On Maximizing the Lifetime of Delay-Sensitive Wireless Sensor Networks with Anycast

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1 On Maxmzng the Lfetme of Delay-Senstve Wreless Sensor Networks wth Anycast Joohwan Km, Xaojun Ln, Ness B. Shroff, and Prasun Snha School of Electrcal and Computer Engneerng, Purdue Unversty Departments of ECE and CSE, The Oho State Unversty Emal:{ jhkm, Abstract Sleep-wake schedulng s an effectve mechansm to prolong the lfetme of energy-constraned wreless sensor networks. However, t ncurs an addtonal delay for packet delvery when each node needs to wat for ts next-hop relay node to wake up, whch could be unacceptable for delay-senstve applcatons. Pror work n the lterature has proposed to reduce ths delay usng anycast, where each node opportunstcally selects the frst neghborng node that wakes up among multple canddate nodes. In ths paper, we study the jont control problem of how to optmally control the sleep-wake schedule, the anycast canddate set of next-hop neghbors, and anycast prortes, to maxmze the network lfetme subject to a constrant on the expected end-to-end delay. We provde an effcent soluton to ths jont control problem. Our numercal results ndcate that the proposed soluton can substantally outperform pror heurstc solutons n the lterature, especally under the practcal scenaros where there are obstructons n the coverage area of the wreless sensor network. Index Terms Anycast, Sleep-wake schedulng, Sensor network, Energy-effcency, Delay I. INTRODUCTION Sleep-wake schedulng s an effectve mechansm to prolong the lfetme of energy-constraned sensor networks. In ths paper, we are partcularly nterested n event-drven wreless sensor networks, where events occur occasonally. Therefore, by puttng nodes to sleep when there are no events, the energy consumpton of the sensor nodes can be sgnfcantly reduced. In the lterature, synchronzed sleep-wake schedulng protocols have been proposed n [1] [3]. In these protocols, sensor nodes perodcally or aperodcally exchange synchronzaton nformaton wth neghborng nodes. However, these synchronous protocols could ncur addtonal communcaton overhead, and consume a consderable amount of energy. In ths work, we are nterested n asynchronous sleep-wake schedulng protocols such as those proposed n [4], [5]. In these protocols, the sleep-wake schedule at each node s ndependent of that of other nodes, and thus no synchronzaton s requred. However, due to the lack of knowledge of the sleepwake schedule of other nodes, t ncurs addtonal delays for packet delvery when each node needs to wat for ts next-hop node to wake up. Ths delay could be unacceptable for delay- Ths work has been partally supported by the Natonal Scence Foundaton through awards CNS , CNS , CNS , CCF , and ARO MURI Award No. W911NF (SA08-03). senstve applcatons, such as fre detecton or tsunam alarm, whch requre that the event reportng delay be small. Pror work n the lterature has proposed the use of anycast to reduce ths event reportng delay [6] [10]. In contrast to tradtonal sleep-wake schedulng, where each sendng node wakes up a partcular next-hop relay node, n anycast each sendng node tres to wake up a group of neghborng nodes n a canddate set, and the sendng node then pcks the frst node that wakes up to relay packets. Roughly speakng, f each neghborng node wakes up once every T tme, by selectng a canddate set of n nodes, the tme needed before the frst node wakes up s on average around T n (assumng that the sleepwake schedules of the n nodes are ndependent). Thus, the delay to wake up the next-hop neghbors can be sgnfcantly reduced. On the other hand, the end-to-end delay not only depends on the per-hop delay, but also the end-to-end path that packet traverses. Hence, the set of canddate nodes must be carefully chosen because t wll also affect the possble routng paths. The exstng anycast schemes n the lterature have manly focused on the so-called MAC-layer anycast problem,.e., they try to fnd the canddate set at each node such that some local measure of delay s mnmzed. For the routng path, they ether use a separate routng algorthm [8], [9], or rely on geographcal nformaton [6], [7], [10]. Thus, the nteractons between the choce of the canddate set and the routng path was not systematcally studed, and t s then unclear whether such approaches wll mnmze the actual end-to-end delay. In ths paper, we drectly optmze the system wth respect to the end-to-end delay. In partcular, we formulate the jont control problem of how to optmally control the sleep-wake schedule, the anycast canddate set of neghborng nodes, and anycast prortes among neghborng nodes, to maxmze the network lfetme subject to a constrant on the end-to-end delay. We provde an effcent soluton to ths jont control problem, and as a part of soluton, we also show how to optmally choose the canddate set n order to mnmze the end-to-end delay for all nodes. The rest of ths paper s organzed as follows. In Secton II, we descrbe the system model and ntroduce the lfetmemaxmzaton problem that we ntend to solve. In Secton III, we analyze the end-to-end delay under anycast, and we develop an optmal dstrbuted anycast algorthm that mn-

2 Report Documentaton Page Form Approved OMB No Publc reportng burden for the collecton of nformaton s estmated to average 1 hour per response, ncludng the tme for revewng nstructons, searchng exstng data sources, gatherng and mantanng the data needed, and completng and revewng the collecton of nformaton. Send comments regardng ths burden estmate or any other aspect of ths collecton of nformaton, ncludng suggestons for reducng ths burden, to Washngton Headquarters Servces, Drectorate for Informaton Operatons and Reports, 1215 Jefferson Davs Hghway, Sute 1204, Arlngton VA Respondents should be aware that notwthstandng any other provson of law, no person shall be subject to a penalty for falng to comply wth a collecton of nformaton f t does not dsplay a currently vald OMB control number. 1. REPORT DATE REPORT TYPE N/A 3. DATES COVERED - 4. TITLE AND SUBTITLE On Maxmzng the Lfetme of Delay-Senstve Wreless Sensor Networks wth Anycast 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) School of Electrcal and Computer Engneerng, Purdue Unversty 8. PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR S ACRONYM(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for publc release, dstrbuton unlmted 13. SUPPLEMENTARY NOTES The orgnal document contans color mages. 14. ABSTRACT 15. SUBJECT TERMS 11. SPONSOR/MONITOR S REPORT NUMBER(S) 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT UU a. REPORT unclassfed b. ABSTRACT unclassfed c. THIS PAGE unclassfed 18. NUMBER OF PAGES 9 19a. NAME OF RESPONSIBLE PERSON Standard Form 298 (Rev. 8-98) Prescrbed by ANSI Std Z39-18

3 mzes the end-to-end delay of all nodes. In Secton IV, we solve the lfetme-maxmzaton problem. In Secton V, we provde smulaton results that llustrate the performance of our proposed algorthm compared to other heurstc algorthms n the lterature. II. SYSTEM MODEL We consder a wreless sensor network wth N nodes. Let N denote the set of all nodes n the network. Each sensor node s n charge of both detectng events and relayng packets. If a node detects an event, the node packs the event nformaton nto a packet, and delvers the packet to a snk s va multhop relayng. We assume n ths paper that there s a sngle snk, however, the analyss can be generalzed to the case wth multple snks. We assume that the sensor network employs sleep-wake schedulng to mprove energy-effcency and to prolong the network lfetme. Wth sleep-wake schedulng, nodes sleep for most of the tme and occasonally wake up for a short perod of tme t actve. When a node has a packet for node j to relay, t wll send a beacon sgnal followed by an ID sgnal (carryng sender nformaton). Let t B and t C be the duraton of the beacon sgnal and the ID sgnal, respectvely. When node j wakes up and senses a beacon sgnal, t keeps awake, watng for the followng ID sgnal to recognze the sender. When node j wakes up n the mddle of an ID sgnal, t keeps awake, watng for the next ID sgnal. If node j successfully recognzes the sender, and t s the next-hop node of node, t then communcates wth node to receve the packet. If a node wakes up and does not sense any beacon sgnal or any ID sgnal, t wll then go back to sleep. In ths paper, we assume that the tme nstants that a node j wakes up follow a Posson random process wth rate λ j. We also assume that the wake-up processes of dfferent nodes are ndependent. The ndependence assumpton s sutable for the scenaro where nodes do not synchronze ther wakeup tmes, whch s easer to mplement than other schemes that requre global synchronzaton [1] [3]. The advantage of Posson sleep-wake schedulng s that, due to ts memoryless property, sensor nodes are able to use a tme-nvarant optmal polcy to maxmze the network lfetme (see the dscusson n Secton III-B). Whle the analyss n ths paper focuses on the case when the wake-up tmes follow a Posson process, we expect that the methodology n the paper can also be extended to the case wth non-posson wake-up processes, wth more techncally-nvolved analyss. A well-known problem of usng sleep-wake schedulng n sensor networks s the addtonal delay ncurred when a node has to wat for ts next-hop node to wake up. To reduce ths delay, we use an anycast forwardng scheme descrbed n Fg. 1. Let C denote the set of nodes n the transmsson range of node. Suppose that node has a packet, and t needs to pck up a node n C to relay the packet. Each node mantans a lst of nodes that node ntends to use as a forwarder. We call the set of such nodes a forwardng set, whch s denoted by F. In addton, each node j s also assumed to mantan a lst of nodes that use node j as a forwarder (.e., j F ). As shown n Fg. 1, node starts sendng a beacon sgnal and an ID sgnal, successvely. All nodes n C hear these sgnals regardless of whom these sgnals are ntended for. A node j that wakes up durng the beacon sgnal or the ID sgnal wll check f t s n the forwardng set of node. If t s, node j sends one acknowledgement after the ID sgnal ends. After each ID sgnal, node checks whether there s any acknowledgement from nodes n F. If no acknowledgement s detected, node repeats the beacon- ID-sgnalng and acknowledgement-detecton processes untl t hears one. On the other hand, f there s an acknowledgement, t may take some addtonal tme for node to dentfy whch node acknowledges the beacon-id sgnals, especally when there are multple nodes that wake up at the same tme. We let t R denote the resoluton perod, durng whch tme node dentfes whch nodes have sent acknowledgements, and, f there are multple awake nodes, chooses one node among them that wll forward the packet. After the resoluton perod, the chosen node receves the packet from node durng the packet transmsson perod t P, and then starts the beacon- ID-sgnalng and acknowledgement-detecton processes to fnd the next forwarder. Snce nodes consume energy when awake, t actve should be as small as possble. However, t actve has to be larger than t A because otherwse a neghborng node could wake up after an ID sgnal and could turn to sleep before the next beacon sgnal. In ths paper, we set t actve = t A + t C so that the node that wakes up rght before the frst beacon sgnal also has the same chance of detectng the beacon sgnal as nodes that wake up between two beacon sgnals. A. Sleep-wake Schedule, Forwardng Set, and Prorty In ths model, there are three control varables that affect the network lfetme and the end-to-end delay experenced by a packet. 1) Sleep-Wake Schedule: The sleep-wake schedule s determned by the rate λ j of the Posson process wth whch each node j wakes up. If λ j ncreases, the expected one-hop delay wll decrease, and so wll the end-to-end delay of any routng paths that pass though node j. However, t leads to hgher energy consumpton at node j so that the network lfetme may decrease. In the rest of the paper, t s more convenent to work wth the noton of awake probablty whch s a functon of λ j. Suppose that node sends the frst beacon sgnal at tme 0, as n Fg. 1. If no nodes n F have heard the frst m 1 beacon and ID sgnals, then node transmts the m-th beacon and ID Fg. 1. System Model

4 sgnals n the tme-nterval [(t B +t C +t A )(m 1), (t B +t C + t A )(m 1)+t B +t C ]. For a neghborng node j to hear the m- th sgnals and to recognze the sender, t should wake up durng [(t B +t C +t A )(m 1) t A t C, (t B +t C +t A )m t A t C ]. Therefore, provded that node s sendng the m-th sgnals, the probablty that node j C wakes up and hears these sgnals s p j =1 e λj(t B+t C +t A ). We call p j the awake probablty of node j. It should be noted that, due to the memoryless property of a Posson random process, p j s same for each beacon-id sgnalng teraton, m. Note that there s a one-to-one mappng between the awake probablty p j and wakng-up frequency λ j. Hence, an awake probablty s also closely related to both delay and energy consumpton. Let p =(p, N) represent the global awake probablty vector. 2) Forwardng Set: The forwardng set F s the set of canddate nodes chosen to forward a packet at node. In prncple, the forwardng set should contan nodes that can quckly delver the packet to the snk. However, snce the end-to-end delay depends on the forwardng set of all nodes, choosng the correct forwardng set s not easy. We use a matrx A to represent the forwardng set of all nodes collectvely, as follows: A =[a j,=1,..., N, j =1,..., N] where a j =1f j s n node s forwardng set, and a j = 0 otherwse. We call ths matrx A the forwardng matrx. Recprocally, we defne F (A) as the forwardng set of node under forwardng matrx A,.e., F (A) ={j C a j =1}. We let A denote the set of all possble forwardng matrces. Wth anycast, a forwardng matrx determnes the paths that packets can potentally traverse. Let g(a) be the drected graph G(V, E(A)) wth vertces V = N, and edges E(A) ={(, j) j F (A)}. If there s a path n g(a) that leads from node to node j, we say that node s connected to node j. Otherwse, we call t dsconnected to node j. For convenence, we call a node that s connected (dsconnected) to snk s smply as connected (dsconnected). An acyclc path s the path whch does not traverse any node more than once. If g(a) has any cyclc path, we call t a cyclc graph, otherwse we call t an acyclc graph. 3) Prorty: When multple nodes send an acknowledgement after the same ID sgnal, the source node needs to pck one of them as a forwarder. We assume that node assgns prortes to all nodes n C, and wll pck the node wth the hghest prorty among these nodes that wake up. Clearly, the prorty assgnment wll also affect the expected delay. We use a matrx B to represent the global prorty decson, as follows: B =[b j,=1,..., N, j =1,..., N] where b j {1,, C } f j C, and b j =0otherwse. Ths b j represents the prorty of node j from the vewpont of node. We call ths matrx B the prorty matrx. The prorty matrx B further satsfes b j1 b j2 for all dstnct nodes j 1,j 2 C. Among the awake nodes, the node j that satsfes b j >b k for all the other nodes k wll be chosen as a forwarder. Note that even though only the nodes n a forwardng set need prortes, we here assgn prortes to all nodes to make prorty matrx B an ndependent control varable from forwardng matrx A. We let B denote the set of all possble prorty matrces. B. Performance Metrcs We next descrbe the performance metrcs that we are nterested n. 1) End-to-End Delay: We assume that the end-to-end delay for event delvery s domnated by the cumulatve sum of the delay for each hop to wake up and to relay a packet to ts next-hop neghbor. Ths s a reasonable approxmaton n many event-drven networks. Note that when an event occurs n an event-drven sensor network, the frst packet generated by the event usually suffers most of the delay because at every hop t has to wat for nodes to wake up so that t can be relayed. Once the frst packet goes through, the sensor nodes can stay awake for a whle, hence the delay of the subsequent packets are often much smaller. Therefore, n ths paper, we defne the end-to-end delay as the delay ncurred by the frst packet, whch s the sum of the delay for each hop to wake up and to relay the packet to ts next-hop neghbor. Gven A, B, and p, the stochastc process wth whch a packet traverses the network from the source node to the snk s completely specfed, and can be descrbed by a Markov process. We defne D ( p,a, B) as the expected end-to-end delay for a packet from node to reach snk s, when awake probablty vector p, forwardng matrx A, and prorty matrx B are gven. Snce snk s s the destnaton of all packets, the delay of packets from snk s s regarded as zero,.e., D s ( p,a, B) =0, regardless of p, A, and B. If node s dsconnected to snk s under forwardng matrx A, packets from the node cannot reach snk s. Therefore, the end-toend delay from such a node s regarded as nfnte,.e., D ( p,a, B) =. From now on, we call the expected endto-end delay from node to snk s smply as the delay from node. 2) Network Lfetme: We assume that node consumes u unt of energy each tme t wakes up. Let Q be the energy avalable to node. Then, the expected lfetme of node s Q u λ. By ntroducng the power consumpton rato e = u /Q, we can express the lfetme of node as T ( p) = 1 = t B + t C + t A e λ 1. (1) e ln (1 p ) Here we have used the defnton of the awake probablty p = 1 e λj(t B+t C +t A ) from (1). Note that n ths defnton of lfetme we have chosen not to account for the energy consumpton by data transmsson. Ths s a reasonable approxmaton for those event-drven sensor networks where events occur very rarely, n whch case the energy consumpton of the sensor nodes s domnated by the energy consumed durng the sleep-wake schedulng. We assume that the network lfetme s determned by the shortest lfetme of all nodes. In other words, the network

5 lfetme for a gven awake probablty vector p s gven by T ( p) = mn N T ( p). The methodology of the paper may be extended to handle other defntons of lfetme, e.g., when the sensor network s consdered operatonal f less than a certan percentage of nodes are alve. However, we leave ths more general defnton of lfetme for future work. 3) Problem Formulaton: The objectve of ths paper s to choose awake probablty vector p, forwardng matrx A, and prorty matrx B to maxmze the network lfetme, subject to the constrant that the expected delay from each node to snk s s below the maxmum allowable delay,.e., (P) max p,a,b T ( p) subject to D ( p,a, B) ξ, N p (0, 1] N, A A, B B. where ξ s the maxmum allowable delay. III. MINIMIZATION OF END-TO-END DELAYS FOR GIVEN AWAKE PROBABILITIES In ths secton, we consder how each node should choose ts forwardng set and assgn prortes to neghborng nodes to mnmze the delay D ( p,a, B), when the awake probabltes are gven. Then, n Secton IV, we relax the fxed-probablty assumpton to solve problem (P). A. Local Delay Relatonshp We frst derve a recursve relatonshp for the delay, D ( p,a, B). When node has a packet, the probablty P j,h that node j n C becomes a forwarder rght after the h-th beacon-id sgnals s equal to the probablty that no nodes n C have woken up for the past h 1 beacon-id-sgnalng teratons, and node j wakes up at the h-th beacon-id sgnals whle all nodes wth a hgher prorty than node j reman sleepng at the h-th teraton,.e., P j,h = k F (A) (1 p k ) h 1 p j k F (A):b j<b k (1 p k ). Condtoned on ths event, the expected delay from node to snk s s gven by (t B + t C + t A )h + t R + t P + D j ( p,a, B). For ease of notaton, we defne the teraton perod t I t B + t C + t A and the data transmsson perod t D t R + t P. We can then calculate the expected delay D ( p,a, B) of node for gven awake probablty vector p, forwardng matrx A, and prorty matrx B as follows: = D ( p,a, B) [(t I h + t D + D j ( p,a, B))P j,h ] h=1 j F (A) = t D + + t I 1 j F (A) j F (A) (1 p j ) D j ( p,a, B)p j (1 p k ) k F (A):b j<b k 1.(2) (1 p j ) j F (A) We call (2) the local delay relatonshp, whch must hold for all nodes except the snk s. (Recall D s ( p,a, B) = 0 regardless of the delay of neghborng nodes.) B. The Optmal Forwardng Set and Prorty Assgnment In ths subsecton, we frst consder the hypothetcal scenaro where a node knows the delays D j from ts neghborng nodes j to snk s, and such delays D j are fxed. Under ths hypothess, we wll study how node should adjust ts own forwardng set and prorty assgnment to mnmze the expected delay from node to snk s. Then, n the next subsecton, we wll use the nsght from the result to determne the optmal forwardng matrx A, and prorty matrx B. Consder that a node has multple neghborng nodes wth fxed delay. Smlar to (2), we can calculate the expected delay from node to snk s for a gven neghborng delay vector π =(D j,j C ), forwardng set F, and prorty assgnment b as f( π, F, b ) t D + t I + j F D j p j k F :b j <b k (1 p k ) 1. (3) j F (1 p j ) We call the functon f(,, ) the local delay functon. We frst show that, n order to mnmze f(,, ), the optmal prorty assgnment b can be completely determned by the neghborng delay vector π. Proposton 1: Let b be the prorty assgnment that gves hgher prortes to neghborng nodes wth smaller delay,.e., for each par of nodes j and k satsfyng b j < b k, the nequalty D k D j holds. Then, for any gven F, f( π, F, b ) f( π, F, b ) (4) for all possble b. The detaled proof s provded n Appendx A n our on-lne techncal report [11]. The ntuton behnd Proposton 1 s that when multple nodes sends acknowledgements, selectng the node wth the smallest delay should mnmze the expected delay. Therefore, prortes must be assgned to neghborng nodes accordng to ther (known) delays D j, ndependent of awake probabltes and forwardng sets. In the sequel, we use b ( π ) to denote the optmal prorty assgnment for gven

6 neghborng delay vector π,.e., for all nodes j and k n C, f b j ( π ) <b k ( π ), then D k D j. For ease of notaton, we defne the value of the local delay functon wth ths optmal prorty assgnment as ˆf( π, F ) f( π, F, b ( π )). The followng propertes characterze the structure of the optmal forwardng set. Proposton 2: For a gven π, let J 1, J 2, and J 3 be mutually dsjont subsets of C satsfyng b j 2 ( π ) <b j 1 ( π ) for all nodes j 1 J k and j 2 J k+1 (k =1, 2). Let j J k D j p j k J k :b D Jk = j ( π )<b k ( π )(1 p k) 1, j J k (1 p j ) denote the weghted average delay n J k for k =1, 2, 3. Then, the followng propertes related to ˆf( π, ) hold (a) ˆf( π, J 1 J 3 ) < ˆf( π, J 1 ) D J3 + t D < ˆf( π, J 1 ) D J3 + t D < ˆf( π, J 1 J 3 ). (b) ˆf( π, J 1 J 3 )= ˆf( π, J 1 ) D J3 + t D = ˆf( π, J 1 ) D J3 + t D = ˆf( π, J 1 J 3 ). (c) If ˆf( π, J 1 J 3 ) < ˆf( π, J 1 ), then ˆf( π, J 1 J 2 J 3 ) < ˆf( π, J 1 J 3 ). (d) If ˆf( π, J 1 J 3 )= ˆf( π, J 1 ), then ˆf( π, J 1 J 2 J 3 ) ˆf( π, J 1 J 3 ), and the equalty holds only when D j2 = D j3 for all j 2 J 2 and j 3 J 3. Proof: Ths proposton can be shown by notng that each node set J k (k =1, 2, 3) can be regarded as a node wth delay D Jk and awake probablty P Jk =1 j J k (1 p j ),.e., the probablty that any node n J k wakes up. Then, the local delay functon can be expressed as ˆf( π, K k=1j k )=t D + t I + K k=1 D k 1 J k P Jk l=1 (1 P J l ) 1 K k=1 (1 P J k ) for K = 1, 2, 3. Then, by algebrac manpulaton, we can establsh Propertes (a)-(d). Detals are agan avalable n Appendx B n [11]. The nterpretaton of Proposton 2 s straghtforward. For example, Property (a) mples that addng lower prorty nodes of J 3 nto the current forwardng set F = J 1 decreases the delay f and only f the weghted average delay n J 3 plus t D s smaller than the current delay. Usng Proposton 2, we can obtan the followng man result. Proposton 3: Let F = arg mn F C ˆf( π, F ). Then, F has the followng structural propertes. (a) F must contan all nodes j n C that satsfy D j < ˆf( π, F ) t D. (b) F cannot contan any nodes j n C that satsfy D j > ˆf( π, F ) t D. (c) For all nodes j n C that satsfy D j = ˆf( π, F ) t D, the followng relatonshp holds, ˆf( π, F )= ˆf( π, F \{j}) = ˆf( π, F {j}). We can prove Proposton 3 by usng the result of Proposton 2. (See the detal provded n Appendx C n [11].) From Proporton 3, we can characterze the optmal forwardng set as F = {j C D j < ˆf( π, F ) t D} G, where G s a subset of {j C D j = ˆf( π, F ) t D}. Ths means that f there exsts a node j such that D j = ˆf( π, F ) t D, F s not unque. In other words, f such a node j wakes up frst, there s no dfference n the overall delay whether node transmts a packet to ths node or wats for the other nodes n F to wake up. Snce the optmal forwardng set conssts of nodes whose delay s smaller than or equal to some threshold value, the smplest soluton to fnd the optmal forwardng sets s to run an exhaustve search from the hghest prorty,.e., k = C, to the lowest prorty,.e., k =1, to fnd the k that mnmzes ˆf( π, F,k ) where F,k = {j C b j ( π ) k}. If there are multple optmal forwardng sets, we only need to fnd one of them. In ths paper, we chose to use the set F = {j C D j < ˆf( π, F ) t D} as the optmal forwardng set because t s the frst one that we can obtan n the exhaustve search. Therefore, we redefne the optmal forwardng set F as the forwardng set that satsfes both F = arg mn F C ˆf( π, F ) and F = {j C D j < ˆf( π, F ) t D}. Note that wth ths defnton, the optmal forwardng set s unque. Then, the followng lemma helps us to fnd the optmal forwardng set more quckly. Lemma 1: For all F C that satsfes F = {j C D j < ˆf( π, F) t D }, F F. Proof: From Proposton 3 (a) and the defnton of F, all nodes k F satsfy D k < ˆf(π, F ) t D ˆf(π, F ) t D, for any subset F C. Snce F C, we obtan D k < ˆf(π, F) t D for all nodes k F. Hence, F F. Lemma 1 mples that when we exhaustvely search for the optmal forwardng set from k = C to k = 1, we can stop searchng f we fnd the frst (largest) k such that for all nodes j F,k, D j < ˆf( π, F,k ) t D, and for all nodes l/ F,k, D l ˆf( π, F,k ) t D. Snce all neghborng nodes are prortzed by ther delays, we do not need to compare the delays of all neghborng node wth the threshold value. Hence, the stoppng condton can be further smplfed as follows: node searches the largest k such that for node j wth b j ( π )=k, D j < ˆf( π, F,k ) t D, and for node l wth b l ( π )=k 1, D l ˆf( π, F,k ) t D. It should be noted that the optmal forwardng set s tmenvarant due to the memoryless property of a Posson random process. Specfcally, the expected tme for each node j n C to wake up s always the same as t I /p j regardless of how long the source node have wated. Therefore, the strategy to mnmze the expected delay s also tme-nvarant. C. Globally Optmal Forwardng and Prorty Matrces We next use the nsght of Secton III-B to develop an algorthm computng the globally optmal forwardng and prorty matrces for gven p. Ths algorthm has the flavor of the dstrbuted Bellman-Ford s algorthm for fndng the shortest paths. At each teraton, each node uses the delay estmates from the prevous teraton to update the forwardng set and the prorty assgnment. We wll show that the algorthm converges n N teratons, and the resultng A and B mnmze the expected delay D ( p,a, B).

7 The algorthm s presented next. The OPT-DELAY Algorthm Step (1) At teraton 0, each node sets D (0) = { 0 f = s, otherwse. and F (0) =. Each node arbtrarly assgns prortes to neghborng nodes. Step (2) At teraton h ( 1), each node sets b (h) b ( π (h 1) ), where π (h 1) =(D (h 1) j,j C ). Step (3) Each node updates F (h) forwardng set for π (h 1) D (h) = and also updates D (h) = by fndng the optmal as follows (h 1) ˆf( π, F (h) ). (5) Step (4) If D (h) = D (h 1) for all nodes N, ths algorthm termnates. Otherwse, each node ncreases h by one and goes back to Step (2). To analyze the OPT-DELAY algorthm, we wll use the followng notatons. We defne the subgraph g (A) = G(V (A),E (A)) as the graph wth vertces V (A) = {j V (A) s connected to j n g(a)} and edges E (A) = {(j, k) E(A) {j, k} V (A)}. By conventon, node s connected to tself,.e., V (A) for all A A. Ths subgraph g (A) shows all possble paths from node under forwardng matrx A. For any forwardng matrx A, the number of dstnct acyclc paths n g(a) s fnte when the total number of nodes s fnte. Let g(a) be the maxmum length of acyclc paths n g(a). Then, the followng proposton states an mportant property for analyzng the OPT-DELAY algorthm. Proposton 4: For any p, A A, and B B such that g (A) s cyclc, there exst A A and B B such that g (A ) s acyclc, g (A ) g (A), and D ( p,a, B ) D ( p,a, B). The detaled proof s provded n Appendx D n [11]. Proposton 4 mples that for any forwardng and prorty matrces that cause a cyclc path from any node to snk s, there always exst other forwardng and prorty matrces wth whch all paths from node to snk s are acyclc, and the delay from node wth the new matrces s equal to or smaller than the delay wth the orgnal matrces. Ths s ntutvely true because t wll ncur hgher delay f the packets have to traverse loops. Let A (h) be the forwardng matrx that corresponds to F (h) =1f j F (h), or a (h) j =0, for all nodes N,.e., a (h) j otherwse. Smlarly, let B (h) be the prorty matrx n whch the transpose of the -th row s b (h). Let A ( p) and B ( p) be the forwardng and prorty matrces when the OPT-DELAY algorthm converges. (Note that p s fxed and gven.) The followng proposton provdes the key propertes of the algorthm. Proposton 5: The algorthm has the followng propertes: 1) At teraton h, g(a (h) ) s an acyclc graph. 2) The OPT-DELAY algorthm converges wthn N teratons. 3) For gven p, (A ( p), B ( p)) = arg mn A,B D ( p,a, B) for all nodes N. Proof: In ths paper, we show the basc deas of the proof. The detaled verson of proof s provded n Appendx E n [11]. We frst show that D (h+1) D (h) for h 1 and all nodes. Suppose n contrary that there exsts node such that D (h) < D (h+1). Then, from [11], we can show that < D (h) 1. there must exst node 1 C such that D (h 1) 1 Repeatng ths procedure, we can fnd a sequence of nodes 2,, k,, h 1 such that k C k 1 and D (h k) k < D (h k+1) k. For node h 1, D (1) h 1 < D (2) h 1. However, f snk s s n h 1 s transmsson range,.e., s C h 1, then D (k) h 1 = D (1) h 1 = t D +t I /p s at any teraton k, because node h 1 can delver the packet drectly to snk s. If snk s s not n C h 1, D (2) h 1 D (1) h 1 =. Ths leads to a contradcton. Thus, D (h+1) D (h) for all nodes N and teraton h>1. We now prove the frst property. Suppose n contrary that there s a cyclc path n g(a (h) ). Let the sequence of nodes along ths cyclc path be 1, 2,, K, and K+1 = 1,.e., k+1 F (h) k show that the delays along the cyclc path satsfy for k =1, 2,..., K. Then, from [11], we can D (h) 1 >D (h) 2 > >D (h) K >D (h) 1. Ths s a contradcton. Therefore, g(a (h) ) s an acyclc graph. Let A (h) = {A A g (A) h}. A (h) denotes the set of forwardng matrces wth whch the maxmum number of hops along acyclc paths from node to snk s s less than h. We now show that D (h) mn D ( p,a, B). (6) A A (h),b We prove by nducton. At teraton 1, F (1) = {s} and D (1) = t D +t I /p s f snk s C. Otherwse, F (1) = and D (1) =. Now consder the rght-hand-sde of (6). If A (1) s not an empty set, ths means that node has a drect path to snk s, whch mples that ts expected delay s t D + t I /p s. If A (1) s an empty set, ths means that there s no path for node to reach snk s wthn 1 hop, whch mples that the expected delay to reach snk s wthn 1 hop s nfnte. Therefore, (6) holds at teraton 1. Next, assume that the nducton hypothess (6) holds at teraton h,.e., D (h) mn D ( p,a, B) N. (7) A A (h),b Then, usng Proposton 4, we can show that (7) also holds at teraton h +1. (See the detal n Appendx E n [11].) Hence, (6) holds for all nodes. We next prove that D ( p,a (h), B (h) ) D (h). At teraton 1, F (1) = {s} f s C. Otherwse, F (1) =. Hence, { D ( p,a (1), B (1) td + t )= I /p s f s C, otherwse.

8 Thus, D ( p,a (1), B (1) ) = D (1) for all nodes, and so D ( p,a (h), B (h) ) D (h) holds at teraton 1. Next assume that the nducton hypothess D ( p,a (l), B (l) ) D (l) N (8) holds for all l h. Then, from [11], we can show that (8) also holds for l = h +1. Hence, D ( p,a (h), B (h) ) D (h) holds for all and h. From the prevous results, we conclude that D ( p,a (h), B (h) ) D (h) mn D ( p,a, B) (9) A A (h),b The maxmum length of an acyclc path s equal to or less than N. Therefore, A (N) = A for all nodes. Snce A (h) A, at teraton N, we obtan D ( p,a (N), B (N) )=D (N) = mn D ( p,a, B) A,B from (9). Hence, the algorthm must converge n at most N teratons, and A (h), B (h), and D (h) converge to the optmal forwardng matrx, the optmal prorty matrx, and the mnmum expected delay from node to snk s, respectvely. Proposton 5 shows that there always exsts (A, B) that can mnmze the delay from all nodes at the same tme, and (A( p), B( p)) corresponds to such a soluton. Furthermore, the graph g(a ( p)) s acyclc. The complexty of ths algorthm s gven by O(N). Moreover, ths algorthm can be mplemented n a fully dstrbuted fashon. IV. SOLUTION TO THE LIFETIME-MAXIMIZATION PROBLEM In ths secton, we solve the orgnal lfetme-maxmzaton problem (P), usng the results n prevous sectons. By lettng q = ln(1 p ) e, we can rewrte problem (P) as t I (P1) max mn, q,a,b N q subject to D ( p,a, B) ξ, N p =1 e q/e, N (10) q (0, ), N A A, B B. Snce for any gven p, A ( p) and B ( p) are the optmal forwardng matrx and the optmal prorty matrx, respectvely, that mnmze the delay from all nodes, we have D ( p,a ( p), B ( p)) D ( p,a, B) for all A and B. Hence, we can rewrte problem (P1) as follows: t I (P2) max mn, q N q subject to D ( p,a ( p), B ( p)) ξ, N p =1 e q /e, N q (0, ), N Problem (P2) can be further smplfed wth the followng proposton. Proposton 6: If q s the optmal soluton to problem (P2), then so s q such that q =(q = max k qk, N ),.e., we can let every node have the same q. Proof: Snce both solutons have the same objectve value, t s suffcent to show that f q s n the feasble set, so s q. Let p and p be the awake probablty vectors that correspond to q and q, respectvely, by (10). Snce p s monotoncally ncreasng as q ncreases, and q q, we have p p. (The symbol denotes componentwse nequalty,.e., f q p, then q p for all, where q and p are the -th components of q and p, respectvely.) Note that the delay D ( p,a ( p), B ( p)) from each node s a non-ncreasng functon wth respect to each component of p. (See Appendx F n [11].) Snce p p, for all nodes, we have D ( p,a ( p), B ( p)) D ( p, A ( p ), B ( p )). Hence, f q s n the feasble set, so s q. Usng the above proposton, we can rewrte problem (P2) nto a problem wth one varable q, (P3) mn q, subject to max D ( p,a( p), B( p)) ξ N p =1 e q/e, N q (0, ). If q s the soluton to problem (P3), then ( p, A( p ), B( p )) (p =1 /e e q ) corresponds to the soluton of the orgnal problem (P). Note that max N D ( p,a( p), B( p)) s a non-ncreasng functon of p. (See the proof of Proposton 6.) Snce p s an ncreasng vector of q, the smplest soluton to Problem (P3) s to lnearly search q such that max N D ( p,a( p), B( p)) = ξ where p =1 e q/e. We develop an effcent bnary search algorthm for computng the optmal value of q. The Bnary Search Algorthm for Problem (P3) Step (1) Intally, snk s sets p (1) =0.5 and k =1. Step (2) Snk s sets q (k) = ln(1 p (k) ) max N e. Step (3) Nodes run the OPT-DELAY algorthm for gven p (k) =(p (k) =1 e q(k) /e, N). Step (4) After N teratons, the optmal forwardng set and the optmal prorty assgnment under p (k) are found. Nodes j that are not n the other node s forwardng set,.e., j / F (A ( p (k) )) for all nodes, send feedback of ther delays D j ( p (k), A ( p (k) ), B ( p (k) )) to snk s. Step (5) Let D max be the maxmum feedback delay arrved at snk s. If D max >ξ +ɛ, then snk s sets p (k+1) = p (k) +0.5 k+1, ncreases k by one, and goes back to Step (2). If D max <ξ ɛ, then snk s sets p (k+1) = p (k) 0.5 k+1, ncreases k by one, and goes back to Step (2). If D max [ξ ɛ,ξ + ɛ], then the algorthm termnates, and returns q (k) as the optmal soluton to Problem (P3). The reason that we take q (k) wth respect to the maxmum e n Step (2) s because ths makes all p (k) less than or equal to p (k). (Note that we only search p (k) over (0, 1].) In

9 Step (4), only such a node j that does not belong to any other forwardng set needs to send the feedback delay to the snk s because the node wth the maxmum delay does not belong to any other forwardng set accordng to Property (a) n Proposton 3. Snce snk s only needs to know the maxmum delay, there s no need for the other nodes to feedback ther delays. V. SIMULATION RESULTS In ths secton, we provde smulaton results to llustrate the performance advantage of our optmal anycast algorthm. We smulate a wreless sensor network wth 400 nodes deployed randomly over a 10-by-10 area wth unform dstrbuton, and the snk s s located at (0, 0). We assume that the transmsson range from each node s a dsc wth radus 1.5,.e., j C, f the dstance between node j and node s less than 1.5. The parameters t I and t D are set to 1 and 5, respectvely. We also assume that power consumpton rato e s dentcal for all nodes. A. Exstng Algorthms Proposed n the Lterature In ths subsecton, we revew some exstng algorthms that we wll compare wth our optmal algorthm. Normalzed-latency Anycast Algorthm: The normalzedlatency algorthm proposed n [10] s an anycast-based heurstc that explots geographc nformaton to reduce the delay from each node. Let d be the dstance from node to snk s, and let r j be the progress from node to node j toward snk s,.e., r j = d d j. If a node has a packet, let D be the one-hop delay from node to a next-hop node, and let R be the progress between two nodes. Snce node selects the next-hop node probablstcally, both D and R are random varables. The objectve of the normalzed latency algorthm s to fnd the forwardng set that mnmzes the expectaton of normalzed one-hop delay,.e., E[ D R ]. The dea behnd ths algorthm s to mnmze the expected delay per unt dstance, whch mght help to reduce the actual end-to-end delay. Nave Anycast Algorthm: The nave algorthm proposed n [10] s also an anycast-based heurstc algorthm that explots geographc nformaton. Under ths algorthm, each node ncludes all neghborng nodes wth postve progress n the forwardng set. Determnstc Routng Algorthm: By determnstc routng, we mean that each node has only one desgnated nexthop forwardng node. Therefore, determnstc routng can be vewed as a specal case of anycast, n whch the sze of the forwardng set at each node s restrcted to one. Therefore, nstead of fndng the optmal forwardng set F (h) = (h 1) arg mn F C ˆf( π, F) n Step (3) of the OPT-DELAY algorthm, we update F (h) accordng to F (h) = arg mn ˆf( π (h 1), F),. (11) F C : F =1 After the above modfcaton, the OPT-DELAY algorthm becomes one that fnds the optmal next hop under determnstc routng. Note that ths modfed algorthm s equvalent to the well-known Bellman-Ford shortest path algorthm, n whch the length of each lnk (, j) s gven by t I /p j +t D. Let D ( p) denote the mnmum delay from node under determnstc under the modfed algorthm converges to D ( p). In ths smulaton, n order to compare the network lfetme under the dfferent algorthms, we run the bnary search algorthm for Problem (P3), replacng the OPT-DELAY algorthm n Step (3) wth the above mentoned algorthms. routng. Then, D (h) B. Performance Comparson In Fg. 2, we compare the network lfetme under the dfferent algorthms, where x-axs represents dfferent maxmum allowable delays ξ n our orgnal Problem (P), and y- axs represents the maxmum lfetme for each ξ. The curve labeled Anycast (optmal) represents the lfetme under the optmal anycast algorthm,.e., the OPT-DELAY algorthm. The curves labeled Anycast (norm) and Anycast (nave) represent the lfetme under the normalzed-latency anycast algorthm, and under the nave anycast algorthm, respectvely. The curve labeled Determnstc routng represents the lfetme under the determnstc routng algorthm. Network Lfetme Anycast (optmal) Anycast (norm) 2 Anycast (nave) Determnstc Routng Maxmum Allowable Delay ξ * Fg. 2. The network lfetme accordng to dfferent allowable delay ξ when nodes are unformly deployed. From Fg. 2, we observe that all anycast algorthms sgnfcantly extend the lfetme compared to the determnstc routng algorthm. We also observe that the performance of the optmal and the normalzed-latency algorthm s very close. Note that the normalzed-latency algorthm gves preference to nodes wth larger progress, whle our optmal algorthm gves preference to nodes wth smaller delays. The results n Fg. 2 seem to suggest that there s a correlaton between progress and delay when nodes are deployed unformly. Fnally, the reason for the performance gap between the optmal and the nave algorthms s that transmttng a packet to a neghbor wth small progress s often not a good decson f a node wth hgher progress s expected to wake up soon. We next smulate a topology where there s a hole n the sensor feld as shown n Fg. 3. Ths s motvated by practcal scenaros, where there are obstructons n the sensor feld, e.g.,

10 a lake or a mountan where sensor nodes cannot be deployed. The smulaton result based on ths topology s provded n Fg Maxmum Delay Node Lake Fg. 3. Node deployment and routng paths under dfferent forwardng algorthms when p =0.5: The dotted lnes llustrate all routng paths under the optmal anycast algorthm, the thck sold lnes llustrate the unque routng path under the determnstc routng path, and thn sold lnes llustrate all routng paths under the normalzed-latency anycast algorthm Network Lfetme Anycast (optmal) Anycast (norm) 2 Anycast (nave) Determnstc Routng Maxmum Allowable Delay ξ * Fg. 4. The network lfetme accordng to dfferent allowable delay ξ when nodes are not unformly dstrbuted. From ths fgure, we observe that the optmal anycast algorthm substantally outperforms the other algorthms ncludng the normalzed-latency anycast algorthm. Fg 3 provdes us wth the ntuton for ths performance gap. We plot the routng paths from the nodes wth the largest delay. The dotted lnes (above the hole) llustrate all routng paths under the optmal anycast algorthm. The thck sold lnes (above the hole) llustrate the unque routng path under the determnstc routng algorthm. The thn sold lnes (above the hole) llustrate all routng paths under the normalzed-latency anycast algorthm. The routng paths under the nave anycast algorthm are omtted because they are smlar to those under the normalzedlatency anycast algorthm. In our optmal algorthm, n order to reduce the delay, a packet s frst forwarded to neghbors wth negatve progress but smaller delay. However, under the normalzed-latency algorthm, all packet are forwarded only to nodes wth postve progress, and hence they take longer detours. Therefore, the result of Fg 3 shows that when the node dstrbuton s not unform, there may not be a strong correlaton between progress and delay. Thus, the anycastbased heurstc algorthms dependng only on geographcal nformaton could perform poorly. VI. CONCLUSION In ths paper, we study how to use anycast to reduce the end-to-end delay and to prolong the lfetme of wreless sensor networks employng asynchronous sleep-wake schedulng. In partcular, we study the jont control problem of how to optmally control the sleep-wake schedule, the anycast canddate set of next-hop neghbors, and the anycast prortes, n order to maxmze the network lfetme subject to a upper lmt on the expected end-to-end delay. We provde an effcent soluton to ths jont control problem, and as a part of the soluton, we also show how to optmally choose the anycast canddate set to mnmze the end-to-end delay from all sensor nodes. Our numercal results suggest that the proposed soluton can substantally outperform pror heurstc solutons n the lterature under practcal scenaros where there are obstructons n the coverage area of the wreless sensor network. The algorthms that we have developed can be easly appled to energy-constraned event-drven wreless sensor networks. In future work, we plan to extend the result to the case wth non-posson sleep-wake patterns, and to handle more general notons of network lfetme. REFERENCES [1] W. Ye, H. Hedemann, and D. Estrn, Medum Access Control wth Coordnated Adaptve Sleepng for Wreless Sensor Networks, IEEE/ACM Transactons on Networkng, vol. 12, no. 3, pp , June [2] T. van Dam and K. Langendoen, An Adaptve Energy-Effcent MAC Protocol for Wreless Sensor Networks, n Proc. SenSys 03, November 2003, pp [3] G. Lu, B. Krshnamachar, and C. S. Raghavendra, An Adaptve Energy-Effcent and Low-Latency MAC for Data Gatherng n Wreless Sensor Networks, n Proc. IPDPS 04, Aprl 2004, pp [4] J. Polastre, J. Hll, and D. Culler, Versatle Low Power Meda Access for WIreless Sensor Networks, n Proc. SenSys 04, November 2004, pp [5] J. Polastre, J. Hll, P. Levs, J. Zhao, D. Culler, and S. Shenker, A Unfyng Lnk Abstracton for Wreless Sensor Networks, n Proc. SenSys 05, November 2005, pp [6] M. Zorz and R. R. Rao, Geographc Random Forwardng (GeRaF) for Ad Hoc and Sensor Networks: Energy and Latency Performance, IEEE transactons on Moble Computng, vol. 2, no. 4, pp , October [7], Geographc Random Forwardng (GeRaF) for Ad hoc and Sensor Networks: Multhop Performance, IEEE transactons on Moble Computng, vol. 2, no. 4, pp , October [8] R. R. Choudhury and N. H. Vadya, MAC-Layer Anycastng n Ad Hoc Networks, SIGCOMM Computer Communcaton Revew, vol. 34, no. 1, pp , January [9] S. Jan and S. R. Das, Explotng Path Dversty n the Lnk Layer n Wreless Ad Hoc Networks, n Proc. WoWMoM, June 2007, pp [10] S. Lu, K.-W. Fan, and P. Snha, CMAC: An Energy Effcent MAC Layer Protocol Usng Convergent Packet Forwardng for Wreless Sensor Networks, n Proc. SECON 07, San Dego, CA, June [11] J. Km, X. Ln, N. B. Shroff, and P. Snha, On Maxmzng the Lfetme of Delay-Senstve Wreless Sensor Networks wth Anycast, Techncal Report, Purdue Unversty, km309/km08tech.pdf, 2007.