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Maximizing delay and minimizing life time of wireless sensor with anycast.


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Maximizing delay and minimizing life time of wireless sensor with anycast.

  1. 1. 1 Minimizing Delay and Maximizing Lifetime for Wireless Sensor Networks With Anycast Joohwan Kim, Student Member, IEEE, Xiaojun Lin, Member, IEEE, Ness B. Shroff, Fellow, IEEE, and Prasun Sinha Abstract—In this paper, we are interested in minimizing the systems, there are four main sources of energy consumption:delay and maximizing the lifetime of event-driven wireless sensor energy required to keep the communication radios on; energynetworks, for which events occur infrequently. In such systems, required for the transmission and reception of control packets;most of the energy is consumed when the radios are on, waitingfor an arrival to occur. Sleep-wake scheduling is an effective energy required to keep sensors on; and energy required formechanism to prolong the lifetime of these energy-constrained actual data transmission and reception. The fraction of totalwireless sensor networks. However, sleep-wake scheduling could energy consumption for actual data transmission and receptionresult in substantial delays because a transmitting node needs is relatively small in these systems, because events occurto wait for its next-hop relay node to wake up. An interesting so rarely. The energy required to sense events is usually aline of work attempts to reduce these delays by developing“anycast”-based packet forwarding schemes, where each node constant and cannot be controlled. Hence, the energy expendedopportunistically forwards a packet to the first neighboring node to keep the communication system on (for listening to thethat wakes up among multiple candidate nodes. In this paper, medium and for control packets) is the dominant componentwe first study how to optimize the anycast forwarding schemes of energy consumption, which can be controlled to extendfor minimizing the expected packet-delivery delays from the the network lifetime. Thus, sleep-wake scheduling becomessensor nodes to the sink. Based on this result, we then provide asolution to the joint control problem of how to optimally control an effective mechanism to prolong the lifetime of energy-the system parameters of the sleep-wake scheduling protocol constrained event-driven sensor networks. By putting nodesand the anycast packet-forwarding protocol to maximize the to sleep when there are no events, the energy consumption ofnetwork lifetime, subject to a constraint on the expected end- the sensor nodes can be significantly packet-delivery delay. Our numerical results indicate that Various kinds of sleep-wake scheduling protocols havethe proposed solution can outperform prior heuristic solutionsin the literature, especially under the practical scenarios where been proposed in the literature. Synchronized sleep-wakethere are obstructions, e.g., a lake or a mountain, in the coverage scheduling protocols have been proposed in [2]–[6]. In thesearea of wireless sensor networks. protocols, sensor nodes periodically or aperiodically exchange Index Terms—Anycast, Sleep-wake scheduling, Sensor net- synchronization information with neighboring nodes. How-work, Energy-efficiency, Delay ever, such synchronization procedure could incur additional communication overhead, and consume a considerable amount of energy. On-demand sleep-wake scheduling protocols have I. I NTRODUCTION been proposed in [7], [8], where nodes turn off most of Recent advances in wireless sensor networks have resulted their circuitry and always turn on a secondary low-poweredin a unique capability to remotely sense the environment. receiver to listen to “wake-up” calls from neighboring nodesThese systems are often deployed in remote or hard-to- when there is a need for relaying packets. However, this on-reach areas. Hence, it is critical that such networks operate demand sleep-wake scheduling can significantly increase theunattended for long durations. Therefore, extending network cost of sensor motes due to the additional receiver. In thislifetime through the efficient use of energy has been a key work, we are interested in asynchronous sleep-wake schedul-issue in the development of wireless sensor networks. In this ing protocols such as those proposed in [9], [10]. In thesepaper, we will focus on event-driven asynchronous sensor protocols, the sleep-wake schedule at each node is independentnetworks, where events occur rarely. This is an important of that of other nodes, and thus the nodes do not requireclass of sensor networks and has many applications such as either a synchronization procedure or a secondary low-powerenvironmental monitoring, intrusion detection, etc. In such receiver. However, because it is not practical for each node to have complete knowledge of the sleep-wake schedule of other This work has been partially supported by the National Science Foundation nodes, it incurs additional delays along the path to the sinkthrough awards CNS-0626703, CNS-0721477, CNS-0721434, CCF-0635202, because each node needs to wait for its next-hop node to wakeand ARO MURI Award No. W911NF-07-10376 (SA08-03). An earlier versionof this paper has appeared in the Proceedings of IEEE INFOCOM, 2008 [1]. up before it can transmit. This delay could be unacceptable for J. Kim and X. Lin are with School of Electrical and Com- delay-sensitive applications, such as fire detection or tsunamiputer Engineering, Purdue University, West Lafayette, IN 47907, USA alarm, which require that the event reporting delay be small.(Email:{jhkim,linx} N. B. Shroff is with Department of Electrical and Computer Engineering Prior work in the literature has proposed the use of anycastand Department of Computer Science and Engineering, The Ohio State packet-forwarding schemes to reduce this event reportingUniversity, Columbus, OH 43210 (Email: delay [11]–[15]. Under traditional packet-forwarding schemes, P. Sinha is with Department of Computer Science and Engineering,The Ohio State University, Columbus, OH 43210 (Email: prasun@cse.ohio- every node has one designated next-hop relaying node in neighborhood, and it has to wait for the next-hop node to
  2. 2. 1 1 1 1 rate of each node). Note that the latter will directly impact both network lifetime and the packet-delivery delay. Hence, to Source Sink optimally tradeoff network lifetime and delay, both the wake- K hops up rates and the anycast packet-forwarding policy should be (a) Data forwarding without anycast: the expected delay at jointly controlled. However, such interactions have not been each hop is λ . 1 systematically studied in the literature [11], [12], [15]. In this paper, we address these challenges. We first in- vestigate the delay-minimization problem: given the wake- up rates of the sensor nodes, how to optimally choose the Source Sink anycast forwarding policy to minimize the expected end-to- n candidate end delay from all sensor nodes to the sink. We develop a n 1 n 1 n 1 n 1 forwarders low-complexity and distributed solution to this problem. We then formulate the lifetime maximization problem: given a K hops constraint on the expected end-to-end delay, how to maximize (b) Data forwarding with anycast: the expected delay at each the network lifetime by jointly controlling the wake-up rates hop is nλ . 1 and the anycast packet-forwarding policy. We show how to useFig. 1. Example of anycast date-forwarding: anycast can reduce the expected the solution to the delay-minimization problem to constructone-hop delay and the expected end-to-end delay by n times. an optimal solution to the lifetime-maximization problem for a specific definition of network lifetime. Before we present the details of our problem formulationwake up when it needs to forward a packet. In contrast, under and the solution, we make a note regarding when the anycastanycast packet-forwarding schemes, each node has multiple protocols and the above optimization algorithms are relaying nodes in a candidate set (we call this set a We can view the lifetime of an event-driven sensor networksforwarding set). A sending node can forward the packet to the as consisting of two phases: the configuration phase and thefirst node that wakes up in the forwarding set. The example operation phase. When nodes are deployed, the configurationin Fig. 1 well illustrates the advantage of anycast in sensor phase begins, during which nodes optimize the control parame-networks with asynchronous sleep-wake scheduling. Assume ters of the anycast forwarding policy and their wake-up rates. Itthat each node wakes up independently according to a Poisson is during this phase that the optimization algorithms discussedprocess with a common rate λ, i.e., the intervals between every above will be executed. In this phase, sensor nodes do nottwo successive wake-up events of any given node are i.i.d. and even need to follow asynchronous sleep-wake patterns. Afterexponentially distributed with mean λ . In the figure, a packet 1 the configuration phase, the operation phase follows. In thegenerated at a source node needs to be relayed by K hops operation phase, each node alternates between two sub-phases,in order to reach a sink. Without anycast, the expected one- i.e., the sleeping sub-phase and the event-reporting sub-phase.hop delay is λ , and the expected end-to-end delay is K (see 1 λ In the sleeping sub-phase, each node simply follows the sleep-Fig. 1(a)). Now, if we employ an anycast packet-forwarding wake pattern determined in the configuration phase, waitingscheme, assume that each node has n nodes in the forwarding for events to occur. Note that since we are interested in asyn-set as shown in Fig. 1(b), i.e., each node has n candidates chronous sleep-wake scheduling protocols, the sensor nodesto be the next-hop node. Then, the expected one-hop delay do not exchange synchronization messages in this sleepingand the end-to-end delay decrease to nλ and nλ , respectively. 1 K sub-phase. Finally, when an event occurs, the informationHence, anycast reduces the event-reporting delay by n times needs to be passed on to the sink as soon as possible, whichin this example. becomes the event-reporting sub-phase. It is in this event Although anycast clearly reduces the event-reporting delay, reporting sub-phase when the anycast forwarding protocol isit leads to a number of challenging control and optimization actually applied, using the control parameters chosen duringproblems. The first challenge is for each node to determine its the configuration phase. Note that the configuration phaseanycast forwarding policy to minimize the end-to-end packet- only needs to be executed once because we assume that thedelivery delay. Note that in practice sensor networks are often fraction of energy consumed due to the transmission of datanot laid out with a layered structure as in Fig. 1(b). Therefore, is negligible. However, if this is not the case, the transmissioneach individual node must choose its anycast forwarding energy will play a bigger role in reducing the residual energypolicy (e.g., the forwarding set) distributively, with minimal at each node in the network. In this case, as long as the fractionknowledge of the global network topology. The existing of energy consumed due to data transmission is still small (butanycast schemes in the literature [11], [12], [15] address not negligible), the practical approach would be for the sink tothis problem using geographical information. However, these initiate a new configuration phase after a long time has passed.heuristic solutions do not minimize the end-to-end delay (We The rest of this paper is organized as follows. In Sec-provide comparisons in Section V). tion II, we describe the system model and introduce the delay- The second challenge stems from the fact that good perfor- minimization problem and the lifetime-maximization problemmance cannot be obtained by studying the anycast forwarding that we intend to solve. In Section III, we develop a distributedpolicy in isolation. Rather, it should be jointly controlled with algorithm that solves the delay-minimization problem. In Sec-the parameters of sleep-wake scheduling (e.g., the wake-up tion IV, we solve the lifetime-maximization problem using the 2
  3. 3. policy to maximize the network lifetime (see the discussion at the end of Section III-B). While the analysis in this paper focuses on the case when the wake-up times follow a Poisson process, we expect that the methodology in the paper can also be extended to the case with non-Poisson wake-up processes, with more technically-involved analysis.Fig. 2. System Model A well-known problem of using sleep-wake scheduling in sensor networks is the additional delay incurred in transmitting a packet from source to sink, because each node along thepreceding results. In Section V, we provide simulation results transmission path has to wait for its next-hop node to wakethat illustrate the performance of our proposed algorithm up. To reduce this delay, we use an anycast forwarding schemecompared to other heuristic algorithms in the literature. as described in Fig. 2. Let Ci denote the set of nodes in the transmission range of node i. Suppose that node i has II. S YSTEM M ODEL a packet and it needs to pick up a node in its transmission We consider a wireless sensor network with N nodes. Let range Ci to relay the packet. Each node i maintains a list ofN denote the set of all nodes in the network. Each sensor node nodes that node i intends to use as a forwarder. We call theis in charge of both detecting events and relaying packets. If set of such nodes as the forwarding set, which is denoteda node detects an event, the node packs the event information by Fi for node i. In addition, each node j is also assumedinto a packet, and delivers the packet to a sink s via multi- to maintain a list of nodes i that use node j as a forwarderhop relaying. We assume in this paper that there is a single (i.e., j ∈ Fi ). As shown in Fig. 2, node i starts sending asink, however, the analysis can be generalized to the case with beacon signal and an ID signal, successively. All nodes inmultiple sinks (see Section III-D). Ci hear these signals, regardless of whom these signals are We assume that the sensor network employs sleep-wake intended for. A node j that wakes up during the beacon signalscheduling to improve energy-efficiency. We first introduce or the ID signal will check if it is in the forwarding set ofa basic sleep-wake scheduling protocol as follows. For ease node i. If it is, node j sends one acknowledgement after theof exposition, in this basic protocol we assume that there is ID signal ends. After each ID signal, node i checks whethera single source that sends out event-reporting packets to the there is any acknowledgement from the nodes in Fi . If nosink. This is the most likely operating mode because when acknowledgement is detected, node i repeats the beacon-ID-nodes wake up asynchronously and with low duty-cycles, the signaling and acknowledgement-detection processes until itchance of multiple sources generating event-reporting packets hears one. On the other hand, if there is an acknowledgement,simultaneously is small. (We can extend this basic protocol it may take additional time for node i to identify whichto account for the case of collisions by multiple senders node acknowledges the beacon-ID signals, especially when(including hidden terminals) or by multiple receivers. The there are multiple nodes that wake up at the same time. Letdetailed protocol is provided in Section IV of our online tR denote the resolution period, during which time node itechnical report [16].) The sensor nodes sleep for most of identifies which nodes have sent acknowledgements. If therethe time and occasionally wake up for a short period of time are multiple awake nodes, node i chooses one node amongtactive . When a node i has a packet for node j to relay, it them that will forward the packet. After the resolution period,will send a beacon signal followed by an ID signal (carrying the chosen node receives the packet from node i during thethe sender information). Let tB and tC be the duration of the packet transmission period tP , and then starts the beacon-beacon signal and the ID signal, respectively. When node j ID-signaling and acknowledgement-detection processes to findwakes up and senses a beacon signal, it keeps awake, waiting the next forwarder. Since nodes consume energy when awake,for the following ID signal to recognize the sender. When tactive should be as small as possible. However, tactive has tonode j wakes up in the middle of an ID signal, it keeps be larger than tA because otherwise a neighboring node couldawake, waiting for the next ID signal. If node j successfully wake up after an ID signal and could return to sleep beforerecognizes the sender, and it is the next-hop node of node the next beacon signal. In this paper, we set tactive = tA soi, it then communicates with node i to receive the packet. that nodes cannot miss on-going beacon-ID signals and alsoNode j can then use a similar procedure to wake up its own can reduce the energy consumption for staying node. If a node wakes up and does not sense abeacon signal or ID signal, it will then go back to sleep.In this paper, we assume that the time instants that a node A. Anycast Forwarding and Sleep-Wake Scheduling Policiesj wakes up follow a Poisson random process with rate λj . In this model, there are three control variables that affectWe also assume that the wake-up processes of different nodes the network lifetime and the end-to-end delay experienced byare independent. The independence assumption is suitable for a packet: wake-up rates, forwarding set, and priority.the scenario in which the nodes do not synchronize their 1) Wake-up rates: The sleep-wake schedule is determinedwake-up times, which is easier to implement than the schemes by the wake-up rate λj of the Poisson process with which eachthat require global synchronization [3]–[5]. The advantage of node j wakes up. If λj increases, the expected one-hop delayPoisson sleep-wake scheduling is that, due to its memoryless will decrease, and so will the end-to-end delay of any routingproperty, sensor nodes are able to use a time-invariant optimal paths that pass though node j. However, a larger wake-up 3
  4. 4. rate leads to higher energy consumption and reduced network acknowledgement after the same ID signal, the source nodelifetime. i will pick the highest priority node among them as a next- In the rest of the paper, it is more convenient to work with hop node. Although only the nodes in a forwarding set needthe notion of awake probability which is a function of λj . priorities, we assign priorities to all nodes to make the prioritySuppose that node i sends the first beacon signal at time 0, as assignment an independent control variable from forwardingin Fig. 2. If no nodes in Fi have heard the first m − 1 beacon matrix A. Clearly, the priority assignments of nodes will alsoand ID signals, then node i transmits the m-th beacon and ID affect the expected delay. In order to represent the globalsignal in the time-interval [(tB + tC + tA )(m − 1), (tB + tC + priority decision, we next define a priority matrix B as follows:tA )(m−1)+tB +tC ]. For a neighboring node j to hear the m-th signals and to recognize the sender, it should wake up during B = [bij , i = 1, ..., N, j = 1, ..., N ][(tB + tC + tA )(m − 1) − tA − tC , (tB + tC + tA )m − tA − tC ]. We let B denote the set of all possible priority matrices.Therefore, provided that node i is sending the m-th signal, the Among the three control variables, we call the combinationprobability that node j ∈ Ci wakes up and hears this signal is of the forwarding and priority matrices (A, B) the anycast pj = 1 − e−λj (tB +tC +tA ) . (1) packet-forwarding policy (or simply an anycast policy) be- cause these variables determine how each node chooses itsWe call pj the awake probability of node j. It should be noted next-hop node. We also call the awake probability vectorthat, due to the memoryless property of a Poisson process, p j the sleep-wake scheduling policy because this variable affectsis the same for each beacon-ID signaling iteration, m.1 when each node wakes up. Note that there is a one-to-one mapping between the awakeprobability pj and the wake-up frequency λj . Hence, the B. Anycast Objectives and Performance Metricsawake probability is also closely related to both delay andenergy consumption. Let p = (pi , i ∈ N ) represent the global In this subsection, we define the performance objectives ofawake probability vector. the anycast policy and the sleep-wake scheduling policy that 2) Forwarding Set: The forwarding set Fi is the set of we intend to optimize. We remind the readers that, althoughcandidate nodes chosen to forward a packet at node i. In the sleep-wake patterns and the anycast forwarding policy areprinciple, the forwarding set should contain nodes that can applied in the operation phase of the network, their controlquickly deliver the packet to the sink. However, since the parameters are optimized in the configuration phase.end-to-end delay depends on the forwarding set of all nodes, 1) End-to-End Delay: We define the end-to-end delaythe optimal choices of forwarding sets at different nodes are as the delay from the time when an event occurs, to thecorrelated. We use a matrix A to represent the forwarding set time when the first packet due to this event is received atof all nodes collectively, as follows: the sink. We motivate this performance objective as follows: for applications where each event only generates one packet, A = [aij , i = 1, ..., N, j = 1, ..., N ] the above definition clearly captures the delay of reportingwhere aij = 1 if j is in node i’s forwarding set, and aij = the event information. For those applications where each0 otherwise. We call this matrix A the forwarding matrix. event may generate multiple packets, we argue that the eventReciprocally, we define Fi (A) as the forwarding set of node reporting delay is still dominated by the delay of the firsti under forwarding matrix A, i.e., Fi (A) = {j ∈ Ci |aij = 1}. packet. This is the case because once the first packet goesWe let A denote the set of all possible forwarding matrices. through, the sensor nodes along the path can stay awake for With anycast, a forwarding matrix determines the paths that a while. Hence, subsequent packets do not need to incur thepackets can potentially traverse. Let g(A) be the directed wake-up delay at each hop, and thus the end-to-end delay forgraph G(V, E(A)) with the set of vertices V = N , and the subsequent packets is much smaller than that of the firstthe set of edges E(A) = {(i, j)|j ∈ Fi (A)}. If there is a packet.path in g(A) that leads from node i to node j, we say that When there is only one source generating event-reportingnode i is connected to node j under the forwarding matrix A. packets, the end-to-end delay of the first packet can beOtherwise, we call it disconnected from node j. An acyclic determined as a function of the anycast policy (A, B) andpath is the path that does not traverse any node more than the sleep-wake scheduling policy p. One may argue that itonce. If g(A) has any cyclic path, we call it a cyclic graph, may be desirable to design protocols that can potentiallyotherwise we call it an acyclic graph. reduce the end-to-end delay by adjusting the anycast policy 3) Priority: Let bij denote the priority of node j from the dynamically after the event occurs, e.g., according to trafficviewpoint of node i. Then, we define the priority assignment density. However, this dynamic adjustment is not possibleof node i as bi = (bi1 , bi2 , · · · , biN ), where each node for the first packet, because when the first packet is beingj ∈ Ci is assigned a unique number bij from 1, · · · , |Ci |, and forwarded, the sensor nodes have not woken up yet. Hence, tobij = 0 for nodes j ∈ Ci . When multiple nodes send an / forward the first packet to the sink, the sensor nodes must use some pre-configured policies determined in the configuration 1 To hear the first ID signal, the neighboring node j should wake-up phase (We remind the readers about the discussion of differentduring [−tA , tB ], which results in a smaller awake probability pj = 1 − phases at the end of the introductory section.) After the firste−λj (tB +tA ) than (1). For simplicity of analysis, we can set the duration ofthe first beacon signal to tB + tC so that the awake probability is consistent packet is delivered to the sink, the sensor nodes along theat all beacon-ID signals. path to the sink have woken up. Thereafter, they are able to 4
  5. 5. adapt their control policies dynamically, e.g, according to the mW), which is only 1 percent of the energy consumption duetraffic density. In this paper, since we are mostly interested in to waking up.reducing the delay of the first packet, these dynamic adaption By introducing the power consumption ratio ei = µi /Qi ,policies are outside the scope of our paper. In other words, we can express the lifetime of node i aswe mainly focus on the optimization of the anycast and sleep- 1 tB + t C + t Awake scheduling policies at the initial configuration phase. Ti (p) = = 1 . (2) e i λi ei ln (1−pi ) Based on the preceding discussion, we define the end-to-enddelay as the delay incurred by the first packet. Given A, B, and Here we have used the definition of the awake probabilityp, the stochastic process with which the first packet traverses pi = 1 − e−λj (tB +tC +tA ) from (1).the network from the source node to the sink is completely We define network lifetime as the shortest lifetime of allspecified, and can be described by a Markov process with nodes. In other words, the network lifetime for a given awakean absorbing state that corresponds to the state that a packet probability vector p is given byreaches the sink. We define Di (p, A, B) as the expected end-to-end delay for a packet from node i to reach sink s, when T (p) = min Ti (p). (3) i∈Nthe awake probability vector p and anycast policy (A, B) are Based on the above performance metrics, we present thegiven. Since sink s is the destination of all packets, the delay of lifetime-maximization problem (which is the second problempackets from sink s is regarded as zero, i.e., Ds (p, A, B) = 0, we intend to solve in this paper) as follows:regardless of p, A, and B. If node i is disconnected from sink sunder the forwarding matrix A, packets from the node cannot (P) max T (p)reach sink s. In this case, the end-to-end delay from such a p,A,Bnode i is regarded as infinite, i.e., Di (p, A, B) = ∞. From subject to Di (p, A, B) ≤ ξ ∗ , ∀i ∈ Nnow on, we call ‘the expected end-to-end delay from node i p ∈ (0, 1]N , A ∈ A, B ∈ B,to sink s’ simply as ‘the delay from node i.’ where ξ ∗ is the maximum allowable delay. The objective of Our first objective is to solve the following delay- the above problem is to choose the anycast and sleep-wakeminimization problem: minA,B Di (p, A, B). This problem scheduling policies (p, A, B) that maximize the networkis to find the optimal anycast forwarding policy (A, B) that lifetime and also guarantee that the expected delay from eachcan minimize the delay from node i for given asynchronous node to sink s is no larger than the maximum allowable delaysleep-wake scheduling policy (i.e., given wake-up rates p). In ξ∗.Section III, we develop an algorithm that completely solves Remarks: The lifetime definition in (3) is especially usefulthis problem for all nodes i, i.e., our solution minimizes the in dealing with the most stringent requirement for networkdelays from all nodes simultaneously. 2) Network Lifetime: We now introduce the second per- lifetime such that all nodes must be alive to carry out the func-formance metric, the network lifetime, and the corresponding tionality of the sensor network [18]–[20]. In Section IV, welifetime-maximization problem (subject to delay constraints). solve the lifetime-maximization problem (P) with the lifetimeLet Qi denote the energy available to node i. We assume that definition in (3), using the solution of the delay-minimizationnode i consumes µi units of energy each time it wakes up. problem as a component. Specifically, for any given p, it wouldWe define the expected lifetime of node i as µQλi . Note that i be desirable to use an anycast policy (A, B) that minimizes the iimplicitly in this definition of lifetime we have chosen not delay. Hence, the solution to the delay-minimization problemto account for the energy consumption by data transmission. will likely be an important component for solving the lifetime-As mentioned in the introduction, this is a reasonable ap- maximization problem, which is indeed the case in the solutionproximation for event-driven sensor networks in which events that we provide in Section IV. There are application scenariosoccur very rarely because the energy consumption of the where alternate definitions of network lifetime could be moresensor nodes is dominated by the energy consumed during the suitable, e.g., when the sensor network can be viewed assleep-wake scheduling. For example, the IEEE 802.15.4-based operational even if a certain percentage of nodes are dead.low-powered sensor modules on IRIS sensor nodes consume We believe that a similar methodology can also be used for19.2mW of power while awake (i.e., in an active mode) and other lifetime definitions, which we leave for future work.40.8mW when transmitting at 250kbps [17]. Assume thatnodes stay awake only 1 percent of the time, and each node has III. M INIMIZATION OF E ND - TO -E ND DELAYSto deliver 50M bytes of information per year on average. Then, In this section, we consider how each node should choosein a year, the total amount of energy consumed by a sensor its anycast policy (A, B) to minimize the delay Di (p, A, B),node to just wake up is about 6054.9W · s (365 days/year when the awake probabilities p are given. Then, in Section IV,× 24 hours/day × 60 minutes/hour × 60 seconds/minute we relax the fixed awake-probability assumption to solve×0.01 × 19.2mW ).2 In contrast, the energy consumption due Problem (P).to packet transmission for a year is about 66.8W · s (50 M The delay-minimization problem is an instance of thebyte × 1024 Kbytes/Mbyte × 8 bits/byte / 250Kbps × 40.8 stochastic shortest path (SSP) problem [21, Chapter 2], where 2 Note that this amount of energy is within the range of capacity of two the sensor node that holds the packet corresponds to thetypical 1500 mAh AA batteries (1500mA · h × 2 × 1.5V = 4.5W · h = “state”, and the delay corresponds to the “cost” to be mini-16200W · s). mized. The sink then corresponds to the terminal state, where 5
  6. 6. the cost (delay) is incurred. Let i0 , i1 , i2 , · · · be the sequenceof nodes that successively relay the packet from the source ∞node i0 to sink node s. Note that the sequence is random Di (p, A, B) = [(tI h + tD + Dj (p, A, B))Pj,h ]because at each hop, the first node in the forwarding set h=1 j∈Fi (A)that wakes up will be chosen as a next-hop node. If the (6)packet reaches sink s after K hops, we have ih = s for = di (p, A, B) + qi,j (p, A, B)Dj (p, A, B). (7)h ≥ K. Let dj (p, A, B) be the expected one-hop delay j∈Fi (A)at node j under the anycast policy (A, B), that is, the We call (7) the local delay relationship, which must hold forexpected delay from the time the packet reaches node j to all nodes i except the sink s. Recall that Ds (p, A, B) = 0the time it is forwarded to the next-hop node. Then, the end- regardless of the delay of the neighboring nodes. Note thatto-end delay Di (p, A, B) from node i can be expressed as from (4) and (5), the anycast policies of other nodes doDi (p, A, B) = E [ ∞ dik (p, A, B)] . k=0 not affect the one-hop delay di (p, A, B) and the probability In this section, we solve the delay minimization problem qi,j (p, A, B) of node i. Hence, we can rewrite the local delayusing a Dynamic Programming approach. Our key contribution relationship using the following delay function f that is onlyis to exploit the inherent structure of the problem to greatly affected by the anycast policy of node i: for given delay valuesreduce the complexity of the solution. We start with the πi = (Dj , j ∈ Ci ), the forwarding set Fi , and the priorityrelationship between the delays of neighboring nodes. assignment bi , let f (πi , Fi , bi )A. Local Delay Relationship tI + pj k∈Fi :bij <bik (1 − pk )Dj . (8) j∈Fi tD + We first derive a recursive relationship for the delay, 1− j∈Fi (1 − pj )Di (p, A, B). When node i has a packet, the probability Pj,h We call the function f (·, ·, ·) the local delay function. Withthat the neighboring node j becomes a forwarder right after the local delay function, we can express the local delaythe h-th beacon-ID signals is equal to the probability that no relationship (7) as Di (p, A, B) = f (πi , Fi (A), bi ), wherenodes in Fi (A) have woken up for the past h − 1 beacon- πi = (Dj (p, A, B), j ∈ Ci ).ID-signaling iterations, and that node j wakes up at the h-th Let Di (p) be the minimal expected delay from node i ∗beacon-ID signals while all nodes with a higher priority than to the sink for given awake probabilities p, i.e., Di (p) =∗node j remain sleeping at the h-th iteration, i.e., minA,B Di (p, A, B). Then, we can find the optimal anycast  h−1 policy that achieves Di (p) for all nodes using value-iteration ∗ [21, Section 1.3] as follows. Start from some appropriatelyPj,h =  (1 − pk ) pj (1 − pk ). chosen initial values of the delay Di . At each iteration (0) k = 1, 2, · · · , each node i takes the delay value of other nodes k∈Fi (A) k∈Fi (A):bij <bikConditioned on this event, the expected delay from node i to from the (k − 1)-th iteration, and updates its own policy andsink s is given by (tB + tC + tA )h + tR + tP + Dj (p, A, B), delay value usingwhere the sum of the first three terms is the one-hop delay, (9) (k) Di = min f (πi , Fi , bi ),and the last term is the expected delay from the next-hop node F i , bij to the sink (see Fig. 2). For ease of notation, we define the where πi = (Dj , j ∈ Ci ). As we will show later, the (k−1)iteration period tI tB + tC + tA and the data transmissionperiod tD tR + tP . We can then calculate the probability value iterations will converge to the minimum delay valuesqi,j (p, A, B) that the packet at node i will be forwarded to Di (p), and we can obtain the stationary optimal anycast ∗node j as follows: policy. For this value iteration method to work, we will need an efficient methodology to solve (9). Note that since there pj (1 − pk ) are 2|Ci | possible choices of Fi , where |Ci | is the number of ∞ k∈Fi (A):bij <bik neighboring nodes of node i, an exhaustive search to solve qi,j (p, A, B) Pj,h = . (4) (9) will have an exponential computational complexity. In the 1− j∈Fi (A) (1 − pj ) h=1 next subsection, we will develop a procedure with only linearSimilarly, we can also calculate the expected one-hop delay complexity.di (p, A, B) at node i as follows: ∞ B. The Optimal Forwarding Set and Priority Assignment di (p, A, B) [(tI h + tD )Pj,h ] In this subsection, we provide an efficient algorithm for the h=1 j∈Fi (A) value-iteration. For ease of exposition, let Dj denote the delay tI value of node j at the (k−1)-st iteration, and let πi = (Dj , j ∈ = tD + . (5) Ci ). Our goal is to find the anycast policy (Fi , bi ) of node i 1− j∈Fi (A) (1 − pj ) that minimizes f (πi , Fi , bi ). Using the above notations, we can express the expected We first show that, in order to minimize f (·, ·, ·), the optimaldelay Di (p, A, B) of node i for given awake probability vector priority assignment b∗ can be completely determined by the ip, forwarding matrix A, and priority matrix B as follows: neighboring delay vector πi . 6
  7. 7. Proposition 1: Let b∗ be the priority assignment that gives i the nodes of J2 (that have higher priorities than the nodeshigher priorities to neighboring nodes with smaller delays, i.e., of J3 ) together with the nodes of J3 will further reduce thefor each pair of nodes j and k that satisfy b∗ < b∗ , the ij ik current delay. Finally, Property (d) implies that if adding theinequality Dk ≤ Dj holds. Then, for any given Fi , we have lower priority nodes of J3 do not change the delay, and thef (πi , Fi , b∗ ) ≤ f (πi , Fi , bi ) for all possible bi . i weighted average delay of the nodes in J2 is smaller than thatThe detailed proof is provided in Appendix A. The intuition of the nodes in J3 , then adding the nodes of J2 together withbehind Proposition 1 is that when multiple nodes send ac- the nodes of J3 will decrease the current delay. Otherwise, ifknowledgements, selecting the node with the smallest delay adding the lower priority nodes of J3 do not change the delay,should minimize the expected delay. Therefore, priorities and the weighted average delay of the nodes in J2 is equal tomust be assigned to neighboring nodes according to their that of the nodes in J3 , adding the nodes of J2 together with(given) delays Dj , independent of the awake probabilities and the nodes of J3 will not change the current delay.forwarding sets. In the sequel, we use b∗ (πi ) to denote the i Using Proposition 2, we can obtain the following mainoptimal priority assignment for given neighboring delay vector result.πi , i.e., for all nodes j and k in Ci , if b∗ (πi ) < b∗ (πi ), then ij ik Proposition 3: Let Fi∗ = arg minFi ⊂Ci f (πi , Fi ). Then, ˆDk ≤ Dj . For ease of notation, we define the value of the Fi has the following structural properties. ∗local delay function with this optimal priority assignment as (a) Fi∗ must contain all nodes j in Ci that satisfy Dj < ˆ f(πi , Fi ) f (πi , Fi , b∗ (πi )). (10) f(πi , Fi∗ ) − tD . ˆ (b) Fi∗ cannot contain any nodes j in Ci that satisfy Dj > i The following properties characterize the structure of the f(πi , Fi∗ ) − tD . ˆoptimal forwarding set. (c) If there is a node j in Fi∗ that satisfies Dj = f (πi , Fi∗ )− ˆ Proposition 2: For a given πi , let J1 , J2 , and J3 be tD , the following relationship holds,mutually disjoint subsets of Ci satisfying bij2 (πi ) < b∗ 1 (πi ) ∗ ij ˆ ˆ f (πi , Fi∗ ) = f(πi , Fi∗ {j}).for all nodes j1 ∈ Jk and j2 ∈ Jk+1 (k = 1, 2). Let j∈Jk Dj p j k∈Jk :bij (πi )<bik (πi ) (1 ∗ ∗ − pk ) Proof: We prove this proposition by contradiction. In DJ k = , order to prove Property (a), assume that there exists a node j 1− j∈Jk (1 − pj ) such that Dj < f (πi , Fi∗ ) − tD and j ∈ Fi∗ . There are two ˆ /denote the weighted average delay for the nodes in Jk for cases. Case 1: if b∗ (πi ) < b∗ (πi ) for all nodes k in Fi∗ , let ij ikk = 1, 2, 3. Then, the following properties related to f (πi , ·) ˆ J1 = Fi∗ and J3 = {j}. Then, since Dj < f(πi , J1 ) − tD , ˆhold we have f ˆ(πi , J1 ∪ J3 ) < f (πi , J1 ) by Property (a) in Propo- ˆ (a) f(πi , J1 ∪ J3 ) < f (πi , J1 ) ⇔ DJ3 + tD < f(πi , J1 ) ⇔ ˆ ˆ ˆ sition 2. This contradicts to the fact that Fi∗ is the optimal DJ 3 + t D < f ˆ(πi , J1 ∪ J3 ). forwarding set. Case 2: if there exists node k in Fi∗ such (b) f(πi , J1 ∪ J3 ) = f (πi , J1 ) ⇔ DJ3 + tD = f(πi , J1 ) ⇔ ˆ ˆ ˆ that b∗ (πi ) < b∗ (πi ), Let J1 = {l ∈ Fi |b∗ (πi ) > b∗ (πi )}, ik ij il ij DJ 3 + t D = f ˆ(πi , J1 ∪ J3 ). J2 = {j}, and J3 = {l ∈ Fi |b∗ (πi ) < b∗ (πi )}. Note that il ij (c) If f (πi , J1 ∪J3 ) < f (πi , J1 ), then f (πi , J1 ∪J2 ∪J3 ) < ˆ ˆ ˆ J1 ∪J3 = Fi∗ . If J1 is an empty set, we assume a virtual node ˆ i , J1 ∪ J3 ). f(π 0 such that b∗ (πi ) = b∗ (πi ) + 1, D0 < Dj , and p0 → 0, and i0 ij (d) If f (πi , J1 ∪J3 ) = f (πi , J1 ), then f (πi , J1 ∪J2 ∪J3 ) ≤ ˆ ˆ ˆ let J1 = {0}. The idea behind this assumption is that introduc- ˆ i , J1 ∪ J3 ), and the equality holds only when Dj = f(π 2 ing a hypothetical node that wakes up with infinitesimal prob- Dj3 for all j2 ∈ J2 and j3 ∈ J3 . ability do not change the delay analysis. Since Fi∗ = J1 ∪ J2The proof is provided in Appendix B. While Proposition 2 is optimal, we must have f(πi , J1 ∪ J3 ) ≤ f(πi , J1 ). Case ˆ ˆlooks complex, its interpretation is actually quite straightfor- 2-1: if f ˆ(πi , J1 ∪ J3 ) < f (πi , J1 ), then by Property (c) in ˆward. Note that nodes in J1 (or J2 , correspondingly) have Proposition 2, f(πi , J1 ∪ J2 ∪ J3 ) < f (πi , J1 ∪ J3 ). This is a ˆ ˆlower delay (and higher priority) than nodes in J2 (or J3 , contradiction because J1 ∪ J3 = Fi is by assumption the op- ∗correspondingly). Properties (a) and (b) provide a test to decide timal forwarding set. Case 2-2: if f (πi , J1 ∪ J3 ) = f(πi , J1 ), ˆ ˆwhether to include the higher delay nodes of J3 into the then by Property (b) in Proposition 2, DJ3 = f ˆ(πi , J1 )−tD =forwarding set. Specifically, Property (a) implies that adding f(πi , J1 ∪ J3 ) − tD > Dj = DJ2 . By Property (d) in ˆthe lower priority nodes of J3 into the current forwarding Proposition 2, f (πi , J1 ∪ J2 ∪ J3 ) < f (πi , J1 ∪ J3 ). This ˆ ˆset Fi = J1 decreases the delay if and only if the weighted is also a contradiction. Therefore, such node j must be in Fi∗ .average delay of the neighboring nodes in J3 plus tD is In order to prove Property (b), suppose in contrary thatsmaller than the current delay. Similarly, Property (b) implies there exists a node j in Fi∗ such that Dj > f (πi , Fi∗ ) − tD . ˆthat adding the lower priority node of J3 does not change the Let J1 = {l ∈ Fi∗ |Dl ≤ f(πi , Fi∗ ) − tD } and J3 = {l ∈ ˆcurrent delay if and only if the weighted average delay for the Fi∗ |Dl > f(πi , Fi∗ ) − tD }. Then, the weighted average delay ˆnodes in J3 plus tD is equal to the current delay. in J3 is larger than f(πi , Fi∗ ) − tD , i.e.,f(πi , J1 ∪ J3 ) − tD < ˆ ˆ On the other hand, Properties (c) and (d) states that if DJ3 . By Properties (a) and (b) in Proposition 2, We havethe forwarding set already includes the higher delay nodes f (πi , J1 ) < f (πi , J1 ∪ J3 ). Since Fi∗ = J1 ∪ J3 , this leadsin J3 , then it should also include the lower delay nodes in to a contradiction. Therefore, such a node j must not be inJ2 . Specifically, Property (c) implies that, if adding the lower Fi∗ .priority nodes of J3 decreases the current delay, then adding To prove Property (c), let node j in Fi∗ have the highest 7
  8. 8. priority among nodes that satisfy Dj = f (πi , Fi∗ ) − tD . We ˆ The detail is provided in the LOCAL-OPT algorithm.then need to show that f ˆ(πi , F ∗ ) = f (πi , F ∗ {j}). We let i ˆ i The complexity for finding the optimal forwarding setJ1 = Fi∗ {j} and J3 = {j}. From Property (b), J1 does is O(|Ci | log |Ci | + |Ci |) ≈ O(|Ci | log |Ci |), where thenot contain any nodes with a higher delay than f (πi , Fi∗ ) − ˆ complexities for sorting the delays of |Ci | neighboring nodestD , which implies node j is the highest priority node in Fi∗ . and for running the linearly search are O(|Ci | log |Ci |) andThen, by Property (b) in Proposition 2, we have f (πi , J1 ) = ˆ O(|Ci |), respectively.fˆ(πi , J1 ∪ J3 ), where J1 ∪ J3 = F ∗ and J1 = F ∗ {j}. i i The LOCAL-OPT AlgorithmTherefore, the result of Property (c) follows. Step (1) Node i sorts the delays πi of its neighboring nodes. Proportion 3 implies that there must be a threshold value Step (2) Node i assigns different priorities b∗ (1 ≤ bi,j ≤ i,j ∗such that the nodes whose delay is smaller than the value |Ci |) to the neighboring nodes j such that for any neighboringshould belong to the optimal forwarding set Fi∗ , and the other nodes j1 and j2 , if b∗ 1 > b∗ 2 , then Dj1 ≤ Dj2 . i,j i,jnodes should not. Hence, we can characterize the optimal Step (3) Let b−1 (k) be the index of the neighboring node withforwarding set as Fi∗ = {j ∈ Ci |Dj < f (πi , Fi∗ ) − tD } ∪ G, ˆ priority k, i.e., bi,b−1 (k) = k.where G is a subset of {j ∈ Ci |Dj = f(πi , Fi∗ ) − tD }. Note ˆ Step (4) Initial Setup: k ← |Ci |, prod ← 1, and sum ← 0.that if there exists a node j such that Dj = f(πi , Fi∗ ) − tD , ˆ Step (5) Compute f in the following order:then Fi∗ is not unique. Intuitively, this means that, if such a 5a) sum ← sum + Db−1 (k) · pb−1 (k) · prod,node j wakes up first, there is no difference in the overalldelay whether node i transmits a packet to this node or waits tI + sum 5b) prod ← prod · (1 − pb−1 (k) ), and 5c)f ← tD + .for the other nodes in Fi∗ to wake up. On the other hand, it prodwould be desirable to use the smallest optimal forwarding set, Step (6a) If k > 1 and Db−1 (k−1) < f − tD , then decrease ki.e., {j ∈ Ci |Dj < f(πi , Fi∗ ) − tD }, in order to reduce the ˆ by one and go back to Step (5).possibility that multiple nodes send duplicated acknowledge- Step (6b) Else, this algorithm terminates and returnsments. Hence, in this paper, we restrict our definition of theoptimal forwarding set Fi∗ to the following Fi∗ (πi ) = {b−1 (l); l = k, · · · , |Ci |} and min f (πi , F) = f. ˆ F ⊂Ci Fi∗ ˆ = {j ∈ Ci |Dj < min f (πi , F) − tD }. F ⊂Ci It should be noted that the optimal forwarding set is time- invariant due to the memoryless property of the Poisson(Recall that f(πi , Fi∗ ) = minF ⊂Ci f(πi , F).) Under this ˆ ˆ random wake-up process. Specifically, the expected time fordefinition, the optimal forwarding set is unique. each node j in Ci to wake up is always tI /pj regardless of Since the optimal forwarding set consists of nodes whose how long the sending node has waited. Therefore, the strategydelay is smaller than some threshold value, the simplest to minimize the expected delay is also time-invariant, i.e., thesolution to find the optimal forwarding set is to run a linear forwarding set is not affected by the sequence number of thesearch from the highest priority, i.e., k = |Ci |, to the lowest current beacon signal.priority, i.e., k = 1, to find the k that minimizes f (πi , Fi,k ) ˆwhere Fi,k = {j ∈ Ci |bij (πi ) ≥ k}. The following lemma ∗provides a stopping condition, which means that we do notneed to search over all k = 1, · · · , |Ci |. C. Globally Optimal Forwarding and Priority Matrices Lemma 1: For all F ⊂ Ci that satisfies F = {j ∈Ci |Dj < f(πi , F) − tD }, the optimal forwarding set Fi∗ must ˆ We next use the insight of Section III-B to develop anbe contained in F, i.e., Fi∗ ⊂ F. algorithm for computing the optimal anycast policy (A, B) Proof: From Proposition 3 (a), all nodes k ∈ Fi∗ satisfy for given p. This algorithm can be viewed as performing theDk < f (πi , Fi∗ ) − tD . By the definition of Fi∗ , we have ˆ value-iteration [21, Section 1.3] as discussed at the beginningfˆ(πi , F ∗ ) ≤ f (πi , F ), for any subset F of neighboring ˆ of this section. Initially (iteration 0), each node i sets its delay value Di to infinity, and only the sink sets Ds to i (0) (0)nodes, i.e., F ⊂ Ci . Since F (that satisfies F = {j ∈ Ci |Dj <f (πi , F) − tD }) is also a subset of Ci , the threshold values of ˆ zero. Then, at every iteration h, each node i uses the delay values Dj of neighboring nodes j from the previous (h−1)F and Fi∗ satisfy f (πi , Fi∗ ) − tD ≤ f(πi , F) − tD . Hence, ˆ ˆ iteration (h − 1) to update the current forwarding set Fi ∗ (h)Fi ⊂ F.Lemma 1 implies that when we linearly search for the and the current priority assignment bj according to the value (h)optimal forwarding set from k = |Ci | to k = 1, we can stop iteration (9). We will show that when the value iterationssearching if we find the first (largest) k such that for all are synchronized, the algorithm converges in N iterations,nodes j ∈ Fi,k , Dj < f(πi , Fi,k ) − tD , and for all nodes ˆ and it returns the optimal anycast policy that minimizes thel ∈ Fi,k , Dl ≥ f / ˆ(πi , Fi,k ) − tD . Since all neighboring nodes delays of all nodes simultaneously. Furthermore, we will showare prioritized by their delays, we do not need to compare that even if the value iterations and the policy updates ofthe delays of all neighboring node with the threshold value. nodes are not synchronized, the anycast policy of each nodeHence, the stopping condition can be further simplified as still converges to the optimal anycast policy (although, as isfollows: node i searches for the largest k such that for the case with most asynchronous algorithms, the convergencenode j with b∗ (πi ) = k, Dj < f(πi , Fi,k ) − tD , and ij ˆ within N iterations is not guaranteed).for node l with bil (πi ) = k − 1, Dl ≥ f (πi , Fi,k ) − tD . ∗ ˆ The complete algorithm is presented next. 8
  9. 9. The GLOBAL-OPT Algorithm We next show the convergence of the GLOBAL-OPT Step (1) At iteration h = 1: sink node s sets Ds to 0, and (0) algorithm within N iterations, i.e., (A(N ) , B(N ) ) = other nodes i set Di to ∞. (0) (A∗ (p), B∗ (p)). To show this, we need the following result Step (2) At iteration h, each node i runs the LOCAL-OPT on the structure of the optimal forwarding set. algorithm with the input πi (h−1) (h−1) = (Dj , j ∈ Ci ). Proposition 5: For any awake probability vector p, there Step (3) Using the output of the LOCAL-OPT algorithm (6b), exists some optimal anycast policy (A , B ) that solves the each node i updates the anycast forwarding policy and the delay-minimization problem for all nodes and does not incur delay value as follows: any cycle in routing paths, i.e., g(A ) is acyclic. Proof: It suffices to show that for any p, A ∈ A, and POLICY ITERATION: (h) (h−1) Fi ← Fi∗ (πi ), B ∈ B such that g(A) is cyclic (i.e., there is a cyclic path (h) bi ← b∗ (πii (h−1) ) from a node to the sink), we can find A ∈ A and B ∈ B that satisfies the following properties: VALUE ITERATION: Di ← min f(πi , F). (11) (h) ˆ (h−1) F ⊂Ci (a) g(A ) is acyclic, and Step (4a) If h = N , this algorithm terminates and returns (b) Di (p, A , B ) ≤ Di (p, A, B) for all nodes i. We first show how to construct such a policy (A , B ) for and Di (p) = Di . (N ) (N ) Fi∗ = Fi ∗ given A and B, and then show that such A and B satisfy Step (4b) If h < N , h ← h + 1 and goes back to Step (2). Properties (a) and (b). Let A(h) be the forwarding matrix that corresponds to Assume that all nodes are connected to the sink under theFi for all nodes i ∈ N , i.e., aij = 1 if j ∈ Fi , or forwarding matrix A. (We will consider the other case later.) (h) (h) (h)aij = 0, otherwise. Similarly, let B(h) be the priority matrix Then, every node i has at least one neighboring node in the (h)in which the transpose of the i-th row is bi . Then, the forwarding set Fi (A). Now, let every node i select its new (h)following proposition shows the convergence of the GLOBAL- forwarding set among the nodes in Fi (A). Each node thenOPT algorithm: finds the optimal forwarding set Fi∗ and the optimal priority assignment b∗ (πi ) for the delays πi = (Dj (p, A, B), j ∈ Proposition 4: For given p, the delay values Di converge (h) i Fi (A)) using the LOCAL-OPT algorithm. Let A and B beto Di (p), i.e., Di −→ Di (p) as h → ∞. Furthermore, the new global forwarding matrix and the new global priority ∗ (h) ∗the anycast policy (A(h) , B(h) ) also converges to an anycast matrix, respectively, that correspond to the forwarding set F ∗policy (A∗ , B∗ ) as h → ∞ such that Di (p, A∗ , B∗ ) = Di (p) ∗ i and the priority assignment b∗ (πi ) of all nodes i ∈ N .for all nodes i. i We next prove that the new anycast policy (A , B ) satisfies Proof: The GLOBAL-OPT algorithm is a classic value- Property (a). Let bi be the priority assignment of node iiteration for solving Shortest Stochastic Problem (SSP) prob- under the priority matrix B. By the local-delay relationshiplems. Specifically, we map the delay-minimization problem to in (6), we have f (πi , Fi (A), bi ) = Di (p, A, B). Since F ∗ ithe SSP problem as follows. Consider the following Markov is the optimal forwarding set for given πi , we also havechain that corresponds to the process with which a packet f(πi , F ∗ ) ≤ f (πi , Fi (A), bi ). Combining the above, we have ˆ iis forwarded under a given anycast policy. There are states1, 2, · · · , N , and s, where state i represents that a packet ˆ f(πi , Fi∗ ) ≤ Di (p, A, B). (12)is in node i. A state transition occurs from state i to state According to Proposition 3, for a neighboring node j ∈ Fi (A)j when node i forwards the packet to node j. (We do not to be in the new forwarding set F ∗ , the delay Dj (p, A, B) iconsider self-transitions.) State s is the absorbing state (it is must be smaller than f (πi , F ∗ )−tD . From (12), it follows that ˆalso called the termination state in [21]), where state transition all nodes j in F ∗ must satisfy Dj (p, A, B) < Di (p, A, B) − i iends. Under the anycast forwarding policy (A, B), a packet tD . Hence, the delay value Dj (p, A, B) decreases by at leastat node i will be forwarded to neighboring node j with tD along all paths under the new forwarding matrix A . Thisprobability qi,j (p, A, B) given by (4). The cost associated with implies that there could not exist any cyclic path. Hence,the transition from node i to any neighboring node corresponds Property (a) the expected one-hop delay di (p, A, B) given by (5). The We next prove by contradiction that the new anycast policytotal cost, which is the expectation of the accumulated costs (A , B ) also satisfies Property (b), i.e., Di (p, A , B ) ≤from initial state i to sink s, corresponds to the expected end- Di (p, A, B) for all nodes i. From (12), it suffices to showto-end delay Di (p, A, B) from node i to the sink s. Then, the that Di (p, A , B ) ≤ f(πi , F ∗ ) holds for all nodes i. Assume ˆ ievolution of Di corresponds to the value iteration on page in contrary that there exists a node i such that (h)95 of [21]. Hence, according to Proposition 2.2.2 of [21], wehave Di → Di (p) and (A(h) , B(h) ) → (A∗ , B∗ ) such that (h) ∗ ˆ Di (p, A , B ) > f (πi , Fi∗ ). (13)Di (p, A , B ) = Di (p) for all nodes i, as h → ∞. ∗ ∗ ∗ From (10), we can rewrite the right hand side f(πi , Fi∗ ) ˆ Let (A (p), B (p)) be the converged anycast policy, i.e., as f (πi , Fi∗ , b∗ (πi )). Let πi be the delays of nodes ∗ ∗ i(A∗ (p), B∗ (p)) limh→∞ (A(h) , B(h) ). Then, Proposi- in Fi (A) under the new anycast policy (A , B ), i.e.,tion 4 shows that this converged anycast policy corre- πi = (Dj (p, A , B ), j ∈ Fi (A)). Then, comparingsponds to an optimal anycast policy that minimizes the de- (8) and (7), we can also express the left hand sidelays from all nodes simultaneously, i.e., (A∗ (p), B∗ (p)) = Di (p, A , B ) as f (πi , Fi∗ , b∗ (πi )). Since the local delay func- iarg minA,B Di (p, A, B) for all nodes i. tion f (·, Fi∗ , b∗ (πi )) is monotonic with respect to each element i 9
  10. 10. of the first argument, there must exist at least one neighboring that requires all nodes to execute the value-iteration in lockednode j in Fi∗ such that Dj (p, A, B) < Dj (p, A , B ) to satisfy steps. In fact, an asynchronous version of GLOBAL-OPT algo-(13). Since f (πj , Fj ) ≤ Dj (p, A, B) from (12), such node j ˆ ∗ rithm can also be shown to converge, although the convergencesatisfies will typically require more than N iterations. ˆ ∗ Dj (p, A , B ) > f (πj , Fj ). (14) Proposition 7: Assume that at each iteration, every node i independently chooses either to follow Step (2) and StepBy applying the same method iteratively, we can find a (3) in the GLOBAL-OPT algorithm or to maintain its cur-sequence of such nodes j such that all of which satisfy (14). rent policy and delay value, i.e., Fi (h) = Fi (h−1) , bi = (h)Since all paths under A are acyclic and connected to the sink bi (h−1) , and Di (h) = Di (h−1) . Further, assume that everys, the sequence must converge to sink s. However, the delay of node updates its policy and delay value infinitely often.sink s is always zero, i.e., Ds (p, A , B ) = Ds (p, A, B) = 0, Then, the policy and the delay value converge to the op-which is a contradiction to (14). Hence, Property (b) follows. timal policy and the minimum delay, respectively, as h → We have shown that if all nodes are connected to the sink ∞, i.e., limh→∞ (A(h) , B(h) ) = arg minA,B Di (p, A, B),under the forwarding matrix A, there exists an alternative (h) limh→∞ Di = minA,B Di (p, A, B).policy (A , B ) that satisfies Properties (a) and (b). In thecase where some nodes are disconnected from the sink under Proof: The proof again follows from the standard resultsthe forwarding matrix A, we simply exclude these nodes from of Proposition 1.3.5 in [21].the node set N and then use the above procedure to construct Since the asynchronous version of GLOBAL-OPT algo-the alternative anycast policy (A , B ) for the connected nodes rithm does not converge in N steps, for practical implementa-only. Then, for the connected nodes, the policy (A , B ) must tion, we use an expiration time texp to force the algorithmsatisfy Properties (a) and (b) according to the earlier proof. to terminate. The expiration time texp leads to a tradeoffSince the disconnected nodes are still disconnected under between optimality and the duration of the configuration phase.the alternative policy, their delays will remain infinite, and If texp increases, the latest anycast policy will be closer tothere are no cyclic paths from these node to the sink. Hence, the optimal policy, but the execution time will also increase.Properties (a) and (b) also hold for the entire network. (Recall that the asynchronous GLOBAL algorithm runs in the From the existence of the optimal cycle-free anycast policy, configuration phase that occurs at the very beginning of thewe can show the following proposition: deployment of sensor nodes.) Hence, the parameter texp must Proposition 6: For given p, be chosen carefully, with this tradeoff in mind.(a) The GLOBAL-OPT algorithm converges within N itera-tions, i.e., (A(N ) , B(N ) ) = (A∗ (p), B∗ (p)) D. The case with multiple sink nodes(b) The GLOBAL-OPT algorithm does not incur any cyclicpaths, i.e., g(A∗ (p)) is acyclic. Our results can be easily generalized to the case when there Proof: To show that convergence occurs in N iterations, are multiple sink nodes and the event-reporting packets canwe need Proposition 5, which states that there must exist at be collected by any sink nodes. In this case, we can simplyleast one optimal anycast policy that does not incur any cycles. set Ds (p, A, B) = 0 for all sink nodes s. Then the sameHence, based on the proof on page 107 of [21], for all nodes delay-minimization algorithm GLOBAL-OPT can be used.i, the delay value Di converges to min(A,B) Di (p, A, B) (h) The resulting anycast policy will minimize the end-to-endwithin N iterations. Then, according to Property 2.2.2 (c) on delay to reach any sink 99, the policy (A(h) , B(h) ) also converges to the optimalanycast policy (A∗ (p), B∗ (p)) within N iterations. IV. M AXIMIZATION OF N ETWORK L IFETIME We next prove that Property (b) holds. According to Prop- In the previous section, we solved the delay-minimizationerty (a), all nodes i satisfy Fi and Di (N +1) (N ) (N +1) = Fi = problem. In this section, we use the result to develop aDi . Note that Fi is the optimal forwarding set for (N ) (N +1) solution to the lifetime-maximization problem (P). From (2),the delays πi = (Dj , j ∈ Cj ), and Di is the (N ) (N ) (N +1) the lifetime Ti and the awake probability pi have a one-to-onecorresponding optimized value f(πi , Fi ). According ˆ (N ) (N +1) mapping. Hence, we convert Problem (P) to the following → −to Proposition 3, neighboring nodes j in Fi must satisfy (N +1) equivalent problem that controls T = (T1 , T2 , · · · , TN ), A, − tD , which leads to Dj < Di − tD and B: (N ) (N +1) (N ) (N )Dj < D ifor all neighboring nodes j in Fi . This implies that under (N ) (P1) max min Ti , (15)anycast policy (A , B ), the delay value Di decreases (N ) (N ) (N ) → − T ,A,B i∈Nby at least tD along each possible routing path. Hence, subject to Di (p, A, B) ≤ ξ ∗ , ∀i ∈ N (16)Property (b) follows. tI (17) −e Proposition 6 shows that the anycast policy (A(N ) , B(N ) ) pi = 1 − e , ∀i ∈ N i Tithat the GLOBAL-OPT algorithm returns corresponds to Ti ∈ (0, ∞), ∀i ∈ Nthe optimal policy (A∗ (p), B∗ (p)). Furthermore, the graph A ∈ A, B ∈ B.g(A∗ (p)) is acyclic. The complexity of this algorithm at eachnode i is given by O(N · |Ci | log |Ci |) For any given p, (A∗ (p), B∗ (p)) is the optimal any- The GLOBAL-OPT algorithm is a synchronous algorithm cast policy that minimizes the delay from all nodes, i.e., 10