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USC Computer Science Technical Reports, no. 905 (2009)
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USC Computer Science Technical Reports, no. 905 (2009)
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Dynamic Data Compression in Multi-hop Wireless Networks Abhishek B. Sharma y , Leana Golubchik yx , Ramesh Govindan y , and Michael J. Neely x y Computer Science, x Electrical Engineering, University of Southern California, Los Angeles. email: fabsharma, leana, ramesh, mjneelyg@usc.edu Data compression can save energy and increase network capacity in wireless sensor networks. However,thedecisionofwhetherandwhentocompressdatacandependuponplatformhardware, topology, wireless channel conditions, and application data rates. Using Lyapunov optimization theory,wedesignanalgorithmcalledseecthatmakesjointcompressionandtransmissiondecisions with the goal of minimizing energy consumption. A practical distributed variant, dseec, is able toachievemorethan 30%energy savings and adaptsseamlessly acrossa wide range ofconditions, without explicitly taking topology, application data rates, and link quality changes into account. Categories and Subject Descriptors: C.4 [Performance of Systems]: Design studies, Perfor- mance attributes General Terms: Theory, Algorithms, Performance Additional Key Words and Phrases: Stochastic Network Optimization, Energy e±ciency 1. INTRODUCTION Wirelesssensornetworks,inwhichnodescollectsensordataandtransmitthemover multiple hops to a sink, have enabled unprecedented visibility in many natural and builtenvironments. Datacompressionhasbeenconsideredforthesenetworksintwo contexts. First, compressioncanreducetransmissioncostsandtherebysaveenergy resources. Second, for applications that generate signi¯cant data, compression can increase the e®ective network capacity. Until recently, it was assumed that compression was always energy-e±cient so that the decision to compress could be made statically. However, Sadler and Martonosi [2006] showed that, for many commonly used platforms, whether com- pressing data at a node is energy-e±cient or not depends upon the platform's components, as well as the position of the node in the topology. Speci¯cally, they showed that the balance between the energy required for compression and the en- ergyrequiredfortransmissionissuchthatcompressionwinsonlyifanodeisseveral hops away from the sink (the actual distance depends upon topology). One can This material is supported in part by: NSF Grant Nos. CNS-0540420 and OCE 0520324, the DARPA IT-MANET program grant W911NF-07-0028, and the NSF Career grant CCF-0747525. Permission to make digital/hard copy of all or part of this material without fee for personal or classroom use provided that the copies are not made or distributed for pro¯t or commercial advantage, theACMcopyright/servernotice, thetitleofthepublication, anditsdateappear, and notice is given that copying is by permission of the ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior speci¯c permission and/or a fee. c ° 20YY ACM 0000-0000/20YY/0000-0001 $5.00 ACM Journal Name, Vol. V, No. N, Month 20YY, Pages 1{0??. 2 ¢ extend the arguments in Sadler and Martonosi [2006] to show that the decision to compress at a node can also depend upon the wireless channel conditions along the path from the node to the sink: a noisy channel, because it consumes energy in retransmissions, can make it favorable to compress at nodes closer to the sink. Moreover,thisdecisioncanalsodependupontheapplicationdatarate|whenthe networkisoperatingclosetocapacity,itmightbemoreenergy-e±cienttocompress a fraction of packets, rather than all of them. Inpractice,thismeansthatthedecisionofwhether(andwhen)tocompressdata for energy-e±cient operation must be made dynamically. It is possible, of course, to devise a heuristic decision algorithm for this task, that explicitly takes platform energy consumption, topology, channel characteristics and application data rates into consideration. In this paper, we take a more principled approach. Using tools from Lyapunov optimization theory (Georgiadis et al. [2006]), which explores the design of sta- ble transmission schedulers that optimize a given objective function, we design an algorithm called seec (Stable Energy-E±cient joint Compression and transmis- sion scheduling, Section 4). seec ensures network stability (queues at nodes are ¯nite) and minimizes time average energy expenditure, while dynamically deciding whether and when to compress data in order to achieve energy-e±ciency. It is par- simonious, in the sense that a single parameter V governs the performance of the algorithm. We are able to provide a performance bound for seec. Speci¯cally, we show that seec's energy consumption is within an additive factor of the optimal. It can achieve an energy consumption arbitrarily close to the optimal at the expense of increased queueing delay. seec is a centralized algorithm and makes idealized assumptions about wireless medium access (Section 2), so we also explore a more practical distributed vari- ant called dseec (Section 5). This variant runs on CSMA-based MACs (such as those used with 802.11 and 802.15.4 radios). It uses the same compression de- cision algorithm as seec (which requires only local information), and uses queue backlog information from its local neighborhood to implement seec's transmission scheduling algorithm. dseec's chief advantage is that it is able to achieve energy-savings, while adapt- ing dynamically to platform changes, channel variability, diverse application data rates, and topology diversity. We demonstrate (Section 6) its adaptivity through extensive simulations in Qualnet [2008] (a widely-used high-¯delity wireless simu- lator), and show that dseec can achieve more than 30% energy savings in some settings. dseec's elegance stems from the fact that it is able to make the com- pression decisions without explicitly considering topology, application data rates, or channel qualities. Instead, these characteristics manifest themselves as changes in queue backlog at a node, which triggers compression decisions. Finally, we show thatdseec's performance is relatively insensitive to the choice of its parameter V: changingV bythreeordersofmagnitudea®ectsoverallenergyconsumptionbyless than 10%. ACM Journal Name, Vol. V, No. N, Month 20YY. ¢ 3 2. SYSTEM MODEL In this section we describe our model for a wireless (sensor) network deployed for data collection. This model provides a framework for describing our joint compres- sionandtransmissionschedulingalgorithmandanalyzingitsperformance. Forease ofexposition, weassumethattimeisslotted; oursystemmodelandalgorithmscan be easily generalized to the continuous time case. Dataacquisition. ConsideranetworkofN nodes. LetN =f1;2;:::Ngrepresent the set of nodes. Each node n 2 N has a sensor attached to it, which collects measurementsperiodically. LetA n [t]representthenumberofmeasurementsamples collected by the sensor attached to node n during timeslot t. We assume that the size of each sample is b bits. In a wide range of sensing applications, data is collected at a ¯xed sampling frequency: vibration (Paek et al. [2006]), and imaging (Hicks et al. [2008]). For these applications, A n [t]=a for all timeslots t. However, if the sensing application adapts the sampling frequency in response to an event Caron et al. [2007], then A n [t] may vary across time slots. We model A n [t] as a discrete random variable with maximum value m. Each node has to deliver the measurement data collected by its sensor to a sink (possibly) over multiple wireless hops. In the design and analysis of seec, we assume a single sink but possible dynamicroutingoverdi®erentoutgoinglinks. However, theevaluationinSection6 is done for the special case of a ¯xed routing tree rooted at the sink. DataCompression. Thedatacollectedbyanodeis¯rstprocessedbya compres- sion module. The compression module can compress the data using one of the K compression algorithms available to it. Every timeslot t, the compression module picks a compression option k n [t]. Option k n [t]=0 represents no data compression. This is the central focus of the paper: the decision about whether to compress, and which compression option to use, is made jointly with transmission scheduling (discussed below), taking channel conditions and interference into account. The compressed data is then delivered to the queue at the transmission module (dis- cussed below). For each node n, we model the size of the data after compression, R n [t], as a random variable. In practice, the output of the compression module will depend on the values of measurement samples being compressed. However, for analytical tractability, we assume that R n [t] are i.i.d. conditionally on the value of A n [t] and k n [t]. As we show in our trace-based evaluations (in Section 6), our results hold even for real datasets, indicating that this assumption is not restrictive in practice. Let E n C [t] denote the energy consumed at node n while compressing data using option k n [t]; with E n C [t] = 0 for k n [t] = 0. We model E n C [t] as a random variable whosevalueisdependentuponthecompressionoptionk n [t]. WeassumethatE n C [t] are i.i.d. for all t with the same compression option k n [t]. CompressionAlgorithms. Wecharacterizethedi®erentcompressionalgorithms in terms of their expected compression ratios and energy consumption. For every compressionalgorithmk2K, wede¯netheexpectedsizeofthecompressedoutput ACM Journal Name, Vol. V, No. N, Month 20YY. 4 ¢ m(a;k) and the energy consumption Á(a;k) for A n [t]=a as follows: m(a;k) M = EfR n [t] j A n [t]=a;k n [t]=kg (1) Á(a;k) M = EfE n C [t] j A n [t]=a;k n [t]=kg (2) Given our de¯nition of m(a;k), the expected compression ratio achieved by option k is m(a;k) ab . We assume that m(a;k) and Á(a;k) are known for all compression algorithms k 2 K. In practice, these quantities can be empirically determined from a large enough sample dataset prior to network operation. If such quantities are not available prior to network operation they can be estimated on-line during an initial learning period during which data is always compressed to collect statistics. In our evaluations, we use the former approach for simplicity. Scheduling Transmissions. Let L denote the set of all wireless links in the network. Each link l 2 L is characterized by a pair of nodes (n 1 ;n 2 ) where n 1 transmits data to n 2 over the link l. For each node n, let n represent the set of all outgoing links on which node n can transmit data and £ n represent the set of all incoming links on which node n can receive data. The transmission rate over a wireless link depends on the transmit power and the channel state. The channel state of a wireless link can be characterized using several di®erent metrics: the Signal to Interference plus Noise Ratio (SINR) for the link, the expected transmission count (ETX) metric (De Couto et al.[2003]), the LQI (Link Quality Indicator threshold) (Srinivasan et al. [2006]) and the RSSI (Received Signal Strength Indicator) values, etc. In this paper, we use the ETX metric to capture link quality in our evaluation (Section 6). If we model each packet transmission attempt as a Bernoulli trial, then ETX=1/(p f £ p r ), with p f and p r equal to the successful transmission probability for a packet and its acknowledgment, respectively. We represent the current channel state at time t as a vector ~ S[t] = fS l [t]g l2L where S l [t] denotes the channel state at time t for link l. For all links l 2 L, we assume that S l [t] takes values from a ¯nite set ~ S for all t. We represent the (¯nite) set of all possible channel state vectors by S = ~ S jLj , and hence, ~ S[t]2S for all t. Acentralized 1 transmissionschedulerdeterminesthesetofnodesthatcantrans- mit at time t based on the current channel state ~ S[t]. For each link l 2 L, it determines a transmit power P l tran [t]. We assume that P l tran [t]2f0;P max g for all linkslandtimet(itispossibletoextendourframeworktomultipletransmitpower levels). A node n transmits at time t if and only if for at least one of its outgoing links l 2 n is assigned P l tran [t] > 0. We represent the transmit power allocation by the scheduler at time t as a vector ~ P tran [t]. We can incorporate di®erent scheduling constraints into the transmission sched- uler. For example, if nodes have a single radio, and they cannot transmit and receive (or transmit on more than one link) simultaneously, we can impose the 1 In a later section, we discuss a decentralized scheduler whose decisions approximate this centralized scheduler. ACM Journal Name, Vol. V, No. N, Month 20YY. ¢ 5 following feasibility constraint on ~ P tran [t]: X l2n I(P l tran [t])+ X l2£n I(P l tran [t])·1 8 nodes n (3) where I(P l tran [t]) is equal to 1 if P l tran [t]>0 and zero otherwise. Transmission capacity-power curve Let ¹ l [t] denote the number of bits that can be transmitted over link l during time slot t. We model ¹ l [t] as a function, ¹ l [t]=C l ( ~ P tran [t]; ~ S[t]); (4) where ~ P tran [t]=fP l tran [t]g l2L , ~ S[t]isthecurrentchannelstate,andC l ( ~ P tran [t]; ~ S[t]) isthecapacity-power curveforlinkl de¯nedbythemodulationandcodingschemes used for transmission on link l. We assume that there exists a constant ¹ max , such that ¹ l [t] · ¹ max for all t. We also assume that, for all l, C l ( ~ P;~ s) is piecewise continuous in ~ P, and is monotonically increasing in the transmission power for link l, i.e., C l ((P l ;fP j g j6=l );~ s)¸C l (( ~ P l ;fP j g j6=l );~ s) for P l > ~ P l for each channel state ~ s. Additionally, for every link l, C l ((0;P j j6=l );~ s) = 0 for every channel state ~ s, i.e., zero transmission power always yields zero transmission rate. Thus, at time t, ¹ l [t]>0 only for those links l that are allocated P l tran [t]>0 by the transmission scheduler. Example of Capacity-Power Curve: For a channel subject to additive white Gaus- sian noise, the capacity-power curve for link l, C l ( ~ P tran ;~ s) can be approximated using a log() formula for channel capacity. Hence, we can de¯ne C l ( ~ P tran ;~ s) for link l as follows: C l ( ~ P;~ s)=log Ã 1+ ® l P l tran N l + P ^ l6=l ® ^ l P ^ l tran ! where N l and ® l represent the Noise and fading coe±cients associated with the particular channel state ~ s. Queueing dynamics. The output of the compression module, R n [t], is held in a queue at the transmission module awaiting transmission. Once data is passed on to the transmission module, it is not considered for compression again. Let U n [t] denote the number of bits in queue at node n. The following equation captures the dynamics of the queue backlog: U n [t+1] · max Ã U n [t]¡ X l2 n ¹ l [t];0 ! + R n [t]+ X l2£n ¹ l [t] (5) where ¹ l [t] is given by equation (4) and P l2£ n ¹ l [t] represents the exogenous ar- rivalsatnodenduetoothernodesroutingtheirdatathroughnoden. Theequality in (5) achieved only when the actual endogenous arrivals from other nodes is equal to P l2£n ¹ l [t]; the actual endogenous arrival can be less than P l2£n ¹ l [t] if the other nodes have little or no queue backlog. ACM Journal Name, Vol. V, No. N, Month 20YY. 6 ¢ Symbol Description A n [t] Samples collected; (·m) b size of each sample in bits kn[t] Compression decision R n [t] Data size after compression m(:;k) Expected size of compressed data for compression algorithm k (eq. 1) E n C [t] Energy consumed by data compression Á(:;k) Expected energy consumption for compression algorithm k (eq. 2) L All links in the network n Set of outgoing links £n Set of incoming links ~ S[t] Current channel state; ~ S[t] =fS l [t]g l2L ~ Ptran[t] Transmit power allocation; ~ P trant [t]=fP l tran [t]g l2L ¹ l [t] # bits that can be transmitted on link l during slot t (eq. 4) U n [t] Queue backlog E l T [t] Energy consumed at node transmitting data on link l E l R [t] Energy consumed at node receiving data on link l E l B [t] Energy consumed due to nodes overhearing the transmission on link l TableI.SystemModelNotation: symbolstandndenotethecurrenttimeandanode,respectively. Table I summarizes the notation used to describe our model. Ourframeworkcannaturallyincorporateseveralextensionstothismodel, which we discuss brie°y as part of future work in Section 8. 3. OPTIMIZATION GOAL In this paper, our aim is to design a compression and transmission scheduling algorithm that minimizes the total system power expenditure while maintaining network stability. Using our system model (Section 2), we now formally de¯ne total system power expenditure and network stability. The main sources of energy expenditure in our system are data compression, data transmission, and data reception. At time t, the energy consumed by the compression module at node n is E n C [t], with E n C [t] = 0 if the \no compression" option is chosen (k n [t]=0). If the transmission scheduler allocates power P l tran [t]=P max for link l =(n;~ n), then node n has the opportunity to transmit data for the duration of one time slot, T slot . Now, at time t, node n can transmit ¹ l [t] bits over link l. However, as later described in Section 4, in our scheme transmission schedules are determined based on the queue backlog at di®erent nodes. In most cases, when it is the turn of node n to transmit, its queue will have more data than it can possibly transmit during one slot, i.e., U n [t] ¸ ¹ l [t]. Thus, the energy consumed (at node n) by the data transmission on link l during slot t is E l T [t]=P l tran [t]=P max £T slot (6) When U n [t] < ¹ l [t], we assume (for analytical tractability) that the transmitter stays up for the entire slot duration. In practice, of course, the transmitter does not have to do this. ACM Journal Name, Vol. V, No. N, Month 20YY. ¢ 7 Node ~ n consumes energy while receiving the data packet transmitted by node n. Suppose the wireless interface at each node dissipates a constant amount of power P recv wheninreceivemode. Theenergyconsumed(atnode ~ n)forpacketreception on link l during time slot t, under the same assumptions as above, is: E l R [t]=P recv £T slot (7) We assume that the wireless interface at a node does not consume any energy when it is not transmitting or receiving data. This is an idealized assumption, and requires the existence of a well-designed network duty-cycling protocol. Such a protocolcoordinatespackettransmissionsinawaythatnodescanturntheirradios o® in order to conserve energy. In general, these protocols (Razvan et al. [2008], Burri et al. [2007]) do not achieve the ideal we have assumed, but incur a very small amount of overhead in determining whether it is safe to go back to sleep. We examine this deviation from the ideal in our experiments in Section 6. Finally, nodes can expend signi¯cant energy overhearing packets, when omnidi- rectionalradiosareused(aswehaveassumed). Astandardmethod(Yeetal.[2006]) for overhearing-avoidance is to use the RTS/CTS message exchange to determine whether a node can go to sleep or not. Before transmitting a data packet for node ~ n, a node n broadcasts an RTS control packet that identi¯es ~ n as the intended re- ceiver and the data packet's transfer duration. All the nodes that are not identi¯ed as the receiver can then safely turn o® their radios for the duration of the data packet transfer. We account for this overhearing-avoidance in our analysis in the following man- ner. Nodenbroadcastingapacketisequivalenttonode ntransmittingthatpacket on all its outgoing links l2 n . We assume that a fraction ®<1 of the ¹ l [t] bits is consumed by a control message. The energy consumed at all the neighbors of node n, except ~ n, due to these nodes receiving a control message is E l B [t] = X ~ l2 n : ~ l6=l ®(P recv £T slot ) = ®£(j n j¡1)£(P recv £T slot ) (8) In our analysis, we assume that all timeslots are of the same duration. Without loss of generality, we de¯ne T slot to be of unit length for the rest of the paper. Since node n transmits ¹ l [t] total (control and data) bits on link l, the total network-wide energy consumed due to node n's compression and transmission ac- tivity during time slot t, E n tot , can be written as, E n tot [t]=E n C [t]+ X l2 n ¡ E l T [t]+E l R [t]+E l B [t] ¢ I(P l tran [t]) (9) where I(P l tran [t]) is the indicator function. We de¯ne the total system energy expenditure during slot t as the sum of the energyexpenditureateachnode. Thetimeaveragetotalsystempowerexpenditure is given by liminf t!1 1 t t¡1 X ¿=0 N X n=1 E n tot [¿] (10) ACM Journal Name, Vol. V, No. N, Month 20YY. 8 ¢ Our goal is to design an algorithm for making compression and transmission scheduling decisions to minimize the time average total system power expenditure (10) while ensuring network stability 2 . Formally, we de¯ne a network to be stable if: limsup t!1 1 t t¡1 X ¿=0 N X n=1 EfU n (¿)g<1 (11) Note that network stability implies ¯nite average queue backlog, and hence, ¯nite average delay at each node. 4. THE SEEC ALGORITHM Inthissection,wepresentour¯rstcontribution: thedesignandanalysisofacentral- ized algorithm,seec that minimizes the system energy expenditure while adapting to topology, network dynamics, application data rates, and platform di®erences. In subsequent sections, we explore the design and performance of a distributed variant, dseec. In designing seec, we impose an additional requirement: our algorithm must re- sult in a stable network (Equation 11). This requirement suggests a starting point, theLyapunovoptimizationframework(Neely[2006],Georgiadisetal.[2006]). This framework can incorporate performance metrics such as energy expenditure, fair- ness,etc.,intotheLyapunovdrift,awell-knowntechniquefordevelopingstabilizing control algorithms. The key idea is to de¯ne a non-negative, scalar function, called a Lyapunov function, that measures the aggregate congestion of all the queues in the network during timeslot t. The Lyapunov drift represents the expected change in the Lyapunov function from one timeslot to the next. Under the Lyapunov op- timization framework, control algorithms designed to minimize the Lyapunov drift over time are guaranteed to stabilize the network and achieve near-optimal perfor- mance for a given optimization objective (energy expediture in case of seec). 4.1 Design of seec We ¯rst describe seec, designed using the Lyapunov drift technique. We present the details of the derivation and then, in next subsection, analyze seec's optimal- ity characteristics. seec decouples the choice of the compression algorithm and the transmission power allocation into two separate algorithms. Both algorithms involve a single parameter V > 0 that controls the trade-o® between energy con- sumption and delay. Compression Algorithm. Every timeslot t, each node n2N observes the data collected by its sensor, A n [t], and its current queue backlog, U n [t]. It then chooses a compression option k n [t]2K as follows: k n [t]=argmin k2K (U n [t]m(A n [t];k)+VÁ(A n [t];k)) (12) 2 It may make sense, from the perspective of our application, to consider adding other constraints (for example, constraining average energy usage on individual nodes). We have left this to future work. ACM Journal Name, Vol. V, No. N, Month 20YY. ¢ 9 We break ties arbitrarily if multiple compression options satisfy Equation (12). We describe the intuition for this algorithm below. Transmission Algorithm. For each link l 2L, we de¯ne the di®erential queue backlog U l [t] during timeslot t as U l [t] = U n [t]¡U ^ n [t], where l = (n;^ n) is a link from node n to node ^ n. Let ~ U[t] = (U n [t]) be the vector of queue backlog at all nodes. Every timeslot t, the transmission scheduler observes the current queue backlogs ~ U[t] and the channel state ~ S[t], and allocates a power vector ~ P tran [t]=(P l tran [t]) l2L that solves the following optimization problem: Maximize X n X l2 n U l [t]¹ l [t]¡V ~ P[t] (13) subject to: ~ P[t]=P l tran [t]+I(P l tran [t])[1+®(jnj¡1)]Precv ~ P tran [t]=(P l tran [t]) l2L 2P In Section 2, we assumed that P is ¯nite and that the capacity-power curves C l ( ~ P; ~ S) (that determine ¹ l [t]) are piecewise continuous. Hence, there exists a maximizing power allocation. We break ties arbitrarily if multiple maximizing power vectors exist. Atahighlevel,thesetwoalgorithmsworktogetherasfollows. Foreachtimeslot, thecompressionalgorithmchoosestheoptionk n [t]thatminimizesaweightfunction which depends on the current queue backlogat a node (Equation (12)). In order to maximizethesummationinEquation(13),thetransmissionalgorithmwillschedule transmissionsonlyonlinksforwhichthelink-weight d l [t]=(U l [t]¹ l [t]¡V ~ P[t])>0. If transmission on two links with positive link-weight cannot be scheduled simulta- neously, then the transmission algorithm picks the link with larger link-weight. Next, we describe how we deriveseec (which attempts to jointly make compres- sionandschedulingdecisionsinastablefashion)usingtheLyapunovdrifttechnique (Georgiadis et al. [2006]). Lyapunov analysis. Before we embark upon our analysis, we need a closed form expression for the time evolution of queue lengths in the system. Equation (5) provides this, but does not include the overhead 3 of RTS/CTS control messages (Section 3). The modi¯ed equation is as follows: Un[t+1] · max Un[t]¡ X l2 n ~ ¹ l [t];0 ! + Rn[t]+ X l2£ n ~ ¹ l [t] (14) where ~ ¹ l [t]=(1¡®)£¹ l [t]. 3 Accounting for such overhead may not be strictly necessary in case of seec since it uses a centralized transmission scheduler in a time-slotted system (i.e., ® may be 0). However, it is needed in the case of a distributed transmission scheduler (dseec, Section 5), so we introduce it here and carry it forward in the derivations that follow. ACM Journal Name, Vol. V, No. N, Month 20YY. 10 ¢ We de¯ne a quadratic Lyapunov function of queue backlogs as: L( ~ U[t]) M = 1 2 N X n=1 (U n [t]) 2 (15) and the one-step conditional Lyapunov drift ¢( ~ U[t]) as: ¢( ~ U[t]) M = E n L( ~ U[t+1])¡L( ~ U[t])j ~ U[t] o (16) The Lyapunov drift for our system is given by the following lemma. (See Geor- giadia et al [2006] for further details on Lyapunov drift.) Lemma 1 Suppose the r.v. A n [t] at each node n, and the channel states ~ S[t] are i.i.d. over timeslots. For the queue evolution, given in Equation (14), and the Lyapunov function, de¯ned in Equation (15), the one-step Lyapunov drift for our system satis¯es the following constraint for all t and all ~ U[t]: ¢( ~ U[t])·BN ¡ X n E ( X l2 n ~ ¹ l [t]U l [t]j ~ U[t] ) + N X n=1 U n [t]E n R n [t]j ~ U[t] o (17) with the constant B de¯ned as follows. B M = (R max +¹ in max ) 2 +(¹ out max ) 2 ; R max M = max n (EfR n [t]g) ¹ in max M = max (n;s2S;P2P) X l2£ n ~ ¹ l [t]; ¹ out max M = max (n;s2S;P2P) X l2 n ~ ¹ l [t] Proof. See Appendix A. Incorporating the energy constraint. Recall that our goal is to design an al- gorithm that makes joint compression and transmission decisions while minimizing energy usage. To do this, we use the Lyapunov optimization framework (Geor- giadis et al. [2006], Neely [2006]). We add a weighted cost (total system energy consumed during slot t) to the Lyapunov drift, in Equation (17), to get: ¢( ~ U[t])+V N X n=1 E n E n tot [t]j ~ U[t] o · BN¡ X n E ( X l2 n ~ ¹ l [t]U l [t]j ~ U(t) ) + N X n=1 U n [t]E n R n [t]j ~ U[t[ o +V N X n=1 E n E n tot [t]j ~ U[t] o (18) Inthisinequality, wecanexpandthetermE n E n tot [t]j ~ U[t] o usingEquations(6), ACM Journal Name, Vol. V, No. N, Month 20YY. ¢ 11 (7), and (8), and substitute the value ofE n R n [t]j ~ U[t] o (Equation (1)) to get: ¢( ~ U[t])+V N X n=1 E n E n tot [t]j ~ U[t] o · BN¡ X n E ( X l2 n ~ ¹ l [t]U l [t]¡V ~ P[t]j ~ U[t] ) + N X n=1 E n U n [t]m(A n [t];k n [t])+VÁ(A n [t];k n [t])j ~ U[t] o (19) where ~ P[t]=P l tran [t]+I(P l tran [t])[1+®(j n j¡1)]P recv . seec is designed to minimize the RHS of (19). There are three salient points to notefrom19. First,becausetheinequalityincorporatestheLyapunovdrift,seecis stable. Second, comparing the second term on the RHS of (19) and Equation (13), seec's transmission algorithm contributes to minimizing the RHS of (19) by max- imizing this negative term. Finally, comparing the third term on the RHS of (19) and Equation (12), we see that seec's compression algorithm minimizes this posi- tive term (in order to minimize the Lyapunov drift). Taken together, seec ensures stable, joint compression and transmission scheduling, with the goal of minimizing energy consumption. 4.2 Performance Bounds on seec Next, we provide an analytical bound on the system energy expenditure achieved by seec compared to an optimum value. The optimum value is characterized by theclassofstationaryrandomized algorithmsthatmakethecompressionandtrans- missionschedulingdecisionsinamulti-hopnetworkaccordingtoa¯xedprobability distribution. We then make the following contributions. For this restricted class of algorithms, Lemma 2 describes the minimum system power consumption, P ¤ av , required to achieve network stability. Theorem 1 shows that any joint compression andtransmissionschedulingalgorithmformulti-hopnetworks(andthereforeseec) thatstabilizesthesystemwillrequireasystempowerexpenditureofatleastP ¤ av . In Theorem 2, we show that seec can achieve an average system power consumption arbitrarily close to P ¤ av . Stationary randomized algorithms. Consider the class of stationary random- ized algorithms for making compression and transmission scheduling decisions. Such algorithms choose a compression option (based only on A n [t]) and the trans- mit power allocation (based only on ~ S[t]) for each time slot t according to a ¯xed probability distribution. For example, one policy can be to pick a compression option uniformly at random as well as, based on the current channel state ~ S[t], similarly choose the transmit power vector ~ P tran [t] according to a ¯xed probability distribution. Note that these algorithms do not consider the queue backlogs U n [t] while making their decisions. Condition for achieving stability. Suppose that the data arrival process, i.e., the sequence A n [t], t ¸ 0, is ergodic with a steady state distribution p A 4 . For a 4 For ease of exposition, we assume that p An =p A for all n. ACM Journal Name, Vol. V, No. N, Month 20YY. 12 ¢ given stationary randomized compression decision policy, let the output data rate (inbits/slot)ofthecompressionmoduleber n =EfR n [t]g·EfbA n [t]g. Wede¯ne a lower bound on r n , denoted by r n min as follows: r n min M = E ½ min k2K m(A n [t];k) ¾ for all n2N (20) where the expectation is taken over A n [t]. Thus, r n min is the minimum average bits/slot delivered to the output queue by the compression module at node n, assuming that the compression option that results in the largest expected data size reduction is used in every timeslot. Clearly, r n ¸r n min . Suppose the process representing the time varying channel state, i.e., the se- quence ~ S[t], t¸0, is ergodic with a steady state probability distribution ¼ s . Under agivenstationaryrandomizedtransmissionscheduling(andtransmitpoweralloca- tion)policy,let ~ f =(f l ) l2L wheref l =Ef¹ l [t]g. Acombinationofcompressionand transmission scheduling policy will stabilize the network if and only if ~ f = (f l ) l2L de¯ne a network-°ow satisfying the following conditions 5 . f l ¸ 08 l2L; ²>0 (21) X l2n f l ¡ X l2£n f l = r n +²8 n6=sink (22) N X n=1 rn = X l2£ sink f l (23) Let ¤ denote the set of rates for which there exists an achievable network-°ow ~ f. Clearly, if ~ r min = (r n min ) does not belong to ¤, then for the arrivals A n [t], it is not possible to stabilize the network, even by always compressing data. In our subsequent analysis, we assume that ~ r min 2 ¤. A formal de¯nition of ¤ can be found in Neely et al. [2003], where it is de¯ned as the Network Capacity Region. For the class of stationary randomized policies, we de¯ne the minimum-energy compression function h ¤ n (r)(forallnodesn)andtheminimum energy transmission function g ¤ ( ~ f) as follows. De¯nition 1 Foreachnoden,foranyvalueofr n suchthatr n min ·r n ·EfbA n [t]g, the minimum-energy compression function h ¤ n (r n ) is de¯ned as the in¯mum value h for which there exist probabilities (° a;k ) for a2f1;2;:::;mg, k2K, such that: EfE n C [t]g= m X a=0 K X k=1 p A (a)° a;k Á(a;k)=h (24) rn =EfR n [t]g= m X a=0 K X k=1 pA(a)° a;k m(a;k) (25) ° a;k ¸0 8 a;k; K X k=1 ° a;k =1 8 a 5 ² in Equation (22) is needed to produce an appropriate randomized policy; we omit the details due to lack of space. ACM Journal Name, Vol. V, No. N, Month 20YY. ¢ 13 Given ~ S[t]withdistribution¼ s andcapacity-power curves ~ C( ~ P; ~ S)=(C l ( ~ P; ~ S)) l2L , Neely et al. de¯ne the Network Graph Family, ¡, as the set of average transmission rates ~ w =(w l ) that can be achieved (Neely et al. [2003]. Di®erent power allocation algorithms will lead to di®erent ~ w. De¯nition 2 For any ~ w = (w l ) l2L 2 ¡, we de¯ne the minimum energy trans- mission function g ¤ (~ w) as the in¯mum value g for which there exists a stationary randomized power allocation policy that chooses transmit power vector ~ P tran [t] as a random function of the observed channel state vector ~ S[t], and independent of the current queue backlog, such that: X l2L E n ~ P l [t] o =g; Ef¹ l [t]g=w l 8 l2L where ~ P l [t] = P l tran [t]+I(P l tran [t])[1+®(j n j¡1)]P recv and P l tran [t] is the power allocated for transmission on link l during timeslot t. Thefollowinglemmashowsthatthein¯mumvaluesh ¤ n (r n )andg ¤ (~ w)areachiev- able. Lemma 2 Consider ~ r = (r n ) with r n min · r n ·EfbA n [t]g 8 n and ~ f = (f l )2 ¡ satisfying conditions (21)-(23). For such a scenario, there exists a stationary ran- domizedpolicythatchoosescompressionoptionk ¤ n [t]andtransmitpowerallocation ~ P ¤ [t]fortimeslottbasedonlyonA n [t]and ~ S[t](andindependentofqueuebacklogs) such that: EfE n C [t]g=EfÁ(An[t];k ¤ n [t])g = h ¤ n (r)8 n (26) EfR n [t]g=Efm(A n [t];k ¤ n [t])g = r n (27) X l2L E n ~ P l¤ [t] o = g ¤ ( ~ f) Ef¹ l [t]g = f l 8 l2L Proof. See Appendix B. Optimum system power consumption. The minimum-energy functions h ¤ n and g ¤ de¯ned for stationary, randomized algorithms can be used to characterize the minimum system energy consumption for the larger class of joint compression and transmission scheduling algorithms. Theorem 1 Ateachnoden,letther.v.A n [t],t¸0,bei.i.d.andthecorresponding data arrival process be ergodic with a steady state probability distribution p A . Let the stochastic process representing the time varying channel state ( ~ S[t]) be ergodic with a steady state probability distribution ¼ s . We assume that ~ r min = (r n min ) is within the network capacity region. Then any joint compression and transmission scheduling algorithm that stabilizes the queues ~ U[t] = (U n [t]) requires an average system power expenditure that satis¯es: liminf t!1 1 t t¡1 X ¿=0 N X n=1 E n tot [¿]¸P ¤ av (28) ACM Journal Name, Vol. V, No. N, Month 20YY. 14 ¢ where P ¤ av is the optimal solution to the following optimization problem: Minimize: N X n=1 (h ¤ n (r n ))+g ¤ ( ~ f) (29) subject to: r n min ·r n ·bEfA n [t]g 8 n2N ~ f =(f l ) l2L de¯nes a valid network °ow Proof. See Appendix C. In practice, solving the optimization problem in Equation (29) might not be pos- sible as it requires exact knowledge of functions h ¤ n and g ¤ , which in turn requires complete a priori knowledge of the distributions p A and ¼ s . However, this for- mulation is useful in showing the optimality characteristics of seec, as discussed next. seec Performance. The following theorem shows that seec can achieve near- optimal performance, i.e., achievepowerconsumption arbitrarilycloseto P ¤ av while maintaining network stability and trading-o® delay (as described below). Theorem 2 Suppose the arrival process A n [t] and the channel state ~ S[t] are i.i.d. across timeslots with distributions p A and ¼ s , respectively. We assume that it is possibletostabilizethenetwork,i.e.,~ r min isstrictlyinteriortothenetworkcapacity region ¡. For any control parameter V > 0, the compression and transmit power allocationalgorithms(Equations(12)and(13))achieveaveragepowerconsumption and queue backlogs that satisfy the following constraints: Ptot = liminf t!1 1 t t¡1 X ¿=0 N X n=1 E n tot [¿] ! ·P ¤ av + BN V (30) X n Un M = limsup t!1 1 t t¡1 X ¿=0 X n EfUn[¿]g · BN +VN(Á max + ~ P max ) ² max (31) wheretheconstantB isde¯nedinEquation(18),and ~ P max =P max +P recv (1¡®+ ® ~ max )with ~ max M = max n n . The constantÁ max denotesthemaximum expected power consumptionfor data compression acrossall nodes when, for eachtimeslot t, the compression algorithm that is expected to consume maximum power is chosen. It is de¯ned as: Á max M = max n µ E ½ max k2K [Á(A n [t];k)] ¾¶ (32) wheretheexpectationisovertherandomarrivalprocess A n [t]. ² max isthelargest² suchthat ^ r n =(r n min +²)2¤,i.e.,ifweincreasedtheminimumpossiblecompressed data rate at node n, r n min , by ² max , the resulting rate vector lies on the boundary of the network capacity region ¤. Proof. See Appendix F. ACM Journal Name, Vol. V, No. N, Month 20YY. ¢ 15 Powerconsumptionvs.delaytrade-o®. Wecanchoosealargevalueforcontrol parameterV tomakeB=V arbitrarilysmall,andhence,achievetimeaveragepower consumption P tot arbitrarily close to the optimal value P ¤ av . However, the total average queue backlog P n U n grows linearly in V. Thus, reducing the average power expenditure by choosing a large value for V causes larger queue backlogs resulting in longer delay in delivering data to the base-station. This O(1=V;V) power consumption vs. delay trade-o® is inherent in control algorithms designed based on Lyapunov optimization techniques (Georgiadis et al. [2006]). 5. DSEEC: DISTRIBUTED ALGORITHM seec's performance bounds are derived for a timeslotted system. Under gen- eral SINR constraints, ¯nding the optimal transmission schedule for a timeslot is NP-hard (Georgiadis et al. [2006]). However, for certain scenarios, for ex- ample, a cell partitioned network, ¯nding the optimal transmission schedule is equivalent to ¯nding a maximum weight matching in a graph with link-weights d l [t] = U l [t]¹ l [t]¡ V ~ P[t] (Georgiadis et al [2006]). Given the knowledge of the complete network topology and the link-weights, a maximum weight matching can be found in polynomial time. Inpractice,mostwirelesssystems(e.g.,802.11or802.15.4-basedsystems)arenot time-slotted. In the rest of this paper, we consider a distributed variant of seec, called dseec, for multi-hop wireless networks based on practical MACs. dseec uses the same compression algorithm as seec, since that algorithm only requires local information. The key di®erence between dseec and seec is the transmission algorithm, in which a node uses only information about queue backlogs from its neighbors. Speci¯cally, our transmission heuristic lets only nodes n with positive link-weights, i.e., d l [t]>0forsomelinkl =(n;~ n), contendforthewirelesschannel. Severalother heuristics havebeenproposedimplementingsuchbackpressure-based scheduling with CSMA-based MACs (Umut et al. [2008], Warrier et al. [2007]); Sridharan et al. [2008] show that the positive link-weight based heuristic performs as well as others. Because dseec is not analytically tractable, we evaluate its performance through simulation. 6. EVALUATION In this section we evaluate the performance of our algorithm in simulation, on topologies derived from real wireless testbeds and deployments. This section ¯rst discusses our experimental methodology, then presents our results. 6.1 Methodology We begin by describing our experimental methodology. Implementation in Qualnet. Qualnet [2008] is a widely-used, high-¯delity, packet-level wireless simulator. It has been used extensively for the evaluation of mobile ad-hoc networks. In this paper, we use it to mimic a multi-hop wireless data gathering network. Aconstantbitrate(CBR)applicationinQualnetdrivesourevaluationof dseec. The CBR application generates data periodically. These data bytes are passed on ACM Journal Name, Vol. V, No. N, Month 20YY. 16 ¢ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Cluster 1 Cluster 2 Cluster 3 Cluster 4 Fig. 1. Cluster-tree 6 5 4 3 2 7 8 11 12 21 22 24 15 14 13 10 9 16 17 18 19 1 20 26 23 25 CLIQUE 1 CLIQUE 2 Fig. 2. Shallow-tree 20 24 1 2 7 5 9 8 10 11 12 13 3 4 6 14 15 18 16 17 19 21 22 23 25 26 CLIQUE 1 CLIQUE 2 CLIQUE 3 Fig. 3. Deep-tree to the compression module that decides whether to compress the data or not. The output of the compression module is queued in a bu®er at a wireless link/interface awaiting transmission. Hardware platform. The power consumption for data transmission and/or com- pression depends on the hardware platform. In this paper, we model each node as a LEAP2 (Stathopoulos et al. [2008]), an embedded networked sensor platform 6 optimized for low power processing. The two main reasons behind our choice were: (1) the LEAP2 platform can provide detailed, ¯ne-grained, real-time energy usage information, and (2) it provides signi¯cant computing resources (fast CPU speeds up to 624 MHz, as well as a large memory and storage subsystem). Hence, it is an ideal platform for developing energy aware applications and system components such as the one described in this paper. Data compression model parameters. To compute the average compression energy Á(a;k), and the average data size after compression m(a;k), we use the 6 We expect our results to generalize to other sensor platforms like the motes, although we have not evaluated these platforms. Indeed, Sadler and Martonosi [2006] use these platforms to point out that always compressing data is not necessarily energy-e±cient. ACM Journal Name, Vol. V, No. N, Month 20YY. ¢ 17 zlib compression libraries to compress the data collected during a real-world sensor network deployment (Paek et al. [2006]). This deployment measured vibrations on a large suspension bridge. In all our simulations, we use a single compression algorithm (K =1) (we have left an evaluation of K >1 to future work). We model the data arrival process at each node n, A n [t], as a CBR application generating 600 bytes of data periodically. With each node n in our simulation, we associate a sensor node ~ n from the deployment in Paek et al. [2006]. We partition the data collected by the node ~ n during this deployment into slices of size 600 bytes, and compress each slice separately using the compression function in zlib. We then set m(A n [t];1) to be equal to the average data size achieved after compression across these slices (with values ranging from 430 to 470 bytes). Each simulation run is trace-driven, where each node periodically transmits suc- cessive 600 byte slices from the corresponding trace in Paek et al. [2006]. If it decides to compress a slice at time t, in simulation we actually compress the slice, instead of (say) using the average. To estimate Á(a;1), we ran our compression program on a LEAP2 node, with our traces as input. We measured the average CPU and memory (SDRAM) energy consumption across several runs. We did not notice any signi¯cant variability in theenergyconsumedfordatacompressionontheLEAP2nodeacrossdi®erentruns with data from di®erent nodes. For example, with default CPU speed settings for the LEAP2 node, the average energy consumed for compressing 600 bytes of data, by the CPU and the memory combined, was 3:35 mJ. Hence, in our simulations, we model the energy consumed for data compression as a constant equal to 3:35 mJ for each node n (thus, Á(A n [t];1) is equal to 3:35 mJ for each node n). Data Transmission model parameters. The LEAP2 platform supports 802.11 radios, with a measured radio power consumption of 400 mW, regardless of the wirelessinterfacestate(transmitting, receivingoridle)(Stathopoulosetal.[2008]). We use the 802.11b MAC and physical layer implementation in Qualnet in our simulations. We set the maximum transmission rate to 2 Mbps, and the radio power consumption to 400 mW when the wireless interface is in transmit or receive state. We assume the existence of a scheme for radio duty cycling that turns o® the radio whenever possible, and discuss its impact in Section 6.2. We activate RTS/CTS in 802.11. In our energy accounting, we assume simple overhearing avoidance using RTS/CTS. Before transmitting a data packet, a node n sends a request-to-send (RTS) control packet that identi¯es the intended des- tination ^ n. We assume that all nodes other than ^ n turn o® their radios and do not waste energy trying to receive the data packet transmitted by node n. Simi- larly, all the nodes receiving the clear-to-send (CTS) packet sent by node ^ n to n indicating that ^ n is expecting a packet from node n, turn o® their radios as well. In our simulations, each data packet is 684 bytes (600 bytes of payload plus 84 bytes of headers). RTS and CTS packets are 20 and 18 bytes, respectively. In addition, the 802.11b implementation in Qualnet de¯nes a 192 ¹s synchronization time overhead associated with each transmission. For these values of packet sizes and synchronization time overhead, the energy consumption due to receiving RTS or CTS packets is within 10% of the energy consumed by the wireless interface for receiving a packet. Thus, with RTS/CTS enabled, we set the parameter ® in our ACM Journal Name, Vol. V, No. N, Month 20YY. 18 ¢ transmission decision algorithm (Equation (13)) to 0:1. Network topologies. For a ¯xed energy consumption by the radio and for data compression, the network topology determines whether compressing data saves en- ergy or not. Two network topology dependent factors can have an impact on the energy savings due to data compression: (a) multi-hop routing and (b) the average neighborhood size. Intuitively, compressing data at a node that is multiple hops away from the sink can reduce the total energy consumption in the network. Compression reduces the total size of the data (hence, the number of packets) a node needs to trans- mit. The energy saved from transmitting fewer packets over multiple hops can o®set the energy consumed for data compression, leading to a lower total energy consumption (compared to not compressing data before transmission) (Salder and Martonosi [2006]). In addition, each data packet transmission incurs an energy consumption not only at the intended receiver but also at nodes that are within the radio range of the sender. A node with a large number of neighbors can reduce the total energy consumption in the network by compressing data { the energy consumed for data compression can be o®set by the lower energy consumption for packet reception (due to fewer packets being transmitted). We evaluate the performance of our algorithm using three qualitatively di®erent data collecting trees: a cluster-tree (Figure 1), a shallow-tree (Figure 2), and a deep-tree (Figure 3). These topologies represent di®erent combinations of the two factors { multi-hop routing and neighborhood sizes, as summarized in Table II. # source avg. # Multi-hop nodes neighbors paths Cluster-tree 20 small (4) short (· 2) Shallow-tree 25 large (11) short (· 2) Deep-tree 25 medium (8) long (· 4) Table II. Data Collection Trees 6.2 Energy Savings Inthissection,weprovideevidencewhichillustratestheneedfor dynamic compres- sion decisions. We ¯rst show that realistic data collection topologies exist where statically always compressing data may not be the most energy e±cient strategy. Then, we demonstratedseec's adaptability to varying wireless channel conditions, application data rate changes, and diverse platform settings. Parameters, Metrics, Alternatives. In our simulations, a CBR application at each node generates 600 bytes of data every 5 seconds. In this regime, the data rate is low enough that nodes compress data only to minimize energy, not to stabilize queues. Incidentally, this is the rate at which data is generated at each node in the original deployment. Each simulation run is 100 minutes long, so the CBR application at each node generates ¼ 1200 data packets. Unless otherwise speci¯ed, thecontrolparameter V issetto10 6 (wediscussdseec'ssensitivityto V in Section 6.4), and the compression and data transmission parameters are set to ACM Journal Name, Vol. V, No. N, Month 20YY. ¢ 19 the values given in Section 6.1. Finally, each simulation is averaged over 10 runs; the 95% con¯dence intervals are too small to see on the graphs, so we omit those. We compare dseec against two baseline strategies: (1) always compress and (2) never compress. These are both static compression decisions that do not take the (energy) cost of data compression relative to the cost of transmission into account. To equalize the e®ect of transmission scheduling on energy consumption, we use the same transmission decision algorithm (Equation (13)) for all strategies. We use the time averaged total system power consumption as the performance metric. This is computed as the total energy consumption across all nodes divided by the duration of the experiment. MotivatingDynamicCompression. Figure4depictsthetotalpowerconsumed bythedi®erentcompressiondecisionstrategies,wherethealwayscompress strategy dissipates more power than dseec for all three topologies { 34% for cluster-tree, 17% for shallow-tree, and 10% for deep-tree. In these topologies, the energy con- sumed for compressing data is higher than the savings resulting from having to transmit fewer packets as a result of data compression, so with dseec nodes never compress data, and its system power consumption is the same as never compress. Across topologies, two facts stand out. First, always compress consumes less power in transmitting and receiving packets. This is expected since compressing data reduces the number of packets each node has to transmit. Second, the power savings achieved by dseec for the cluster-tree is signi¯cantly higher than for the other two topologies. Nodes in the cluster-tree have both smaller neighborhood sizes and shorter multi-hop paths than in the other topologies. In the shallow-tree each node has a large number of neighbors. Thus, a signi¯cant fraction of the energy consumed for data compression by always compress is o®set by the lower energyconsumedforreceivingdataandcontrol(RTS/CTS)packets. Thedeep-tree topologyhasnodeswithalargenumberofneighborsaswellasnodesthataremore than 2 hops away from the sink, and the bene¯ts of dseec are the least in this case. Finally, recall that dseec di®ers from seec as follows: it uses only local infor- mation for transmission scheduling. To understand the performance hit caused by local scheduling, we implemented seec without time slots (SEEC-WT), which makes global scheduling decisions on top of a CSMA MAC. For our topologies, the total energy consumed by dseec is comparable to that of SEEC-WT. The di®erence is highest (about 9%) for the deep-tree topology, where transmissions over multiple-hops can be better scheduled centrally. In practice, most wireless de- ployments more closely match cluster-tree or shallow-tree, where dseec performs extremely well. Overall, this experiment demonstrates the e®ect of network topology on power consumption and the bene¯ts of using a dynamic algorithm for compression de- cisions over statically always compressing data. dseec is able to determine this without any explicit information about the multi-hop path used for delivering the data to the sink or about the application data arrival process. In order to determine a priori that nodes should not compress data, a system would need complete infor- mation about dynamic quantities such as the network topology, the routing tree, the data arrival rate, and the wireless link qualities. ACM Journal Name, Vol. V, No. N, Month 20YY. 20 ¢ 0 10 20 30 40 50 60 Cluster−tree Shallow−tree Deep−tree System Power Consumption (mW) Transmit Receive Compress Fig.4. Energyconsumption: nevercompress (¯rst),SEEC-WT(second),DSEEC(third), always compress (fourth) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 5 10 15 20 25 30 35 Idle time (%) Power Savings (%) Fig. 5. Duty-cycling overhead Duty-cyclingoverhead. TheresultsshowninFigure4assumeideal duty-cycling, i.e., that the radio is turned o® whenever it is not transmitting or receiving. In practice, due to the overhead incurred by the duty-cycling protocols, the radio will remain in idle-state from time-to-time before it is turned o®. Figure 5 shows how the power savings achieved by dseec over always compress varies as a function of the overhead of duty-cycling. This overhead is measured by the fraction of the total experiment time that a node could have slept but stayed awake to determine if it was appropriate to sleep, averaged over all nodes. For a carefully-designed duty-cycling protocol, this overhead is topology-dependent but is typically about 1-1.5%(Burri et al. [2007]). As the ¯gure shows, dseec achieves 12-17% savings over the static decision in this regime. 6.3 Adapting to Dynamics In practice, a static compression decision is undesirable not only because of the detailedinformationneededformakingsuchadecision,butalsobecausethesystem needstoadapttochangesinwirelesschannelconditions,applicationtra±cdemand or platform settings. If the wireless link qualities and/or the application data rate change signi¯cantly, then a static decision would have to be recomputed. In contrast, dseec can dynamically adapt compression decisions. Change in link quality. To show that dseec can adjust to changes in wireless link quality we use the deep-tree topology and vary the quality of the link between node 13 and the sink node 1 in two experiments. In the ¯rst experiment, termed \good link", none of node 13's transmissions encounter channel errors. In the second experiment, termed \bad link", 20% of the transmissions by node 13 get corrupted and not received successfully at node 1 (requiring node 13 to retransmit these packets). TableIIIdepictsthefractionofpacketscompressed,bydseec,atallthenodesin thedeep-tree. Fornodesincluster3(nodes19,22-26),thereisasigni¯cantincrease in the fraction of packets compressed in the \bad" link scenario. The rest of the ACM Journal Name, Vol. V, No. N, Month 20YY. ¢ 21 Node ID Light Heavy Good link Bad link Clique 1 2-12,14 0 0 0 0 Clique 2 13 0 0 0 0 15 0 2.7% 0 0 16 0 0.35% 0 0 17 0 31% 0 0 18 0 39.7% 0 0 20 0 1.8% 0 0 21 0 76.8% 0 0.7% Clique 3 19 0 46.8% 0 0.5% 22 0 99.8% 0.1% 24.9% 23 0 99.8% 0.45% 34.3% 24 0 99.7% 3.2% 40.2% 25 0 99.3% 0 9.3% 26 0 42.4% 0.87% 2.4% TableIII.DSEEC,deep-tree: %ofpacketscompressed;changeinapplicationloadandlinkquality nodes did not compress any packet in either scenario. The packet drops at node 13 result in retransmissions at the MAC layer. Packet retransmissions decrease the e®ective transmission rate on the 13!1 link, which increases transmission energy cost for all nodes routing data through that link. This increase in transmission energy cost results in higher compression at nodes 22¡26 that are 3 or more hops away from the sink. However, the increase in transmission energy consumption is notlargeenoughtotriggerdatacompressionatnodesthatare1or2hopsawayfrom the sink. Note that dseec makes these decisions without any explicit information about the location of the \bad" link and the extent of packet drops on it. Compression for queue stability. If the application data rate at a node is greater than the rate R at which that node can transmit data to its parent, then, in order to maintain a stable queue size, the node should compress data to match the transmit rate. If R is small enough that every application packet must be compressed, or if compression is energy-e±cient, then the only possible decision is to compress every packet. Otherwise, when compression is not energy-e±cient and Rissmallerthantheapplicationratebutlargerthantherateresultingfromalways compressing the data, then a node can decide to compress a fraction of packets. Moreover, nodes further away from the sink should compress a greater fraction of packets than those closer to the sink. In the following simulations, we show that dseec is able to make such ¯ne-grain decisions dynamically on a per-packet basis. We steadily increased the application data rate at the nodes in the deep-tree topology until dseec started compressing each packet. We do this by decreasing the packet inter-arrival time. To illustrate dseec's dynamic adaptation to tra±c rates, we consider two data points: (1) Light load (300 milliseconds packet inter- arrival time), and (2) Heavy load (180 milliseconds packet inter-arrival time). Table III gives the percentage of application data compressed at each node for the Light and Heavy load scenarios. As the application tra±c load changes from light to heavy, nodes in cliques 2 and 3 start compressing data, and more packets are compressed by nodes in clique 3. Nodes in clique 1 do not compress data in the two scenarios. The level of data compression varies across di®erent nodes due to the following ACM Journal Name, Vol. V, No. N, Month 20YY. 22 ¢ 2 7 12 17 3 4 5 6 8 9 10 11 13 14 15 16 18 19 20 21 0 2 4 6 8 10 12 14 16 18 Node ID Avg. Queue backlog (# of packets) One hop away Two hops away Fig. 6. Avg. Queue backlog: Cluster-tree reasons. The 13 ! 1 link in the deep-tree topology is a bottleneck link for all the nodes in cliques 2 and 3 trying to send data to the sink. Nodes in clique 1 are outsidethetransmissionrangeofnodesinclique2and3. Hence,theirtransmission rateisnota®ectedbytheincreasedcontentionamongstthenodesinclique2and3 when the application data rate is increased. As a result, the application data rate that can be supported without compression is smaller for nodes in cliques 2 and 3 compared to the nodes in clique 1. This is the reason why nodes in clique 1 never compress the data in both scenarios. In the Heavy load scenario, nodes in clique 3 compress more data than nodes in clique 2 since they are further away from the sink, and hence, their data is transmitted over more hops as compared to the nodes in clique 2. Insight into dseec's adaptability. To understand why dseec is able to adapt to topology, link quality, and application data rate, without explicitly monitoring each of these, it helps to look at the heart of the algorithm: the queue backlog at a node determines the decisions made by dseec. As we discuss below, changing any of these quantities manifests itself as a change in the queue backlog, which drives the scheduling and compression decisions. dseec'stransmissionalgorithmusesthe di®erential queuebacklogonalink(be- tween a transmitter and its intended receiver) to schedule transmissions (Section 4, Equation (13)). Under this scheduling policy, the queue backlog at a node is di- rectlyproportionaltoitshop-countdistancefromthesink. Figure6depictsaverage queue sizes at the nodes in the cluster-tree topology, for dseec. The cluster head nodes 2;7;12, and 16 that are one hop away from the sink have an average queue backlog of 6 packets. The rest of the nodes that are two hops away from the sink haveanaveragequeuebacklogbetween12¡14packets. Withdseec'scompression algorithm (Section 4, Equation (12)), nodes with higher queue backlog are more likely to compress data. Hence, with dseec, nodes that are multiple hops away from the sink have a higher likelihood of compressing data (due to their larger queue sizes) compared to the nodes that are only 1 or 2 hops away . A degradation in link quality and/or increase in application data rate can also result in larger queue backlogs at all the nodes in network. An increase in queue backlogs (for example, when the application tra±c load changes from Light to ACM Journal Name, Vol. V, No. N, Month 20YY. ¢ 23 Heavy in Section 6.3) can cause dseec to change its compression decision. More- over,theimpactofanychangeinnetworkconditionswillalwaysbemoresigni¯cant atnodesthatarefartherawayfromthesinkbecausethesenodeshavealargerqueue backlog compared to the nodes that are closer to the sink. 12 7 17 2 3 4 5 6 8 9 10 11 13 14 15 16 18 19 20 21 0 0.2 0.4 0.6 0.8 1 Cluster−Tree: Node ID Fraction of packets compressed One hop Two hops Fig. 7. Di®erent Platform Di®erent Platform Settings. dseec does not need to be modi¯ed in order to support a di®erent platform or di®erent settings of the same platform. In these cases, all we need to do is to specify the compression and transmission model parameters (Á(a;k), m(a;k), P max ) to dseec, and it will adjust its decisions to optimize for that platform. Figure 7 shows the fraction of packets compressed by the di®erent nodes in the cluster-tree for the LEAP2 processor running at its lowest speed (104 Mhz) with the radio operating at 1 Mbps. While dseec did not compress any data packets earlier (cluster-tree, Figure 4), it now adapts its compression decision to optimize for the di®erent platform. Nodes that are 2 hops away from the sink compress almost all their data packets, while nodes that are 1 hop away compress fewer data packets. 6.4 Sensitivity to V The control parameter V impacts the energy-vs-delay trade-o®. As discussed in Section 4, increasing V reduces the system power consumption but increases the average queue backlog in the system. We have found (results omitted for brevity) that changing V even by 3 orders of magnitude results in a small impact on the total system power consumption. Moreover, relative to a static strategy like always compress, when V is varied from 10 3 to 3£10 6 , dseec consumes from 8.7% to 10% less power. Intuitively, this relative insensitivity results from using the same transmission decision strategy for all three compression strategies. However, changing V does have an impact on the compression decisions made at individualnodes. For V =10 3 andV =10 4 , nodes22¡26inclique3thatare3¡4 hops from the sink compress more than 90% of their data packets, while nodes in clique 2 that are 2¡3 hops from the sink compress between 50¡70% of their data packets. For V ¸ 8£10 5 , none of the nodes compress any data packets. Figure 8 shows the cumulative distribution of the average queue backlog at each node in the deep-tree for di®erent values of V. It is evident that increasing V leads to higher queue backlogs. For example, for V =10 6 and V =3£10 6 node 26 has an average ACM Journal Name, Vol. V, No. N, Month 20YY. 24 ¢ 0 10 20 30 40 50 60 70 80 90 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Avg. Queue Size (packets) CDF(x) Empirical CDF V=10e3 V=10e5 V=8x10e5 V=10e6 V=2x10e6 V=3x10e6 Fig. 8. Avg. Queue backlog vs V queue backlog equal to 29 and 86 packets, respectively. We obtained similar results for the cluster-tree and the shallow-tree topologies, and do not present them due to lack of space. 7. RELATED WORK Most relevant to our work is Neely's [2008] Lyapunov optimization based joint compression and transmission scheduling algorithm for a single node transmitting data to a base station over a single wireless hop. It does not consider energy consumption due to packet receptions, overhearing and duty-cycling overhead that must be accounted for in a multi-hop scenario, as we do. Also, that paper only presentsatheoreticalanalysisofthealgorithm. Inthispaper,weprovideanalytical boundsontheperformanceof seecandalsoevaluateadistributedversion,dseec, using simulations. Two other works related to ours, focusing on compression in wireless networks are Barr and Asanovic [2003], and Sadler and Martonosi [2006]. Speci¯cally, Barr and Asanovic [2003] consider a single wireless hop setting with ¯xed transmission cost (equivalent to static channel condition), and focus on estimating the commu- nication to computation energy ratios for several compression algorithms. Sadler andMartonosi[2006]consideramulti-hopstatic setting,wheretheyexperimentally demonstrate that compression can save signi¯cant amount of energy when data is transmitted over multiple hops. They also develop variants of popular compression algorithms more suited for sensor nodes with limited CPU processing speed and memory. In contrast to both these papers, we discuss a principled technique for making on-line compression and transmission decisions in a dynamic environment. Our approach, based on the Lyapunov optimization techniques, also enables us to provide performance guarantees for the centralized version of our algorithm. Several papers have considered the complementary problem of joint routing and datacompressioninwirelesssensornetworks(Pattemetal.[2006],Pattemetal.[2004], Dang et al. [2007]). The main focus of these papers is to design data correlation aware routing trees that enable nodes to achieve better compression e±ciency (via in-network aggregation and compression) compared to routing trees that are ag- ACM Journal Name, Vol. V, No. N, Month 20YY. ¢ 25 nostic to data correlation. A version of seec that incorporates compression after in-network data aggregation can be used on top of these joint routing and data compression schemes to enable the nodes to adapt their compression decisions to changes in the routing tree, in addition to link quality and application data rate. Lyapunov optimization based techniques have also been used to design stable algorithms that optimize di®erent performance metrics (Georgiadis et al. [2006]). For example, a joint transmit power allocation and transmission scheduling al- gorithm (EECA) that minimizes the system energy expenditure is discussed in Neely [2006]. seec's transmission scheduling algorithm is similar to EECA in its use of a di®erential queue backlog associated with links for making the transmis- sion decision. However, unlike that work, we consider the energy consumption due to data transmission, reception, and overhearing. Georgiadis et al. present Lyapunov optimization based algorithms that maximize the total throughput or achieve fair rate allocation across °ows in addition to achieving network stability (Georgiadis et al. [2006]). An alternative technique for designing algorithms that can optimize a perfor- mance metric and achieve network stability is discussed in Umut et al. [2008]. It uses primal-dual gradient descent techniques to design a joint scheduling and con- gestion control algorithm that also maximizes a utility function. Finally,severalrecentpapershaveusedbackpressurebasedtransmissionschedul- ingfordistributedcongestioncontrol(Sridharanetal.[2008],Warrieretal.[2007]). Thekeycontributioninthesepapersistodesignmechanismsthatenablebackpres- sure based scheduling for CSMA based MACs (802.11 and 802.15.4). As discussed in Section 5, their heuristics can be used in dseec. 8. CONCLUSIONS AND FUTURE WORK In this paper, we have described the design of seec, a stable energy-e±cient compression and scheduling algorithm for multi-hop wireless networks. seec can achievenear-optimalenergyperformance, anditsdistributedvariantdseec adapts to topology, link dynamics, platform settings, and application data rates without explicitly taking those factors into account. We intend to pursue several future directions, including extending seec to consider spatially correlated data, choice of routing paths, compression at intermediate nodes (not just at the source), mul- tiple radio transmit power levels, and bounded-distortion lossy compression. We also intend to evaluate these extensions on dseec, and also evaluate dseec more extensively, using multiple compression levels (K > 1), di®erent platforms, larger networks, and di®erent compression algorithms. REFERENCES 2008. Qualnet. http://www.scalable-networks.com/products. Barr, K. and Asanovi¶ c, K. 2003. Energy Aware Lossless Data Compression. In Proceedings of MobiSys. Burri, N., von Rickenbach, P., and Wattenhofer, R. 2007. Dozer: ultra-low power data gathering in sensor networks. In Proceedings of IPSN (2007-05-02). ACM, 450{459. Caron, D., Das, A., Dhariwal, A., Golubchik, L., Govindan, R., Kempe, D., Oberg, C., Sharma, A. B., Stauffer, B., Sukhatme, G., and Zhang, B. 2007. AMBROSia: An Au- tonomous Model-Based Reactive Observing System. In Proceedings of ICCS, Invited paper. ACM Journal Name, Vol. V, No. N, Month 20YY. 26 ¢ Ciancio, A., Pattem, S., Ortega, A., and Krishnamachari, B. 2006. Energy E±cient Data- Representation and Routing for Wireless Sensor Networks Based on a Distributed Wavelet Compression Algorithm. In Proceedings of the IPSN. D.P.BertsekasandA.NedicandA.E.Ozdaglar .2003. ConvexAnalysisandOptimization. Boston: Athena Scienti¯c. Dang, T., Bulusu, N., and chi Feng, W. 2007. RIDA: A Robust Information-Driven Data CompressionArchitectureforIrregularWirelessSensorNetworks. InProceedingsoftheEWSN. De Couto, D. S. J., Aguayo, D., Bicket, J., and Morris, R. 2003. A High-Throughput Path Metric for Multi-Hop Wireless Routing. In Proceedings of Mobicom. Hicks,J.,Paek,J.,Coe,S.,Govindan,R.,andEstrin,D.2008. AnEasilyDeployableWireless Imaging System. In Proceedings of ImageSense Workshop. L. Georgiadis and M. J. Neely and L. Tassiulas. 2006. Resource Allocation and Cross-Layer Control in Wireless Networks. Foundations and Trends in Networking. M. J. Neely. 2006. Energy Optimal Control for time varying wireless networks. IEEE Transac- tions on Information Theory 52(7), 2915{2934. Neely, M. J. 2003. Dynamic Power Allocation and Routing for Satellite and Wireless Networks with Time Varying Channels. Ph.D. thesis, Massachusetts Institute of Technology. Neely,M.J.2008. DynamicDataCompressionforWirelessTransmissionoveraFadingChannel. In Proceedings of the Conference on Information Sciences and Systems. Neely, M. J., Modiano, E., and Rohrs, C. E. 2003. Dynamic Power Allocation and Routing for Time Varying Wireless Networks. In Proceedings of the INFOCOM. Paek, J.,Gnawali, O.,Jang, K.-Y.,Nishimura, D.,Govindan, R.,Caffrey, J.,Wahbeh, M., and Masri, S. 2006. A Programmable Wireless Sensing System for Structural Monitoring. In Proceedings of the 4th World Conference on Structural Control and Monitoring(4WCSCM). Pattem, S., Krishnamachari, B., and Govindan, R. 2004. The Impact of Spatial Correlation on Routing with Compression in Wireless Sensor Networks. In Proceedings of the IPSN. Razvan, M.-E., Liang, C.-J., and Terzis, A. 2008. Koala: Ultra-Low Power Data Retrieval in Wireless Sensor Networks. In Proceedings of IPSN. Sadler, C. and Martonosi, M. 2006. Data Compression Algorithms for Energy-constrained devices in Delay Tolerant Networks. In Proceedings of the ACM Sensys. Sridharan, A., Moeller, S., and Krishnamachari, B. 2008. Investigating Backpressure based Rate Control Protocols for Wireless Sensor Networks. Tech. Rep. CENG-2008-7, University of Southern California. July. Srinivasan, K. and Levis, P. 2006. RSSI is Under Appreciated. In Proceedings of EmNets Workshop. Stathopoulos, T., McIntire, D., and Kaiser, W. J. 2008. The Energy Endoscope: Real-Time Detailed Energy Accounting for Wireless Sensor Nodes. In Proceedings of the IPSN. Umut, A., Andrews, M., Gupta, P., Hobby, J., Sanjee, I., and Stolyar, A. 2008. Joint scheduling and congestion control in mobile ad-hoc networks. In Proceedings of INFOCOM. Warrier,A.,Le,L.,andRhee,I.2007. Cross-layeroptimizationmadepractical. InProceedings of Broadnets, Invited paper. Ye,W.,Silva,F.,andHeidemann,J.2006. Ultra-LowDutyCycleMACwithScheduledChannel Polling. In Proceedings of the ACM Sensys. ACM Journal Name, Vol. V, No. N, Month 20YY. ¢ 27 A. ALGORITHM DERIVATION We provide the proof for Lemma 1 Proof. From (14) for each node n, we have: 1 2 (Un(t+1)) 2 = 1 2 (max[Un[t]¡ X l2 n ~ ¹ l [t];0]+Rn[t]+ X l2£ n ~ ¹ l [t]) 2 · 1 2 [(U n (t)) 2 +( X l2 n ~ ¹ l [t]+ X l2£ n ~ ¹ l [t]) 2 ] + 1 2 (Rn[t]) 2 ¡Un[t] X l2 n ~ ¹ l [t]¡ X l2£ n ~ ¹ l [t] ! +U n [t]R n [t] and hence, the one-step Lyapunov drift for U n [t] is: ¢(U n [t]) · B¡U n [t]E ( X l2 n ~ ¹ l [t]¡ X l2£ n ~ ¹ l [t]jU n [t] ) +U n [t]EfR n [t]jU n [t]g (33) with the constant B de¯ned as follows. B M = (Rmax +¹ in max ) 2 +(¹ out max ) 2 R max M = max n (EfR n [t]g) ¹ in max M = max (n;s2S;P2P) X l2£ n ~ ¹ l [t] ¹ out max M = max (n;s2S;P2P) X l2 n ~ ¹ l [t] Using (33), the one-step Lyapunov drift for our system is ¢( ~ U(t))· BN¡ N X n=1 Un[t]E ( X l2 n ~ ¹ l [t]¡ X l2£ n ~ ¹ l [t]j ~ U[t] ) + N X n=1 U n [t]E n R n [t]j ~ U(t) o =BN¡ X n E ( X l2n ~ ¹ l [t]U l [t]j ~ U(t) ) + N X n=1 Un[t]E n Rn[t]j ~ U(t) o where for link l = (n;^ n), U l [t] is the di®erential queue backlog, i.e. U l [t] = U n [t]¡U ^ n [t]. ACM Journal Name, Vol. V, No. N, Month 20YY. 28 ¢ B. PROOF: LEMMA 2 The claims in Lemma 2 can be separated into two parts: the ¯rst part claims the existence of a stationary randomized algorithm that makes compression decisions basedonlyonA n [t]and ~ S[t](andindependentofthequeuebacklogs), andachieves the minimum possible energy consumption for compression, h ¤ n (r n ), at each node; the second part claims that the minimum possible energy consumption due to data transmissions is also achievable by a stationary randomized algorithm that makes transmit power allocation decisions based only on A n [t] and ~ S[t]. In the following proof, we ¯rst show the existence of a stationary randomized algorithm for making compression decision k n [t] that achieves energy consumption h ¤ n (r n ) at each node (Part 1), and then prove a similar result for transmit power allocation ~ P[t] (Part 2). Proof. Part 1. The function h ¤ n (r n ) is de¯ned as the in¯mum of EfE n C [t]g over all stationary randomized policies for making the compression decision that yieldEfR n [t]g· r n . This de¯nition implies that there exists an in¯nite sequence of stationary randomized policies, indexed by i2f1;2;::::g that satisfy: E n R (i) n [t] o ·r n 8 i2f1;2;:::g (34) lim i!1 E n E n(i) C [t] o =h ¤ n (r n ) (35) Eachstationaryrandomizedpolicy iischaracterizedbyacollectionofprobabilities (° (i) a;k ) where a = A n [t] · m and k 2 K. These probabilities can be viewed as a ¯nite dimensional vector belonging to a compact set ¨ de¯ned by the constraints ° (i) a;k ¸08 a;k; X k ° (i) a;k =18 k: Itfollowsfromthepropertiesofacompactsetthatthein¯nitesequencef(° (i) a;k )g 1 i=1 contains a convergent subsequence that converges to (° ¤ a;k )2 ¨. The probabilities (° ¤ a;k ) de¯ne a stationary randomized policy k ¤ [t] with expectationsEfR ¤ n [t]g and EfE n¤ C [t]g. TheexpectationsEfR n [t]gandEfE n C [t]garelinearfunctionsofproba- bilities (° a;k ) as shown in equations (24)-(25). Hence, the properties (34)-(35) hold in limit yieldingEfR ¤ n [t]g· r andEfE n¤ C [t]g = h ¤ n (r). IfEfR ¤ n [t]g = r, then the proof is complete. IfEfR ¤ n [t]g<r, we can de¯ne an alternate policy k 0 (t) that chooses policy k ¤ [t] with probability µ and with probability (1¡µ) chooses not to compress data. The value µ is chosen in such a way that E n R 0 n [t] o =µEfR ¤ n [t]g+(1¡µ)bEfAn[t]g=r: The stationary randomized policy k 0 [t] cannot use more energy than the pol- icy k ¤ [t] (because it consumes no energy when it chooses not to compress data with probability (1¡µ)), and henceE n E n 0 C [t] o · h ¤ n (r n ). But h ¤ n (r n ) is the in¯- mum of energy consumption over all possible stationary randomized policies with compressed data output rate at most r n . This implies that h ¤ n (r n )·E n E n 0 C [t] o . ACM Journal Name, Vol. V, No. N, Month 20YY. ¢ 29 Hence,E n E n 0 C [t] o =h ¤ n (r n ), and thus we have proved the existence of a stationary randomized policy k 0 [t] satisfying the constraints (26)-(27) in Lemma 2. Part 2. Our claim that there exists a stationary randomized algorithm that makes transmit power allocation based only on A n [t] and ~ S[t] such that: Ef¹ l [t]g = f l 8 l2L X l2L E n ~ P l¤ [t] o = g ¤ ( ~ f) follows from the Graph Family Achievability lemma (Lemma 8, Chapter 4.3.2) in Neely's PhD thesis (Neely [2003]). C. PROOF: THEOREM 1 Our proof of Theorem 1 is a generalization of the proof of a similar theorem proved inNeely[2008]. Neelyconsidersasinglenodewantingtodeliverdatatoasinkover a wireless link while our proof holds for the general case of multiple nodes trying to deliver data to a sink over multiple hops. Consider a policy that stabilizes the queues at all the nodes in the network. Let ~ k[t]=fk n [t]g n2N and ~ P tran [t]=fP l tran [t]g l2L bethecompressionandtransmission power decisions for this policy where k n [t] 2 K for all n;t and P tran [t] 2 P for all t. Let R n [t] and P n comp [t] be the compression module output process and the power expenditure, respectively, at node n due to compression decision k n [t]. Let ¹ l [t]=C l ( ~ P tran [t]; ~ S[t]) be transmission rate process for link l. We want to prove that Ptot M = liminf t!1 1 t t¡1 X ¿=0 N X n=1 E n tot [¿] (36) = liminf t!1 1 t t¡1 X ¿=0 N X n=1 E n C [¿]+ X l2 n (E l T [¿]+E l R [¿]+E l B [¿])I(P l tran [¿]) ! (37) = liminf t!1 1 t t¡1 X ¿=0 0 @ N X n=1 P n comp [¿]+ X l=(n;^ n)2L ~ P l [¿] 1 A (38) ¸ P ¤ av (39) where ~ P l [¿]=P l tran [¿]+I(P l tran [¿])[1+®(j n j¡1)]P recv foreachlinkl =(n;^ n),P ¤ av is the minimum time average power consumption required to stabilize the queues (de¯ned in Theorem 1). Equation (37) follows from (9), and (38) follows from (6)-(8). The following two lemmas are needed for our proof. Lemma 3 If there exist vectors of constants ~ r = fr n g n2N and ~ P c = fP n c g n2N ACM Journal Name, Vol. V, No. N, Month 20YY. 30 ¢ with an in¯nite sequence of timeslotsft i g 1 i=1 such that: lim i!1 1 t i ti¡1 X ¿=0 EfR n [¿]g=r n (40) lim i!1 1 t i ti¡1 X ¿=0 E © P n comp [¿] ª =P n c (41) then P n c ¸h ¤ n (r) for all n2N, and hence P N n=1 P n c ¸ P N n=1 h ¤ n (r). Proof. See Appendix D. Lemma 4 If there exist vector of constants ~ w =fw l g l2L and ~ P tr =fP l tr g l2L with an in¯nite sequence of timeslotsft i g 1 i=1 such that: lim i!1 1 t i ti¡1 X ¿=0 Ef¹ l ([¿]g=w l (42) lim i!1 1 t i ti¡1 X ¿=0 E © P l [¿] ª =P l tr (43) then P l2L ~ P l ¸g ¤ (~ w), where for ~ P l =P l tr +P recv (1+®(j n j¡1))I(P l tr ) for each link l =(n;^ n). Proof. See Appendix E. Let the lim inf total power expenditure P tot de¯ned in equation (36) be achieved over an in¯nite sequence of timeslots ft i g 1 i=1 , i.e. liminf i!1 1 t i ti¡1 X ¿=0 N X n=1 E n tot [¿]=P tot (44) For any timeslot t, we de¯ne: ^ R n [t]= 1 t t¡1 X ¿=0 EfR n [¿]g; ^ P n comp [t]= 1 t t¡1 X ¿=0 E © P n comp [¿] ª ^ ¹ l [t]= 1 t t¡1 X ¿=0 Ef¹ l [¿]g; ^ P l tran [t]= 1 t t¡1 X ¿=0 E © P l tran [¿] ª Note that for all timeslots t, we have: 0· ^ R n [t]·bEfA[t]g; 0· ^ P n comp [t]·Á max (45) 0· ^ ¹ l [t]·¹ max ; 0· ^ P l tran [t]·P max (46) We de¯ne a 2£(N +jLj) dimensional vector as follows: V[t i ]= ³ f ^ R n [t i ]; ^ P n comp [t i ]g n2N ;f^ ¹ l [t i ]; ^ P l tran [t i ]g l2L ´ Itfollowsfromconstraints(45)and(46)thatthevectorsfV[t i ]g 1 i=1 formanin¯nite sequenceina2£(N+jLj)dimensionalcompactset,andhence,thisin¯nitesequence ACM Journal Name, Vol. V, No. N, Month 20YY. ¢ 31 a convergent subsequence. Let f ~ t i g 1 i=1 represent this convergent subsequence of timeslots such that there exists a vector of constants v = ¡ fr n ;P n c g n2N ;fw l ;P l tr g l2L ¢ satisfying: lim i!1 1 ~ t i ~ ti¡1 X ¿=0 EfR n [¿]g = r n 8 n2N lim i!1 1 ~ t i ~ t i ¡1 X ¿=0 E © P n comp [¿] ª = P n c 8 n2N lim i!1 1 ~ t i ~ ti¡1 X ¿=0 Ef¹ l [¿]g = w l 8 l2L lim i!1 1 ~ t i ~ ti¡1 X ¿=0 E © P l tran [¿] ª = P l tr 8 l2L Thus, we have vectors ~ r =fr n g n2N and ~ P c =fP n c g n2N , and an in¯nite sequence of timeslotsf ~ t i g 1 i=1 satisfying constraints (40) and (41). Hence, from Lemma 3, we have X n2N P n c ¸ X n2N h ¤ n (r n ) Similarly, for vectors ~ w =fw l g l2L and ~ P tr =fP l tr g l2L , from Lemma 4, we have X l2L ~ P l ¸g ¤ (~ w) where ~ P l =P l tr +P recv (1+®(j n j¡1))I(P l tr ). Due to the fact that f ~ t i g 1 i=1 is an in¯nite subsequence of the original sequence ft i g 1 i=1 , from equation (36), we have P tot = liminf i!1 1 ~ t i ~ t i ¡1 X ¿=0 N X n=1 E n tot (¿) = X n2N P n c + X l2L ~ P l ¸ X n2N h ¤ n (r n )+g ¤ (w) (47) To compare the P ¤ av value against the value on the right hand side of the inequality (47), we use the fact that the queues are stable. Since the queues are stable for rate vector ~ r, it follows from the de¯nition of the network capacity region ¤ that ~ r2¤. Furthermore, fromthediscussionontheconditionsforachievingstabilityin Section 4.2, ~ r2¤ implies that there exist variables ~ f =(f l ) l2L de¯ning a network °ow(satisfyingconstraints(21)-(23)). Since,thetransmissionschedulingalgorithm achieves rate w l on link l, in order to achieve stability, we must have f l ·w l for all ACM Journal Name, Vol. V, No. N, Month 20YY. 32 ¢ l2L. This implies g ¤ ( ~ f)·g ¤ (~ w). Therefore, from constraint (47), we have P tot ¸ X n2N h ¤ n (r n )+g ¤ (~ w) ¸ X n2N h ¤ n (r n )+g ¤ ( ~ f) (48) Using the de¯nition of r n min , it is straight forward to show that r n min ·r n ·bEfA[t]g 8 n2N (49) Since P ¤ av is de¯ned as the minimum time average power expenditure required to stabilize a network with compression module output rates fr n g n2N (see Theorem 1), it follows that P tot ¸ X n2N h ¤ n (r n )+g ¤ (w)¸P ¤ av (50) We have shown that the inequality (50) holds for any joint compression and trans- mission power control algorithm that stabilizes all the queues in the network, and hence, proved the claim in Theorem 1. D. PROOF OF LEMMA 3 We prove the Lemma 3 in this section. Proof. Assume that equations (40) and (41) are satis¯ed for all n2N. For all timeslots t, using iterated expectations, for all n2N, we have EfR n [t]g = EfEfR n [t]jA n [t];k n [t]gg = Efm(A n [t];k n [t])g E © P n comp [t] ª = E © E © P n comp [t]jA n [t];k n [t] ªª = EfÁ(A n [t];k n [t])g Equations (40) and (41) can be re-written as lim i!1 1 t i ti¡1 X ¿=0 Efm(A n [¿];k n [¿])g=r n (51) lim i!1 1 t i t i ¡1 X ¿=0 EfÁ(A n [¿];k n [¿])g=P n c (52) For any timeslot t, we can write 1 t t¡1 X ¿=0 Efm(A n [¿];k n [¿])g = m X a=0 X k2K 1 t t¡1 X ¿=0 m(a;k)p A (a)Pr[k n [¿]=kjA n [¿]=a)] = m X a=0 X k2K m(a;k)p A (a)° a;k [t] (53) ACM Journal Name, Vol. V, No. N, Month 20YY. ¢ 33 where the probabilities (° a;k [t]) are de¯ned as ° a;k [t] M = 1 t t¡1 X ¿=0 Pr[k n [¿]=kjA n [¿]=a)] Similarly, for any timeslot t, we can write 1 t t¡1 X ¿=0 EfÁ(A n [¿];k n [¿])g= m X a=0 X k2K Á(a;k)p A (a)° a;k [t] (54) It is straightforward to show that (° a;k [t]) satis¯es the following constraints for all timeslots t. X k2K ° a;k [t]=1 8 a (55) ° a;k [t]¸0 8 a;k (56) By substituting (53) and (54) into equations (51) and (52), we get lim i!1 1 t i m X a=0 X k2K m(a;k)p A (a)° a;k [t i ]=r n (57) lim i!1 1 t i m X a=0 X k2K Á(a;k)p A (a)° a;k [t i ]=P n c (58) The constraints (55)-(56) imply that the probability vector (° a;k [t]) is contained in a ¯nite dimensional compact set ¨ for all timeslots t. It follows that the vec- tors f° a;k [t i ]g i=1 1 form an in¯nite sequence in a compact set, and hence, there must exist a (in¯nite) subsequence of timeslots f ~ t j g 1 j=1 for which the subsequence f° a;k [ ~ t j ]g 1 j=1 converges to a point (° ¤ a;k ) in the set ¨. The following constraints must hold for (° ¤ a;k ). ° ¤ a;k 8 a;k (59) X k2K ° ¤ a;k =1 8 a (60) m X a=0 X k2K m(a;k)p A (a)° ¤ a;k =r n (61) m X a=0 X k2K Á(a;k)p A (a)° ¤ a;k =P n c (62) Theconstraints(59)-(60)follow(° ¤ a;k )belongstothecompactset¨de¯nedby(55)- (56). The equalities (61)-(62) hold becausef ~ t j g 1 j=1 is a subsequence of the original sequence of timeslotsft i g 1 i=1 , and hence the limits in (57)-(58) are preserved when taken over the subsequencef ~ t j g 1 j=1 . It follows from the de¯nition of h ¤ n (r n ) (see De¯nition 1 in Section 4.2) that (° ¤ a;k ) and P n c satisfying constraints (59)-(62) form a particular a solution of the optimization problem of De¯nition 1. Therefore, P n c ¸h¤ n (r n ) because h ¤ n (r n ) is ACM Journal Name, Vol. V, No. N, Month 20YY. 34 ¢ the in¯mum value over all solutions. Since P n c ¸ h ¤ n (r n ) holds for all n2N, this also implies P n2N P n c ¸ P n2N h ¤ n (r n ). E. PROOF OF LEMMA 4 Assume that equations (42) and (43) hold for all links l 2L. As in the proof for Lemma 3, for all links l2L, we can use iterated expectations to write Ef¹ l [¿]g = E n E n ¹ l [¿]j ~ S[¿]=~ s oo = E n E n C l ( ~ P[¿];~ s)j ~ S[¿]=~ s oo = X ~ s2S ¼ ~ s E n C l ( ~ P[¿];~ s)j ~ S[¿]=~ s o (63) Similarly, we can use iterated expectations to write E © P l tran [¿] ª = X ~ s2S ¼ ~ s E n P l tran [¿]j ~ S[¿]=~ s o (64) By substituting (63) and (64) into equations (42) and (43), respectively, we have the following equalities for any timeslot t 1 t t¡1 X ¿=0 Ef¹ l [¿]g= X ~ s2S ¼ ~ s ¹ ~ s l [t] (65) 1 t t¡1 X ¿=0 E © P l tran [¿] ª = X ~ s2S ¼ ~ s P ~ s l [t] (66) where for each ~ s2S, ¹ ~ s l [t] and P ~ s l [t] are de¯ned for all l2L as ¹ ~ s l [t] M = 1 t t¡1 X ¿=0 E n C l ( ~ P[¿];~ s)j ~ S[¿]=~ s o P ~ s l [t] M = 1 t t¡1 X ¿=0 E n P l tran [¿]j ~ S[¿]=~ s o For each channel state~ s and timeslot t, the values (¹ ~ s l [t];P ~ s l [t]) l2L de¯ned above belong to convex hull of the set ¨ ~ s de¯ned as follows: ¨ ~ s M = f(¹ l ;P l ) l2L j ~ P =(P l ) l2L 2P;¹ l =C l ( ~ P;~ s)g The set ¨ ~ s is jLj£2-dimensional. For each l 2 L, ^ ¹ ~ s l and ^ P ~ s l are real valued, and hence, we can think of the set ¨ ~ s as consisting of vectors inR 2jLj . It follows from Carath¶ eodory's theorem [2003] that any element (¹ ~ s l ;P ~ s l ) l2L contained in the convex hull of ¨ ~ s can be represented as a convex combination of at most 2jLj+1 elements of ¨ ~ s . Thus, there exist (transmit) power vectors ~ P ~ s j [t]=(P ~ s l;j [t]) l2L 2P ACM Journal Name, Vol. V, No. N, Month 20YY. ¢ 35 and probabilities (® ~ s j [t]) for j =1;2;:::;2jLj+1 such that for each link l2L: ^ P ~ s l [t] = 2jLj+1 X j=1 ® ~ s j [t]P ~ s l;j [t] (67) ^ ¹ ~ s l [t] = 2jLj+1 X j=1 ® ~ s j [t]C l ( ~ P ~ s j [t];~ s) (68) 2jLj+1 X j=1 ® ~ s j [t] = 1 (69) ® ~ s j [t] ¸ 0 8 j (70) The constraints (69) and (70) imply that the probabilities (® ~ s j [t]) are a vector of values contained in a ¯nite (2jLj+1) dimensional compact set for all timeslots t. Similarly, for all timeslots t, the power vector ~ P ~ s j [t]=(P ~ s l;j [t]) l2L corresponding to theprobabilities(® ~ s j [t])belongstothecompactsetP forallj2f1;2;:::;2jLj+1g. For the in¯nite sequence of timeslots ft i g 1 i=1 , for each j = 1;2;:::;2jLj + 1, the probability vectorsf(® ~ s j [t i ])g 1 i=1 form an in¯nite sequence contained in a com- pact set ¨. Hence, there exists a subsequence of timeslots f ~ t i g 1 i=1 for which the subsequence f(® ~ s j [ ~ t i ])g 1 i=1 converges to a probability vector (~ ® ~ s j ) 2 ¨ for each j = 1;2;:::;2jLj+1. Similarly, for each j = 1;2;:::;2jLj+1, the subsequence of (transmit) power vectors ~ P ~ s j [ ~ t i ] = (P ~ s l;j [ ~ t i ]) l2L converges to a transmit power vector ~ ^ P ~ s j =( ^ P ~ s l;j ) l2L belonging toP. Using equations (67) and (68), and the properties of convergent sequences, for the subsequence of timeslotsf ~ t i g 1 i=1 , we can write lim i!1 P ~ s l [ ~ t i ] = 2jLj+1 X j=1 ~ ® ~ s j ^ P ~ s l;j (71) lim i!1 ¹ ~ s l [ ~ t i ] = 2jLj+1 X j=1 ~ ® ~ s j C l ( ~ ^ P ~ s l ;~ s) (72) where equation (72) follows from the fact that the rate-power functions C l (p;s) are continuous, and hence, lim i!1 C l ( ~ P ~ s j [t i ];~ s)=C l (lim i!1 ~ P ~ s j [t i ];~ s)=C l ( ~ ^ P ~ s j ;~ s) ACM Journal Name, Vol. V, No. N, Month 20YY. 36 ¢ Using equations (43), (66), (67), and (71), for each link l2L, we can write P l tr = lim i!1 1 t i ti¡1 X ¿=0 E © P l tran [¿] ª = lim i!1 1 ~ t i ~ ti¡1 X ¿=0 E © P l tran [¿] ª (73) = lim i!1 X ~ s2S ¼ ~ s P ~ s l [ ~ t i ] = lim i!1 X ~ s2S ¼ ~ s 0 @ 2jLj+1 X j=1 ® ~ s j [ ~ t i ]P ~ s l;j [ ~ t i ] 1 A = X ~ s2S ¼ ~ s 0 @ 2jLj+1 X j=1 ~ ® ~ s j ^ P ~ s l;j 1 A (74) where the equality (73) holds because f ~ t i g 1 i=1 is an in¯nite subsequence of the original sequence of timeslotsft i g 1 i=1 , and hence the limit is preserved when taken over this subsequence. Similarly, using equations (42), (65), (68), and (72), for each l2L, we have w l = lim i!1 1 t i t i ¡1 X ¿=0 E © ¹ l tran [¿] ª = X ~ s2S ¼ ~ s 0 @ 2jLj+1 X j=1 ~ ® ~ s j C l ( ~ ^ P ~ s j ;~ s) 1 A (75) We can use the probability vector (~ ® ~ s j ), j =1;:::;2jLj+1, to de¯ne a stationary randomized policy that chooses transmit power vector ~ ^ P ~ s j during any timeslot t with probability ~ ® ~ s j . It follows from (73)-(74), and (75) that this policy achieves an expectedtransmissionrate w l foreachlinkl2Lwithanexpectedpowerconsump- tion ~ P l associated link l = (n;^ n), where ~ P l = P l tr +P recv (1¡®+®j n j). Thus, (® ~ s j ) and ( ~ P l ) form a particular solution to the optimization problem of De¯nition 2. Since g ¤ (~ w) is de¯ned as the in¯mum value of the total transmission power consumption over all stationary randomized policies that satisfy the constraints in De¯nition 2, it follows that P l2L ~ P l ¸g ¤ (~ w). F. ALGORITHM PERFORMANCE In this section, we prove Theorem 2. Proof. The following lemma from Georgiadis et al. [2006] is needed for our proof. Lemma 5 Let L( ~ U[t]) be a non-negative function of ~ U[t] with Lyapunov drift ¢( ~ U[t]) de¯ned in (16). If there are stochastic processes °[t] and ¯[t] such that for every timeslot t and for all possible values of ~ U[t], the Lyapunov drift satis¯es: ¢( ~ U[t])·E n ¯[t]¡°[t] j ~ U[t] o (76) then, limsup t!1 1 t t¡1 X ¿=0 Ef°[t]g·limsup t!1 1 t t¡1 X ¿=0 Ef¯[t]g ACM Journal Name, Vol. V, No. N, Month 20YY. ¢ 37 seec minimizes the right hand side of the Lyapunov drift expression (19). Hence, for any other (possibly randomized) algorithm that makes compression decisions k ¤ n [t] and chooses transmit power P ¤ tran [t], we have: ¢( ~ U(t))+V N X n=1 E n E n tot [t]j ~ U(t) o · BN¡ X n E ( X l2 n ~ ¹ l [t]U l [t]¡V ^ P[t]j ~ U(t) ) + N X n=1 E n U n [t]m(A n [t];k ¤ n [t])+VÁ(A n [t];k ¤ n [t])j ~ U[t] o (77) with ^ P[t]=P l¤ tran [t]+I(P l¤ tran [t])[1+®(j n j¡1)]P recv . Let ~ r = (r n ), and ~ f = (f l ) l2L represent compressed data rate and transmission rate vector de¯ning a network-°ow (satisfying equations (21)-(23)). Let k ¤ n [t] and ~ P ¤ n [t] represent the compression and transmit power allocations made by a station- ary randomized policy that yields: EfE n C [t]g=EfÁ(A n [t];k ¤ n [t])g = h ¤ n (r)8 n (78) EfRn[t]g=Efm(An[t];k ¤ n [t])g = rn (79) X l2L E n ~ P l¤ [t] o = g ¤ ( ~ f) (80) Ef¹ l [t]g = f l 8 l2L (81) Lemma 2 guarantees the existence of such a policy. The decision k ¤ n [t] and ~ P ¤ [t] made by a stationary randomized algorithm are independent of the queue backlogs ~ U[t]. Thus, the expectations in (78)-(81) are the same when conditioned on ~ U[t]. Substituting (78)-(81) into the right hand side of (77), and re-arranging the terms, we get: ¢( ~ U(t)) + V N X n=1 E n E n tot [t]j ~ U(t) o · BN¡ X n U n [t] ( X l2n f l ¡ X l2£n f l )¡r n ! +V X n (h ¤ n (r n ))+g ¤ ( ~ f) ! (82) The above inequality holds for all ~ r and ~ f. In particular, it also holds for the ~ r ¤ and ~ f ¤ thatoptimize(28)-(29)inTheorem1suchthatP ¤ av = P n (h ¤ n (r ¤ n ))+g ¤ ( ~ f ¤ ), and P l2 n f ¤ l ¡ P l2£ n f ¤ l =r ¤ n . Plugging this into (82), we have: ¢( ~ U(t))+V N X n=1 E n E n tot [t]j ~ U(t) o ·BN +VP ¤ av (83) Using the drift inequality (83) in Lemma 5 with °[t] = V( P n E n tot [t]) and ¯[t] = BN +VP ¤ av yields P tot ·P ¤ av +BN=V. Similarly, choosing ~ r = (r n min ) and the corresponding °ow vector ~ f such that ACM Journal Name, Vol. V, No. N, Month 20YY. 38 ¢ P l2 n f l ¡ P l2£ n f l =r n min +², from (83), we get ¢( ~ U(t)) + V N X n=1 E n E n tot [t]j ~ U(t) o · BN¡ X n U n [t]²+V Ã X n (h ¤ n (r n min ))+g ¤ ( ~ f) ! (84) Since E n tot ¸0 for all n, from (84), we have ¢( ~ U(t))·BN¡ X n U n [t]²+V Ã X n (h ¤ n (r n min ))+g ¤ ( ~ f) ! (85) From the de¯nition of Á max , we have h ¤ n (r n min ) · Á max for all n. Also, since P l tran [t] · P max for all links l during any timeslot t, we have g ¤ ( ~ f) · jLj ~ P max where ~ P max =P max +P recv (1¡®+®j max j) with max M = max n n . For the data collectionusingaroutingtreescenarioconsideredinthispaper, eachnodehasonly one outgoing link, and hence, g ¤ ( ~ f)· N ~ P max . Substituting the upper bounds on h ¤ n (r n min and g ¤ ( ~ f) in (86), we get ¢( ~ U(t))·BN¡ X n U n [t]²+VN(Á max + ~ P max ) (86) Using the drift inequality (87) in Lemma 5 with °[t] = P n U n [t]² and ¯[t] = BN +V(NÁ max +jLjP max ), we get X n U n · BN +VN(Á max + ~ P max ) ² (87) The average total queue backlog bound in (87) holds for any value of ² > 0. A particular choice of ² a®ects only the bound and does not a®ect the compression and transmit power allocation decisions in seec or any the sample path of system dynamics. Hence, we can minimize the bound in (87) by choosing the largest feasible ² such that r n min +²2¤ (de¯ned as ² max yielding: X n U n · BN +VN(Á max + ~ P max ) ² max ACM Journal Name, Vol. V, No. N, Month 20YY.
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Abhishek B. Sharma, Leana Golubchik, Ramesh Govindan and Michael J. Neely. "Dynamic data compression in multi-hop wireless networks." Computer Science Technical Reports (Los Angeles, California, USA: University of Southern California. Department of Computer Science) no. 905 (2009).
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Dynamic data compression in multi-hop wireless networks (
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