller. This can be explained by the fact that, besides the mixture of hop levels around the border areas, there exists a large variation in the number of children connected to nodes at the same distance in the simulation. The large variation of workload results in earlier death of some nodes in the second hop area as compared to that in analysis. 42 0 10 20 30 40 50 0 100 200 300 400 500 600 Maximum Energy Consumption 0 10 20 30 40 50 0 10 20 30 40 50 60 70 80 90 100 Percentage of Connected Nodes (%) Time (round) Figure 3.10: The maximum energy consumption (the dashed line) and the percent- age of nodes connected to the sink (the solid line) in the first hop level. Simulationresultsofenergyconsumptionofnodesinthefirsthoplevelasafunc- tion of time are shown in Figure 3.12. Each line represents the energy consumption of a node in the first hop level along time. We see a wide range of variations in the initial energy consumption and the energy depletion time. Energy depletion of nodes with larger workload causes workload shift to remaining nodes. In addition, there exists a sharp increase of maximum energy consumption over time, which is consistent with the analytical result shown in Figure 3.10. The average residual energy over 100 simulation runs with random deployment isshowninFigure3.13, whereeachlinerepresentsacasewhen1%, 5%, and90%of thenodesaredisconnectedfromasink. Furthermore, analytical(indicatedbydash lines) and simulation (indicated by solid lines) results are compared. Generally speaking, simulation and numerical results show similar pattern except for nodes in the border areas between two consecutive hop levels. In the simulation using a realistic wireless link model, some nodes in these border areas can be directly linked to a node in the upper hop level while others at the same distance cannot. Asaresult, theborderareathatisclosetothefurtherhoplevelinsimulationhasa 43 0 25 50 75 100 0 20 40 60 80 100 Distance Residual Energy (%) Figure 3.11: The residual energy distribution at time with 10% connection loss. The solid line represents the average residual energy higher residual energy than that obtained analytically. In contrast, the border area thatisclosetotheupperhoplevelhasaloweraverageresidualenergyinsimulation than that in analysis. To conclude, the mixture of hop levels at the same distance around border areas due to a realistic wireless link model results in a smoother residual energy distribution along the distance from the sink. When the residual energy distributions of nodes in hop levels 2 and 3 are sep- arated as shown in Figure 3.14, we observe a good match between simulation and analysis results. Figure 3.14(c) gives the node distribution in hop levels 2 and 3, which shows the mixture of hop levels in the border area from the distance 35 to 60, and around 90% of nodes in each hop level reside in the theoretical hop level. To conclude, we observe a larger workload variation at the same distance level and a mixture of hop levels in the simulation using a realistic wireless link model. Thesefactorsleadtosmootherenergyconsumptionandresidualenergydistribution ascomparedtoouranalyticalresults. Generallyspeaking,resultsinsimulationand analysis are consistent with each other in the general trend. 44 0 10 20 30 40 50 0 100 200 300 400 500 600 Time (round) Energy Consumption Maximum Energy Consumpation Energy Consumption of Each Individual Node Figure3.12: Energyconsumptionofthefirst-hopnodesasafunctionoftime. Each line represents the energy consumption of a node along time. 0 25 50 75 100 0 20 40 60 80 100 Distance Average Residual Energy (%) 1% connection loss (a) 1% connection loss 0 25 50 75 100 0 20 40 60 80 100 Distance Average Residual Energy (%) 5% connection loss (b) 5% connection loss 0 25 50 75 100 0 20 40 60 80 100 Distance Average Residual Energy (%) 90% connectiion loss (c) 90% connection loss Figure 3.13: The average residual energy with realistic wireless links, where ana- lytical and simulation results are indicated by dash and solid lines, respectively. 3.3.5 First-hop Node Aging Analysis As discussed in Section 3.3, there exist workload differences among nodes in the same hop level using a random parent selection scheme for the data gathering tree construction. This results in gradual energy depletion among nodes and workload shift from energy-depleted nodes to survivor nodes until the complete network con- nection loss is reached as shown in Figure 3.12. Here, workload represents the amount of data packets to be handled by the node from its descendants in the data 45 0 25 50 75 100 0 20 40 60 80 100 Distance (Hop level=2,3) Average Residual Energy (%) 1% connection loss Hop level=2 Hop level=3 Transitional Region (a) 1% connection loss 0 25 50 75 100 0 20 40 60 80 100 Distance (Hop level=2,3) Average Residual Energy (%) 5% connection loss Hop level=2 Hop level=3 Transitional Region (b) 5% connection loss 0 25 50 75 100 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Distance (Hop level=2,3) Percentage of nodes (%) Hop level = 2 Hop level = 3 Transitional Region (c) Node distribution Figure 3.14: (a),(b) The separate average residual energy of nodes in 2nd and 3rd hop levels with realistic wireless links. Analytical and simulation results are indicated by dash and solid lines, respectively, and (c) the distribution of 2nd and 3rd hop level nodes at each distance unit. gathering tree in order to forward them to the sink. In this section, we characterize these aging phenomena, including energy depletion time and workload shift, of the first-hopnodeswithclosed-formexpressions. Inparticular,wewouldliketoaddress the following questions. 1. How does the workload of each survivor node in the first-hop change over timewhensomenodeswithahigherworkloaddepletetheirequippedenergy? 2. How to estimate the energy depletion time of nodes with a given workload distribution? 3. How much longer does a node with the smallest workload portion survive after the initial node death? 4. What is the effect of the node density and the radio range on the aging phenomenon? Through a dynamic data gathering tree construction process, the workload of energy-depleted nodes in the first-hop level is re-distributed to remaining survivor 46 nodes. Based on the observation in Section 3.3.1, the newly added workload is roughly proportional to the initial workload distribution. That is, since energy de- pletion occurs from the furthest distance from the sink in the first-hop, the amount of newly added workload to survivor nodes rises sharply as the distance from the sink increases since the region has more candidate children as given in Eq. (3.6). Thus, to simplify this first-hop analysis, we assume that the newly added workload distribution among survivor nodes due to energy depletion of other nodes follows the initial workload distribution. The system model for the first-hop analysis and several basic properties can be stated below. ² When all nodes survive, the first-hop nodes are ordered by their workload 1,2,..,i,..,f,wheref isthetotalnumberoffirst-hopnodes,andtheirworkload probability distribution is given by p 1 ¸ p 2 ¸ ::: ¸ p f with P f i=1 p i = 1. As discussed in the previous section, workload of a node is approximately proportional to the number of descendants attached to the node in the data gathering tree. Workload probability indicates the ratio of its own workload to total workload that should be handled by all the first-hop nodes. We do not assume any specific workload distribution. It can be uniform for all first- hop nodes or different among nodes as discussed in the distance-level aging analysis in Section 3.3.1. ² LetN be the total number of nodes deployedand W be the total workloadin the first hop per data gathering round. It is assumed that the total workload in the first hop nodes is proportional to the data units generated by their 47 descendants on a data gathering tree, which is identical to the number of descendants. Thus, we have W =®(N¡f); where ® is a proportional constant. ² The workload of node i at data sampling round t can be computed as L i (0)=Wp i ; f X i=1 L i (t)=W: ² The energy depletion time of node i can have the following order: D 1 ·D 2 ·:::·D f ; since the workload of a node determines its energy depletion time; Node with larger workload consumes energy faster. If all the first-hop nodes have the same portion of workload, then the energy depletion time would be the same. Byassumingthattheworkloadofeachsurvivornodekeepsincreasinginpropor- tion to the initial workload distribution by sharing the workload of energy depleted nodes, we can obtain the workload function of node i over t as L i (t)= 8 > > > > > > > > < > > > > > > > > : Wp i ; if 0·t<D 1 ; Wp i 1¡ j X m=1 p m ; if D j ·t<D j+1 ; 1·j <i; 0; if t¸D i ; (3.9) where j indicates a node that has more workload than i. 48 Energydepletiontimeofnode1,whichhasthelargestworkload,canbeobtained by D 1 = B Wp 1 ; where B is the total workload of a battery. Furthermore, for node i¸2, its energy depletion time D i can be obtained from D i¡1 and the remaining battery capacity divided by its total workload that has been increased after D i¡1 as D i =D i¡1 + B¡ i¡1 X m=1 ³ (D m ¡D m¡1 )L i (D m¡1 ) ´ L i (D i¡1 ) ; i¸2: (3.10) Actually,wecanobtaintheenergydepletiontimeofnodeiusingonlytheinitial workloaddistributionwithoutrelyingontherecursiveformulashowninEq. (3.10). That is, D i can be written as D i = B Wp i ³ 1+(i¡1)p i ¡ i¡1 X m=1 p m ´ ; i¸2; (3.11) which can be proved by induction as shown below. First, for i=2, we have D 2 = D 1 + B¡D 1 L 2 (0) L 2 (D 1 ) = B Wp 1 + ¡ B¡(B=Wp 1 )Wp 2 ¢ (1¡p 1 ) Wp 2 = B Wp 2 ³ 1+p 2 ¡p 1 ´ ; (3.12) 49 where the second equality is due to Eq. (3.9). Thus, Eq. (3.11) holds for i = 2. Now, assuming that Eq. (3.11) holds for any integer k¸2, i.e. D k = B Wp k ³ 1+(k¡1)p k ¡ k¡1 X m=1 p m ´ ; (3.13) we would like to prove that Eq. (3.11) also holds for D k+1 . The energy depletion time interval between k and k¡1 is D k ¡D k¡1 = B W ³ 1 p k ¡ 1 p k¡1 ´³ 1¡ k¡1 X m=1 p m ´ : (3.14) Furthermore, the consumed battery energy of k+1 until D k can be obtained using Eq. (3.14) and Eq. (3.9) as k X m=1 ³ (D m ¡D m¡1 )L k+1 (D m¡1 ) ´ = Bp k k X m=2 ³ 1 p m ¡ 1 p m¡1 ´ + Bp k p 1 = B p k+1 p k : (3.15) The same result can also be obtained from the fact that L k+1 is increased while maintaining the ratio of p k+1 =p k to L k till node k reaches energy depletion. Thus, from Eq. (3.15), Eq. (3.9) and Eq. (3.13), we have D k+1 = D k + B¡ k X m=1 ³ (D m ¡D m¡1 )L k+1 (D m¡1 ) ´ L k+1 (D k ) = B Wp k ³ 1+(k¡1)p k ¡ k¡1 X m=1 p m ´ + ³ B¡B p k+1 p k ´³ 1¡ k X m=1 p m ´ Wp k+1 = B Wp k+1 ³ 1+kp k+1 ¡ k X m=1 p m ´ 50 Thus, we conclude that Eq. (3.11) holds for D k+1 . The proof is completed. The energy depletion time of node f, which is the last surviving node with the smallest workload in the first hop level, can be computed by (3.11). That is, we have D f = Bf W = B ®(N=f¡1) ; (3.16) which is the total battery energy divided by the average workload. According to this expression, the workload distribution does not affect the energy depletion time of the last survivor node, which is the time when a whole network is disconnected from the sink. This result leads to an interesting point. That is, by increasing or decreasingthenodedensitywithafixedradiorangetomaintainthemaximumhop levels, the disconnection time of the last first-hop node will remain the same. If we only consider the average workload in a hop level and the average energy depletion time of nodes in the same hop as done in Section 3.2, the average energy depletion time of the first-hop nodes remains the same regardless of the node density. How- ever,thecalculationoftheenergydepletiontimeofthelastsurvivornodepresented inEq. (3.16)incorporatesthedynamicchangesofworkloadanditsincreasingrate, which are dependent on the earlier energy depletion times of all nodes with more workload, recursively as in Eq. (3.10). In addition, the above result holds for any arbitrary workload distribution. Also, we can obtain the time duration of the aging process by measuring the ratio of the lifetime of node f, the last survivor node, and the lifetime of node 1, the first energy depleted node. From Eq. (3.10) and Eq. (3.16), we have D f D 1 =f¢p 1 ; (3.17) 51 which depends on the number of first-hop nodes and the fraction of the largest workload. If perfect workload balancing can be performed on first-hop nodes, where p i = 1=f, then the energy depletion time will be the same, which is de- sirable. Simultaneous energy depletion of first-hop nodes can also be achieved by heterogeneous energy capacity deployments according to workload. 3.4 Conclusion The aging process in a multi-hop data gathering tree with a sink was studied thor- oughly in this work. Both hop-level analysis for sparse node deployment and finer- grained distance-level analysis for dense node deployment were examined. The effect of the workload distribution and the node density on the aging process in the first-hop nodes was also analyzed. Besides theoretical analysis, we provided simulation results using a realistic wireless link model. There are several open problems to be studied in the future. First, we have not yet incorporated the MAC operations and transmission delay in analysis in our current research. If the duration of a sampling round is large, the effect of MAC contention would be relatively small. However, in a high node density, MAC contention, retransmission and queueing delay could be significantly higher, which demands further study. Second, no transmission range is adjusted in our analy- sis. If transmission range adjustment is allowed after a parent is randomly selected among the upper hop level, then faster energy depletion would occur in the outer region of each hop level. A quantitative analysis is worthwhile. Finally, if different localized topology construction schemes are adopted as described in [51], the work- load distribution among the same hop levels will be different. This will affect the 52 aging process. However, the complete network disconnection due to energy deple- tion of all first-hop nodes will occur at a similar time as predicted by Eq. (3.16). Themaindifferencewouldbethetimeoftheinitialenergydepletionofanodewith the largest workload. Thus, we examine this problem in the next Chapter 4. 53 Chapter 4 Impact of Localized Topology Construction on Network Lifetime 4.1 Introduction For data gathering path construction, nodes in longer hop levels (i.e. non-first-hop nodes) have to determine the next node to forward the data to the sink with a parent selection strategy. A localized topology construction scheme allows each node to select a parent node using its one-hop neighboring node information only. Thus,thepurposeoflocalizedschemesistoreducethecommunicationoverheadfor theconstructionofadatagatheringpath, whichisdesirableforenergy-constrained wirelessnetworks. Eventhoughtherewasstudyonlocalizedtopologyconstruction, e.g.,[51],itseffectonthenetworklifetimehasnotbeenextensivelyexamined. Here, weexaminelocalizedtopologyschemeswithdifferentparentselectionstrategiesand analyze their impact on the network lifetime in conjunction with diverse network conditions such as the node density, the network size and the link quality of the radio range. The link quality can be used as a metric for routing path selection. Recently, the expected transmission count (ETX) of a link between two nodes is considered, 54 which can be derived from the packet reception rate (PRR) of the link [14], [16]. Mathematically, we have ETX ij =ETX ji = 1 PRR ij ¢PRR ji ; where ETX ij is the expected number of transmission required for successful trans- mission over a link between nodes i and j. Qualitatively speaking, a low ETX link canreduceenergyconsumptionduetoredundantretransmissionthanahigherETX link. However, the quantitative effect of a link-quality-based path selection scheme on energy consumption and/or network lifetime has not been fully investigated. Besides the link quality, the number of hops (called the hop count) to the destination is widely used for routing path selection. Each link can be counted as one hop. Then, the routing path with the minimum number of hop counts to the sink is the shortest path. The minimum hop routing (MHR) path can be constructed using the currently known hop level of neighboring nodes. Each node selects a neighboring node that provides the minimum number of hops to the sink. If the routing hop count is smaller, it is reasonable to assume that the total communicationenergyconsumptionofthatroutingpathissmaller,too. Rigorously speaking, the link quality and the radio range will also affect energy consumption. Here, weincorporatethelinkqualityintotheenergyconsumptionanalysisofMHR schemes. By using the ETX link quality metric and the hop count to the sink, we will examine the following four localized topology construction schemes. ² The lowest ETX parent selection scheme, where a node selects a neighboring node that provides the lowest ETX link between each other; 55 ² The random parent selection scheme with MHR, where a node randomly chooses a parent among neighboring nodes in the upper hop level, which provides the minimum hop routing to the sink; ² The lowest ETX parent selection scheme with MHR, where a node chooses the neighboring node in the upper hop level that provides the lowest ETX; ² The balanced parent selection scheme with MHR, where a node selects the neighboringnodeintheupperhoplevelthathasthefewestnumberofchildren as a parent in the data gathering tree. The first scheme does not utilize the hop count but the link quality metric only while the other three schemes take the hop count into consideration for parent selection as well. These localized schemes are examined by real empirical data and analysis in the following sections. 4.2 Case Study with Real Empirical Data In this section, with the empirical data in a real deployment, we examine four localized topology construction schemes to understand their impact on the com- munication load and discuss their differences. The data are from the experiments conducted by the UCLA/CENS group [7], where the packet reception rate (PRR) of each node from all other nodes is given. A set of 55 nodes was deployed in the ceiling of the lab in their indoor experiment. With this PRR information, we examine the connectivity between adjacent hop levels and the communication overhead distribution among nodes. Without respect to a target node, any other node that has a packet reception rate (PRR) for bi-directional links higher than the link threshold is called its neighboring node. 56 In other words, every pair of neighboring nodes can directly communicate with each other if the successful packet transmission and reception rates are above the link threshold. Communication to all other nodes requires multi-hop forwarding. The link threshold can be adjusted, which will change the hop level of nodes from the sink. The use of this threshold makes routing more reliable. As the link thresholdincreases,aconstructedtopologywithmorehoplevelscanprovidehigher throughput due to higher successful transmission rate of the link than a simple minimum hop count routing. 4.2.1 Data Gathering Topology Maps Fig. 4.1 shows the deployment map of 55 sensor nodes and hop levels with four different topology construction schemes. A line represents a data gathering link between adjacent hop levels, which will be discussed further. To forward the data toasink, whichisassumedtobelocatedattheupperleftcorner, eachnodeshould select a parent node towards the sink among neighboring nodes to construct a data gatheringtree. Nodesthathaveconnectionwiththesinkwiththepackettransmis- sion and reception rates higher than the link threshold belong to the first hop level and are represented by a diamond shape. For the lowest ETX parent selection, since the main objective of this scheme is to provide a high packet successful trans- mission rate, the link threshold for the first-hop level is set to 0.95. For all other schemes that are based on the minimum hop routing (MHR), the link threshold is set to 0.90. As shown in Fig. 4.1 (a), the lowest ETX parent selection without hop count consideration results in longer hop levels. The longest hop level is 7. Since each nodeusesthelowestETXparentselection,thedistancebetweentheparentandthe 57 childrennodestendstobecloseandthenumberofhoplevelsincreases. Allpossible directlinksbetweenadjacenthoplevelnodesbytherandomparentselectionscheme with MHR are presented in Fig. 4.1 (b). Each node randomly selects one among nodes that are connected with a direct link as its parent node. As the distance from the sink increases, the first hop nodes have more direct links to the second hop level nodes. With the link threshold 0.9, the MHR scheme reduces the hop count significantly as compared with the lowest ETX scheme in Fig. 4.1 (a). Fig. 4.1(c)showstheconnectivitygraphofthelowestETXparentselectionwithMHR. Since each node selects the lowest ETX neighboring nodes in the upper hop level, the selected parent nodes tend to be located at the edge of the hop level, closer to the second hop level nodes. For the balanced scheme shown in Fig. 4.1 (d), data forwarding paths to the sink are almost evenly spread among the first hop level nodes. We can summarize observations from these topology maps produced by four localized schemes as follows. If we exploit only link quality without using the hop count in the parent selection decision, the distance between the chosen link becomes relatively short and hop levels increases accordingly. When the MHR scheme is used, the link quality based selection results in an unbalanced topology where fewer nodes at the border of hop levels handle most data forwarding tasks from larger hop level nodes. 4.2.2 Link Quality and Communication Load There exists trade-off between the link-quality based and the MHR-based schemes, which will be examined in this section. Fig. 4.2 shows the average link quality (ETX) of data forwarding paths selected by four localized topology schemes. The 58 0 2 4 6 8 10 12 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 Sink 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 Hop=1 Hop=2 Hop=3 Hop=4 Hop=5 Hop=6 Hop=7 (a) Lowest ETX parent selection 0 2 4 6 8 10 12 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 Sink 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 Hop=1 Hop=2 Hop=3 (b) Random parent selection (MHR) 0 2 4 6 8 10 12 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 Sink 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 Hop=1 Hop=2 Hop=3 (c) Lowest ETX parent selection (MHR) 0 2 4 6 8 10 12 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 Sink 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 Hop=1 Hop=2 Hop=3 (d) Balanced parent selection (MHR) Figure 4.1: The indoor deployment location of nodes and four data gathering topology maps with localized topology construction schemes using the PRR data obtained by UCLA/CENS group: (a) the lowest ETX parent selection, (b) the random parent selection scheme (all possible links) with MHR, (c) the lowest ETX parent selection scheme with MHR and (d) the balanced parent selection scheme with MHR. 59 link threshold varies from 0.7 to 0.9. Regardless of the link threshold, we observe that the average link quality has the following order from the highest to the lowest: thelowestETXselection,thelowestETXselectionwithMHR,therandomselection and the balanced selection. The reason for the poor link quality for the balanced selection scheme is that it chooses a parent node with the fewest children, which is consequently farfroma selectingnode. Asthe link threshold increases, the average link quality improves for both the random selection and the balanced selection scheme while the lowest ETX selection remains almost the same. 0.7 0.75 0.8 0.85 0.9 1 1.05 1.1 1.15 1.2 Link threshold Expected transmission counts (ETX) Lowest ETX Random + MHR Lowest ETX + MHR Balanced + MHR Figure4.2: Thecomparisonoflocalizedtopologyconstructionschemes: theaverage ETX of data gathering paths with respect to link threshold. The amount of communication energy of a node during a data gathering round isdeterminedbytheamountofdatareceivingfromchildrennodesandtransmitting to the parent node and their link quality (ETX). Basically, the amount of receiving datafroma childnodeistheproductofthelinkETXfromthat childnodeandthe 60 amount of data that is intended to transmit by that child node. Thus, the amount of communication energy per data gathering round by node i can be calculated as E i = X j2C i f ji ETX ji +¯ X k2P i f ik ETX ik ; (4.1) where E i is the normalized energy consumption with respect to the energy con- sumption for receiving denoted by E rx and ¯ =E tx =E rx =1+E amp =E elec d · ; and where E amp and E elec denote the amplifier energy and the electronic energy, respectively, and d is the radio range and · is the path loss exponent. By following the parameters given in [24]; we set E elec = 50nJ=bit and E amp = 100pJ=bit=m 2 . Besides, whend=20m and·=2, ¯ =1:8. We useC i to denote the set of children nodesofiandP i thesetofparentnodesofi. Thelocalizedselectionschemechooses one parent, and f ji consists of data generated by the descendant nodes of node j in addition to the data generated by node j. Thus, f ik consists of P j2C i f ji and data generated by node i. When the amount of generated data by each node per data gathering round is assumed to be one unit, the number of descendants in the data gathering tree constructed by localized topology schemes determines the communication load of each node. For the lowest ETX without MHR, there exists a larger communication load on the first-hop nodes due to longer hop levels and fewer first-hop nodes. The maximum number of descendants obtained from Fig. 4.1 (a) is 33. When MHR is used, the communication load is distributed among a larger number of first- hop nodes. Fig. 4.3 compares the number of children nodes as a function of the 61 distance between the sink and the first hop level nodes for three different topology construction schemes with MHR. For the random parent selection, the expected children number of first-hop node i is calculated as P j2C i 1 n p j , where j is a node belonging to the second hop level neighboring nodes of node i (C i ), and n p j is the number of upper hop level neighboring nodes of node j. Since there are only two nodes in the third hop level, the number of children and descendants are almost same. Overall, the number of descendants tends to increase along the distance in the random selection scheme. The lowest ETX parent selection scheme can provide higher throughput at a given time, but it results in an extremely unbalanced com- munication load. This causes much faster energy depletion of some nodes so as to result in a large gap of energy depletion time among first-hop level nodes. The balanced parent selection scheme provides a similar energy depletion time among nodes. 0 2 4 6 8 10 0 2 4 6 8 10 12 14 Distance from the sink Expected number of nodes No. of descendants No. of children (a) Random parent selection 0 2 4 6 8 10 0 2 4 6 8 10 12 14 Distance from the sink Number of nodes No. of descendants No. of children (b) Lowest ETX parent selec- tion 0 2 4 6 8 10 0 2 4 6 8 10 12 14 Distance from the sink Number of nodes No. of descendants No. of children (c) Balanced parent selection Figure 4.3: The number of descendants for the first hop level nodes as a function of their distance to the sink under three parent selection schemes with MHR. The maximum energy consumption (max i E i ) determines the initial node death time. Since the topology strategies with the same hop distribution would result in 62 the same network complete disconnection time, the time when all first-hop nodes become energy-depleted, we focus on the initial node death time, which is regarded as the network lifetime. Fig. 4.4 compares the maximum energy consumption of different localized topology construction schemes with MHR when the maximum energy consumption of the lowest ETX without MHR is scaled to 1. The link quality based schemes result in significantly faster initial energy depletion while they provide high link quality. The balanced scheme maintains the initial network operationforlongertimeandtherandomselectionschemehasrelativelylongernet- worklifetime,too. However,thisobservationisobtainedfromasmallnetworkwith few hop levels and nodes. We need more general discussion to analyze the trade- off among various localized topology construction schemes with different network parameters in the following section. 0.7 0.75 0.8 0.85 0.9 0 0.1 0.2 0.3 0.4 0.5 Link threshold Maximum energy consumption ratio Random + MHR Lowest ETX + MHR Balanced + MHR Figure 4.4: The ratio of maximum energy consumption of MHR schemes to the maximum energy consumption of the lowest ETX scheme. 63 4.3 Analysis of Localized Topology Construction Schemes In the last section, we examined the effect of different localized topology schemes on communication loads for one real deployment case. It was observed that the effect of link quality is not significant when MHR has a relatively large number of nodes in the first-hop, since the communication load can be distributed and the energyconsumptionofasinglenodeisreducedaccordingly. However,itisnotclear from this empirical data set whether a lower node density with a small number of nodes in the first-hop produces the same result. In this section, we characterize how diverse network conditions (such as the node density and the network size) affect the energy consumption of each localized topology construction scheme in conjunction with the link threshold. In addition, we examine whether a balanced scheme can always produce longer network lifetime than the link quality based scheme for any network conditions. 4.3.1 Energy Consumption of Localized Schemes To capture the effect of topology construction schemes with respect to the node density and the network size, we examine the communication load of the linear topology as given in Fig. 4.5, where nodes are deployed linearly with equidistance. The one-dimensional (1D) linear topology can be extended to the two-dimensional (2D) network by aggregating multiple linear topology networks to one sink. We will also compare results of 1D and 2D topologies. The average link quality (PRR) is the decreasing function of the distance from the transmitting node as presented in [52]. Following the PRR model in [52], we 64 Figure 4.5: Illustration of the linear topology. Table 4.1: Summary of notation N total number of nodes (network size) r number of nodes in one hop level n c i number of children of node i n d i number of descendants of node i E i energy consumption of node i per round d i distance between k and k+i nodes d r radio range, distance threshold based on link threshold ETX ij ;ETX(d) expected transmission count (ETX) between nodes i and j, and distance d ETX(d r );PRR(d r ) link threshold ETX and PRR adopt the approximate PRR as a function of the distance, whose decreasing rate accelerates along the distance until PRR reaches 0.5. The notation used in this linear topology analysis is summarized in Table 5.2. 4.3.1.1 Lowest ETX Parent Selection For the lowest ETX parent selection scheme as shown in Fig. 4.6, since the link to theclosetneighboringnodeprovidesthelowestETX,eachnodeselectsitsadjacent node that is closer to the sink as the parent node, i.e. the next hop to the sink. Accordingly, eachhopconsistsofonenodeandthemaximumhoplevelis N. Thus, node1,whichisnexttothesink,hasthelargestcommunicationloadtohandledata gathering (argmax i n d i = 1). Thus, the energy consumption of node 1 determines the network lifetime, which is defined to be the initial node death time. 65 Figure 4.6: The lowest ETX parent selection scheme. To analyze the energy consumption, we incorporate the link quality between adjacent nodes of node 1 in the data gathering tree. When every node generates and sends one unit of data to the sink, the expected number of receiving data units from children of node 1 is ETX(d 1 )(N ¡ 1), where ETX(d 1 ) is the number of transmission between nodes that are one-node apart and d 1 is the node distance. The child of node 1 is its adjacent node; i.e., node 2. In addition, the expected number of transmission from node 1 to the sink is ETX(d 1 )N. Thus, the energy consumptionbynode1duringadatagatheringround,whichisnormalizedinterms of the reception energy consumption based on the notation in Eq. (4.1) is equal to max i E i =E 1 =ETX(d 1 )(N¡1)+¯ETX(d 1 )N; (4.2) which is the maximum energy consumption by the lowest ETX parent selection scheme. 4.3.1.2 Random Parent Selection with MHR Figure 4.7: The random parent selection scheme with MHR. 66 The link threshold is used to determine the neighboring nodes that can directly communicateinasingle-hopintheMHRschemes. Thus,eachnodeselectsaparent nodeintheupperhoplevelneighboringnodeswithintheradioranged r ,whered r is the maximum distance from the node that satisfies the link threshold. To calculate the maximum energy consumption for the random parent selection scheme with MHR, we first obtain the expected child number of each node since each node selects a parent node randomly with an equal probability among upper hop level neighboring nodes within the radio range as shown in Fig. 4.7. Since the expected number of children attached to node i can be calculated as P j2C i 1 n p j as shown in Section 4.2.2, the ith node in the first hop level, with 1· i· r, has the expected number of children as E[n c i ]= i X j=1 1 r¡j+1 : The rth node, which is furthest from the sink among the first-hop nodes, has the maximum expected child number (argmax i n c i =r) as P r j=1 1 j . The expected number of descendants of node i can be calculated recursively as i X j=1 (1+n d r+j ) 1 r¡i+1 : The largest expected number of transmission from child nodes to a first-hop node, which is to node r, is max i f rx i =f rx r = r X j=1 ETX(d j )(1+n d r+j ) 1 r¡j+1 : 67 The expected number of transmission to a sink from node r is max i f tx i =f tx r =ETX(d r ) Ã 1+ r X j=1 (1+n d r+j ) 1 r¡j+1 ! : Then, the maximum energy consumption by the random parent selection scheme in a data gathering round can be computed via Eq. (4.1), which is the energy consumption of node r during a data gathering round. 4.3.1.3 Lowest ETX Parent Selection with MHR In the lowest ETX parent selection with MHR, each node selects a parent node that provides the lowest ETX among the upper hop level neighboring nodes. As shown in Fig. 4.8, the node that is closest to the next longer hop level nodes is selected. Thus, the maximum number of descendants is N¡r, which is assoicated with node r, and the maximum number of received data in the lowest ETX with MHR can be calculated as max i f rx i =ETX(d r )(N¡2r)+ r X j=1 ETX(d j ): The expected number of data transmitting to the sink from node r is max i f tx i =ETX(d r )(N¡r+1): The maximum energy consumption in the lowest ETX parent selection can be computed via Eq. (4.1) for node r. 68 Figure 4.8: The lowest ETX parent selection scheme with MHR. 4.3.1.4 Balanced Parent Selection with MHR To achieve the balanced load among nodes in the same hop level, each node selects the furthest neighboring node (i.e. closest to the sink) in the upper hop level within the radio range that satisfies the link threshold. The first-hop nodes have an equally distributed number of descendants from the second-hop level, which is N¡r r . The maximum amount of receiving data from the children is ETX(d r ) N¡r r , andthemaximumtransmittingdatatothesinkisETX(d r ) N r . Thus,themaximum number of data communication is equal among first-hop nodes. The maximum energy consumption by the balanced parent select scheme is max i E i =ETX(d r ) N¡r r +¯ETX(d r ) N r : (4.3) Figure 4.9: Teh balanced parent selection scheme. 69 4.3.2 Comparison of Localized Topology Schemes Based on the obtained maximum energy consumption of each localized topology scheme, we study the effects of the network size (the total number of nodes), the node density and the link threshold on the network lifetime. The network size effect is compared in Fig. 4.10. The number of nodes in a hop level is r = 10 in both figures. Since the energy consumption increasing rate as a functionof thenetworksizeisdifferentfor differenttopologyconstructionschemes, the shortest network lifetime among these schemes changes with the network size. The lowest ETX scheme achieves longer network lifetime than the random selec- tion and the lowest ETX with MHR as the network size increases. Among MHR schemes, the difference of the maximum energy consumption between the balanced scheme and other schemes becomes larger. 20 40 60 80 100 0 100 200 300 400 500 600 700 800 900 1000 Number of nodes (N) Maximum energy consumption Lowest ETX Random+MHR Lowest ETX+MHR Balanced+MHR (a) Link threshold (PRR): 0.5 20 40 60 80 100 0 50 100 150 200 250 300 350 400 450 Number of nodes (N) Maximum energy consumption Lowest ETX Random+MHR Lowest ETX+MHR Balanced+MHR (b) Link threshold (PRR): 0.75 Figure 4.10: The maximum energy consumption as a function of the network size (i.e. the total number of nodes, N, in a network). Fig. 4.11 compares the effect of the node density on the maximum energy consumption. Two link thresholds (expressed in terms of PRR) are presented in this figure and the network size (N) is 20. We compare the minimum hop routing 70 (MHR) schemes and the link quality scheme with respect to the node density. As thenodedensityincreases,theenergyconsumptionofthreeMHRschemesdecreases while that of the lowest ETX scheme without MHR remains almost the same. The random selection scheme with MHR and the lowest ETX with MHR can provide longer lifetime than the lowest ETX as the number of nodes in a hop increases since communication loads can be more evenly distributed among the same hop level nodes. The lowest ETX without MHR can provide longer network lifetime whenboththelinkthresholdandthenodedensityarelow. Whenthelinkthreshold is equal to 0.5 as given in Fig. 4.11 (a), the balanced scheme does not have longer network lifetime than the lowest ETX when the number of nodes in a hop level is less than around 3.5. We will discuss the condition for longer network lifetime of the balanced scheme in the next subsection. The energy consumption result from the empirical data as presented in Fig. 4.4 in Section 4.2.2 is consistent with that of the linear topology with a high link threshold, a high node density and a small network size. Under these conditions, the lowest ETX without MHR has the larger maximum energy consumption as compared to MHR-based topology construction schemes. 4.3.3 Conditions for Longer Lifetime of Balanced Scheme As shown in Fig. 4.11, the balanced scheme with MHR does not always have longer network lifetime than the lowest ETX scheme. This is because the balanced parent selection scheme may select a link of poor quality, which results in more data transmission over the link. This is also related to the node density for a given network size. We would like to determine the number of nodes that share the 71 2 3 4 5 6 7 8 9 10 0 50 100 150 200 250 Number of nodes / hop (r) Maximum energy consumption Lowest ETX Random+MHR Lowest ETX+MHR Balanced+MHR (a) Link threshold (PRR): 0.5 2 3 4 5 6 7 8 9 10 0 10 20 30 40 50 60 70 80 90 100 Number of nodes / hop (r) Maximum energy consumption Lowest ETX Random+MHR Lowest ETX+MHR Balanced+MHR (b) Link threshold (PRR): 0.75 Figure 4.11: The maximum energy consumption as the number of nodes in a hop (r). communication load from the further hop level nodes and the link threshold so as to guarantee longer network lifetime of the balance scheme. To obtain conditions for longer network lifetime of the balanced scheme than the lowest ETX, we compare the maximum energy consumption obtained in Eq. (4.2) and Eq. (4.3). The energy consumption of the balanced scheme should be less than that of the lowest ETX scheme. First, we determine the lower bound of the number of nodes in a hop, r, to ensure longer lifetime of the balanced scheme below: r > 1 1¡ ³ 1¡ 1 N(1+¯) ´³ 1¡ ETX(d 1 ) ETX(dr) ´: (4.4) Weseethatthislowerboundisafunctionofthenetworksize,theportionofenergy consumption for transmission (¯) and the link threshold. The effect of the network sizeand¯ isminorsinceN À1. AsN increases,theincreaseofr quicklysaturates andthegapbetweensmallandlarge N valuesisquitesmall. Forthelinkthreshold effect, r decreases as the link threshold improves. 72 We can also obtain the link threshold that guarantees longer lifetime of the balanced scheme regardless of network size and ¯ that depends on the transmitter power. Theorem1. Thebalancedschemeguaranteesthelongerlifetimeregardlessofother network conditions including network size, transmitter power, if the link threshold PRR(d r ) is greater or equal to q 1 r . Proof. Fora givennetworksize and a node density, the condition for link threshold to achieve longer network lifetime of the balanced scheme can be obtained as PRR(d r )>PRR(d 1 ) s 1 r µ 1¡ r¡1 N(1+¯)¡1 ¶ : (4.5) Basically, the lower bound of the link threshold is determined by node density r in a hop. Since 0 < r¡1 N(1+¯)¡1 < 1 with r ¸ 2, N > r, ¯ ¸ 1 and PRR(d 1 )· 1, the right-hand-side of Eq. (4.5), PRR(d 1 ) r 1 r ³ 1¡ r¡1 N(1+¯)¡1 ´ , is always smaller than q 1 r regardless of other parameters. Corollary 1. A link threshold PRR above 0.7 guarantees the longer lifetime of the balanced parent selection scheme regardless of the network size or node density, or any other parameters. This link threshold lower bound comes from the minimum number of nodes in a hop, r =2. 4.4 Comparison to Global Optimum In this section, we compare the network lifetime performance of localized topology construction schemes and the centralized scheme that uses the global knowledge of 73 the network including the quality of all links. We present a linear programming formulation, which is similar to that in [9]. Here, the main difference is that we incorporate the link quality metric ETX into the energy consumption model. The objective is to find the optimal flow for every directional links to maximize the networklifetime, T, whichcorrespondstothemaximumdatagatheringroundwith all nodes operating with equipped energy. The two main constraints are: the flow conservation constraint and the energy constraint. By the flow conservation constraint, we mean that the outgoing flow fromanode(say, P N j=0 f ij fornodei)isthesameastheaggregateofincomingflow to the same node, P N j=1 f ji , plus the amount of data generated by that node, G i . Theenergyconstraintisthatthetotalenergyconsumedbyanodeisboundedbyits equippedenergycapacity, B i . Wefocusonthecommunicationenergyconsumption andthecalculationthatfollowsEq. (4.1)inSection4.2.2. Thus,wecanincorporate the communication load with the link quality, which is represented by ETX, as max T s:t: P N j=1 f ji +G i = P N j=0 f ij i=1:N ³ P N j=1 f ji ETX ji +¯ P N j=0 f ij ETX ij ´ T ·B i i=1:N f ij ¸0; G i ¸0 i=1:N;j =0:N (4.6) In the above, the sink is represented by node 0 and data generating and forwarding nodes are represented by nodes 1 to N. B i is the battery capacity of node i. All flows on links and the generated data by each node is non-negative. To examine the link threshold effect, we consider an exemplary network with topology shown in Fig. 4.12, which consists of 8 nodes with one sink. There are two link quality values; namely, high PRR (low ETX) and low PRR (high ETX) 74 links, for performance comparison. We fix the high PRR link to be 0.95 while the lowPRRlinkvariesfrom0.6to0.9. Wechoosethebesttopologiesfortwolocalized topology schemes (i.e. the lowest scheme without MHR and the balanced scheme) and compare them with the global optimal value. Figure 4.12: The topology of an exemplary network. Fig. 4.13 gives the optimal flow with two link thresholds, where G i is set to 10 for all i. As shown in this figure, the optimal flow is also distributed to the lower PRR link. The amount of flow over the lower PRR link increases as the link threshold increases. (a) Link threshold (PRR) = 0.6 (b) Link threshold (PRR) = 0.9 Figure 4.13: The optimal flow with link thresholds equal to 0.6 and 0.9. 75 Fig. 4.14 compares the maximum energy consumption and the normalized net- work lifetime in terms of the optimal network lifetime parameterized by the low PRR link value equal to 0.6, 0.7, 0.8 and 0.9. We see that the maximum energy consumption of the optimal flow decreases as the link threshold increases since the optimized scheme balances the flow and load by utilizing low PRR links. For the lowest ETX scheme, it does not use the lower PRR link so that the maximum en- ergy consumption remains the same regardless of the change of the link threshold value. When the link threshold for low PRR links is 0.6, the balanced scheme has significantly higher energy consumption as compared with that of the optimal flow and the lowest ETX. This echoes the result in Section 4.3.3, namely, the balanced scheme does not guarantee longer lifetime when the link threshold is below 0.7. As the link threshold increases, the balanced scheme achieves lower maximum energy consumption. However, the decreasing rate of the maximum energy consumption quickly saturates since ETX is an inverse function of the square of PRR. Fig. 4.14 (b) shows the ratio of two localized schemes to the optimal lifetime. ThenetworklifetimeofthelowestETXschemelinearlydecreasestothenormalized optimalvalueasthelinkthresholdincreases. Forthebalancedscheme, almost90% of the optimal network lifetime is achieved when the link threshold is 0.7 or above. 4.5 Conclusion and Future Work Localizedtopologyconstructionschemeswithempiricaldatawerestudiedandtheir performance was analyzed and compared. The link threshold and the node density are the main factors that affect the energy consumption of each localized scheme and their effect can be quite different based on the parent selection criteria. In the dense node deployment with a high link threshold and a small network size, 76 0.6 0.7 0.8 0.9 0 20 40 60 80 100 120 140 Link threshold (PRR) Maximum energy consumption Optimal Lowest ETX Balanced + MHR (a) Maximum energy consumption 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Link threshold (PRR) Network lifetime ratio Lowest ETX Balanced + MHR (b) Network lifetime ratio Figure 4.14: Comparison of the lowest ETX scheme and the balanced scheme in terms of (a) the maximum energy consumption and (b) the network lifetime nor- malized to the optimal network lifetime. the MHR schemes reduces the energy consumption significantly as compared to schemes that uses the link quality only for parent selection. However, for the opposite network conditions, the lowest ETX scheme can achieve longer network lifetime than MHR schemes. Conditions that guarantee longer network lifetime of the balanced parent selection scheme were derived. In the future, we would like to examine a distributed topology establishment algorithm that incorporates the link quality and load balancing to provide longer network lifetime under dynamic network conditions. In addition, we will examine the optimal link threshold that provides maximum lifetime. 77 Chapter 5 Aging in Heterogeneous Deployment In this chapter, we extend to aging analysis of the multiple sink deployment case with heterogeneous devices. On one hand, a number of devices of higher capability are used to increase the quality and quantity of data processing inside a network with heterogeneous deployment, and they can extend the network lifetime through higherenergy capacity. Onthe otherhand, higher capabilities that include alarger amount of equipped energy, higher processing power and memory capacity and a longer sensing range, will increase the cost of the device. Hence, under prescribed budgetconstraints, increasingthenumberofhigh-costdevicescanreducethenum- beroflow-capacitydevices,whichwillaffectthesensingcoverageperformancesuch as the sensor node density or the coverage area. There exist advantages and disad- vantages of deploying high-cost devices under the total budget constraint. Considering these conditions and trade-offs, we examine the aging process in a heterogeneous deployment with both single-hop communication model and multi- hop communication model, and derive expressions for the heterogeneous device deployment that maximizes the lifetime coverage. Coverage performance would be varied with different mixtures of heterogeneous deployments depending on deploy- ment conditions. In addition, the deployment of high capability nodes affects the 78 energy depletion of other low-cost sensors, which results in the different coverage aging processes,i.e.,howcoverageofanetworkdegradesovertimeasnodesbecome energy-depleted, which is also dependent on the communication modes: single-hop and multi-hop. In order to examine the effect of heterogeneity on the coverage, two metrics are used; namely, the total sensing coverage and the information utility. The total sensing coverage represents both the spatial extent and the degree to which the targetphenomenoncanbeobserved. Thesensingcoverageareaisthespatialextent of the network covered by sensors, which indicates the breadth of sensing coverage. Ontheotherhand,thesensingcoveragedegreemeanstheaveragenumberofsensors covering each point in the area, which implies the depth of sensing coverage and node density. The information utility incorporates spatial correlation among data collected by nodes, where the amount of fresh information collected by nodes are determined based on the correlation in the network field. First, we identify the trade-off in deploying heterogeneous devices in terms of coverage performance and network lifetime. Second, we analyze the lifetime cov- erage under various deployment conditions and communication models and derive expressions for the heterogeneous device deployment that maximizes the lifetime coverage. The results of this analysis help understand the effect of several param- eters including the cost ratio, the sensing range ratio, the network size, and the ratio of energy consumption operations. Finally, the effect of spatial correlation in the network field is examined with information utility coverage metrics. The optimaldeploymentcanenhancethelifetimeinformationutilitysignificantlyinthe low correlation case while the difference is small in the high correlation case. In addition, the result indicates that a proper mixture of heterogeneity based on the 79 Table 5.1: Key findings (n H : no. of high cost devices, n H¡opt : the optimum n H ) Findings Section Single-hop 5.2 Coverage area aging process over time w.r.t. n H shows the trade-off between initial coverage and network lifeitme. 5.2.1 n H¡opt increasessub-linearlyw.r.t. thenetworksize(thefixed area case). 5.2.2 Theeffectofsensingrangeratioonlifetimecoverageisminor. 5.2.2 n H¡opt increases as spatial correlation increases. 5.2.3 n H¡opt inlowcorrelationcanachievehighlifetimeinformation utility in a wide range of spatial correlations. 5.2.3 Multi-hop 5.3 n H significantly affects the lifetime of the network with no data aggregation. 5.3.1 n H¡opt isinverselyproportionaltonon-communicationrelated energy consumption. 5.3.2 n H¡opt decreases as spatial correlation increases. 5.3.3 n H¡opt inlowcorrelationcanachievehighlifetimeinformation utility in a wide range of spatial correlations. 5.3.3 low correlation case can provide high lifetime coverage close to the maximum value under a wide range of correlation conditions. 5.1 System Models Heterogeneous devices and their deployment model are introduced and two cov- erage metrics are discussed in this section. The notation used in the proposed heterogeneous deployment and the coverage model are summarized in Table 5.2. 5.1.1 Heterogeneous Deployment Model Heterogeneous deployment consists of two types of devices: high-cost devices (H) and low-cost devices (L). Each sensor is assumed to collect the information from 80 Table 5.2: Summary of notation for a heterogeneous deployment model H Type H sensor: High cost sensor used as a sink L Type L sensor: Low cost sensor C tot Total cost constraint C L ;C H Cost for type L sensor and for type H sensor n L ;n H Number of type L sensor and of type H sensor n ¤ L Number of type L sensors with one sink deployment (network size) R L ;R H Sensing range radius of type L sensor and of type H sensor E L Initial battery energy of type L sensor S d Sensing coverage degree S a Sensing coverage area S tot Total sensing coverage S life Lifetime sensing coverage U Information utility U life Lifetime information utility T d Network lifetime d max Maximum distance from the closest sink its sensing coverage and transmit it to the closest sink (i.e. the high-cost sensor) at each sampling round, which is used as a time unit. For the energy capacity of type-H devices, it is assumed that H is initially equipped with enough energy that allows H to function as a sink until all low- cost devices deplete their battery energy. The number of low-cost sensors per sink decreases as the number of high-cost devices (n H ) increases. Thus, the energy consumed by a type-H device becomes smaller accordingly. The mixture of devices is constrained by the total budget as C tot ¸n H C H +n L C L ; (5.1) where C H and C L are costs for type-H and type-L devices, respectively. We can express the number of low-cost devices with the cost ratio, the number of high-cost 81 devices and the maximum number of low-cost devices when one sink is deployed, which is denoted by n ¤ L , as n L =n ¤ L ¡ C H C L (n H ¡1); (5.2) where the total budget constraint C tot is related with the maximum number of low-cost devices n ¤ L via C tot =C H +n ¤ L C L : (5.3) 5.1.2 Coverage Evaluation Models Themetricsofthetotalsensingcoverageandinformationutilityareusedtoevaluate the impact of the heterogeneous deployment on the lifetime coverage. They are detailed below. 5.1.2.1 Total Sensing Coverage We will evaluate the network performance with the total amount of information that can be extracted from the network field. It is represented by the total sensing coverage (S tot ), which is the sum of the sensing coverage of each live sensor. It includes two types of information: 1. the sensing coverage area (S a ): how large an area can be covered with a fixed node density deployment, and 2. thesensingcoveragedegree(S d ): howmorefine-grainedandmoreinformation can be extracted from the fixed area. The sensing degree can be expressed with the number of sensors in a unit sensing coverage area. It represents the density of sensor nodes as well as the reliability of 82 monitored data from sensors in a certain area. For example, if two deployments have the same size of covered area but have different coverage degrees, the higher coverage degree deployment can extract more fine-grained spatial information from the field. We consider both the fixed coverage area and the fixed coverage degree deploy- ment cases in the single-hop communication analysis as shown in Fig. 5.1 (a) and (b), respectively. In this figure, the one sink deployment and the four sink deploy- ment are compared. In Fig. 5.1 (a), the targeting network area S a to cover is fixed regardless of the number of sinks deployed. Thus, as the number of sinks increases, the density of low-cost nodes becomes lower, which decreases the coverage degree. In Fig. 5.1 (b), the density of low-cost nodes remains the same as the number of sinks increases while the covered network area decreases. The average maximum distance of a node from the nearest sink is denoted as d max . Two deployment cases will result in different life times of nodes, which will be discussed later. (a) fixed area deployment (b) fixed coverage degree deployment Figure 5.1: Heterogeneous deployments with single-hop direct communication: (a) the fixed area deployment and (b) the fixed coverage degree deployment. 83 Mathematically, S tot can be expressed by the number of high cost devices and the sensing ranges of each type of device as S tot (n H ) = ¼ ³ ¡(R 2 L C H C L ¡R 2 H )n H +( C H C L +n ¤ L )R 2 L ´ (5.4) = ¼(¡® 1 n H +® 2 ); (5.5) where R H and R L are the sensing range radii of H and L, respectively, and ® 1 =R 2 L C H C L ¡R 2 H and ® 2 =( C H C L +N ¤ L )R 2 L (5.6) are parameters introduced for notational simplicity. Parameter ® 1 determines the impact of the number of H devices on S tot . Here, we will focus on the case ® 1 > 0 (or, equivalently, C H =C L > (R H =R L ) 2 ) only, where the total sensing coverage decreases as n H increases. Please note that, if ® 1 <0, we are led to the trivial case where all type-L sensors should be replaced by type-H sensors in order to increase S tot . Inthecaseoffixedcoveragedegreedeployment, thechangeofthecoveragearea due to varying n H can be computed as S a (n H )=S tot (n H )=S d ; whichisderivedbasedonadifferentnumberofnodesdeployedinthefieldbutwith the same node density. 5.1.2.2 Information Utility The other metric for sensing coverage is developed by studying spatial correlation in a network field, which can be utilized for data aggregation. For this purpose, 84 the following two characteristics in the information utility function should be con- sidered. 1. As the number of nodes increases, the information utility function increases. However, for the fixed area deployment, the information utility function will saturate with respect to the number of nodes since the closer the distance among nodes, the lower the amount of freshness in the information collected by each node due to higher correlation. 2. With a given number of nodes, as the distance among nodes increases, the information utility function increases. Since the coverage area increases as the distance between neighboring nodes increases, the information collected by each sensor contains more uncorrelated data from its neighboring nodes. To capture these characteristics of the information utility function, we adopt the approximate expression for the total amount of uncorrelated data as presented in [39]. The total amount of uncorrelated data generated by a set of n nodes is expressed as H n = ¡ 1+ n¡1 1+c=d ¢ H 1 ; where d is the distance between neighboring nodes, c is the spatial correlation parameter and H is the entropy of the data from a source [39]. When d = c, the new information at that location is equal to one half of the information gathered at thenode. Alargervalueofcindicateshigherdatacorrelationatthesamedistance. With the budget constraint in Eq. (5.2), the total number of nodes deployed in a network field, which is the function of n H , can be expressed as n(n H )=¡ µ C H C L ¡1 ¶ n H +n ¤ L + C H C L =® 3 ¡® 4 n H ; (5.7) 85 where ® 3 =C H =C L +n ¤ L and ® 4 =C H =C L ¡1 (5.8) are parameters used for notational simplicity. For the fixed coverage area deployment, the approximate distance between neighboring nodes can be expressed as d(n H )= r S a ® 3 ¡® 4 n H : (5.9) Thus, the information utility function, which is the total amount of fresh informa- tion generated by a set of n(n H ) nodes, can be expressed as U(n H ) = 0 @ 1+ ® 3 ¡® 4 n H ¡1 1+c q ® 3 ¡® 4 n H S a 1 A H 1 (5.10) ¼ 0 B B @ C H C L +n ¤ L ¡( C H C L ¡1)n H 1+c r C H C L +n ¤ L ¡( C H C L ¡1)n H S a 1 C C A H 1 ; (5.11) where the approximation is valid under the assumption of n(n H )À1 in large-scale networks. Forthefixedcoveragedegreedeployment,wherenodedensityremainsthesame, the coverage area decreases as n H increases. In this case, the distance among neighboring nodes remains the same as the initial approximate distance between neighboring nodes when n H =1, which is d ¤ = q S a =n ¤ L ; 86 where S a is the initial coverage area with one sink deployment. Then, the information utility function with respect to n H is U(n H )= Ã C H C L +n ¤ L ¡( C H C L ¡1)n H 1+cd ¤ ! H 1 : (5.12) Since the coverage area is maintained as n H increases in the case of the fixed area deployment, the fixed area deployment can obtain higher information utility than the case of the fixed coverage degree. 5.2 CoverageAgingAnalysiswithSingle-hopDirect Communication We examine how a different mixture of heterogeneity affects the coverage aging process in this section. Basically, we evaluate the energy consumption of a low-cost device L at each sampling round as E round (d)=E 1 +E 2 d · ; (5.13) where E 1 consists of the energy required for data processing, sensing and commu- nication (radio electronics) that are independent of the node’s radio range to the sink, E 2 is the amplifier energy that is related to the distance from the sink, and · is the path-loss exponent. One data unit is generated and sent to a sink. The energy operation ratio (E 2 =E 1 ) would be dependent on the application. 87 5.2.1 Coverage Area Aging and Its Initial Time Fortheagingphenomenon,S tot decreasesafterthefirstnodedeath,whichislocated atthefurthestdistancefromasink. Theaveragemaximumdistanceofanodefrom the nearest sink can be obtained as d max (n H )= s S a (n H ) ¼n H : (5.14) Theenergydepletiontimeofanodeat d max (n H ), denotedbyT d (d max (n H )), canbe obtained from Eqs. (5.5), (5.13) and (5.14) by computing the energy consumption per sampling round as T d (d max (n H )) = E L E 1 +E 2 d 2 max (n H ) (5.15) = S d E L n H (S d E 1 ¡® 1 E 2 )n H +® 2 E 2 ; (5.16) whereE L istheinitialbatterycapacityofatype-Lsensorandpathlossexponentis assumedtobe2. Higherpathlossexponentmakestheenergydepletiontimeofthe furthest node faster with the given n H . Thus, more number of sinks deployment would be desirable in terms of network lifetime. However, the decrease of sensing coverage diminishes the above benefit of larger n H . In addition, E 1 value is much largerthanE 2 valueinmostcases. Thus,theeffectofpathlossexponentonlifetime sensing coverage is minor. The network aging process is described by a coverage area contracting process. Thatis,thecoverageareashrinksalongtimefromthenodeatthefurthestdistance 88 from a sink. From Eq. (5.15), the coverage area aging process with a different number of high-cost device deployments can be expressed as S a (n H ;t)= 8 > < > : S a (n H ); if t<T d (d max (n H )); n H ¼ E 2 ¡ E L t ¡E 1 ¢ ; if T d (d max (n H ))·t·T d (d=0): (5.17) In the case of a fixed area deployment where the initial covered area (S a ) is maintained while node density is changed with respect to n H , we can obtain T d (d max (n H ))= ¼E L n H ¼E 1 n H +S a E 2 (5.18) The coverage aging analysis with fixed coverage area follows Eq. (5.17) except that S a (n H ) is replaced with S a when the time is less than T d (d max (n H )). Fig. 5.2 shows the aging process of the total sensing coverage (S tot ) over time under analysis with a different mixture of heterogeneity. Since the coverage degree in the live covered area is preserved uniformly over time, S tot aging follows S a aging in Eq. (5.17). For this figure, we set C H =C L = 15, n ¤ L = 500, E L = 2000, E 2 =E 1 = 0:005, R H = 4 and R L = 2. For the fixed area deployment, S a = 2500, and for the fixed degree deployment, S d =2:5 and the initial covered network area with one sink deployment is 2500. Since we only consider positive ® 1 as discussed with Eq. (5.6), a network with more n H has a smaller total sensing coverage before aging starts, while the sensing operation duration - coverage aging initiation time - isenhanced. Bothfixedareaandfixeddegreecaseshowsimilaragingphenomenon. The coverage aging initiation time of the fixed degree case, T d (d max ), is prolonged more as the number of sinks increases than the fixed area case, which is due to the smaller network area covered by nodes as the number of sinks increases in the case of fixed degree. 89 (a) fixed area deployment (b) fixed coverage degree deployment Figure5.2: Theanalyzedagingprocessofthetotalsensingcoverage, S tot ,overtime with respect to the number of sinks. 5.2.2 LifetimeSensingCoverageandOptimumNo. ofSinks Thelifetimesensingcoverage,S life ,withdifferentn H canbeobtainedbyintegrating S tot over the network lifetime as S life (n H )= Z T d (n H ) t=0 S tot (n H ;t) dt (5.19) We present the optimum number of sinks that maximizes the lifetime sensing cov- erage, S life , in both the fixed degree and the fixed area deployment cases in this section. Inthe case of fixed coveragedegree deployment, S life untilT d (d max (n H ))can be expressed from Eq. (5.16) as S life (n H )= S d E L (® 2 n H ¡® 1 n 2 H ) (S d E 1 ¡® 1 E 2 )n H +® 2 E 2 : (5.20) 90 Then, the optimum number of sinks, n H¡opt , that maximizes the total lifetime sensing coverage can be calculated by setting the derivative of Eq. (5.20) with respect to n H to zero. It is found that n H¡opt = ( C H C L +n ¤ L )R 2 L S d E 1 E 2 ¡ C H C L R 2 L +R 2 H 0 @ v u u t S d E 1 E 2 C H C L R 2 L ¡R 2 H ¡1 1 A ; (5.21) where ® 1 and ® 2 in (5.20) are replaced by device costs and sensing ranges of each type of devices as given by Eq. (5.6). In the case of fixed coverage area deployment, we can obtain S life from Eq. (5.18) as S life (n H )= ¼ 2 E L (® 2 n H ¡® 1 n 2 H ) ¼E 1 n H +S a E 2 : (5.22) The optimum number of sinks (n H¡opt ) that maximizes the total lifetime sensing coverage is given by n H¡opt = v u u u t ³ S a E 2 ¼E 1 ´ 2 +(R 2 H +n ¤ L R 2 L ) S a E 2 ¼E 1 C H C L R 2 L ¡R 2 H + S a E 2 ¼E 1 ¡ S a E 2 ¼E 1 : (5.23) The analyzed lifetime sensing coverage S life with respect to n H in the case of fixed area deployment is shown in Fig. 5.3. The parameter setting is the same as that in Fig. 5.2. We compare two lifetimes (time up to the first node death and time up to all nodes death) in this figure to see the coverage aging effect. It is observedthattheoptimummixtureofheterogeneitycanobtain significantincrease of the lifetime sensing coverage. Since the aging process starts earlier with smaller n H due to the long d max , a larger difference between two lifetimes can be observed in the case of smaller n H . However, it is shown that the optimum number of sinks that maximizes the lifetime sensing coverage for the two lifetimes is quite similar. 91 In addition, we see that the impact of the heterogeneous deployment saturates as n H increases. This is because the decrease of initial total sensing coverage, as n H increases, offsets the advantage of the network duration enhancement. Figure5.3: Theanalyzedlifetimesensingcoverage,S life ,withrespecttothenumber ofsinks, wheretwolifetimesareconsidered(timeto thefirstnode’sdeathandtime to all nodes’ death). The impact of n ¤ L , the initial network size, on the optimum number of sinks in the heterogeneous deployment is shown in Fig. 5.4. In the case of the fixed degree deployment,n H¡opt increaseslinearlywithrespectton ¤ L . However,n H¡opt increases sub-linearlywithrespecttothenetworksizeinthefixedareadeploymentcase. The impact of n ¤ L on the optimal heterogeneous deployment is more significant in the fixed degree case than in the fixed area case. This is because that the network lifetime enhancement with the increased number of sinks in the case of the fixed area deployment is smaller than that in the case of the fixed degree deployment. Fig. 5.5 shows how the cost ratio and the sensing range ratio of heterogeneous devices can affect the optimal heterogeneous deployment presented in Eq. (5.21). AsdiscussedwithEq. (5.4)inSection5.1.2,as® 1 approachestozerofromapositive value, i.e., C H =C L approaches to (R H =R L ) 2 , n H¡opt sharply rises. If C H =C L < 92 0 400 800 1200 1600 2000 0 10 20 30 40 50 60 70 Initial network size (n*L) Optimum number of sinks (nH−opt) Fixed degree Fixed area Figure 5.4: The optimum number of sinks with respect to the initial number of low-cost devices. (R H =R L ) 2 , all low cost devices should be replaced with the high cost devices to maximize the lifetime sensing coverage. This is because the decreasing rate of S tot with respect to n H approaches zero while T d (d max ) increases. Hence, as the cost ratio decreases and the sensing range ratio increases, it is desirable to deploy a larger number of H devices in order to increase the lifetime sensing coverage. However, unless (R H =R L ) 2 is close to C H =C L , the effect of the sensing range ratio on heterogeneous deployments is minor as shown in this figure. The impact of the cost ratio and the sensing range ratio on n H¡opt are similar in both the fixed area and the fixed degree deployments. 5.2.3 Lifetime Information Utility The lifetime information utility, U life , with different n H can be obtained by inte- grating information utility (U) over the lifetime of sensors. The information utility results given in Eq. (5.10) and Eq. (5.12) in Section 5.1.2.2 and the network life- timeresultsgiveninEq. (5.16)andEq. (5.18)canbeusedtocompute U life forthe fixed degree and the fixed area cases, respectively. Since each node is assumed to 93 4 8 12 16 20 1 1.2 1.4 1.6 1.8 0 10 20 30 40 50 RH/RL CH/CL Optimum number of sinks (nH−opt) Figure 5.5: The optimum number of sinks with respect to the cost ratio (C H =C L ) and the sensing range ratio (R H =R L ). send a data unit per sampling round directly to a sink, the energy depletion time for the information utility metric case is the same as that for the sensing coverage metric. For the fixed degree deployment, the lifetime information utility is computed as U life (n H )= S d E L (® 3 n H ¡® 4 n 2 H ) ³ (S d E 1 ¡® 1 E 2 )n H +® 2 E 2 ´³ 1+cd ¤ ´; (5.24) where ® 3 = C H =C L +n ¤ L and ® 4 = C H =C L ¡1 from Section 5.1.2.2. Then, the optimum number of sinks, n H¡opt , that maximizes the lifetime information utility can be obtained by setting the derivative of Eq. (5.24) with respect to n H to zero as n H¡opt = ( C H C L +n ¤ L )R 2 L S d E 1 E 2 ¡ C H C L R 2 L +R 2 H 0 @ v u u t S d E 1 E 2 +R 2 H ¡R 2 L ( C H C L ¡1)R 2 L ¡1 1 A ; (5.25) 94 whichisindependentofspatialcorrelationparameterc. Inthecaseoffixedcoverage area deployment, the lifetime information utility can be found as U life (n H )= ¼E L (® 3 n H ¡® 4 n 2 H ) ³ ¼E 1 n H +S a E 2 ´³ 1+c q ® 3 ¡® 4 n H S a ´: (5.26) It is observed that, in the case of fixed coverage degree deployment, the ob- tained n H¡opt is similar to that in the lifetime sensing coverage (S life ) metric since the different n H value does not affect the the node density. This means that the correlation does not affect the optimal mixture that maximizes the lifetime infor- mation utility. On the other hand, for the fixed area deployment, the deployment of a larger number of high cost devices helps maximize the lifetime information utility in a highly correlated network as shown in Fig. 5.6 as compared with the lifetime sensing coverage metric. This can be explained by the spatial correlation effect since a sparser node density with more n H does not affect the information utility much, especially in a highly correlated network while the network lifetime can be enhanced with increasing n H . For this figure, the same parameter values as in the lifetime sensing coverage metric case in Fig. 5.3 are used. The second observation is that, as the correlation increases, the lifetime infor- mationutilityplotbecomesflat, whichmeansthedifferencebetweenthemaximum value with n H¡opt and the other n H deployments become smaller. Thus, the n H¡opt value obtained with a low correlation value can provide high lifetime information utility close to the maximum value for networks with a wide range of correlation values. 95 Figure 5.6: The lifetime information utility with respect to the number of sinks with various correlation parameters (c=0.1, 1, 5, 10). Fixed area deployment case. 5.3 Coverage Aging Analysis with Multi-hop Communication Inamulti-hopheterogeneousnetwork,allnodesparticipateinconstructingthedata gathering trees rooted at the nearest high-cost device (H), which is functioning as a sink. In this section, we examine the coverage aging phenomenon; namely, how coverage degrades over time as nodes death occurs, through simulation. Then, analysis results of the heterogeneous deployment focused on the maximum lifetime sensing information with two coverage metrics, sensing coverage and information utility, are presented. 5.3.1 Coverage Aging with Simulation Results First, we present simulation results of the lifetime sensing coverage and sensing coverage aging in a multi-hop communication network for periodic data gathering 96 applications. In this simulation, all functioning and connected nodes process the samplinginformationfromtheirownsensingcoverageandforwardthedatatotheir sink for the data gathering purpose at each round. Twoextremedataaggregationmodes(withoutaggregationandwithperfectag- gregation)areusedinthesimulation. Whenperfectdataaggregationisperformed, we assume that all the data from child nodes at each round can be aggregated into one data packet with its own data (e.g., MIN, MAX, SUM, COUNT). The effect of an increasing number of high-cost devices on the lifetime coverage and aging behavior is examined with and without data aggregation. Figure 5.7: Illustration of the multi-hop network model with n H =4 in simulation. In the simulation, we deploy 200 nodes for n ¤ L and set 10 for C H =C L . Type- H and type-L nodes are uniformly and randomly placed in the unit square area. Three sensing range ratios of a high-cost device to a low cost device are considered; namely, R H =R L = 1;1:5;2. The sensing range of the low-cost device L is set to one half of its radio range. Parameter n H increases from 1 to 13 and n L is calculated by Eq. (5.2). 30 different random deployments for each n H , R H =R L ratio, and with/without data aggregation cases are performed and averaged for each simulation result, respectively. 97 In order to monitor the coverage area and the coverage degree by randomly deployedsensorsovertime, 400gridpoints(20by20)areplacedinthisunitsquare area, which allows one low-cost sensor to cover 7 grid points on average. The nearest node in the upper hop level toward the sink is selected for the parent of each node, which follows the data gathering construction method in [51]. Nodes searchforanewparentwhenthecurrentparentdiesorbecomesdisconnectedfrom the data gathering tree. If the node fails to find a new parent, then the node gets disconnected from the tree and is regarded as a non-functioning node even though sufficient energy still remains. 5.3.1.1 Total Sensing Coverage Aging (S tot ) We examine the total sensing coverage aging over time with respect to the number of sinks with and without perfect data aggregation when R H =R L = 2 in Fig. 5.8. Both cases show that a smaller n H provides a higher total sensing coverage in the beginning stage of the network lifetime. As nodes become energy-depleted, the coverage starts to degrade over time. A smaller n H achieves a higher initial total sensing coverage since a larger value of n L is available under the total cost con- straint. However, in the case of no data aggregation, the homogeneous deployment with one sink shows a drastic decrease of coverage in a quite early stage as com- paredtothedeploymentofmultiplesinksasshowninFig. 5.8(a). As n H increases, the network maintains the initial coverage for a longer period of time. When data aggregation is available as shown in Fig. 5.8(b) , the initial coverage is preserved for a much longer period as compared to the case without data aggregation. If all nodes can perform perfect data aggregation, the number of packets transmitted to asinkcanbedrasticallyreduced, whichenhancesthenetworklifetimesignificantly. Inaddition,exceptfortheone-sinkdeployment,themultiple-sinkdeploymentshow 98 a similar total sensing coverage and aging pattern from the middle of the network lifetime, which indicates the gain from increasing of n H saturates. 0 100 200 300 400 500 600 700 800 900 1000 0 500 1000 1500 Time (Round) Total sensing coverage (Stot) nH=1 3 5 7 9 11 13 (a) without data aggregation 0 100 200 300 400 500 600 700 800 900 1000 0 500 1000 1500 Time (Round) Total sensing coverage (Stot) nH=1 3 5 7 9 11 13 (b) with data aggregation Figure 5.8: The total sensing coverage (S tot ) aging over time with respect to the number of sinks (n H ) for multi-hop networks. 5.3.1.2 Coverage Area Loss Rate Fig. 5.9showsthecoveragearealossratewithrespecttothenumberofsinks,which indicatesthenumberofgridpointslossperroundduetonodes’death. Weseefrom the figure that the coverage area loss rates of different sensing range ratios show little difference. Furthermore, we observe a drastic difference of coverage loss rates between one-sink deployment and the multiple-sink deployments in Fig. 5.9(a). As the number of sinks increases, the difference of coverage loss rate decreases. Since themaximumhopdistanceisonlyslightlyreducedasthenumberofsinksincreases, the effect of more sinks on the coverage aging becomes less significant. In contrast, thecoveragearealossratesaresimilarfordifferentn H valuesinnetworkswithdata aggregation. It should be noticed that the y-axis scale of Fig. 5.9(b) is much finer than that of Fig. 5.9(a). This phenomenon can be explained as follows. Since the 99 data communication amount is independent of the cluster size in the perfect data aggregation case, a shorter hop distance resulted by a larger value of n H does not reduce energy consumption as much as that without data aggregation. 1 3 5 7 9 11 13 0 1 2 3 4 5 6 7 8 9 Number of sinks (nH) Coverage area loss rate RH/RL=2 RH/RL=1.5 RH/RL=1 (a) without data aggregation 1 3 5 7 9 11 13 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Number of sinks (nH) Coverage area loss rate RH/RL=2 RH/RL=1.5 RH/RL=1 (b) with data aggregation Figure 5.9: The coverage area loss rate (the averaged grid point loss per round) under a given number of sinks (n H ). 5.3.1.3 Sensing Coverage Degree Aging Fig. 5.10 shows the average sensing coverage degree in the covered area as nodes becomeenergydepletedovertime, wherethetimeissetto1, 400and800sampling rounds. It is measured by the average number of live sensors that cover a grid point, whichiscoveredbyatleastonesensor. Theinitialcoveragedegreedecreases linearly as n H increases since a smaller number of low-cost sensors are deployed in a fixed network area. Deployments with a smaller value of n H have a drastic decrease of the coverage degree over time in the covered area, while deployments with a larger number of high-cost devices can maintain a broader coverage area over time as well as much slower coverage degree loss rate over time. 100 1 3 5 7 9 11 13 1 1.5 2 2.5 3 3.5 4 Number of sinks (nH) Mean coverage degree (Sd) Time=1, RH/RL=2 Time=1, RH/RL=1.5 Time=1, RH/RL=1 Time=400, RH/RL=2 Time=400, RH/RL=1.5 Time=400, RH/RL=1 Time=800, RH/RL=2 Time=800, RH/RL=1.5 Time=800, RH/RL=1 (a) without data aggregation 1 3 5 7 9 11 13 1 1.5 2 2.5 3 3.5 4 Number of sinks (nH) Mean coverage degree (Sd) Time=1, RH/RL=2 Time=1, RH/RL=1.5 Time=1, RH/RL=1 Time=400, RH/RL=2 Time=400, RH/RL=1.5 Time=400, RH/RL=1 Time=800, RH/RL=2 Time=800, RH/RL=1.5 Time=800, RH/RL=1 (b) with data aggregation Figure 5.10: The sensing coverage degree (S d ) aging over time with respect to the number of sinks (n H ). 5.3.1.4 Lifetime Sensing Coverage (S life ) Fig. 5.11showsthelifetimesensingcoveragethroughoutthenetworklifetime(with time set to 400 and 800 sampling rounds) with respect to the number of sinks. We see that the effect of the sensing range ratio becomes more significant as the network ages. As shown in Fig. 5.11(a), the heterogeneous deployment with 11 sinks achieves about 6 times as much lifetime sensing coverage as the homogeneous deployment with one sink can obtain until the time is equal to 800 rounds. If n H goes beyond that point, the increase of n H does not gain more lifetime sensing coverage since the initial coverage becomes smaller. In contrast, as illustrated in Fig. 5.11(b), the lifetime sensing coverage with data aggregation decreases as n H increases when the time is equal to 400 rounds. A smaller initial coverage (especially the coverage degree) reduces the gain of increasing n H for a network withperfectdataaggregation. Thus,whenweconsiderheterogeneousdeployments, we should take into account the degree of data aggregation that can be achieved by an application during data gathering. 101 1 3 5 7 9 11 13 0 1 2 3 4 5 6 x 10 5 Number of sinks (nH) Lifetime sensing coverage information (Slife) Time=800, RH/RL=2 Time=800, RH/RL=1.5 Time=800, RH/RL=1 Time=400, RH/RL=2 Time=400, RH/RL=1.5 Time=400, RH/RL=1 (a) without data aggregation 1 3 5 7 9 11 13 0 1 2 3 4 5 6 7 8 x 10 5 Number of sinks (nH) Lifetime sensing coverage information (Slife) Time=800, RH/RL=2 Time=800, RH/RL=1.5 Time=800, RH/RL=1 Time=400, RH/RL=2 Time=400, RH/RL=1.5 Time=400, RH/RL=1 (b) with data aggregation Figure 5.11: The lifetime sensing coverage (S life ) with respect to the number of sinks (n H ). 5.3.2 LifetimeSensingCoverageandOptimumNo. ofSinks To examine the optimal mixture of heterogeneous deployment, we analyze the net- work lifetime and the lifetime sensing coverage in a multi-hop data gathering tree. Similar to the network lifetime (T d (n H )) computation with a single-hop communi- cation model in Section 5.2.1, the lifetime is obtained from the battery capacity divided by the energy consumption per sampling round (which is used as the time unit) as E L =E round (n H ). Since the network lifetime is determined by the first-hop nodes that have the largest workload in the network, the energy consumption of the first-hop nodes is used to compute the lifetime. The workload of the first-hop nodes with a different number of n H is dependent on the amount of data collected by each sink, which is an expensive device. Thus, energyconsumptionofthefirst-hopnodesisproportionalton(n H )=n H ,thenumber of sensors in each cluster, since the first-hop nodes should receive and forward data 102 of nodes in further hop areas. The average energy consumption of the first-hop nodes per round can be computed as E round (n H )=E 1 +E 2 n(n H ) n H d(n H ) · +E 3 n(n H ) n H : (5.27) wherepath-lossexponent·issetto2,E 1 thenon-communicationenergyforsensing andprocessingoperations,E 2 thecommunicationenergyfortheamplifyingportion and E 3 the communication energy required for distance-independent operations all for one sampling round. The radio range is assumed to be proportional to the approximate distance between neighboring nodes d(n H ), which follows Eq. (5.9). The lifetime sensing coverage, S life , can be obtained by Eq. (5.19) with S tot given in Eq. (5.4). Thus, S life can be expressed as S life = E L ¼(® 2 n H ¡® 1 n 2 H ) (E 1 +E 3 ® 4 )n H +E 2 S a +E 3 ® 3 : (5.28) Furthermore, theoptimumnumberofsinks, n H¡opt , thatmaximizesthelifetime sensing coverage can be calculated by setting the derivative of Eq. (5.28) with respect to n H to zero as n H¡opt = E 2 S a +® 3 E 3 E 1 +® 4 E 3 Ã r 1+ ® 2 ® 1 ³ E 1 +® 4 E 3 E 2 S a +® 3 E 3 ´ ¡1 ! ; (5.29) where ® 1 and ® 2 are given in Eq. (5.6), and ® 3 and ® 4 are in Eq. (5.8), which include the cost ratio (C H =C L ), the sensing range ratio (R H =R L ) and the initial network size (n ¤ L ). From the above derivation, we observe that n H¡opt is inversely proportional to E 1 as illustrated in Fig. 5.12(a) where the x-axis represents the portion of non- communication energy consumption when E 3 = 1. The other network parameters 103 follow the plots with the single-hop communication model in Section 5.2.2. This figure shows that, as the energy consumption for non-communication operations increases, which depends on specific applications, the amount of high cost devices should be reduced accordingly so as to obtain the maximum lifetime sensing cover- age. In addition, the effect of the initial network size (n ¤ L ) can be examined when ® 2 and ® 3 are replaced with the original expression. We see from Fig. 5.12(b) that n H¡opt increases linearly with n ¤ L . 0 100 200 300 400 500 4 6 8 10 12 14 16 18 20 Optimum number of sinks (nH−opt) Non−communication energy consumption portion (E1) (a) Effect of energy consumption ratio 0 500 1000 1500 2000 0 5 10 15 20 25 30 35 40 45 Optimum number of sinks (nH−opt) Network size (nL*) (b) Effect of network size Figure 5.12: The optimum number of sinks with respect to non-communication energy consumption (E1) and the initial network size (n ¤ L ). 5.3.3 Lifetime Information Utility The effect of multi-hop heterogeneous deployment with spatial correlation of the sensing data is examined by the lifetime information utility metric. It is assumed that each node performs data aggregation during data forwarding from its children toward the sink based on the correlation, which produces the approximate joint entropy discussed in Section 5.1.2.2. Since the data are compressed during data gathering toward the nearest sink, the amount of data handled by the first-hop 104 nodes is proportional to U(n H )=n H and, accordingly, the energy consumption per round is given by E round (n H )=E 1 +E 2 U(n H ) n H d(n H ) · +E 3 U(n H ) n H : (5.30) Using the network lifetime in Eq. (5.30) and the information utility in Eq. (5.10), the lifetime information utility can be expressed as U life (n H )= E L (® 3 n H ¡® 4 n 2 H ) E 3 + E 2 S a n H +E 1 (1+c q (® 3 ¡® 4 n H ) S a ) : (5.31) Fig. 5.13 shows the lifetime information utility with respect to the number of sinkswithvariouscorrelationparameters. Thisillustratestheexistenceofthemax- imum lifetime information utility and the effect of spatial correlation. In the case of low spatial correlation, we observe that the optimum heterogeneous deployment can achieve a significant increase in the lifetime information utility as compared with those of other mixture heterogeneity. However, as the correlation increases, the difference in the lifetime information utility among different n H deployments becomessmaller. Eventhoughasmallernumberofsinkscanachievethemaximum U life in highly correlated networks, the difference from n H¡opt with low correlation isquitesmall. Thisresultindicatesthat n H¡opt deploymentbasedon lowercorrela- tion can provide the lifetime information utility quite close to the maximum value even for networks with high non-uniform or time-varying correlation. The small value of n H¡opt in highly correlated networks can be justified by Fig. 5.14. The high spatial correlation will increase the amount of data aggregation be- tween neighboring nodes along the data gathering tree, which significantly reduces the communication load in the first-hop nodes even with a small amount of sink 105 Figure 5.13: The lifetime information utility with respect to the number of sinks with various correlation parameters (c=0.1, 1, 5, 10). deployment and the network lifetime gain saturates faster with respect to n H as spatial correlation increases. 5.4 Conclusion The trade-offs in deploying high-cost devices under a total budget constraint were examined in this research. High-cost devices can function as a cluster-head or sink to collect and process the data from low-cost sensors and this heterogeneous deployment can enhance the duration of the network sensing operation. However, the deployment of multiple high-cost devices can reduce the number of low-cost sensors under a budget constraint, which leads to the decrease of the initial sensing coverage either in terms of coverage area or coverage degree. The expressions of the optimum number of high-cost devices, which maximizes the lifetime coverage, in the single-hop communication model and the multi-hop 106 Figure 5.14: The network lifetime with respect to the number of sinks with various correlation parameters (c=0.1, 1, 20). communication model were also derived by incorporating several factors that affect the initial sensing coverage and the energy consumption of nodes. As far as the sensing range ratio is concerned, unless (R H =R L ) 2 is close to C H =C L , the sensing range ratio effect is not significant. An optimal heterogeneous deployment can achieve a lifetime coverage that is several times as much as that with homogeneous deployment by considering the initial coverage and the sensing operation duration. Thecoverageagingpatternswithtwocommunicationmodelsanddifferentratios of heterogeneity were examined. For the single-hop communication model, the initialcoveragedegreecanbepreservedinthecoveredareaasnodesbecomeenergy- depleted over time since the communication energy consumption is not dependent on neighboring nodes, but on the distance from a sink. As n H increases, a faster aging rate in the single-hop communication model and a slower aging rate in the multi-hop communication model were observed. This is due to the high variance of workload among nodes. From the perspective of re-deployment, a uniform faster 107 aging rate would be preferred since the re-deployment can be done at once for the entire area. The effect of spatial correlation on the heterogeneous deployment was evalu- ated with the information utility coverage metric. In the case of the single-hop communication model, as the correlation increases, n H¡opt increases. On the other hand, n H¡opt decreases in the case of the multi-hop communication model owing to the increase of data aggregation in a highly correlated network, which significantly reduces the communication load in the first-hop nodes even with a small amountof sink deployment. In both communication models, the optimum mixture of hetero- geneitycansignificantlyincreasethelifetimecoverageespeciallyinalowcorrelated network, and this mixture can also provide the high lifetime coverage close to the maximum value for highly correlated networks. 108 Chapter 6 Conclusion and Future Work 6.1 Conclusion The aging process, which goes beyond the initial node death, in a multi-hop data gathering tree as well as with various heterogeneous deployment schemes was stud- ied extensively in this work. In addition, we examine the impact of localized topol- ogy construction schemes on network lifetime. Several important results were presented in Chapter 3. They are summarized below. ² For the hop-level analysis in a sparse node deployment, three conditions of node death; namely, energy depletion without data aggregation, with data aggregation and due to device failure, were examined for the connectivity ag- ing per hop level over time. The existence of multiple alternate paths toward the sink leads to a power law relation, where the probability of connection to the sink decreases in proportion to the hop level with an exponent, as node death occurs. This relation holds in the device failure case as well as in the late stage of the energy depletion case with data aggregation. However, there 109 existsasignificantdifferenceintheconnectivitylossandthehopconnectivity exponent between them in a given portion of node death. ² By incorporating dynamic data gathering tree re-construction, aging analy- sis in densely deployed networks enables fast and accurate prediction of the energy consumption distribution and the residual energy distribution in a large-scale network over time after initial node death. Besides theoretical analysis, we provided simulation results using a realistic wireless link model presented by [52] and demonstrated a good match between analytical and simulation results. was demonstrated. In addition, it was shown by the first- hop analysis that the increased node density with a fixed radio range does not affect the complete network disconnection time due to energy depletion of all first hop nodes regardless of workload distribution among them. InChapter4, weexaminedtheeffectoflocalizedtopologyconstructionschemes on energy consumption and network lifetime by incorporating the link quality met- ric and load distribution based on the empirical data as well as analysis. The comparison with the global optimal strategy was also presented. The four localized topology construction schemes considered are the lowest ETX selection, the ran- dom selection with MHR, the lowest ETX selection with MHR and the balanced selectionwithMHR.Thereexistsatrade-offbetweenthelink-quality-basedscheme and the minimum-hop-routing-based scheme in terms of network lifetime. ² When the network size is small and the node density is high with a high link threshold,theminimumhoproutingschemesachieveslongernetworklifetime than the scheme whose selection is based only on the link quality. However, with the opposite network conditions, the lowest ETX scheme can provide lower maximum energy consumption among nodes. 110 ² We presented the conditions for longer lifetime of the load-balanced scheme thanthelink-quality-basedscheme. WhenthelinkthresholdPRRislessthan 0.7, the balanced-parent selection scheme does not guarantee longer lifetime than the link-quality-based scheme, where the node density is the dominant factor that determines the superior scheme in terms of network lifetime. The heterogeneous deployment was discussed in Chapter 5. The trade-offs in deploying high-cost devices under a total budget constraint were examined in this research. High-cost devices can function as a cluster-head or sink to collect and process the data from low-cost sensors and this heterogeneous deployment can enhance the duration of the network sensing operation. However, the deployment of multiple high-cost devices can reduce the number of low-cost sensors under a budget constraint, which leads to the decrease of the initial sensing coverage either in terms of coverage area or coverage degree. ² The expressions of the optimum number of high-cost devices, which maxi- mizes the lifetime coverage, in the single-hop communication model and the multi-hop communication model were also derived by incorporating several factors that affect the initial sensing coverage and the energy consumption of nodes. As far as the sensing range ratio is concerned, unless (R H =R L ) 2 is close to C H =C L , the sensing range ratio effect is not significant. An opti- mal heterogeneous deployment can achieve a lifetime coverage that is several times as much as that with homogeneous deployment by considering the ini- tial coverage and the sensing operation duration. ² The coverage aging patterns with two communication models and different ratios of heterogeneity were examined. For the single-hop communication model, the initial coverage degree can be preserved in the covered area as 111 nodesbecomeenergy-depletedovertimesincethecommunicationenergycon- sumption is not dependent on neighboring nodes, but on the distance from a sink. As n H increases, a faster aging rate in the single-hop communica- tion model and a slower aging rate in the multi-hop communication model were observed. This is due to the high variance of workload among nodes. From the perspective of re-deployment, a uniform faster aging rate would be preferred since the re-deployment can be done at once for the entire area. ² The effect of spatial correlation on heterogeneous deployment was evaluated with the information utility coverage metric. For the single-hop communi- cation model, n H¡opt increases as the correlation increases. On the other hand, n H¡opt decreases for the multi-hop communication model owing to the increase of data aggregation in a highly correlated network that significantly reduces the communication load in first-hop nodes even with a small amount of sink deployment. In both communication models, the optimum mixture of heterogeneity can significantly increase the lifetime coverage, which is espe- cially true for a low correlated network. Such a mixture can also provide a highlifetimecoveragethatisclosetothemaximumvalueforhighlycorrelated networks. 6.2 Future Research Direction It is interesting to examine distributed topology construction schemes that achieve the maximum network lifetime in a dynamic environment using the link quality information as well as the communication load among nodes. Under dynamic net- work conditions, the link quality among nodes and the communication load of each node can change over time due to the change of the external environment or the 112 network operation. For example, we may consider partial redeployment/recharging and/or the overall network topology change by varying the sink location (mobile sink) and/or mobile nodes. In addition, we will examine the optimal link threshold thatprovidesmaximumlifetime. Inthecaseoffixednodedensitydeployments,the carefuladjustmentoflinkthresholdwilloptimallybalancecommunicationoverhead driven by imperfect link quality and communication load sharing by more nodes in a larger radio range. In addition, it is worthwhile to examine existing topology construction and routing protocols to provide longer network lifetime. The ZigBee standard, which resides on top of the IEEE 802.15.4 PHY/MAC layer, has been recently provided byanallianceofseveralcompanies. Itcontainsprotocolsfornetworktopologycon- struction/management, routing and security and service for the application layer. Thetable-based-routingcanbeperformedintheZigBeeprotocolbesideshierarchi- calroutingwithtree-basedaddresses. Itusesthepathcosttoselectamorereliable path from the source to the destination. The link cost can be estimated based on LQI (link quality indicator) provided by the 802.15.4 MAC/PHY layer or the measurement of successful data transmissions. However, one problem in topology construction (possible unbalanced tree topology) and routing decision is the lack of energy consumption consideration. The ZigBee standard does not specifically address the power consumption and the load balancing issues in its network layer schemes. WecanexaminewaystoextendtheZigbeeprotocolbyconsideringtopol- ogy construction, routing path selection and scheduling to achieve longer network lifetime. This task is feasible by studying the communication load distribution and the link metric in conjunction with the energy cost estimation as presented in our work. 113 Reference List [1] S. Bandyopadhyay and E. Coyle, “An energy efficient hierarchical clustering algorithm for wireless sensor networks,” in Proc. IEEE Infocom, vol. 3, 2003. [2] A. Barroso, U. Roeding, and C. Sreenan, “Maintenance efficient routing in wirelesssensornetowrks,”inIEEE Workshop on EmbeddedNetworkedSensors (EmNetS-II), Sydney, Australia, May 2005. 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Lee, Jae-Joon
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Aging analysis in large-scale wireless sensor networks
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Viterbi School of Engineering
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Doctor of Philosophy
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Computer Engineering
Publication Date
01/26/2007
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11/28/2006
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University of Southern California
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network lifetime,OAI-PMH Harvest,wireless sensor networks
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Krishnamachari, Bhaskar (
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