Sliding Mode Baed Congetion Control and 1 Scheduling for Multi-cla Traffic over Per-link Queueing Wirele Network Zongrui Ding and Dapeng Wu Abtract Thi paper conider the problem of joint congetion control and cheduling in wirele network with quality of ervice QoS) guarantee. Different from per-detination queueing in the exiting work, which i not calable, thi work conider per-link queueing at each node, which ignificantly reduce the number of queue per node. Under per-link queueing, we formulate the joint congetion control and cheduling problem a a network utility maximization NUM) problem and ue a dual decompoition method to eparate the NUM problem into two ub-problem, i.e., a congetion control problem and a cheduling problem. Then, we develop a liding mode SM) baed ditributed congetion control cheme, and prove it convergence and optimality property. Different from the exiting cheme, our congetion control cheme i capable of providing multi-cla QoS under the general cenario of multi-path and multi-hop; in addition, it i robut againt network anomalie, e.g., link failure, becaue it can achieve multi-path load balancing. Index Term Sliding mode control, per-link queue, QoS, multi-path, tochatic optimization, interactive multimedia, robutne againt network anomalie. I. INTRODUCTION The author are with Department of Electrical and Computer Engineering, Univerity of Florida, Gaineville, FL 32601. Email of Zongrui Ding: dingzr@ufl.edu. Correpondence author: Prof. Dapeng Wu, wu@ece.ufl.edu, http://www.wu.ece.ufl.edu. Thi work wa upported in part by National Science Foundation under grant CNS-0643731, CNS-1116970, Office of Naval Reearch under grant N000140810873, and the Joint Reearch Fund for Overea Chinee Young Scholar under grant No. 61228101.
2 F UTURE wirele network are expected to conit of a variety of heterogenou component and aim to upport interactive application uch a voice over IP VoIP), online multi-player game and live haring of multimedia content in term of video, audio, text or image among ditributed uer. In uch a context, a network hould be deigned to provide quality of ervice QoS) to inelatic traffic uch a video and audio, and imultaneouly provide high data rate for elatic traffic, e.g, email and web traffic. Therefore, how to perform congetion control and cheduling efficiently i crucial to achieve thee key feature for a wirele network. Different from wire-line network, the joint congetion control and cheduling in wirele network i particularly challenging due to unreliability, time-varying channel gain, and interference among wirele channel. The joint congetion control and cheduling can be formulated a a network utility maximization NUM) problem [1] [2] [3] [4] [5] [6] [7]. Under the etting that each node ha no buffer, the NUM problem can be olved in a ditributed manner by iteratively updating the link price, which i the um of the per-hop price [1] [5]. Thi mechanim require that each link calculate and feed back the per-hop price to the ource, thu inducing a lot of overhead. In per-detination queueing network, the optimal olution for the NUM problem can be obtained by the primal-dual algorithm within the network capacity region [8] [9], which require each node maintain a queue for each flow. Work uch a [6]and [7] optimize the end-to-end delay performance by the NUM problem. There are two commonly ued queueing tructure: the per-detination queue [2] [3] [9] [10] and the per-link queue [3] [11] [12] [6] [7]. In the per-detination queueing network, every node need to maintain a queue [13] for each flow, thu the number of queue per node can be a large a the number of node in the network. Thi overhead i unbearable for relay node in a large-cale wirele network. It i hown in [11] that the per-link queue tructure involve le overhead, preerve ome good deign feature uch a decompoition [14] and naturally provide multi-path awarene. Therefore, we conider the joint congetion control and cheduling problem under the etting of per-link queueing. In wirele network, ditributed algorithm are deirable due to the unavailability of global information. Generally, the QoS contraint make it difficult to olve the congetion control problem in a ditributed manner. Directly projecting the non-qos-contrained olution to the QoS-contrained region uually reult in a degraded performance. Sliding mode SM) control theory [15] provide a powerful tool for deigning ditributed algorithm for olving convex optimization problem with a large number of contraint, and ha been applied to congetion control problem in the Internet [16] [17] [18]. However, the SM control i not directly applicable to wirele network becaue of time-varying channel. Fortunately, the decompoition between the cheduling problem and the congetion control problem may
3 make it poible to apply the SM control to the congetion control problem which doe not involve the time-varying channel tate, given the Lagrangian multiplier reulted from the decompoition. Different from the exiting work, we conider the congetion control and cheduling of a per-link queueing wirele network with multi-cla QoS requirement under both contant channel and timevarying channel. By per-link-queue per-next-hop queue), we mean that packet detined to the ame next hop node, are put into the ame queue. Thu the number of queue that a node ha to maintain i the number of it neighbor within one hop. Our ytem model hare ome imilaritie with the routedependent cae in [3], where time-varying channel and QoS are not conidered. Some work tudy imilar problem under the per-detination queueing model, uch a [19] [20] [21]. In [22], the author conider the general framework of the time-average penalty or reward optimization, but no ditributed olution i given. Different from the SM controller in [16] [17] deigned for the Internet, the wirele context induce an interaction between congetion control on Layer 4 and cheduling on Layer 2, o that the convergence condition are naturally atified and no network feedback i neceary, which i a good example of layering a optimization decompoition [14] [23]. Under the per-link queueing model, we apply the SM control technique to the congetion control problem and develop a ditributed cheme. We prove that 1) the ditributed control cheme converge and achieve optimality under continuou-time ytem parameter; and 2) it converge to a bounded neighborhood of the optimal olution under dicrete time and time-varying channel. In thi paper, we firt conider the congetion control and cheduling under contant wirele channel for implicity, and then extend the reult to time-varying wirele channel and evaluate the performance by the Lyapunov drift method [22] [24]. The ret of the paper i organized a follow. In Section II, we decribe the ytem model, followed by the formulation and olution to the NUM problem under contant channel in Section III. Section IV extend the olution in Section III to wirele network under time-varying channel. Section V preent performance analyi and Section VI preent imulation reult. Section VII conclude the paper. II. SYSTEM MODEL In thi ection, we preent the model of a multi-hop multi-path per-link queueing wirele network. Section II-A decribe the per-link queueing model. Section II-B preent the QoS requirement for heterogenou ervice and Section II-C decribe the capacity region of a per-link queueing network.
4 A. The Queueing Model For a per-link queueing wirele network, let F denote the et of flow and N denote the et of node. A link l in the network ha a tranmitting node i N and a receiving node j N i, where N i i the et of node i next hop node. L = {l } denote the et of all link. Thee link are directed; we aume connectivity between any two node i ymmetric and the topology i aumed to be tatic. Each flow F ha a total tranmiion rate x over multiple path. Let x denote the tranmiion rate of flow over l with ource node b = i. Then the total tranmiion rate x over multi-path i x = x. 1) i=b j N i The rate power function denoted by µg, P ) i the offered phyical-layer capacity matrix under channel tate matrix G = [g ] and power allocation matrix P = [p ] P, where P i the et of feaible power allocation matrix P. Matrix X = [x ] denote the multi-path tranmiion rate for all flow. Throughout thi paper, A = [a ] i a tenor notation for matrix A. A wirele channel i a hared medium, where link interfered with each other contend for excluive acce to the channel. A conflict graph [20] can capture the contention among link; the conflict graph depend on G and P. The convex hull of the correponding rate vector of the independent et of the conflict graph determine the feaible rate region at link layer. It i aumed that a proper routing algorithm i applied to the network according to information uch a ditance or ignal-to-interference-plu-noie-ratio SINR), which reult in a routing table denoted by R. The ource-detination pair on link layer are conidered a flow. In cheduling and routing, we conider the packet-level operation and the overhead of packetization i aumed to be zero. Since we ue the per-link queueing model, each next hop of l ha a dedicated queue at node i) denoted by q. Therefore, the number of the queue in node i i at mot the ame a the number of it next hop. If l i activated, the data in q i tranmitted. At time t, the output of q conit of the traffic detined to node j and the tranit traffic detined to node other than node j. Thu we have µ t) = k µ,k t) + α t) 2) where α t) i the data rate delivered to node j through l ; and µ t) i a horthand notation for µ Gt), P t)), i.e., the phyical-layer capacity offered on l ; and µ,k t) i the tranmiion rate from q to q j,k through l at time t. Here, a triplet node index i, j, k) i ued to define the two cacading queue q and q j,k involved in a per-link tranmiion.
5 B. QoS Requirement To decribe the QoS requirement of heterogenou application, define the QoS contraint function for flow by: h 1 x ) = θ min x h 2 x ) = x θ max 3a) 3b) where θmin, and θ max are contant. The QoS requirement can be decribed by uing inequality contraint on the function in 3). For example, the ervice that guarantee minimum tranmiion rate for flow can be defined a h 1 x ) 0. For notation implicity, we ue h 1 x ) 0 and h 2 x ) 0 to repreent equality contraint, where θmin = θ max. C. Capacity Region of a Per-link Queueing Network In queueing network, the joint congetion and cheduling problem hould be olved under the contraint of network tability, which i defined a the tability of the queue. Let the exogenou arrival at q be defined by have x = :b =i x. 4) Let F i t) and F o t) denote the teady tate inflow and outflow rate of q at time t. Intuitively, we F i t) = F o t) = Then the average inflow and outflow rate, denoted by F i t 1 τ=0 m N i µ m, τ) t t 1 τ=0 k N j µ,k τ). 5) t F i = lim F i t t) F o = lim F o t t) and F o can be written a Define X a the capacity region of a per-link queueing network under time-varying channel, then the neceary and ufficient condition are derived a x F o F i, i, j) L 6) under the aumption of convergent and bounded arrival and convergent time-varying channel tate imilar to [?]. Note that a network with contant arrival and contant wirele channel i a pecial cae
6 for thi aumption. Uually, 6) i alo known a the flow conervation contraint for queueing network [2] [3], which i recognized a the network tability condition [9]. Note that the capacity region of a network under per-link queueing may be different from that under per-detination queueing. Intuitively, the per-detination queueing yield maller granularity in cheduling than per-link network. In [25], we tudied the capacity region of the per-link queueing network with per-detination backlog information. Pleae refer to [9] [25] for detail about the derivation and dynamic control in queueing network. III. THE NUM PROBLEM UNDER CONSTANT CHANNELS In thi ection, the joint congetion control and cheduling under contant wirele channel i formulated a an NUM problem. By dual decompoition, the NUM problem i decompoed into a congetion control problem and a cheduling problem. We conider an NUM problem with QoS contraint a below max {X X } F U x ) 7a).t. : h 0 7b) x F o + F i 0, i, j) L x = x j N i i=b where U x ) i the utility of flow with tranmiion rate x S); 7c) i the flow conervation contraint; 7b) pecifie the QoS contraint and vector h = [h 1 x 1 ), h 2x 2 ),..., h Ωx Ω )]T, where h ω x ω ) ω S, ω {1, 2,..., Ω}) can be any function defined in 3); and H = {ω h ω x ) i an element of h} i the et of the indice of the contraint function for flow S). For example, in a network where only Flow 1 ha a tranmiion rate requirement 10 x 1 20, then h = [x 1 20, 10 x 1 ] T H 1 = {1, 2}, H = 1). Note that F o, F i in 7) are the teady tate outflow and inflow for link l, which are contant. The olution to the NUM problem i organized a below: in Section III-A, 7) i decompoed into the congetion control problem 9) and the cheduling problem 10), given the Lagrangian multiplier Λ, which i updated iteratively according to Section III-B. Then 9) and 10) are olved in Section III-C and Section III-D, repectively. The optimality and convergence of the continuou time control law are tated in Theorem 1 in Section III-E. 7c) 7d) and
7 A. Decompoition To olve problem 7), we ue the dual decompoition method [3]. The Lagrangian dual function of 7) with repect to w.r.t.) contraint 7c) i DΛ): DΛ) = max {X X } S ) L = max {X X } U x ) λ x F o + F i S max ) L U x ) ) ) L λ x + λ F o F i ), 8) where Λ = [λ ] i the Lagrangian price matrix for contraint 7c) with λ correponding to each node pair i, j) L. It i implied by 8) that given the optimal link price [λ ], the firt and the econd term in 8) can be olved eparately. Define the congetion control problem a max U x ) {X X,h 0} F λ x, 9) ) L where 8) only depend on X given Λ, and the cheduling problem i given by max ) L λ F o F i ), 10) which i independent of X with Λ known. To olve 10), a et of link are choen to tranmit under ome interference contraint and proper routing algorithm to get the minimum. Thu, by dual decompoition, the joint problem i decompoed into two eparate problem correponding to different protocol layer, and they interact with each other through Lagrangian price. Next, we explain how to determine the Lagrangian price [λ ij ]. B. Lagrangian Price To obtain [λ ], conider the dual problem of 7) min Λ DΛ), 11) Since DΛ) i a convex function w.r.t. [λ ] [26], taking the partial derivative w.r.t. λ, we obtain: d DΛ) λ = F o F i x. 12)
8 which i aumed to be uniformly bounded by contant D. According to Slater theorem [26], there i no duality gap between 7) and 11). Thu, the Lagrangian price can be updated iteratively by: λ t) = [ y X, t) x t) Ft) o + Ft) i )] +, i, j) L 13) λ t) where y X, t) 0 i a uer-pecified function and if y X, t) i a contant, the Lagrangian price i proportional to the per-link queue length; and the projection [a] + b i defined by: [a] + b = a, if b 0 or a 0. 0, otherwie C. SM Baed Solution to the Congetion Control Problem In thi ection, the SM controller for the congetion control problem 9) i developed for given [λ ]. The SM control theory [15] [27] provide an optimal ditributed method to iteratively olve convex optimization problem with a large number of contraint. Furthermore, the SM controller involve ome adjutable parameter which correpond to the performance of the controller uch a teady-tate error and rate of convergence. Here, we give a brief introduction to the liding mode control theory. We illutrate the technique of liding mode control by an optimization problem example. Conider a convex optimization problem min Y Y ).t. f 1 Y ) 0 14) f 2 Y ) 0 15) f 3 Y ) 0 16) with Y ) a the objective function and Y R n a the optimization variable. The contraint 14) to 16) repreent region in R n pace and the interection of thee region i the feaible region. The liding mode controller produce a ditributed olution whoe trajectory i like liding along the curve that i at the interection of urface f 1 Y ) = 0 and/or f 2 Y ) = 0 and/or f 3 Y ) = 0; the trajectory finally reache the optimal point. For more detail, we refer the reader to [15] and [17]. Thi motivate the deign of our SM baed controller for the congetion control problem 9) a x t) = z X, t) u t) λ t) + + v m t) hmx ) x, i, j) L 17) m H x t) x t)
where z X, t) 0 i a uer-pecified function; u t) U x ) x i the partial derivative of U ) x t) w.r.t. x t) and evaluated at x t); and h mx ) x x t) b t) i the partial derivative of h m w.r.t. x and v m t) i defined a v m x t)) = γ m t), if h m x t)) > 0 0, if h m x t)) 0 where γ m t) > 0 i a uer-pecified function correponding to h m x ) and we ue γ m t) = v > 0 a an example in our following deign, where v i a contant. The function [z X, t)] and [y X, t)] are called iteration function of the ytem. Here, we do not dicu the parameter deign for SM control. Relevant work can be found in [16] and [17]. 9 18) D. Solution to the Scheduling Problem Similar to [2] [20], our olution to 10) at time t i to elect a et of interference-free link L o that maximize ) L o λ t) F o t) F i t) ), 19) where F i t) and F o t) are the time-averaged inflow rate and outflow rate at time t. According to the definition of F i t) and F o t), each node can etimate thee parameter by tracking the arrival rate, tranmiion rate and receive rate, which are local information. The Lagrangian price λ t) i proportional to the queueing length in the node. Therefore, the olution to the cheduling problem turn out to be an maximum weighted independent et problem with weight λ t) F o t) F ). i t) Alternatively, thi olution can be implified to yield a practical cheduling and routing algorithm, which will be preented in Section IV under time-varying channel. E. Optimality and Convergence of the Joint Scheme In ummary, we olve 7) by iteratively computing the Lagrangian price for each node pair i, j) L, adjuting the tranmiion rate of each ource and cheduling the link according to the network tate. In thi ection, we prove the optimality and convergence property of the joint cheme, i.e., 13), 17) and 19), for the NUM problem 7), a tated below: Theorem 1. Aume that: 1) X0) i a feaible rate matrix; 2) each utility function U ) i concave w.r.t. the tranmiion rate x ;
10 3) 13), 17) and 19) are the governing law of the ytem; 4) y X, t) = y i,k X, t) = z 1 X, t) = z 2 X, t) i, j) L, 1 2, k j), i.e., the iteration function are the ame for node i. Then the ytem tate Xt), Λt)) will converge to the optimal olution, denoted by ˆX, ˆΛ), where ˆX i the optimal rate matrix for 7), and ˆΛ i the optimal price matrix. The proof i ketched in Appendix A. Furthermore, if the optimal olution exit, and the et of all optimal olution i bounded, the SM controller can make the rate matrix converge to an optimal point from any initial rate matrix, which i not necearily within the capacity region [15] [16]. Intuitively, for a given Λ, 17) make the tranmiion rate converge to their optimality imilar to the cae in the Internet [17]. Meantime, the cheduler elect an optimal et of link to tranmit baed on the given Λ a in 19). Thee deciion produce an updated Λ, and finally the tranmiion rate and the Lagrangian price converge imultaneouly to an optimal olution of 7). It i pointed out in Theorem 1 that y X, t) = y i,k X, t) = z 1 X, t) = z 2 X, t) i, j) L, 1 2, j k) i a ufficient condition for the convergence of the joint iteration 13) and 17) under contant channel. Thu, the iteration function z, S are the ame for all flow originating from node i, which are alo the ame a the iteration function y over multipath of thee flow. Thi naturally allow the co-exitence of heterogeneou component in a network. We can ee that the control law given in [11] ha z = y = 1, S, i, j) L, which i a pecial cae for the deign in 13) and 17). IV. EXTENSION TO TIME-VARYING CHANNELS In thi ection, the NUM problem of a wirele network i conidered under lotted time-varying channel. Baed on the control law of 13), 17) and 19), we deign Algorithm 1 and Algorithm 2, which aymptotically olve the NUM problem under time-varying channel. For implicity, we conider a wirele network with lotted time-varying channel, the channel gain of which are i.i.d. convergent [24]. Different from the tranmiion rate under contant channel, the intantaneou tranmiion rate under time-varying channel change from lot to lot, thu the optimized variable are [ x ], defined in a time-average ene. Accordingly, the QoS contraint are alo defined on the time-averaged tranmiion rate. The network utility for a flow i a function of the average tranmiion rate and an optimal dynamic control policy i defined over all poible power control, routing and cheduling that can realize the optimal average tranmiion rate ˆX.
11 Similarly, the NUM problem under time-varying channel i formulated a max {X X,P } F U x ) 20a).t. : P P 20b) x F o + F i 0, i, j) L 20c) h 0 20d) x = i=b 20e) j N i x, where 20b) i the contraint on power and each element of which i ubject to contraint on minimum SINR and maximum tranmiion power. 20) allow many type of traffic uch a contant bit rate CBR), variable bit rate VBR) and elatic traffic. Although 20) eem imilar to 7), it cannot be directly olved by dual decompoition, which may reult in an unfeaible link aignment or tranmiion rate. Intead, we directly extend the control law, i.e., 13), 17) and 19), to the time-varying ytem etting and evaluate it performance in Section V. The SM baed joint congetion control and cheduling algorithm under time-varying channel i given in Algorithm 1 where i the ampling interval for the dicrete time ytem. 1 Initialization: X0) X, F o 0) = 0, F i 0) = 0, Λ0) = 0; 2 At time t + 1, update the ytem tate according to: 3 Price: the Lagrangian price i updated according to λ t + 1) = [y X, t) x t) F o t) + F i t) ) λ t) ] + ; 21) 4 Congetion Control: each ource update it multi-path tranmiion rate by x t + 1) = [x t) + z t) u t) λ t) + vb t) )] + ; 22) 5 Scheduling: perform Alg. 2 to get the optimal power control. Algorithm 1: SM baed congetion control and cheduling The time average inflow rate F i F i t) = F o t) = t) and outflow rate F o t) at time t can be repreented a m R m,t 1) + m µ m, t) t 23a) k D,kt 1) + µ t), 23b) t
12 where R m, t 1) i the total tranmitted data from lot 0 to lot t 1 over q m,i to q and D,k t 1) i the total tranmitted data from q to q j,k ; µ m, t) and µ t) atify 2). Subtituting 23) into 19), we obtain the optimal power control a below P t) = arg max λ t) µ t) ) µ P P m, t) m = arg max λ t)µ t) ) µ P P,k t)λ j,k t), 24) k where 24) i due to the fact that the deciion at time t doe not affect m R m,t 1) and k D,kt 1). It i recognized that 24) i a maximum weighted independent et MWIS) problem with λ t)µ t) k µ,k t)λ j,kt) aigned a the weight W for each l. Then 24) can be olved in three tep baed on a predefined routing algorithm R, a tated in Algorithm 2, where L k mi, j)t) i the length of the mth packet in q at time t, which will be tranmitted from node i to node j and put into q j,k ; and Γ t) = {γ m=γ k,m=1 Lk mi, j) µ t)} i the et of packet index for tranmiion, which alo depend on P and G. 1 At time t for each q i, j) L), decide the next hop node k of each packet that not detined to node j by k = arg min λ j,nt), n:j,n) R then get µ,k t) = m Γ t) Lk mi, j)t); 2 The weight W t) for any link l i calculated by: W t) = µ t)λ t) k µ,k t)λ j,k t); 3 Allocate tranmiion power P according to: P t) = arg max P P W Gt), P ); 25) Algorithm 2: The cheduling and multi-path routing algorithm under per-link queue It i eay to jutify that Algorithm 2 give an optimal olution to 24) baed on routing cheme R. The Lagrangian price 21) and the tranmiion rate 22) at each node are updated in a ditributed manner with low computational complexity and the information ued in Algorithm 1 i local rather than global. Given the link price [λ ], 25) i generally N P hard, even if the power control i on-off with two value. If uboptimal performance i allowed, ome algorithm uch a column generation [28] can
13 be ued to olve the MWIS problem. The MWIS problem can alo be olved in a ditributed manner with low complexity by the algorithm mentioned in [20], which uually achieve a performance within about 4/5 of the optimal performance. There are alo ome work tudying the impact of the imperfect cheduling and it bound uch a [29] [30]. Algorithm 1 cannot be derived directly from the dual decompoition of the NUM problem 20). However, 20) can be ued a a reference ytem to characterize the performance of Algorithm 1. The performance of Algorithm 1 i tudied in the next ection. V. PERFORMANCE ANALYSIS In thi ection, Section V-A how the performance of Algorithm 1. To enhance the performance of Algorithm 1 in term of delay, we propoe a Very Important Packet VIP) queueing tructure in Section V-B. A. Performance of Algorithm 1 The ytem tate under the dicrete-time control policy evolve a a Markov chain and we need to how that thi Markov chain i table. Becaue we apply the floor operation to the Lagrangian price and the tranmiion rate, it i eay to check that the Markov chain ha a countable tate pace, but i not necearily irreducible. Thu, we conider the partition of the tate pace a the tranit tate et and the recurrent tate et. It i defined to be table if all recurrent tate are poitive recurrent and the Markov proce hit the recurrent tate with probability one [8] a t approache infinity. Thi can guarantee that the Markov chain i aborbed/reduced into ome recurrent cla, and the poitive recurrence enure the periodicity of the Markov chain over thi cla. Then we evaluate it tability by the following theorem: Theorem 2. The Markov chain decribed by 21) 22) i table. Theorem 3. The network utility produced by Algorithm 1 converge tatitically to the neighborhood of the optimal utility with radiu [ E {U x ))} ) K2 S V0 2 +D0+Z 2 1 1 +Z 2 ) 2 2K 1, i.e., Û ] 2 K 2 S V 2 0 + D2 0 + Z 1 + Z 2 ) 2K 1 26) where 26) i the reult under y X, t) = y i,k X, t) = z 1 X, t) = z2 X, t) = 1 i, j) L, 1 2, k j) and imilar reult hold for y X, t) = y i,k X, t) = z 1 X, t) = z 2 X, t) 1 i, j) L, 1 2, k j); x ) denote the tranmiion rate of the Markov chain in teady tate, and V and
14 D are contant bound of function u λ + vb )+ and the partial derivative of the dual function d ; and K 2 = max u x )) 2 and K 1 > 0 i the larget lower bound of u û. x t) ˆx The proof of Theorem 2 and 3 are given in Appendix B. Theorem 2 and 3 imply that the expectation of the network utility converge to a mall neighborhood of the optimal utility. Thu, the iteration of tranmiion rate and the Lagrangian price converge imultaneouly. There may be other decompoition poibilitie a tudied in [23]. Here, we provide ome intuition about our liding mode baed joint congetion control and cheduling deign. In queueing network, the NUM problem can be decompoed into a cheduling problem and a congetion control problem, which produce the claic primal-dual algorithm [31]. The dual algorithm turn out to be the back-preure routing/cheduling, which can tabilize the network whenever the ource arrival rate i within the capacity region of the wirele network [8] [9]. Since the QoS requirement in our problem are function of ource tranmiion rate, thee contraint will only affect the olution to the congetion control problem rather than the tability of the network within the capacity region. Since the liding mode control theory provide a ditributed optimal olution to convex optimization problem, it can be applied to olving the congetion control problem. Note the the capacity region under per-link queueing may be different from that under per-detination queueing. Intuitively, the per-detination queueing provide maller granularity for cheduling and it can be eaily proved that the capacity region under per-detination queueing i not le than that under per-link queueing. Little work i done about thi comparion. In Sec. VI, we preent the imulation reult about the throughput comparion of a network under both per-link queueing and per-detination queueing. B. Performance Enhancement Mechanim: VIP Queue In the above ytem model, we do not conider the ervice with tringent delay requirement. In thi ection, we propoe a framework for the co-exitence of the per-link queue non-vip queue) and the VIP queue o that delay-enitive ervice can be upported. The VIP queue and non-vip queue are maintained eparately along the route of delay-enitive application and the VIP queue will be cheduled by ome delay-aware algorithm. Since thee application may have much le number of detination, the per-detination queue can alo be ued. One application i for interactive multi-cla uch a VoIP, online multiplayer game, which have tringent delay requirement. The packet from interactive multimedia application are marked a VIP and will be put into VIP queue and tranmitted with higher priority over non-vip. Another application i the tranmiion of layered video, where
15 Fig. 1. Network topology: a) Dumbbell topology. b) Chain topology. c) Loop topology the bae-layer packet can be placed into the VIP queue and tranmitted with high priority, and the enhancement layer packet are placed into the non-vip queue and tranmitted with low priority. There are ome theoretical analyi of the performance regarding priority queue [32]. VI. SIMULATION RESULTS In thi ection, we demontrate the performance of Algorithm 1 by imulating ad hoc network in three topologie with typical traffic pattern. Section VI-A decribe the imulation etting, and Section VI-B preent the imulation reult of Algorithm 1 under contant and time-varying time diviion multiple acce TDMA) channel. A. Simulation Setting The ad hoc network topology that we imulate i depicted in Fig. 1, where Fig. 1a), 1b) and 1c) how a dumbbell topology, a chain topology and a loop topology repectively. The line in Fig. 1 indicate bi-directional link and the ditance between any two neighboring node i the ame. Each contant wirele channel ha a capacity of 1.4Mb/. The time-varying channel tate can be one of the five tate/rate, i.e., {0.28, 0.84, 1.40, 1.96, 2.52}Mb/, with probability 1/5. Thu, the expectation of the tranmiion rate for a time-varying channel i 1.4Mb/. Without lo of generality, only one-hop interference i conidered, i.e., each node cannot tranmit and receive imultaneouly.
16 TABLE I PARAMETERS Topology Service QoS Parameter 1:{l 1,3 } No QoS = 0.01TVC) 2:{l 1,3, l 3,2 } Dumbbell 3:{l 1,3, l 3,4 } 4:{l 1,3, l 3,4, l 4,5 } 5:{l 1,3, l 3,4, l 4,6 } 1:{l 1,2, l 2,3 } x 2 350kbp = 1 Chain 2:{l 2,3, l 3,4 } x 4 = 100kbp v = 200 CC) 3:{l 3,4} 4:{l 1,2, l 2,3, l 3,4} 1:{l 1,2, l 2,3} x 2 350kbp = 0.01 Chain 2:{l 2,3, l 3,4} x 4 = 100kbp v = 200 TVC) 3:{l 3,4} 4:{l 1,2, l 2,3, l 3,4 } 1:{l 1,2, l 2,3 } x 1 350kbp = 0.01TVC) Loop and {l 1,4, l 4,3 } v = 300 2:{l 2,3 } The utility function of each flow i: U x ) = w lnx + 1) 27) where w i a uer pecified parameter to indicate different weight of the ervice, and we chooe identically w = 2000 here. In Algorithm 1 and 2, deign parameter v and the calar iterative tep-ize are alo uer-pecified parameter and lited in Table I, where CC i horthand notation for contant channel and TVC i horthand notation for time-varying channel. B. Performance of Algorithm 1 1) Throughput Performance: In thi ection, we tudy the throughput performance of Algorithm 1. The et up of the imulation i the following: in a 6-node ad hoc network in dumbbell topology, a hown in Fig. 1a), there are five bet effort BE) ervice from node 1 to node 2, 3, 4, 5, 6 repectively. The timevarying channel tate are i.i.d. uniformly ditributed with tranmiion rate {40, 120, 200, 280, 360}kb/. For comparion, it i imulated under Algorithm 1 with per-link queueing and under the MaxWeight algorithm [8] with per-detination queueing. It i known that the MaxWeight achieve the throughput
17 Average Rate kbp) Average Rate kbp) 25 20 15 10 5 x 1 x 2 x 3 x 4 0 0 100 200 300 400 500 Iteration Step a) 25 20 15 10 5 0 0 100 200 300 400 500 Iteration Step c) x 5 x 1 x 2 x 3 x 4 x 5 Average Queue Length x10 3 b) Average Queue Length x10 3 b) 700 600 500 400 300 200 100 q 1,3 q 2,3 q 3,4 q 4,5 q 4,6 0 0 100 200 300 400 500 Iteration Step b) 700 600 500 400 300 200 100 q 1,3 q 2,3 q 3,4 q 4,5 q 4,6 0 0 100 200 300 400 500 Iteration Step d) Fig. 2. Tranmiion rate and queue length of dumbbell topology: a) The average tranmiion rate under Alg. 1 and per-link queueing. b) The average queue length under Alg. 1 and per-link queueing. c) The average tranmiion rate under MaxWeight and per-detination queueing. d) The average queue length under MaxWeight and per-detination queueing. optimal under per-detination queueing. To get the optimal throughput under Algorithm 1, we chooe tranmiion rate a the utility function for each node. The routing table applied to Algorithm 1 i lited in Table I, while the MaxWeight algorithm ue the back-preure routing. Fig. 2 illutrate the time-averaged rate of data reaching each detination, and queue length v. time under both queueing model. From Fig. 2a), we can ee the time-averaged data rate at each detination converge to about 22kb/, which mean the per-link queue in the dumbbell network i able to upport uch data rate. The time-averaged queue length alo converge fat to finite value a hown in Fig. 2b). Thi data rate can alo be upported by the per-detination queue/maxweight not hown in imulation reult). However, with a tranmiion rate 23kb/ for each ervice, the network under per-detination queueing i untable. It i een from Fig. 2c) that the time-average data rate at ome detination node are obviouly below 23kb/. Accordingly, the queue are not table in 2d) becaue the rate of data entering the network i more than the rate of data reaching the detination. In the per-link queueing,
18 node 3 i the only next hop node of node 1, thu the ource rate of node 1 will converge to the ame rate according to Algorithm 1 under the ame utility function, which i alo validated by Fig. 2a). Since it i very difficult to get an analytical reult of the capacity region of a wirele network, we have to reort to imulation to roughly compare the throughput of a network under the per-link queueing and the per-detination queueing. It i een that with reaonable routing algorithm applied, Algorithm 1 can achieve a throughput performance cloe to MaxWeight but uing much le queue. There are 6 6 = 36 queue under the per-detination queueing wherea only 10 queue under the per-link queueing. 2) Supporting Multi-cla Service: In thi ection, we evaluate whether Algorithm 1 can imultaneouly upport variou QoS required by different ervice. Both elatic and inelatic ervice are imulated in a 4-node ad hoc network in chain topology, a hown in Fig. 1b). There are four ervice in the network. 1 and 3 are BE flow uch a email and file tranfer protocol FTP). 2 ha a minimum tranmiion rate requirement of 350kb/ in time-average ene under time-varying channel), which can provide the VBR flow uch a video over IP. 4 i a CBR flow tranmitting at 100kb/, which repreent 10 VoIP tream each with 10kb/). Fig. 3 how the time average tranmiion rate v. iteration tep without/with QoS requirement under contant channel CC) in the 4-node chain ad hoc network. In Fig. 3a), we compare the convergent tranmiion rate of the ad hoc network under Algorithm 1 per-link queueing) and the MaxWeight algorithm per-detination queueing). The tranmiion rate under MaxWeight are denoted by x1 1, x1 2, x1 3. Since 4 tranmit at a contant rate, it i not hown in the figure. It i hown that the throughput of Algorithm 1 i lightly maller than the Maxweight algorithm which i throughput optimal and ue more queue. It i alo noticed that the QoS contraint under Maxweight are not atified. In addition, imilar imulation reult can be oberved in Fig. 4 under time-varying channel. If QoS requirement i not conidered, the tranmiion rate for x 1 i much lower than 350kbp, a hown in Fig. 3a). In comparion, the QoS requirement are atified in Fig. 3b) for all flow under Algorithm 1. Comparing the network utility under no QoS with that under QoS requirement, the former i larger than the latter, which i reaonable becaue the feaible region i maller under QoS contraint. In thi chain topology, l 1,2 and l 3,4 can be activated imultaneouly. 1, 2 and 4 pa l 2,3, which ha to be activated with no other active link. Thu, l 2,3 i the bottleneck and x 2 i expected to be maller without QoS contraint, which alo matche our imulation reult. Similar reult are oberved in Fig. 4a) and 4b) under time-varying channel where 2 i the VBR flow. In ummary, all thee imulation reult demontrate that Algorithm 1 i capable of upporting traffic with multi-cla QoS requirement. One poible diadvantage of Algorithm 1 i that the convergence of the tranmiion rate may be lower.
19 Average Rate Mbp) 1.2 1 0.8 0.6 0.4 0.2 x 1 x 2 x 3 x 4 x1 1 x1 2 x1 3 0 0 20 40 60 80 100 120 140 160 180 200 Iteration Step a) Average Rate Mbp) 1.2 1 0.8 0.6 0.4 0.2 x 1 x 2 x 3 x 4 0 0 20 40 60 80 100 120 140 160 180 200 Iteration Step b) Fig. 3. Tranmiion rate for chain topology under contant channel: a) The average tranmiion rate under Alg. 1 and MaxWeight without QoS requirement. b) The average tranmiion rate under Alg. 1 with QoS requirement. 3) Robutne againt Link Failure: In thi ection, we tudy the robutne of Algorithm 1 againt link failure under time-varying channel. To emphaize that Algorithm 1 can upport multi-path, a 4- node loop ad hoc network i imulated, a hown in Fig. 1c). 1 i a video-over-ip flow with minimum tranmiion rate requirement of 350kb/ and 2 i BE flow. There i multi-path routing identified for 1 through l 1,2 and l 1,4 repectively. Fig. 5 illutrate the multi-path tranmiion rate v. iteration tep under loop topology hown in Fig. 1c) under time-varying channel. In Fig. 5a) and 5b), the QoS requirement i atified for 1 by the um of the tranmiion rate over two path. The time-average queue length alo converge. Since 1 and 2 have to hare l 2,3, the tranmiion rate x 1 1,4 over l 1,4 i expected to be larger than x 1 1,2 over the other path, which matche the imulation reult in Fig. 5a). For comparion, Fig. 5c) and 5d) how the tranmiion rate and queue length in the cae of the link failure of l 1,4. When encountering a link failure, it i hown in Fig. 5d) that q 1,4 correponding to l 1,4 increae quickly, which decreae the tranmiion rate x 1 1,4 quickly to zero. In thi cae, the QoS requirement i not atified and the
20 0.6 Average Rate Mbp) 0.5 0.4 0.3 0.2 x 1 x 2 x 3 x 4 x1 1 0.1 x1 2 x1 3 0 0 20 40 60 80 100 120 140 160 180 200 Iteration Step a) Average Rate Mbp) 0.5 0.4 0.3 0.2 0.1 x 1 x 2 x 3 x 4 0 0 20 40 60 80 100 120 140 160 180 200 Iteration Step b) Fig. 4. Tranmiion rate for chain topology under time-varying channel: a) The average tranmiion rate under Alg. 1 and MaxWeight without QoS requirement. b) The average tranmiion rate under Alg. 1 with QoS requirement. the tranmiion rate x 1 1,2 of the other path increae according to 22). Finally the QoS requirement i atified a hown in Fig. 5c) and the network i table. Thee imulation reult how that Algorithm. 1 i robut againt link failure becaue the multi-path tranmiion adaptively doe load balancing to atify QoS requirement without changing the routing table. VII. CONCLUSIONS In thi paper, we tudy the joint congetion control and cheduling problem for multi-hop, multipath per-link queueing wirele network with QoS contraint. It i formulated a an NUM problem under both contant and time-varying channel, which i decompoed into a congetion control problem and a cheduling problem. Then, a ditributed SM baed controller i deigned to iteratively olve the congetion control problem, which can provide multi-path rate adaption to atify QoS contraint. The cheduling problem i identified a a maximum weighted independent et problem and can alo be olved in a ditributed manner. We extend thi algorithm to the cae with time-varying channel and prove that the dynamic control law, i.e., Algorithm 1 i table and the network utility converge
21 Average Rate Mbp) Average Rate Mbp) 1 0.8 0.6 0.4 0.2 x 1 1,2 x 1 1,4 x 2 0 0 100 200 300 400 500 Iteration Step a) 1 0.8 0.6 0.4 0.2 0 0 100 200 300 400 500 Iteration Step c) x 1 x 1 1,2 x 1 1,4 x 2 x 1 Average Queue Length x10 6 b) Average Queue Length x10 6 b) 6 5 4 3 2 1 q 1,2 q 1,4 q 2,3 0 0 100 200 300 400 500 Iteration Step b) 6 5 4 3 2 1 q 1,2 q 1,4 q 2,3 0 0 100 200 300 400 500 Iteration Step d) Fig. 5. Tranmiion rate for loop topology: a) The tranmiion rate under multi-path with TVC and QoS requirement. b) The queue length under multi-path with TVC and QoS requirement. c) The tranmiion rate under link failure for l 1,4 with TVC and QoS requirement. d) The queue length under link failure for l 1,4 with TVC and QoS requirement. to a bounded neighborhood of the optimal. Simulation reult how that Algorithm 1 i capable of providing heterogenou multi-cla ervice with different QoS requirement. Becaue of the multi-path load balancing feature of thi algorithm, it i robut againt network anomalie uch a link failure. The application of the SM control theory in wirele network i tightly related to the decompoition property of the protocol tack. Since the joint congetion control and cheduling i a cro-layer optimization, thee two ubproblem interact with each other through the Lagrangian price. Therefore, the ource adapt the tranmiion rate to the price, which mak the link tatu uch a time-varying channel capacitie and dynamic cheduling. At the ame time, the cheduling problem i independent of the ource tranmiion rate. Thi property make it poible to apply SM control theory. Furthermore, the convergence condition of the SM controller are alo relaxed becaue the decompoed ub-problem provide a way of negative feedback to each other in wirele network.
22 APPENDIX A PROOF OF THEOREM 1 Proof: We ketch the proof of Theorem 1 below, and intereted reader can find the deign law of the SM controller in [15] [16]. To prove the convergence and optimality of the algorithm, we contruct the Lyapunov function a W X, Λ) = W 1 X) + W 2 Λ) 28) where W 1 Λ) 1 2 W 2 X) 1 2 ) L S,j=n λ ˆλ ) 2 = 1 2 Λ ˆΛ 2 2 x ˆx ) 2 = 1 2 X ˆX 2 2; 29) and [ˆλ ], [ˆx ]) are the optimal olution to problem 7) aumed to be feaible), thu they are contant. For brevity, we omit the t) of the quantitie below. Differentiating 28) with repect to t, we obtain: Ẇ X, Λ) =,j,j x ˆx ) [ z u + vb )] + λ + x x ˆx ) z u + vb ) λ + λ ˆλ ) [y x + F i F o )] + λ 30) λ ˆλ ) y x + F i F o ), 31) where the lat inequality follow that 30) and 31) are equal if the projection in 13) and 17) are inactive, and if the projection i active, the projection in 13) and 17) are zero, while the expreion in 31) are poitive due to x + F i F o < 0 and u + vb λ < 0. Here, and,j i a horthand notation for ) L and S,i=b,j N i. Since [ˆx ] i the optimal olution to the NUM problem, then the fact come that ˆλ = û, where
û = U x ) x. Subtituting it into 31) yield ˆx 23 Ẇ X, Λ) x ˆx ) z u û λ + ˆλ + vb,j ) + y λ ˆλ ) x ˆx ) + =,j,j =,j y λ ˆλ ) ˆx + F i F o ) z z x ˆx ) u û ) + x ˆx ) vb +,j y λ ˆλ ) ˆx + F i F o ) x ˆx ) z u û ) + z x ˆx ) ˆλ λ ) + y λ ˆλ ) x ˆx ) +,j y λ ˆλ ) ˆx + F i F o z ) x ˆx ) vb + The econd term and the forth term in 33) cancel each other if z = y S, b = i, n = j) becaue flow haring the ame ource node i and the next hop node j relate to the ame λ, then,j z x ˆx ) ˆλ λ ) = ) y ˆλ λ x ˆx ) Becaue U i concave w.r.t. x, and u = u j N i ), where u du x dx. Then we have: z x ˆx ) u û ) where û du dx ˆx,j 32) 33) 34) = z x ˆx u û ) 35) j N i j N i = z x ˆx ) u û ) 36) 0 and 35) follow that u û depend on x in term of x for a ource ; and 36) follow that U i a concave function of x. From contraint 7c), we have ˆx ˆF o ˆF i. Becaue F i, F o and λ are the optimal olution for 10) at time t and ˆF i and ˆF o λ ˆx + F i F o ) are feaible olution to the outflow and inflow, thu λ ˆF o ˆF i + F i F o ) 0. 37)
From the KaruhKuhnKucker KKT) condition, λ ˆx + ˆF i ˆF ) o = 0 hold and ) ˆλ ˆx + F i F o ) = ) ˆλ ˆF o ˆF ) i F o + F i 0 38) becaue the quantitie with hat are the optimal olution to 10). Thu, y 0. We alo have,j x ˆx )vb 0 according to the definition of b. 24 λ ˆλ ) ) ˆx + F i F o Therefore, Ẇ 0, which implie that the joint congetion control and cheduling cheme i optimal and convergent according to LaSalle Invariance Principle [2]. A. Proof of Theorem 2 APPENDIX B PROOF OF THEOREM 2 AND 3 Proof: The Lyapunov function i the ame a defined in Appendix A. Here, for implicity, it i aumed that y X, t) = z t) = 1 F, i, j) L, X, t). Conider the expectation of the one-tep Lyapunov drift E {W t + 1) W t) Xt), Λt)} = E{W 1 t + 1) W 1 t) Xt), Λt)} + E{W 2 t + 1) W 2 t) Xt), Λt)} = E{W 1 Xt + 1)) W 1 Xt)) Xt), Λt)} + E{W 2 Λt + 1)) W 2 Λt)) Xt), Λt)} 39) By ubtituting 21) into 39), the firt term of 39) i E{W 1 t + 1) W 1 t) Xt), Λt)} = E { W 1 [Λt) d] + ) W 1 Λt)) Xt), Λt) } = E { W 1 [Λt) d] + ϵ 1 ) Xt), Λt) } E {W1 Λt)) Xt), Λt)} E {W 1 Λt) d) W 1 Λt)) Xt), Λt)} [ E {ϵ 1 Λt) d) + ˆΛ ] 1 } 2 ϵ 1 2 2 Xt), Λt) E {W 1 Λt) d) W 1 Λt)) Xt), Λt)} + ϵ 1 ˆΛ + 1 2 ϵ 1 2 2, 40) where d = [d t)] with d t) = F o t) F i t) x t) a defined in 12) and ϵ 1 = [Λt) d] + [Λt) d] + ; the firt inequality come from expanding the term W 1 [Λt) d] + ) ϵ 1 and relax
25 the projection in it. Note that 0 ϵ 1 < 2 1, where 0 and 1 are zero and unit matrix repectively. Thu, ϵ 1 ˆΛ + 1 2 ϵ 1 2 2 2 ˆΛ 2 2 + 1 2 4 1 2 2 2 Z 1. Subtitute it into 40), and we have E{W 1 t + 1) W 1 t) Xt), Λt)} E {W 1 Λt) d) W 1 Λt)) Xt), Λt)} + 2 Z 1 = E d λ λ λ ˆ ) Xt), Λt) + 1 2 2 E { d 2 } 2 Xt), Λt) + 2 Z 1 41) where the econd equality follow from expanding the term W 1 Λt) d) W 1 Λt)). Denote the objective function in 9) a V Xt), Λt)) = S U x ) ) L λ x. The partial derivative w.r.t. x i V = u λ and g = [V ], where u i the partial derivative w.r.t. x of U x ). Subtituting 22) into the econd term of 39) and following the ame manipulation a above, we have E{W 2 t + 1) W 2 t) Xt), Λt)} =E { W 2 [Xt) + g + vbt))] + ) Xt), Λt) } E {W2 Xt)) Xt), Λt)} =E { W 2 [Xt) + g + vbt))] + ϵ 2 ) } Xt), Λt) E {W2 Xt)) Xt), Λt)} { E {W 2 Xt) + g + vbt))) Xt), Λt)} E W 2 Xt)) 1 } 2 ϵ 2 2 2 Xt), Λt) [ E {ϵ 2 Xt) + g + vbt))) + ˆX ] } Xt), Λt) E { W 2 Xt) + g + vbt)) + ) Xt), Λt) } E {W2 Xt)) Xt), Λt)} + ϵ 2 ˆΛ + 1 2 ϵ 2 2 2, 42) where Bt) = [b t)] and ϵ 2 = [Xt) + g] + [Xt) + g] +. Further implify 42) by ϵ 2 ˆX + 1 2 ϵ 2 2 2 2 ˆX 2 2 + 1 2 4 1 2 2 2 Z 2 and we get E{W 2 t + 1) W 2 t) Xt), Λt)} E { W 2 Xt) + g + vbt)) + ) } Xt), Λt) E {W2 Xt)) Xt), Λt)} + 2 Z 2 = E g x x ˆx ) Xt), Λt),j + E { vb x ˆx } ) Xt), Λt),j + 1 2 2 E { g + vbt)) + 2 } 2 X, Λ + 2 Z 2 E g x x ˆx ) Xt), Λt) + 1 2 2 E { g + vbt)) + 2 } 2 X, Λ + 2 Z 2 43),j
26 where the econd equality follow the definition of b t). Subtitute 41) and 43) into 39), then E {W t + 1) W t) Xt), Λt)} E dλ ˆλ ) Xt), Λt) + E g x x ˆx ) Xt), Λt) + 2 Z 1 + Z 2 + D 2 0 + V 2 0 ), 44),j where the lat inequality follow that the conditional expectation of d 2 2 and g + vbt))+ 2 2 are uniformly bounded by contant D 0 > 0 and V 0 > 0. For d λ t), the per-link flow variable [F o t)], [F i t)] and the tranmiion rate [x t)] are bounded. For g + vbt)) + 2 2, if g + vbt) 0, g + vbt)) + 2 2 u t) 2 2 + vbt) 2 2 i bounded, otherwie g + vbt))+ 2 2 = 0. Follow imilar manipulation procedure a in the proof of Theorem 1 and take conditional expectation, 44) can be implified to E {W t + 1) W t) Xt), Λt)} E {u û ) x t) ˆx )} + 2 Z 1 + Z 2 + D 2 0 + V 2 0 ), 45) where û i the partial derivative of U x ) w.r.t. x and evaluated at ˆX. Define A, uch that A X E u û ) ˆx x t)) V 2 0 + D 2 0 + Z 1 + Z 2 ) }, which i not empty and ha finite number of element with properly choen. Therefore, we have: E {W t + 1) W t) Xt), Λt)} 1 2 2 V 2 0 + D 2 0 + Z 1 + Z 2 )I X A 1 2 2 V 2 0 + D 2 0 + Z 1 + Z 2 )I X A c 46) where I i the indicator function. Thu, by Theorem 3.1 in [8], which i an extenion of Foter criterion, the Markov chain i table.
27 APPENDIX C PROOF OF THEOREM 3 Proof: Taking expectation of 46) w.r.t. X, Λ), we have E{W t + 1) W t)} 1 2 2 V 2 0 + D 2 0 + Z 1 + Z 2 ) + E {u û ) x t) ˆx )} 47) Summing 47) over τ = 0,..., t 1, it i obtained that ) 1 t 1 E {u û ) x t) ˆx )} E{W X0), Λ0))} + t 2 2 V0 2 + D2 0 + Z 1 + Z 2 ) 48) t t τ=0 If x t) ˆx, 48) can be implified a t 1 1 { u E û }) t x t) ˆx x t) ˆx ) 2 τ=0 E{W X0), Λ0))} + t 2 2 V0 2 + D2 0 + Z 1 + Z 2 ). 49) t Since U x ) i a trictly concave function of x and the tranmiion rate x [0, x m], u û x t) ˆx i non-negative and bounded. Then u û x t) ˆx K1 hold, where K 1 > 0 i the larget lower bound of u û. x t) ˆx Letting t going to infinity in 49) produce Then it i obtained that E { x ) ˆx ) 2} V 2 0 + D2 0 + Z 1 + Z 2 ) 2K 1. 50) [ E {U x ))} ] 2 Û [E {U x ))} Û ] 2 { E U x )) Û ) } 2 E { u x )) 2 x ) ˆx ) 2} K 2 F V 2 0 + D2 0 + Z 1 + Z 2 ) 2K 1, 51) where K 2 = max u x )) 2 i a contant; and F i the cardinality of F.
28 REFERENCES [1] F. Kelly, A. Maulloo, and D. Tan, Rate control for communication network: hadow price, proportional fairne and tability, Journal of the Operational Reearch ociety, vol. 49, pp. 237 252, Mar. 1998. [2] S. Shakkottai and R. Srikant, Network optimization and control. Boton: Now Publiher, 2008. [3] X. Lin and N. Shroff, Joint rate control and cheduling in multihop wirele network, in Proc. 43rd IEEE Conf. on Deciion and Control, Paradie Iland, 2004, pp. 1484 1489. [4] Y. Yi and M. Chiang, Stochatic network utility maximiation-a tribute to Kelly paper publihed in thi journal a decade ago, European Tran. Telecommunication, vol. 19, pp. 421 442, Jun. 2008. [5] S. Low and D. Lapley, Optimization flow control I: baic algorithm and convergence, IEEE/ACM Tran. Netw., vol. 7, pp. 861 874, Dec. 1999. [6] P. Huang, X. Lin, and C. Wang, A low-complexity congetion control and cheduling algorithm for multihop wirele network with order-optimal per-flow delay, in Proc. 32nd IEEE International Conf. on Computer Communication, Shanghai, 2011, pp. 2588 2596. [7] H. Xiong, R. Li, A. Eryilmaz, and E. Ekici, Delay-aware cro-layer deign for network utility maximization in multi-hop network, IEEE J. Sel. Area Commun., vol. 29, pp. 951 959, May 2011. [8] L. Taiula and A. Ephremide, Stability propertie of contrained queueing ytem and cheduling policie for maximum throughput in multihop radio network, IEEE Tran. Autom. Control, vol. 37, pp. 1936 1948, Dec. 1992. [9] M. Neely, E. Modiano, and C. Rohr, Dynamic power allocation and routing for time-varying wirele network, IEEE J. Sel. Area Commun., vol. 23, pp. 89 103, Jan. 2005. [10] M. Chiang, Balancing tranport and phyical layer in wirele multihop network: Jointly optimal congetion control and power control, IEEE J. Sel. Area Commun., vol. 23, pp. 104 116, Jan. 2005. [11] Z. Ding and D. Wu, Sliding Mode Baed Joint Congetion Control and Scheduling in Multi-hop Ad Hoc Network with Multi-cla Service, in Proc. Global Communication Conf., Miami, 2010, pp. 1 5. [12] L. Bui, R. Srikant, and A. Stolyar, Novel architecture and algorithm for delay reduction in back-preure cheduling and routing, in Proc. 25th IEEE International Conf. on Computer Communication, Rio de Janeiro, 2009, pp. 2936 2940. [13] M. Neely, Energy optimal control for time-varying wirele network, IEEE Tran. Inf. Theory, vol. 52, pp. 2915 2934, Jul. 2006. [14] M. Chiang, S. Low, A. Calderbank, and J. Doyle, Layering a optimization decompoition: A mathematical theory of network architecture, Proc. IEEE, vol. 95, p. 255, Jan. 2007. [15] V. Utkin, Sliding mode in control and optimization. Berlin: Springer-Verlag, 1992. [16] C. Lagoa, H. Che, and B. Movichoff, Adaptive control algorithm for decentralized optimal traffic engineering in the internet, IEEE/ACM Tran. Netw., vol. 12, pp. 415 428, Jun. 2004. [17] C. Lagoa and H. Che, Decentralized optimal traffic engineering in the internet, ACM SIGCOMM Computer Communication Review, vol. 30, pp. 39 47, Oct. 2000. [18] B. Movichoff, C. Lagoa, and H. Che, Decentralized optimal traffic engineering in connectionle network, IEEE J. Sel. Area Commun., vol. 23, pp. 293 303, Feb. 2005. [19] M. Neely, Univeral cheduling for network with arbitrary traffic, channel, and mobility, in Proc. 49th IEEE Conf. Deciion and Control, Atlanta, 2010, pp. 1822 1829. [20] L. Chen, S. Low, M. Chiang, and J. Doyle, Cro-layer congetion control, routing and cheduling deign in ad hoc wirele network, in Proc. 25th IEEE International Conf. on Computer Communication, Barcelona, 2006, pp. 1 13.