Cross-Layer Interactions in Multihop Wireless Sensor Networks: A Constrained Queueing Model

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1 Cro-Layer Interaction in Multihop Wirele Senor Network: A Contrained Queueing Model YANG SONG Univerity of Florida and YUGUANG FANG Univerity of Florida Xidian Univerity In thi article, we propoe a contrained queueing model to invetigate the performance of multihop wirele enor network. Specifically, the cro-layer interaction of rate admiion control, traffic engineering, dynamic routing, and adaptive link cheduling are tudied jointly with the propoed queueing model. In addition, the tochatic network utility maximization problem in wirele enor network i addreed within thi framework. We propoe an adaptive network reource allocation cheme, called the ANRA algorithm, which provide a joint olution to the multiple-layer component of the tochatic network utility maximization problem. We how that the propoed ANRA algorithm achieve a near-optimal olution, that i, (1 ɛ of the global optimum network utility where ɛ can be arbitrarily mall, with a trade-off with the average delay experienced in the network. The propoed ANRA algorithm enjoy the merit of elf-adaptability through it online nature and thu i of particular interet for time-varying cenario uch a multihop wirele enor network. Categorie and Subject Decriptor: C.2.1 [Computer-Communication Network]: Network Architecture and Deign Wirele communication 4 General Term: Algorithm, Deign, Performance, Theory Additional Key Word and Phrae: Cro-layer deign, online algorithm, tochatic network optimization, tochatic utility maximization Y. Fang i alo a Changjiang Scholar Chair Profeor with the National Key Laboratory of Integrated Service Network, Xidian Univerity, Xi an, China. Thi work wa upported in part by the U.S. National Science Foundation under Grant CNS , CNS and CNS The work of Y. Fang wa alo partially upported by the National Natural Science Foundation of China order Grant , the Fundamental Reearch Fund for the Central Univeritie under Grant JY , and the 111 Project under Grant B Author addree: Y. Song, Department of Electrical and Computer Engineering, Univerity of Florida, Gaineville, FL 32611; yangong@ufl.edu; Y. Fang, Department of Electrical and Computer Engineering, Univerity of Florida, Gaineville, FL 32611; fang@ece.ufl.edu. Permiion to make digital or hard copie of part or all of thi work for peronal or claroom ue i granted without fee provided that copie are not made or ditributed for profit or commercial advantage and that copie how thi notice on the firt page or initial creen of a diplay along with the full citation. Copyright for component of thi work owned by other than ACM mut be honored. Abtracting with credit i permitted. To copy otherwie, to republih, to pot on erver, to reditribute to lit, or to ue any component of thi work in other work require prior pecific permiion and/or a fee. Permiion may be requeted from Publication Dept., ACM, Inc., 2 Penn Plaza, Suite 701, New York, NY USA, fax +1 ( , or permiion@acm.org. C 2010 ACM /2010/12-ART4 $10.00 DOI / ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December 2010.

2 4:2 Y. Song and Y. Fang ACM Reference Format: Song, Y. and Fang, Y Cro-Layer interaction in multihop wirele enor network: A contrained queueing model. ACM Tran. Model. Comput. Simul. 21, 1, Article 4 (December 2010, 26 page. DOI = / INTRODUCTION Wirele enor network have attracted ignificant attention in both indutrial and academic communitie in the pat few year, epecially with the advance in low-power circuit deign and mall-ize energy upplie which ignificantly reduce the cot of deploying large-cale wirele enor network. The enor network can ene and meaure the phyical environment, for example, temperature, peed, ound, radiation, and the movement of the object, etc. In addition, wirele enor network have become an important olution for military application uch a information gathering and intruion detection. Other implementation of the wirele enor network include the healthcare body enor network, vehicular-to-roadide communication network, multimedia enor network, and underwater communication network. For more dicuion on the wirele enor network, refer to the urvey paper uch a Akyildiz and Kaimoglu [2004] and Yick et al. [2007]. Since the enor node in the network are uually deployed in place where traditional wired networking olution are not feabile, wirele tranmiion among enor node are trongly preferred. In addition, due to the retrained ize of wirele enor node, the computational capability of a ingle node i limited. Therefore, the meaured information i uually tranmitted to a remote Data Proceing Center (DPC for further data analyi. Furthermore, due to the unreliable wirele link, multiple data ink may exit in the network which collect the meaured data and tranmit the packet to the DPC node ecurely and reliably, poibly through the Internet. Before the wide deployment of wirele enor network, a ytematic undertanding on the performance of the multihop wirele enor network i deired. However, finding a uitable and accurate analytical model for wirele enor network i particularly challenging. Firt, the time-varying channel condition among wirele link ignificantly complicate the analyi for the network performance in term of throughput and experienced delay, even in an average ene [Stolyar 2006; Lin 2006; Gupta and Shroff 2009]. Secondly, due to the unpredictability of the behavior of the monitored object, the exogenou traffic arrival to the network, that i, the number of newly generated packet, i a tochatic proce. Therefore, to enure the tability of the network, that i, to keep the queue in the network contantly finite, the analytical model of wirele enor network hould comprie a rate admiion control mechanim which can dynamically adjut the number of admitted packet into the network. Thirdly, due to the hotile wirele communication link, a dynamic routing cheme hould be included in the analytical model. Moreover, the model hould capture the complex iue of wirele link cheduling which i ignificantly challenging due to the mutual interference of wirele tranmiion. Latly, in order to fully explore the network reource and to mitigate ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December 2010.

3 Cro-Layer Interaction in Multihop Wirele Senor Network 4:3 the network congetion, an appropriate analytical network model hould be able to dynamically deliver packet through multiple data ink and thu an automatic load balancing olution can be achieved. In the exiting literature, mot of the propoed model for wirele enor network rely on the fluid model [Kelly et al. 1998], where a flow i characterized by a ource node and a pecific detination node, for example, Kelly et al. [1998], Chiang [2005], Chiang et al. [2007], Low and Lapley [1999]. However, thi model i not applicable to the cae where the generated packet can be delivered to any of the ink node, that i, the detination node i one of the ink and i elected dynamically. Moreover, thi fluid model neglect the actual queue interaction within the wirele enor network. In thi article, to tudy the cro-layer interaction of the multihop wirele enor network, we propoe a contrained queueing model where a packet need to wait for ervice in a data queue. More pecifically, we invetigate the joint rate admiion control, dynamic routing, adaptive link cheduling, and automatic load balancing olution to the wirele enor network through a et of interconnected queue. Due to the wirele interference and the underlying cheduling contraint, at a particular time lot, only a ubet of queue can be cheduled for tranmiion imultaneouly. To demontrate the effectivene of the propoed contrained queueing model, we invetigate the Stochatic Network Utility Maximization (SNUM problem in multihop wirele enor network. Baed on the propoed queueing model, we develop an Adaptive Network Reource Allocation (ANRA cheme which i a cro-layer olution to the SNUM problem and yield a (1 ɛ near-optimal olution to the global optimum network utility where ɛ>0 can be arbitrarily mall. The propoed ANRA cheme conit of multiple-layer component uch a joint rate admiion control, traffic plitting, dynamic routing, a well a adaptive link cheduling. In addition, the ANRA cheme i eentially an online algorithm which only require the intantaneou information of the current time lot and hence ignificantly reduce the computational complexity. The ret of the article i organized a follow. Section 2 briefly ummarize the related work in the literature. The contrained queueing model for the cro-layer interaction of wirele enor network i propoed in Section 3. The tochatic network utility maximization problem of the wirele enor network i invetigated in Section 4, where a cro-layer olution, called the ANRA cheme, i developed. The performance analyi of the ANRA cheme i provided in Section 5. An example which demontrate the effectivene of the ANRA cheme i given in Section 6 and Section 7 conclude thi article. 2. RELATED WORK To capture the cro-layer interaction of multihop wirele enor network, everal analytical model have been propoed in the literature. For example, in Chiang [2005], Chiang et al. [2007], Eryilmaz and Srikant [2005], Kelly et al. [1998], Low and Lapley [1999], Song and Fang [2007], the multihop network reource allocation problem ha been tudied through a fluid model. Each flow, or eion, i characterized by a ource and a detination node where ingle path routing or multipath routing cheme are implemented. Mot of the ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December 2010.

4 4:4 Y. Song and Y. Fang work rely on the dual optimization framework which decompoe the complex cro-layer interaction into eparate ublayer problem by introducing dual variable. For example, the flow injection rate, controlled by the ource node of the flow, i calculated by olving an optimization problem with the knowledge of the dual variable, called hadow price [Kelly et al. 1998; Low and Lapley 1999], of all the link that are utilized. However, there are everal drawback for the fluid-baed model. Firt, to calculate the optimum flow injection rate, the information along all path hould be collected in order to implement the rate admiion control mechanim. In a dynamic environment uch a wirele enor network, thi proce of information collection may take a ignificant amount of time which inevitably prolong the network delay. Secondly, the optimization-baed olution uually purue fixed operating point which are hardly optimal in dynamic wirele etting with tochatic traffic arrival and time-varying channel condition. Thirdly, the fluid model uually aume that the change of the flow injection rate are perceived" by all the node along it path intantaneouly. The actual queue dynamic and interaction are neglected. In contrat, following the eminal paper of Taiula and Ephremide [1992], many olution have been focued on the queueing model for tudying the complex interaction of communication network. Neely et al. extend the reult of Taiula and Ephremide [1992] into wirele network with time-varying channel condition [Neely 2003]. For a more complete urvey of thi area, refer to Georgiadi et al. [2006]. The key component of the queue-baed olution in thee paper i the MaxWeight cheduling algorithm [Taiula and Ephremide 1992; Neely 2003]. Intuitively, at a time lot, the network pick the et of queue which: (1 can be active imultaneouly and (2 have the maximum overall weight. It i well-known that the MaxWeight algorithm i throughputoptimal in the ene that any arrival rate vector that can be upported by the network can be tabilized under the MaxWeight cheduling algorithm. In addition, the MaxWeight algorithm i an online policy which require only the information about current queue ize and channel condition. However, one notoriou drawback of the MaxWeight algorithm i the delay performance. The reaon i that in order to achieve the throughput-optimality, the MaxWeight algorithm explore a dynamic routing olution where long path are utilized even under a light traffic load. Thi phenomenon i ubtantiated via imulation by a recent work of Ying et al. [2009]. In Ying et al. [2009], the author propoe a variant of the MaxWeight algorithm where the average number of hop of tranmiion i minimized. Therefore, when the traffic i light, the propoed olution provide a much lower delay than the traditional MaxWeight algorithm. However, a a trade-off, the induced network capacity region in Ying et al. [2009] i noticeably maller than that of the original MaxWeight algorithm. Conequently, it i difficult to provide a minimum rate guarantee on all the eion in the network. Our work i inpired by Ying et al. [2009]. With repect to Ying et al. [2009], however, our article innovate in the following way. Firt, we focu on a heavy-loaded wirele enor network. Therefore, our olution incorporate a rate admiion control mechanim which i not conidered in Ying et al. [2009]. Secondly, rather than minimizing the overall number of hop, we ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December 2010.

5 Cro-Layer Interaction in Multihop Wirele Senor Network 4:5 maximize the overall network utility which can alo enure the fairne among competitive traffic eion. Thirdly, we pecifically provide a minimum average rate guarantee for every eion to enure the QoS requirement. Fourthly, intead of a ingle detination cenario a conidered in Ying et al. [2009], we extend the model to cae where multiple data ink node are available. Each ource node can deliver the packet to any of the ink. Moreover, the dynamic routing and the iue of automatic load balancing i realized by the network onthe-fly. Finally, while Ying et al. [2009] treat different eion equally when minimizing the overall number of hop, our model prioritize all the eion with different QoS requirement. Therefore, a more flexible olution with ervice differentiation can be achieved. We will preent the contrained queueing model in the next ection. 3. A CONSTRAINED QUEUEING MODEL FOR WIRELESS SENSOR NETWORKS 3.1 Network Model We conider a multihop wirele enor network repreented by a directed graph G ={N, L} where N and L denote the et of vertice and the et of link, repectively. We will ue the notation of A to repreent the cardinality of et A, for example, the number of node in the network i N and L i the number of link. Time i lotted a t = 0, 1,... and at a particular time lot t, the intantaneou channel data rate of link (m, n L i denoted by μ m,n (t. In other word, link (m, n can tranmit a number of μ m,n (t packet during time lot t. We aume that during one time lot, the channel condition of link will remain contant. However, the value of μ m,n (t i ubject to change at the boundarie of time lot. Denote μ(t athenetwork link rate vector at time lot t. In thi article, we aume that μ(t remain contant within one time lot but i ubject to change at time lot boundarie. The value of μ(t i aumed to be evolving following an irreducible and aperiodic Markovian chain with arbitrarily large yet finite number of tate. 1 However, the teady tate ditribution are unknown to the network. At time lot t, the network elect a feaible link chedule, denoted by I(t = {I 1 (t, I 2 (t,, I L (t} where I l (t = 1 if link l i elected to be active and I l (t = 0 otherwie. The et of all feaible link chedule i denoted by (t which i determined by the underlying cheduling contraint uch a interference model and duplex contraint. Therefore, electing an interference-free link chedule in the network graph G i equivalent to the proce of attaining an independent et in the aociated conflict graph G, where the vertice are the link in G and a link exit in G if the two original link in G cannot tranmit imultaneouly. 1 It hould be noted that the Markovian aumption i for the eae of analyi. Our propoed model can be extended to more general cenario where the time average of an arbitrary link rate tate i well defined, a in Neely et al. [2005]. ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December 2010.

6 4:6 Y. Song and Y. Fang Fig. 1. Topology of wirele enor network. 3.2 Traffic Model There are a number of S ource node in the wirele enor network which conitently monitor the urrounding and inject exogenou traffic to the network. For example, in a wildlife monitoring cenario, the enor, uually placed with camera, need to meaure the animal movement and behavior and tranmit the generated packet to the remote Data Proceing Center (DPC in a multihop fahion. To implify analyi, we aume that each ource node i aociated uniquely with a eion. The et of ource node i denoted by S ={n 0 1, n0 2,...,n0 S } where n0, = 1,..., S i the ource node of eion. It i worth noting that the following analyi can be extended traightforwardly to the cenario where each ource node may generate multiple eion. There are D number of ink in the network which are connected to the remote data proceing center via the Internet. In other word, the ink node can be viewed a the gateway of the wirele enor network. Denote the et of ink a D ={d 1, d 2,...,d D }. In thi article, we conider a general cenario where the data packet from a ource node can be delivered to the DPC via any oftheinknodeind. Therefore, different from the exiting literature uch a Chiang [2005], Song and Fang [2007], and Ying et al. [2009], the ource node do not pecify the particular detination node for the generated packet. The election of the detination node i achieved by the network via dynamic routing cheme. The network topology conidered in thi article i illutrated in Figure 1. ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December 2010.

7 Cro-Layer Interaction in Multihop Wirele Senor Network 4:7 For a particular node in the network, ay node n, we denote φn d a the number of minimum hop from node n to the dth data ink in et D. Define φ n = min(φn d, d = 1,..., D (1 d a the minimum value of φn d for node n, that i, the minimum number of hop from node n to a ink node in et D. We aume that node n i aware of the value of φ n a well a thoe value of the neighboring node, which are attainable via precalculation by traditional routing mechanim uch a Dijktra algorithm. At time lot t, the exogenou arrival of eion, that i, the number of new packet 2 generated by the ource node of eion, i denoted by A (t. We aume that there i an upper bound for the number of new packet within one time lot, that i, A (t A max,, t. For eae of expoition, we aume that A (t i independently and identically ditributed over time lot with an average rate of λ. However, the data rate from multiple ource node can be arbitrarily correlated. For example, if the wirele enor network i deployed for monitoring purpoe, it i very likely that a movement of the object will trigger everal concurrent update of the nearby enor. Denote the vector λ ={λ 1,...,λ S } a the network arrival rate vector. The network capacity region i thu defined a all the feaible 3 network arrival vector that can be upported by the network via certain policie, including thoe with the knowledge of futuritic traffic arrival and channel rate condition. In thi article, we conider a heavy-loaded traffic cenario where the network arrival vector λ i outide of the network capacity region. Therefore, in order to achieve the network tability, a rate admiion control mechanim i implemented at the ource node. More pecifically, at time lot t, we only admit a number of X (t packet into the network from the ource node of eion, that i, n 0. Apparently, we have X (t A (t,, t. (2 In addition, we aume that each eion ha a continuou, concave, and differentiable utility function, denoted by U (X (t, which reflect the degree of atifaction by tranmitting X (t number of packet. It i worth noting that by electing proper utility function, the fairne among competitive eion can be achieved. For example, if U (X (t = log(x (t, a proportional fairne among multiple eion can be enforced [Chiang et al. 2007; Srikant 2003; Shakkottai and Srikant 2008]. 3.3 Queue Management For each node n in the network, there are N φ n number of queue that are maintained and updated. The queue are denoted by Q n,h, where 2 We aume that the packet have a fixed length. For cenario with variable packet length, the unit of data tranmiion can be changed to bit per lot and the following analyi till hold. 3 Note that additional contraint may be impoed. For example, the contraint on the minimum average rate and the maximum average power expenditure can be enforced. For more dicuion, pleae refer to Georgiadi et al. [2006]. ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December 2010.

8 4:8 Y. Song and Y. Fang h = φ n,..., N 1. Note that N 1 i the maximum number of hop for a loop-free routing path in the network. The packet in the queue of Q n,h are guaranteed to reach one of the ink node in et D within h hop, a will be hown in Section 4. It i intereting to oberve that for a newly generated packet by eion, the ource node, that i, n 0, can place it in any of the queue of Q n 0,h, where h = φ n 0,..., N 1, for further tranmiion. That i to ay, conecutive packet from the ource node n 0 may travere through different number of hop before reaching a detination ink node in et D. Therefore, when a new packet i generated, the ource node need to make a deciion on which queue the packet hould be placed, namely, traffic plitting deciion. In addition, the deciion hould be made promptly on an online bai with low computational complexity. With a light abue of notation, we ue Q n,h to denote the queue itelf and Q n,h (t to repreent the number of queue backlog 4 in time lot t. For a ingle queue, ay Q n,h, it i table if [Neely et al. 2005; Neely 2003] lim g(b 0, (3 B where g(b = lim up T 1 T T 1 Pr(Q n,h (t > B, t=0 where B i a poitive number. The network i table if all the individual queue in the network are table. For a link (n, j L, we require that the packet from Q n,h can be only tranmitted to Q j,h 1, if exit. Therefore, the queue updating dynamic for Q n,h i given by Q n,h (t + 1 [ Q n,h (t (n, j L u n,h n, j (t ] + + (m,n L um,n m,h+1 (t + X h (tδ n=n 0, (4 where [A] + denote max(a, 0 and u n,h n, j (t repreent the allocated data rate for the tranmiion of Q n,h Q j,h 1 on link (n, j, at time lot t, and N 1 h= φ n u n,h n, j (t = u n, j(t, where u n, j (t = μ n, j (t ifi n, j (t = 1, that i, link (n, j i cheduled to be active during time lot t, andu n, j (t = 0 otherwie. The notation of X h (t denote the number of packet that are admitted to the network for eion and are tored in queue Q n,h for future tranmiion. The indicator function δ A = 1 if event A i true and δ A = 0 otherwie. Note that the inequality in (4 incorporate the cenario where the tranmitter of a particular link ha le packet in the queue than the allocated data rate. We aume that during one time lot, the number of packet that a ingle queue can tranmit and receive are upper 4 In the unit of packet. ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December 2010.

9 Cro-Layer Interaction in Multihop Wirele Senor Network 4:9 bounded. Mathematically peaking, we have um,n m,h+1 (t u in, n, h, t, (5 and (m,n L (n, j L u n,h n, j (t u out, n, h, t. (6 3.4 Seion-Specific Requirement In thi article, we conider a cenario where each eion ha a pecific rate requirement α. Therefore, to enure the minimum average rate, we need to find a policy that 1 lim T T T 1 t=0 X (t α,. (7 In addition, we aume that each eion in the network ha an average hop requirement β. More pecifically, define where N 1 M (t = hx h (t, (8 h= φ n 0 N 1 We require that for each eion, h= φ n 0 X h (t = X (t,, t. 1 lim T T T 1 t=0 M (t β,. (9 Note that the average hop for a particular eion i related to the average delay experienced and the average energy conumed for the packet tranmiion of eion. Therefore, by aigning different value of α and β, a prioritized olution among multiple competitive eion can be achieved for the network reource allocation problem. 4. STOCHASTIC NETWORK UTILITY MAXIMIZATION IN WIRELESS SENSOR NETWORKS In the previou ection, we propoe a contrained queueing model to invetigate the performance of multihop wirele enor network. The model conit of everal important iue from different layer, including the rate admiion control problem, the dynamic routing problem, a well a the challenge of adaptive link cheduling. To better undertand the propoed contrained queueing model, in thi ection, we will examine the Stochatic Network Utility Maximization Problem (SNUM in multihop wirele enor network. A a crolayer olution, an Adaptive Network Reource Allocation (ANRA cheme i ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December 2010.

10 4:10 Y. Song and Y. Fang propoed to olve the SNUM problem jointly. The propoed ANRA cheme i an online algorithm in nature which provably achieve an aymptotically optimal average overall network utility. In other word, the average network utility induced by the ANRA cheme i (1 ɛ of the optimum olution, where ɛ>0 i a poitive number that can be arbitrarily mall, with a trade-off with the average delay experienced in the network. 4.1 Problem Formulation Recall that every eion poee a utility function U (X (t which i continuou, concave, and differentiable. Without lo of generality, in the ret of thi article, we will aume that U (X (t = log(x (t. Therefore, in light of the tochatic traffic arrival a well a the time-varying channel condition, our objective i to develop a policy which maximize Stochatic Network Utility Maximization (SNUM Problem uch that: E(U (X (t (10 The network remain table. The average rate requirement of all S eion, denoted by α = {α 1,...,α S }, are atified. The average hop requirement of all S eion, denoted by β ={β 1,...,β S }, are atified. Note that if the underlying tatitical characteritic of the tochatic traffic arrival and the time-varying channel condition are known, the SNUM problem i inherently a tandard optimization problem and thu i eay to olve. However, due to the unawarene of the teady tate ditribution, the SNUM problem i remarkably challenging. In addition, in wirele enor network, dynamic algorithmic olution with low computational complexity are trongly deired. In the following, we propoe an ANRA cheme to olve the SNUM problem aymptotically. The ANRA cheme i a cro-layer olution which conit of joint rate admiion control, traffic plitting, dynamic routing, a well a adaptive link cheduling component. Moreover, the ANRA algorithm can achieve an automatic load balancing olution by utilizing different ink node correponding to the variation of the network condition. The ANRA algorithm i an online algorithm in nature which require only the tate information of the current time lot. We how that the ANRA algorithm achieve a (1 ɛ optimal olution where ɛ can be arbitrarily mall. Therefore, the propoed ANRA algorithm i of particular interet for dynamic wirele enor network with time-varying environment. ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December 2010.

11 Cro-Layer Interaction in Multihop Wirele Senor Network 4: The ANRA Cro-Layer Algorithm Before preenting the propoed ANRA cheme, we introduce the concept of virtual queue [Neely 2006; Georgiadi et al. 2006; Stolyar 2005] to facilitate our analyi. Specifically, for each eion, we maintain a virtual queue Y, which i initially empty, and the queue updating dynamic i defined a Y (t + 1 = [Y (t X (t] + + α,, t. (11 Similarly, we define another virtual queue for every eion, denoted by Z, and the queue dynamic i given by Z (t + 1 = [Z (t β ] + + M (t,, t, (12 where M (t i defined in (8. Note that the virtual queue are oftware-baed counter which are eay to maintain. For example, the ource node of each eion can calculate the value of virtual queue Y (t andz (t and update the value accordingly following (11 and (12. In addition, we introduce a poitive parameter J which i tunable a a ytem parameter. The impact of J on the algorithm performance will be dicued hortly. The propoed ANRA cro-layer algorithm i given a follow. Adaptive Network Reource Allocation (ANRA Scheme: Joint Rate Admiion Control and Traffic Splitting (at time t: For each ource node, ay n 0, there are a number of queue, that i, Q n 0,h, h = φ n 0,..., N 1. Find the value of h which minimize Z (th + Q n 0,h(t, (13 where tie are broken arbitrarily. Denote the optimum value of h a h. The ource node n 0 admit a number of new packet a X (t = min ( X (t, A (t, (14 where [ X (t = J 2(Z (th + Q n 0,h (t 2Y (t ] +. (15 For traffic plitting, the ource node n 0 will depoit all X (t packet in Q n 0,h. Joint Dynamic Routing and Link Scheduling (at time t: For each link (m, n L, define a link weight denoted by W m,n (t, which i calculated a [ W m,n (t = max h= φ m,..., N 1 ( Qm,h (t Q n,h 1 (t ] +. (16 Note that if Q n,h 1 doe not exit, the tranmiion from queue Q m,h to Q n,h 1 are prohibited. At time lot t, the network elect an interference-free link chedule I(t which olve max I(t (t μ m,n (tw m,n (t. (17 ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December 2010.

12 4:12 Y. Song and Y. Fang If link (m, n i active, that i, I m,n (t = 1, the queue of Q m, h i elected for tranmiion where ( h = argmax h= φ m,..., N 1 Qm,h (t Q n,h 1 (t. (18 End Note that (17 i imilar to the original MaxWeight algorithm introduced in Taiula and Ephremide [1992] and generalized in Neely et al. [2005], Neely [2003], and Stolyar [2005]. The dynamic routing and link cheduling are addreed jointly by olving (17, which require centralized computation. However, following Taiula and Ephremide [1992], many work have been focued on the ditributed olution of (17. Although the ditributed computation iue i not the focu of thi article, we emphaize that our propoed ANRA cheme can be approximated well by exiting ditributed olution uch a Joo [2008], Akyol et al. [2008], Radunovic et al. [2008], Modiano et al. [2006], Jiang and Walrand [2008], Gupta et al. [2007], Stolyar [2008], and Wu et al. [2007]. For example, in Akyol et al. [2008], each node in the network utilize an IEEE MAC protocol where the contention window ize, or equivalently, the channel acce probability in Stolyar [2008], i adjuted conitently to approximate the link weight. The accuracy of uch random-acce-baed ditributed approximation are tudied and evaluated extenively in Akyol et al. [2008]. The cheduling component, that i, (17, of our propoed ANRA cheme can be approximated well by the olution uggeted in the aforeaid paper. For the packet placed at queue Q n 0,h, atmoth hop of tranmiion are needed in order to reach one of the ink node in et D. Thi can be verified traightforwardly due to the requirement that a tranmiion from Q m,h to Q n,h 1 can occur if and only if h 1 φ n. Moreover, the joint rate admiion control and the optimum traffic plitting component of ANRA can be implemented by the ource node in a ditributed fahion. Note that in order to calculate the intantaneou admitted rate, the ource node of eion need only to know the local queue backlog information. Moreover, the deciion of traffic plitting require only local queue information a well. Therefore, at every time lot, the joint rate admiion control and traffic plitting deciion can be made on an online bai in accordance to the time-varying condition of local queue. Furthermore, we will how that thi imple adaptive trategy doe not incur any lo of optimality. The achieved network utility induced by the ANRA cheme can be puhed arbitrarily cloe to the optimum olution. Next, we will characterize the global optimum utility in the network and provide the main performance reult of the propoed ANRA cheme. 4.3 Performance of the ANRA Scheme In thi ection, we firt characterize the global optimum olution of the SNUM problem in (10. Define U a the global maximum network utility that any cheme can achieve, that i, the optimum olution of (10. In order to achieve U, it i naturally to conider more complicated policie uch a thoe with the knowledge of futuritic arrival and channel condition. However, in the following theorem, we how that, omewhat urpriingly, the global optimum ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December 2010.

13 Cro-Layer Interaction in Multihop Wirele Senor Network 4:13 olution of the SNUM problem can be achieved by certain tationary policie, that i, the reponive action i choen regardle of the current queue ize in the network and the time lot that the deciion i made. Recall that we have aumed that for each eion, A (t i independently and identially ditributed over time lot. Denote A(t a the vector of intantaneou arrival rate of all eion, at time lot t.leta be the et of all poible value of A(t. Note that for every element in A, thati,a a, a = 1,..., A, wehave0 A a A max where denote the element-wie comparion. We ue π a to repreent the teady tate ditribution of A a. THEOREM 1. If the contraint in the SNUM problem are atified, the maximum network utility, denoted by U, can be achieved by a cla of tationary randomized policie. Mathematically, the value of U i the olution of the following optimization problem, with the auxiliary variable pa k and Rk a,a uch that: max a π a ( S +1 U k=1 p k a Rk a (19 The contraint in (10 are atified. 0 Ra k A a. pa k 0, a, k. S +1 k=1 pa k = 1, a. PROOF. We prove Theorem 1 by howing that for arbitrary policy which atifie the contraint in the SNUM problem, the overall network utility i at mot U, which i the optimum utility attained by a cla of tationary randomized policie. In other word, we need to how that ( T U P 1 1 = lim up U (X (t U, (20 T T t=0 where U P i the average network utility under a policy P. For each tate in A, aya a, define R a a the et of nonnegative rate vector that are element-wie maller than A a. Define CR a a the convex hull of et R a. Therefore, any point in CR a can be conidered a a feaible network admitted rate vector given that the current arrival rate vector i A a. Note that every point in CR a i a vector with a dimenion of S -by-1. Therefore, it can be repreented by a convex combination of at mot S +1 point, denoted by Ra k, k = 1,..., S +1, according to Caratheodory theorem. In light of thi, we firt conider a time interval from 0 to T 1. Denote N a (T atheetoftime lot that A(t = A a. Therefore, we can rewrite (20 a U P N a (T = lim up T T a ( S +1 U k=1 p k a Rk a ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December

14 4:14 Y. Song and Y. Fang Due to the tationary aumption, we have U P = a π a lim up T ( S +1 U k=1 p k a Rk a Note that we alo aume that the utility function i continuou and bounded. Therefore, the compactne of the utility function i aured. Next, we focu on a ubequence of time duration, denoted by T i, i = 1,...,. Denote Unet P (T i = ( S +1 π a U pa k (T ira k. a k=1 It i traightforward to verify that U P = lim up Unet P (T i. i Due to the compactne of the utility function, following Bolzano-Weiertra theorem [Trench 2003], we claim that there exit a ubequence of T i, i = 1,,, uch that ( S +1 lim U pa k (T ira k Ũ a. i k=1 Denote p a k a the value which generate Ũ a,thati, ( S +1 Ũ a = U. We have k=1 U P = lim up U net (T i = i a p k a Rk a π a ( S +1 U k=1. p k a Rk a According to the definition of U in (19, we conclude that U P U. Intuitively, Theorem 1 indicate that the global maximum network utility can be achieved by certain randomized tationary policie. However, to calculate U, the tationary policy need to know the teady tate ditribution which are difficult to obtain in practice. In light of thi, we propoe an adaptive network reource allocation cheme, namely, ANRA, which i an online olution and doe not require uch tatitical information a a priori. For notation uccinctne, denote U A (t = E ( U (X (t a the expected network utility induced by the ANRA cheme. The performance of the ANRA algorithm, with a parameter J, i given a the following theorem. THEOREM 2. For a given ytem parameter J, we have lim inf T 1 T T 1 t=0. U A (t U B J, (21 ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December 2010.

15 Cro-Layer Interaction in Multihop Wirele Senor Network 4:15 where B i a contant and i given by B = N ( N 1((u out 2 + (u in + A max 2 + ((α 2 + (β 2 + S (A max 2 (( N In addition, the contraint in the SNUM problem, that i, (10, are atified imultaneouly. PROOF. The proof of Theorem 2 i deferred to Section 5. The value of contant Bi determined by the number of node in the network, the number of eion, and the value of eion requirement, etc. It i worth noting that if we let J, the performance induced by the ANRA algorithm can be arbitrarily cloe to the global optimum olution U. However, a a trade-off, a larger value of J alo yield a longer average queue ize in the network. According to Little Law, a larger queue ize correpond to a longer average delay experienced in the network. Therefore, by electing the value of J properly, a trade-off between the network optimality and the average delay in the network can be achieved. We will dicu more about thi iue in the next ection. 5. PERFORMANCE ANALYSIS In thi ection, we provide a proof to Theorem 2 in the previou ection. Recall that in (11 and (12, we introduce two virtual queue, that i, Y (tandz (tfor each eion. Therefore, the average rate and the average hop requirement from all eion are converted into the tability requirement for the virtual queue. For example, the virtual queue update of Y (t i given by (11. If the virtual queue Y i table, the average arrival rate hould be le than the average departure rate of the queue, that i, 1 α lim T T T 1 t=0 X (t, which i exactly the minimum average rate requirement impoed by eion. By the ame token, the average hop requirement of eion i converted to the tability problem of the virtual queue Z. Define Q(t = ( Q(t, Y (t, Z(t, namely, all the data queue and the virtual queue in the network. Our objective i to find a policy which tabilize the network with repect to Q while maximizing the overall network utility. We firt take the quare of (4 and have (Q n,h (t (Q n,h (t (n, j L u n,h n, j (t 2 + 2Q n,h (t (n, j L (m,n L u n,h n, j (t um,n m,h+1 (t + (m,n L um,n m,h+1 (t X h (tδ n=n 0 X h (tδ n=n 0 ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December

16 4:16 Y. Song and Y. Fang Since we aume that each node generate at mot one eion, we have X h (tδ n=n 0 A max, t, n. Note that if we allow that a node can initiate multiple eion, we have X h (tδ n=n 0 S A max, t, n, where S i the number of eion in the network. In light of (5 and (6, we have (Q n,h (t (Q n,h (t 2 (u out 2 + (u in + A max 2 2Q n,h (t u n,h n, j (t um,n m,h+1 (t X h (tδ n=n 0. (22 (n, j L (m,n L We next um (22 over all the data queue in the network, that i, Q n,h,and have (Q n,h (t 2 B 1 (Q n,h (t n,h n,h 2Q n,h (t u n,h n, j (t (n, j L (m,n L um,n m,h+1 (t where B 1 = N ( N 1((u out 2 + (u in + A max 2. Note that M (t, defined in (8, atifie M (t ( N 1 2 A max. Next, we take the quare of (11 and (12 and thu have and (Y (t (Y (t 2 + (X (t 2 + (α 2 2Y (t(x (t α (Z (t (Z (t 2 + (M (t 2 + (β 2 2Z (t(β M (t. Similarly, we um over all the eion and have (Y (t (Y (t 2 B 2 2 Y (t(x (t α where B 2 = S (A max 2 + (α 2. X h (tδ n=n 0, (23 Alo, we obtain (Z (t (Z (t 2 B 3 2 Z (t(β M (t, ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December 2010.

17 Cro-Layer Interaction in Multihop Wirele Senor Network 4:17 where B 3 = (β 2 + S ( N 1 4 (A max 2. Define the ytem-wide Lyapunov function a L(Q(t = (Q n,h (t 2 + n,h (Y (t 2 + (Z (t 2. Next, we define the Lyapunov drift [Neely 2003] of the ytem a = E ( L(Q(t + 1 L(Q(t Q(t. (24 Define B = B 1 + B 2 + B 3, we have B 2 Q n,h (te u n,h n, j (t u m,h m,n (t X h (tδ n=n 0 Q(t S n,h (n, j L (m,n L ( 2E Y (t(x (t α Q(t 2E Z (t(β M (t Q(t. Next, we ubtract both ide by JE U (X (t Q(t and have JE U (X (t Q(t B 2 Q n,h (te u n,h n, j (t u m,h+1 m,n (t Q(t n,h (n, j L (m,n L ( + 2E Q n 0,h(tX h Q(t (t 2E Y (tx (t Q(t + 2 Y (tα h + 2E Z (tm (t Q(t 2 Z (tβ JE U (X (t Q(t. (25 We rewrite the R.H.S. of (25 a R.H.S. = B + 2 Y (tα 2 Z (tβ 2 Q n,h (te u n,h n, j (t u m,h+1 m,n (t Q(t n,h (n, j L (m,n L E 2Y (tx (t 2Z (tm (t 2Q n 0,h(tX h (t h + J U (X (t. Q(t ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December 2010.

18 4:18 Y. Song and Y. Fang We oberve that the dynamic routing and cheduling component of the ANRA cheme i actually maximizing Q n,h (te u n,h n, j (t um,n m,h+1 (t Q(t. (26 n,h (n, j L (m,n L In addition, the joint rate admiion control and traffic plitting component of the ANRA cheme i eentially maximizing E 2Y (tx (t 2Z (tm (t 2Q n 0,h(tX h (t+ J U (X (t Q(t h (27 with the contraint of X h (t = X (t,, t. (28 h To ee thi, we can decompoe (27 to how that each eion only maximize JU ( X (t + 2Y (tx (t 2Z (t h hx h (t h 2Q n 0,h(tX h (t. (29 Therefore, the propoed ANRA algorithm indeed minimize the R.H.S. of (25 over all policie. Conider a reduced network capacity region, denoted by ɛ, parameterized by ɛ>0, a {λ λ n,h + ɛ }, (30 where i the original network capacity region and 1 λ n,h = lim T T T 1 X h (tδ n=n 0. (31 t=0 Define Uɛ a the global optimum network utility achieved in the reduced capacity region. Apparently, we have lim ɛ 0 Uɛ U. In addition, denote X,ɛ h (0, X,ɛ h (1,..., Xh,ɛ (t,... a the optimum rate equence which yield U ɛ. Define X ɛ a the average of the optimum rate equence of eion, in the reduced capacity region. It i traightforward to verify that Xɛ + ɛ i in the original network capacity region. Therefore, following a imilar analyi a in Neely [2003], Neely et al. [2005], Neely et al. [2008], and Sharma et al. [2009], we claim that there exit a randomized policy, denoted by R, which generate E u n,h n, j (t um,n m,h+1 (t X,ɛ h + ɛ (32 (n, j L (m,n L if n i one of the ource node and E u n,h n, j (t (n, j L (m,n L um,n m,h+1 (t ɛ (33 ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December 2010.

19 Cro-Layer Interaction in Multihop Wirele Senor Network 4:19 for other node. Furthermore, policy R enure E X,ɛ h (t α c + ɛ and h E(M,ɛ (t + ɛ β, where M,ɛ (t i generated by Xh,ɛ (t. Due to the fact that the propoed ANRA cheme minimize the R.H.S. of (25 overall all policie, including R, wehave JE U (X (t Q(t B 2ɛ Q n,h (t + n,h Y (t + Z (t JE U h We next take the expectation with repect to Q(t and obtain L(Q(t + 1 L(Q(t JE U (X (t B 2ɛE n,h JE U h Q n,h (t + Y (t + Z (t X h,ɛ (t + ɛ Q(t. X h,ɛ (t + ɛ. (34 We um over time lot 0,...,T 1 and have T 1 L(Q(T L(Q(0 JE U (X (t ince T T B 1 t=0 t=0 JE U h E Q n,h (t + n,h Y (t + X h,ɛ (t + ɛ Z (t i alway nonnegative. Next, we divide the both ide of (35 by T and rearrange term to have T 1 1 JE U (X (t 1 T 1 JE U X,ɛ h T T (t + ɛ B L(Q(0, T t=0 t=0 h where the nonnegativity of the Lyapunov function i utilized. Since we aume that the initial queue backlog in the ytem are finite and the virtual queue are initially empty, taking ɛ 0 and lim inf T yield the performance reult of the ANRA algorithm tated in Theorem 2. ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December (35

20 4:20 Y. Song and Y. Fang We next how that the contraint of the SNUM problem are alo atified. To illutrate thi, we how that the queue in the network, including real data queue and virtual queue, are table. Baed on (34, we um over time lot 0,...,T 1 and have T 1 2ɛE Q n,h (t + t=0 n,h T 1 L(Q(0 + t=0 JE Z (t Y (t + U (X (t + T B. (36 Due to X (t A max and the aumption on the utility function, we claim that U (t i upper bounded and denote the maximum utility within one time lot a U max,thati, U (t U max,, t. (37 Divide the both ide of (36 by T and we have T 1 1 2ɛE Q n,h (t + Y (t + T t=0 n,h By taking lim up T,wehave lim up T 1 T T 1 t=0 E n,h Q n,h (t + Z (t L(Q(0 + J S U max + T B. Y (t + Z (t J S U max + B. (38 2ɛ Note that the preceding analyi hold for any feaible value of ɛ. Denote ϕ a the maximum value of ɛ uch that ϕ i not empty. Finally, we conclude that lim up T 1 T T 1 t=0 E n,h Q n,h (t + Y (t + Z (t J S U max + B <. (39 2ϕ The tability of the network follow immediately from Markov Inequality and thu complete the proof. It i worth noting that a hown in (39, a large value of J induce a longer average queue ize in the network. Therefore, a trade-off between the algorithm performance of the ANRA cheme and the average delay experienced in the network can be controlled effectively by tuning the value of J. 6. CASE STUDY In thi ection, we demontrate the effectivene of the ANRA algorithm numerically through a imple network hown in Figure 2. We tre that, however, thi exemplifying tudy cae reproduce all the challenging problem involved in the complex cro-layer interaction in time-varying environment, uch a tochatic traffic arrival, random channel condition, and dynamic routing and cheduling, etc. A hown in Figure 2, the ource node in the network are node A and B wherea the detination ink node are denoted by E and F. ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December 2010.

21 Cro-Layer Interaction in Multihop Wirele Senor Network 4:21 Fig. 2. Example network. There are ix node and twelve link in the network. Therefore, node A and B each maintain four queue, from hop 2 to hop 5, and node C and D each maintain five queue, from hop 1 to hop 5, in the buffer. At each time lot, a wirele link i aumed to have three equally poible data rate 5, 2, 8 and 16. The traffic arrival are independently and identically ditributed with three equally poible tate, that i, 0, 10, and 20. The minimum rate requirement of the two eion are 5 and 8 and the average hop requirement of the eion are 30 and 10. Without lo of generality, we aume that at a given time lot, two link with a common node cannot be active imultaneouly. For example, if link A B i active, link B A, A C, C A, B D and D B cannot be elected. Figure 3 depict the average network utility achieved by the ANRA cheme for different value of J where each experiment i executed for time lot. We can oberve from Figure 3 that the overall network utility rie a the value of J increae. However, the peed of utility improvement decreae and the achieved network utility converge to the global optimum utility U gradually. It i worth noting that in practice, the value of U cannot be attained efficiently without knowing the underlying tatitical characteritic. However, the propoed ANRA cheme can achieve a olution which i arbitrarily cloe to the global optimum olution with no uch information required. To demontrate the trade-off of different value of J, in Figure 4, we how the average queue ize in the network for J = 20, 50, 200, 500, 1000, 2000, 5000, 10000, and We can ee that, a expected, the average queue ize increae a the value of J get larger. Note that the average queue ize i related to the average delay in the network. Therefore, a trade-off between the network optimality and the average experienced delay can be achieved by tuning the value of J. In Figure 5, we illutrate the ample trajectorie of the admitted rate of two eion with J = 5000, for the firt 50 time lot. We can oberve that each eion admit different amount of packet into the network adaptively 5 Note that the unit of data tranmiion i packet per lot. ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December 2010.

22 4:22 Y. Song and Y. Fang Fig. 3. Average network utility achieved by ANRA for different value of J. Fig. 4. Average network queue ize by ANRA for different value of J. following the time-varying condition of the network. In addition, we depict the trajectorie of the four virtual queue with the ame etting, in Figure 6, for the firt 100 time lot. By comparing Figure 5 and Figure 6 jointly, we can oberve that for the minimum rate virtual queue, ay Y 1, whenever there i the tendency that the virtual queue i accumulating, a depicted in Figure 6, the correponding admitted rate by eion 1 increae in Figure 5. ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December 2010.

23 Cro-Layer Interaction in Multihop Wirele Senor Network 4:23 Fig. 5. Sample trajectorie of the admitted rate of two eion for J = Fig. 6. Sample trajectorie of the virtual queue for J = By the definition of the virtual queue, a larger backlog of Y 1 indicate that the average departure rate of the virtual queue, that i, the average admitted rate, i inufficient. Therefore, the ource node of eion 1 will attempt to increae the admitted rate and thu the backlog of the virtual queue will decreae accordingly where the tability of the virtual queue can be aured. In Figure 7, the traffic plitting deciion of the two ource node, that i, the hop election of the ource node, are illutrated. We can oberve that both ource node incline to utilize the queue with the maller number of hop. The queue with longer hop, for example, h = 3 or 4, are ued only when the queue backlog in the queue with maller hop are overwhelmed. In addition, we can ee that on average, eion 2 utilize a maller number of average hop ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December 2010.

24 4:24 Y. Song and Y. Fang Fig. 7. Sample trajectorie of the hop election for J = than eion 1. Recall that eion 2 ha a much more tringent contraint on the average number of hop than eion 1, that i, 10 veru 30. Therefore, the ource node of eion 2, that i, n 0 2, incline to depoit more packet on the queue with maller hop count. A a conequence, by aigning different value of rate and hop requirement, a ervice differentiation olution can be achieved by the ANRA cheme among multiple competitive eion in the network. In addition, a near-optimal network utility can be attained imultaneouly. 7. CONCLUSIONS AND FUTURE WORK In thi article, we propoe a contrained queueing model to capture the crolayer interaction in multihop wirele enor network. Our model conit of component from multiple layer uch a rate admiion control, dynamic routing, and wirele link cheduling. Baed on the propoed model, we invetigate the tochatic network utility maximization problem in wirele enor network. A a cro-layer olution, an adaptive network reource allocation cheme, called the ANRA algorithm, i propoed. The ANRA algorithm i an online mechanim which yield an overall network utility that can be puhed arbitrarily cloe to the global optimum olution. A a future work, energy-aware ditributed cheduling algorithm are to be tudied and evaluated. In addition, the extenion of our model to wirele enor network with network coding eem intereting and need further invetigation. REFERENCES AKYILDIZ, I. F. AND KASIMOGLU, I. H Wirele enor and actor network: Reearch challenge. Elevier Comput. Network. AKYOL, U., ANDREWS, M., GUPTA, P., HOBBY, J., SANIEE, I., AND STOLYAR, A Joint cheduling and congetion control in mobile ad-hoc network. In Proceeding of the Annual Joint Conference of the IEEE Computer and Communication Societie (InfoCom. ACM Tranaction on Modeling and Computer Simulation, Vol. 21, No. 1, Article 4, Pub. date: December 2010.