Downlink Power Control for Variable Bit Rate Videos over Multicell Wireless Networks

Transcription

1 This paper was presented as part of the ain technical progra at IEEE INFOCOM 211 Downlink Power Control for Variable Bit Rate Videos over Multicell Wireless Networks Yingsong Huang and Shiwen Mao Departent of Electrical and Coputer Engineering, Auburn University, Auburn, AL Abstract We investigate the proble of downlink power control for streaing ultiple variable bit rate (VBR) videos in a ulticell wireless network, where downlink capacities are liited by inter-cell interference. We adopt a deterinistic odel for VBR traffic that considers video frae sizes and playout buffers at the obile users. The proble is to find the optial transit powers for the base stations, such that VBR video data can be delivered to obile users without causing playout buffer underflow or overflow. We forulate a nonlinear nonconvex optiization proble and prove the condition for the existence of feasible solutions. We then develop a centralized branch-andbound algorith incorporating the Reforulation-Linearization Technique, which can produce (1-ε)-optial solutions. We also propose a low-coplexity distributed algorith with fast convergence. Through siulations with VBR video traces under fading channels, we find the distributed algorith can achieve a perforance very close to that of the centralized algorith. I. INTRODUCTION With the draatic advances in wireless networking technology and wireless counication devices, there is an exponentially increasing deand for wireless video service. According to a recent study by Cisco [1], obile data traffic will double every year through 214, increasing 39 ties between 29 and 214 globally. In addition, video will account for 66% of the obile data traffic by 214. This trend is driven by the copelling need for ubiquitous access to video content over wireless access networks, and will significantly stress the capacity of existing wireless networks and strongly influence the design of future wireless networks. Significant effort is needed in wireless video research to eet this treendous deand. While it is iportant to develop new wireless architectures and technologies for higher spectral efficiency, it is equally iportant to investigate how to support video in existing wireless networks, since the infrastructure will still last for a considerable period of tie. In this paper, we consider video streaing over a ulticell wireless network, a wireless network architecture widely deployed all over the world. We consider the typical case of downlink video transissions. For the ulticell syste, generally intracell interference can be effectively controlled with precise synchronization or the use of guard ties. The capacities of the downlinks are ainly liited by the inter-cell interference due to siultaneous base station (BS) transissions using the sae channel. Therefore, effective downlink power control is necessary to support concurrent videos. We consider the proble of streaing concurrent variablebit-rate (VBR) videos in the ulticell wireless network. This is otivated by the superior perceived quality of VBR videos over constant-bit-rate (CBR) videos. VBR video has stable visual quality for the fraes, but at the cost of large variations in the bit rate, while CBR video aintains a stable bit rate, but the fraes have large variations in visual quality. Due to this reason, any stored videos are VBR. We ai to investigate how to provide ubiquitous access to such stored VBR videos through existing cellular networks. It is a challenging proble to support VBR video traffic, which is found to exhibit both strong long-range and short-range-dependence [2], [3]. It is nontrivial to develop parsionious traffic odels that can accurately capture the auto-correlation structure. The large frae size variations ay cause frequent playout buffer underflow or overflow. To address this issue, we adopt a deterinistic traffic odel for stored VBR video, which considers frae size, frae rate, and playout buffers [4] [7]. Unlike prior work that is focused on a single video session over a given CBR or VBR channel, we exploit power control, a unique capability in wireless networks, to adjust the downlink capacities based on prior knowledge of frae sizes and playout schedules. Usually large fraes are rarely transitted siultaneously. Thus jointly optiizing the BS transit powers is, in soe sense, analogous to statistical ultiplexing VBR videos in the ulticell network. We presented a proble forulation that considers downlink power control, inter-cell interference, VBR video characteristics, and playout buffer requireents. The objective is to achieve high playout buffer utilization, under playout buffer underflow and overflow constraints and peak power constraint. This is a nonlinear nonconvex proble to which traditional convex optiization techniques [8] and low- or high-sinr approxiations [8], [9] do not directly apply. In this paper, we first derive the condition of the existence of feasible power assignents, which can achieve downlink capacities to guarantee no buffer underflow and overflow. We then develop a centralized algorith that can produce solutions with bounded optiality gap. Specifically, we use the Linearization-Reforulation Technique (RLT) to obtain a linear prograing (LP) relaxation of the original proble. Solving this LP relaxation yields an upper bound to the original proble. Interestingly, since the constraints are preserved in the relaxation procedure, the upper-bounding solution is also feasible to the original proble; the corresponding objective value with this solution provides a lower bound to the global optiu. The LP relaxation is then incorporated into the branch-and-bound fraework to obtain a centralized /11/$ IEEE 2561

2 Fig. 1. A ulticell wireless network with concurrent VBR video sessions. The inter-cell interference experienced by the central cell user is illustrated. algorith, which can produce a solution within the (1-ε) range of the global optial. To siplify coputation and control, we also develop a distributed algorith based on distributed constrained power control (DCPC) [1], where each BS iteratively updates transit power based on feedback of easured SINR at the target receiver. It is shown that with DCPC, the power vector converges to a unique power vector that can achieve the goal of axiizing playout buffer utilization and avoiding playout buffer underflow and overflow. We evaluate the proposed algoriths with siulations using VBR video traces [11] and fading channels. The distributed algorith is shown to achieve a perforance very close to that of the centralized algorith. Both algoriths are deonstrated to be highly effective for streaing VBR videos over ulticell wireless networks. In the reinder of this paper, we present the proble forulation in Section II. We describe a centralized algorith in Section III and a distributed algorith in Section IV. Siulation results are presented in Section V and related work is discussed in Section VI. Section VII concludes this paper. II. PROBLEM STATEMENT A. Network and Video Syste Model We consider the downlinks of an M-cell wireless network as shown in Fig. 1. In each cell, a BS streas video to obile users in the cell, each allocated with a downlink channel. A channel is a spectral resource slot, the nature of which depends on the specific ultiple access technique adopted for the ulticell network. Without loss of generality, we assue that the downlink channels within a cell are orthogonal (e.g., due to perfect synchronization of spreading codes or use of guard ties). The ain interference at a user stes fro the concurrent downlink transissions in neighboring cells that use the sae channel. There is a need for the BS s to adopt power control to itigate such inter-cell interference. We consider the proble of streaing ultiple VBR videos in the ulticell network. We assue the wired segent of a video session path is reliable with sufficient bandwidth, while the last-hop wireless link is the bottleneck [12]. Thus the corresponding video data is always available at the BS before the scheduled transission tie. It is non-trivial to accurately odel VBR video traffic, which exhibits both strong asyptotic self-siilarity and short-range correlation [2]. A stochastic odel capturing the auto-correlation structure often requires a large nuber of paraeters, and is thus hard to be incorporated for scheduling real-tie video data. To this end, we adopt a deterinistic odel that considers frae sizes, playout buffers, and schedule [5]. Let D i (t) denote the cuulative consuption curve of the i-th user, representing the cuulative aount of bits consued by the decoder at tie t. The cuulative consuption curve is deterined by video characteristics such as frae sizes and rates, and playout schedule. Assue user i has a playout buffer of size b i bits and its video has L i fraes. We can derive a cuulative overflow curve for user i as B i (t) =in{d i (t 1) + b i,d i (L i )}, t L i. (1) B i (t) is the axiu nuber of cuulative received bits at tie t without overflowing user i s playout buffer. Finally we define cuulative transission curve X i (t) as the cuulative aount of bits transitted to user i at tie t. To siplify notation, we assue the video sessions have identical frae rate and the frae intervals are synchronized. Thus a tie slot t is equal to the t-th frae interval, denoted as τ, for t ax i {L i }. 1 Since D i (t), B i (t) and X i (t) are cuulative curves, they are all nondecreasing functions of tie. The three curves are illustrated in Fig. 2. A feasible transission schedule will produce a cuulative transission curve X i (t) that lies within D i (t) and B i (t), i.e., causing neither underflow nor overflow at the playout buffer. In practice, D i (t) s are known for stored videos and are delivered to the BS s (or a centralized video scheduler) during the session setup phase, and B i (t) s are then derived as in (1). B. Proble Foration For the ulticell wireless video network, consider a specific channel and let U = {un 1,un 2,,un M } denote the set of users sharing the channel, where un is the user in cell. 2 Let the BS transit power vector be P (t) = [P 1 (t),p 2 (t),,p M (t)] T in tie slot t. The capacity of the downlink fro BS to user un, denoted as C (t), depends on the SINR at un, which can be written as γ ( P (t)) = G P (t) k = G k P k(t)+η, (2) 1 For exaple, if the frae rates are different, we can use a tie slot duration that is equal to the greatest coon divisor of all the frae intervals (if not too sall). If the frae intervals are not synchronized, a tie slot can be a fraction of a frae interval within which the D i(t) s of all the videos reain constant. In fact, the tie slot duration could be arbitrary as in [13] (i.e., equal to ultiple frae intervals). Since the cuulative overflow and consuption curves are known, we can still deterine the upper and lower bounds for the transission rate in each tie slot. The proble forulation and proposed solution procedures to be discussed in the following sections apply to these cases. 2-1 index variables can be used to odel the case where no user uses the channel in soe cells, but are oitted for brevity. 2562

3 Bits Fig. 2. Cuulative overflow curve B i (t) Infeasible transission schedule playout buffer overflow Playout buffer size b i Feasible transission schedule D i (t) X i (t) B i (t) Frae interval Frae size Cuulative consuption curve D i (t) Infeasible transission schedule playout buffer underflow Tie Feasible and infeasible transission schedules for video session i. where G k is the path gain fro BS k to user un and η is the noise power at un. We assue slow-fading channels such that the path gains do not change within each tie slot [13], but vary over different tie slots following a certain distribution. The downlink capacity C (t) also depends on the channel bandwidth B w and the transceiver design, such as odulation and channel coding. Without loss of generality, we use the upper bound as predicted by Shannon theore: C ( P ( (t)) = B w log 1+γ ( P ) (t)). (3) The ipact of fading channels is incorporated in the SINR in (3). For practical systes, the achievable capacity ay be a fraction of C ( P (t)), but this part is oitted for brevity. Once the link capacity is deterined, C (t)τ video bits will be delivered to user un in that tie slot. The cuulative transission curve X (t) can be written as X () = ; X (t) =X (t 1) + C (t)τ. (4) Assue peak power constraint P P, for all. The proble is to deterine the transit power vector P (t), for <t ax i {L i }, such that the resulting cuulative transission curves satisfy D (t) X (t) B (t), for all, t, (5) i.e., without causing playout buffer underflow or overflow. Since the video fraes have variable sizes and the video sessions have rando phases, large fraes fro different sessions are less likely to occur in the sae tie slot. Jointly considering power control for the downlinks is, in soe sense, analogous to statistical ultiplexing of VBR video flows. Fro (3) (5), the feasible SINR range at user un is e ax{,d(t) X(t 1)} Bw τ 1 γ e B(t) X(t 1) Bw τ 1. (6) In (6), the lower bound is the SINR that just epties the buffer without causing underflow. The upper bound is the SINR that just fills up the buffer without causing overflow. Generally, the feasible transit power vector P (t) is not unique for a given set of VBR video sessions. Aong the set of feasible solutions, a schedule that transits ore data is ore desirable since it provides a larger search space for optiizing transit power vectors for future tie slots. Oitting the constant B w, we forulate the optial power control proble for VBR videos, tered Proble OPT-VBR, as axiize U log(1 + γ (t)) (7) G subject to: γ (t) = P (t) k = G k P, (8) k(t)+η γ in (t) γ (t) γ ax (t), (9) P P,, (1) where γ ax (t) is the upper bound in (6) and γ in (t) is the larger one between the lower bound in (6) and γ th, a iniu SINR requireent iposed by the transceiver design. In Proble OPT-VBR, the total aount of video data delivered in tie slot t is axiized, under playout buffer underflow and overflow constraints and peak transit power constraints. This is a nonlinear nonconvex proble, to which traditional convex optiization techniques do not directly apply. Furtherore, to achieve the objective of avoiding playout buffer underflow and overflow, the SINRs ay assue values ranging fro very low to very high. Thus the existing high SINR approxiation [8] and low SINR approxiation [9] techniques cannot be used. In the following, we first prove the existence of feasible solutions. We then derive effective centralized and distributed algoriths to solve Proble OPT- VBR in Sections III and IV. C. Existence of Feasible Solutions Due to the wide range of VBR video frae sizes, the corresponding SINR requireents also assue a wide range of values. Under conditions where any video sessions coincidently transit their large fraes in the sae tie slot, Proble OPT-VBR ay not have a feasible power assignent to deliver all the fraes. In this section, we derive the conditions for the existence of feasible power assignents. We assue a centralized scheduler in the ulticell network, which has prior knowledge of all the path gains and the cuulative consuption and overflow curves. We define the iniu required rate for user un in tie slot t, denoted as C in (t), as the bit rate such that the playout buffer is just eptied, but without underflow, at the end of tie slot t. We have the following result for C in (t). Lea 1: The largest value for the iniu required rate C in in (t) is C (t) =[D (t) D (t 1)] /τ. Proof: According to the definition of X (t) in (4), we have C (t) =[X (t) X (t 1)] /τ. Fro the definition of C in (t), the playout buffer is eptied at the end of tie slot t, i.e., X (t) = D (t). Therefore, we can derive the iniu required rate as C in (t) =ax{,d (t) X (t 1)} /τ. (11) Fro the feasibility condition (5), we have X (t 1) D (t 1). Substituting it into (11), we have C in (t) [D (t) D (t 1)] /τ C in (t). (12) 2563

4 in Rate C (t) occurs when the playout buffer is epty at both the beginning and end of tie slot t, but without buffer overflow during the entire tie slot. We have the following condition for the existence of a feasible power assignent for Proble OPT-VBR. Theore 1: There exits a feasible power assignent for Proble OPT-VBR for tie slot t, if there exits a feasible power assignent that can achieve the rate vector [ Cin 1 (t), C 2 in (t),, Cin M (t)]. Proof: Recall that γ in is the SINR corresponding to the iniu required rate C in (t). Let γ in (t) be the in SINR corresponding to C (t). Since (3) is a onotonically increasing function, we have γ in (t) γ in (t). We now consider the power assignent that achieves rates C in (t), or, the corresponding SINRs γ in (t). Fro (8) and (9), the iniu SINR constraint is: γ (t) = G P (t) k = G k P k(t)+η γ in (t),. (13) Eqn. (13) is a syste of linear equations of the power vector P (t), which can be written in the atrix for as: ( I Γ in A ) P (t) ર Γ in ν, (14) where I is the identity atrix, A is an M M atrix with {, = k A k = G (15) k /G, = k, Γ in = diag{ γ 1 in (t), γ 2 in (t),, γ M in (t)} is a diagonal atrix, and ν =[η 1 /G 1 1,η 2 /G 2 2,,η M /G M M ]T. Define Γ in = diag{γ1 in (t),γ2 in (t),,γm in (t)} and Δ = Γ in Γ in ર. Assue P is a power assignent that achieves γ in (t) for all, which satisfies (14). Substituting Γ in = Δ + Γ in into (14), we have ( I Γ in A ) ( P ર Γ in ν + Δ ν + AP ). Since ( Δ, ν, A and P all have non-negative eleents, we have Δ ν + AP ) ર, and therefore, ( I Γ in A ) P ર Γ in ν. (16) That is, P can also achieve γ in (t) for all and it satisfies the iniu SINR constraint in (9). Once the iniu SINR constraint in (9) (i.e., no buffer underflow) is satisfied, the axiu SINR constraint in (9) (i.e., no buffer overflow) can be satisfied since BS can stop transission when the playout buffer at user un is full. Theore 1 allows us to evaluate, for a given set of videos, if there is a feasible power assignent for each tie slot. There is no need to consider the transission schedules and playout buffer occupancies in previous tie slots. At (t) fro the cuulative consuption curve D(t) and channel gains. If the linear syste (14) is solvable and the resulting P satisfies constraint (1), then there is a feasible power assignent for Proble OPT-VBR for this tie slot. The following fact fro [14] can be used for the feasibility test. the beginning of tie slot t, we obtain γ in Fact 1: The following stateents are equivalent: (i) there exits a feasible power assignent satisfying (14); (ii) the axiu odulus eigenvalue of ( Γ in A ) is less than 1; (iii) the reciprocal atrix (I Γ in A) 1 = k= ( Γ in A ) k exists and is positive coponent-wise. III. CENTRALIZED ALGORITHM As discussed, Proble OPT-VBR is a nonlinear nonconvex proble, to which traditional convex optiization techniques do not directly apply. In this section, we present a centralized algorith to provide solutions with bounded optiality gap. We first use RLT to obtain a linear prograing (LP) relaxation of Proble OPT-VBR [15]. We then incorporate the linear relaxation into a branch-and-bound fraework, which can produce (1-ε)-optial solutions. A. Reforulation and Linearization We first apply polyhedral outer approxiation for the logarith functions in Proble OPT-VBR to obtain a Polynoial Prograing Proble OPT-VBR(p) [16]. We then use RLT bound-factor product constraints to relax the quadratic ters to obtain an LP relaxation OPT-VBR(l). The tie slot index (t) is dropped in the following to siplify notation. We first process the logarith functions in the objective function. Letting u = log(1+γ ), we obtain a linear objective function U u and new constraints u = log (1 + γ ). We deal with the new constraints using polyhedral outer approxiation. Since γ in γ γ ax,we choose H points, denoted as {γ}, h within this range as ( ) h 1+γ γ h =(1+γ in ax H 1 ) 1,h=,,H 1, 1+γ in (17) where γ = γ in and γ H 1 = γ ax. We can obtain a convex envelop for the logarith function in [γ in, γ ax ], which consists of H tangent lines at the H points given in (17) and the line segent connecting the two end points. We relax the logarith constraint by using its convex envelop, represented by the following new linear constraints: u log (1+γin ) γ ax γin (γ ax γ )+ log (1+γax ) γ ax γin (γ γ in ) u log(1 + γ)+ h γ γh,h=, 1,,H 1. 1+γ h (18) The first line is for the segent connecting the two end points, and the second line is for the tangent lines at the H points. A four-point approxiation is illustrated in Fig. 3. With the polyhedral outer approxiation, we obtain a polynoial prograing proble OPT-VBR(p), as given in (19) (26). We can rewrite the last constraint (26) as k = G k γ P k G P + η γ =, (27) which contains quadratic ters in the for of γ P k. We next introduce RLT bound-factor product constraints to reove such ters and to obtain an LP relaxation. Define substitution variables v k = γ P k, for all, k. Since γ and P k are bounded by their respective lower and 2564

5 u =log(1+ ) Tangent lines in 1 2 ax 3 Fig. 3. Four-point polyhedral outer approxiation for u =log(1+γ ), 1 <γ in γ γ ax. axiize U u (19) subject to: ( ) G P k = G k P k + η γ in, (2) ( ) G P k = G k P k + η γ ax, (21) P P, (22) log (1 + γin ) u γ ax γ in (γ ax log (1 + γ ax) γ ax γin (γ γ in ), (23) u log(1 + γ h )+ γ γh 1+γ h,, h (24) ( 1+γ γ h =(1+γ in ax ) h H 1 ) 1,, h (25) 1+γ in G γ = P k = G k P,. (26) k + η upper bounds as γ in γ γ ax and P k P,we obtain the following RLT bound-factor product constraints (γ γ in ) (P k ) (γ ax γ ) (P k ) (γ γ in ) ( P (28) P k ) (γ ax γ ) ( P P k ). Substituting γ P k = v k, we obtain the following four linear constraints for v k : v k γ in P k γ ax P k v k γ P vk γ in P + γ in (29) γ ax P P k γ ax P k γ P + vk. The quadratic ters P k γ are thus replaced with v k with the above linear RLT bound-factor constraints, and an LP relaxation OPT-VBR(l) is obtained as given in (3) (41). The LP relaxation OPT-VBR(l) can be effectively solved with an LP solver in polynoial tie. The optial solution to the LP relaxation consists of { P, u, γ, v }.Itis worth noting that during the reforulation and linearization procedure, we ainly relax the logarith function in the objective function of OPT-VBR. The original constraints of OPT-VBR are preserved in OPT-VBR(l). Therefore, we have the following theore regarding the feasibility of the solution, which greatly siplifies the local search procedure of the branch-and-bound algorith to be presented in Section III-B. axiize U u (3) subject to: ( ) G P k = G k P k + η γ in, (31) ( ) G P k = G k P k + η γ ax, (32) P P, (33) log (1 + γin u γ ax (γ γin ax γ )+ log (1 + γ ax ) γ ax γ in (γ γ in ), (34) u log(1 + γ)+ h γ γh,, h (35) γ h =(1+γ in ) 1+γ h ( 1+γ ax 1+γ in ) h H 1 1,, h (36) v k γ in P k,, k = (37) (γ γ in v k + γ in k,, k = (38) γ ax P k v k,, k = (39) (γ ax γ ) P γ ax P k + v k,, k = (4) k = v kg k G P + ηγ =,. (41) Theore 2: The optial transit power vector P to the LP relaxation OPT-VBR(l) is a feasible solution to the original proble OPT-VBR. B. Branch-and-Bound Algorith According to Theore 2, we can substitute the optial power assignent P for the LP relaxation into Proble OPT-VBR to obtain a lower bound, while the LP solution itself provides an upper bound. We next incorporate the LP relaxation into a branch-and-bound fraework to obtain an algorith that can produce (1-ε)-optial solutions. Branch-and-bound is an iterative ethod for solving optiization probles, especially for discrete and cobinatorial probles. A branch-and-bound procedure has two key coponents. The first one, called branching, is to partition a proble into subprobles. The procedure is repeated recursively to each of the subprobles and all produced subprobles naturally for a tree structure, i.e., the branch-and-bound tree. Its nodes are the constructed subprobles. The leaves of the tree is also call the Proble List. The other coponent is bounding, which is a fast way of finding upper and lower bounds for the optial solution for each subproble. For a axiization proble, an infeasible upper bound (UB) can be found by solving a relaxed proble. A local search algorith is then used to explore the neighborhood, to find a feasible lower-bounding solution (LB). As discussed, we can easily derive upper and lower bounds by solving the LP relaxation (no need for local search). The core of the approach is an observation that, for a axiization task, if the upper bound for a subproble l 1 is saller than the lower bound for any other subproble l 2, then l 1 and the branch rooted at l 1 can be safely discarded fro the tree, such that the coputational coplexity can be reduced. This procedure is called pruning. The algorith terinates when the upper bound reaches (1 + ε) of the lower bound. Let the optial object value be 2565

6 TABLE I BRANCH-AND-BOUND ALGORITHM Initialization: 1 Obtain LP relaxation OPT-VBR(l) asprob1; 2 Set optial solution sol = φ, Proble list S = {Prob 1}, UB =, andlb =; 3 Solve Prob 1 for solution { P, u, γ, v } and upper bound UB 1 ; 4 Use P, (7), and (8) to get lower bound LB 1 ; 5 Set UB = UB 1 and LB = LB 1 ; Iteration & pruning: 6 Select Prob l with the largest UB l in S and set UB = UB l ; 7 IF (LB l >LB) { 8 Set sol = P l and LB = LB l; 9 IF (UB (1 + ε)lb) stop with solution sol; 1 ELSE reove all probs k in S with UB k (1 + ε)lb; } Partition: 11 For Prob l, find the axiu relaxation error aong all RLT variables, e.g., ax,k { γ P k v k }; 12 Evaluate the following condition: (γ ax γ in ) in{γ γ in,γ ax γ } (P ax P in ) in{p P in,pax P }; 13 IF (true) partition [γ in,γ ax ] into [γ in,γ ] and [γ,γax ]; 14 ELSE partition [P in,p ax ] into [P in,p ] and [P,P ax ]; Bounding: 15 Solve the partitioned probs l 1 and l 2 to get solutions sol l1, sol l2 and bounds UB l1, UB l2, LB l1, LB l2 ; 16 Reove Prob l fro S; 17 IF ((1 + ε)lb < UB l1 ) add Prob l 1 into S; 18 IF ((1 + ε)lb < UB l2 ) add Prob l 2 into S; 19 IF (S = φ) stop; 2 ELSE go to Step 6; O UB,wehaveLB 1 1+ε UB 1 1+ε O =(1 ε + ε2 ε 3 + )O (1 ε)o, for ε 1. The pseudo code for the branch-and-bound algorith is given in Table I. C. Enhanceent We introduce a heuristic to accelerate the convergence of the branch-and-bound algorith. At the beginning of tie slot t,if the playout buffer occupancy is above a certain threshold, say, 8%, and X (t 1) D (t) at user un,wesetp (t) = and reove the link fro the optiization process. Generally the playout buffer size should at least be greater than the largest frae size. Given the large variations in VBR frae sizes, there could be ultiple fraes stored when the buffer is close to full. When the above conditions are satisfied, there is little chance of buffer underflow at the end of tie slot t even if we do not transit anything to user un.onthe other hand, if we schedule a non-zero power P (t) for this link, only a sall aount of bits can be transitted due to the buffer overflow constraint, but at the cost of reduced SINRs at all other links. Excluding such links fro transission not only greatly speeds up the convergence of the branch-andbound algorith, but also increases the SINR and capacity of other active links. IV. DISTRIBUTED ALGORITHM Although the RLT-based branch-and-bound algorith can provide a (1 ε)-optial solution, it requires a centralized ipleentation. A centralized controller is needed to collect network, link and video related inforation, and to update transit power for each downlink. In this section, we develop a distributed algorith for Proble OPT-VBR that can be ipleented in each BS and operate with local inforation. We assue each BS obtains video cuulative consuption curves and playout buffer sizes for its users during the video session initiation phase. At the beginning of tie slot t, each BS coputes for user un the iniu rate as [D (t) X (t 1)]/τ, i.e., the data rate that epties the playout buffer at the end of tie slot t but without underflow, and the axiu rate as [B (t) X (t 1)]/τ, i.e., the data rate that akes the playout buffer full at the end of tie slot t but without overflow. BS then translates the iniu and axiu rates to iniu and axiu SINRs, i.e., (t) and γ ax (t) as given in (6). In the following, we again drop the tie slot index (t) to siplify notation. γ in To axiize objective function (7), BS sets a target SINR as γ tar = γ ax, and tries to achieve the target SINR by adjusting its transit power. The proble then becoes a Distributed Constrained Power Control (DCPC) proble [1]. BS first randoly sets its initial transit power as < P P.Letγ i be the i-th SINR easureent at user un, which is fed back to BS. BS then uses the following DCPC algorith to update its power after receiving the i-th SINR feedback: P i =in { } P, γtar γ i P i 1,i=1, 2,. (42) If the γ tar s are feasible (see Section II-C), the power vector series { P, P 1,, P i, } is proved to converge to a unique positive power vector satisfying the following equation [1] P =in{ P,Γ tar (AP } + ν), (43) where Γ tar = diag{ γ tar } = diag{γ1 tar,γ2 tar,,γm tar }. Furtherore, the converged power vector P (t) also achieves the target SINR γ tar (t) for each BS. The convergence result is suarized as the following fact fro [1]. Fact 2: With the DCPC algorith (42), the transit power vector converges to a unique positive power vector P satisfying (43). After convergence, either P achieves γ tar or at least one of the coponents in P is equal to P. The pseudo code for the distributed DCPC algorith is given in Table II, where α is a fraction in (,1) and β is a positive integer. If BS s transit power reains at the axiu power P for β iterations, while the target is still not achieved, we reset the target SINR as γ tar = γ in +α (γ tar γ in ) and restart the iterative update process. We choose α =.618, the reciprocal of the golden ratio, and β fro 2 to 5 in our siulations. In practice, the path gains vary over tie due to channel fading. It is possible that during soe tie slot, the transission is not feasible even for the iniu required rate. It is nontrivial to test the feasibility of the target SINR vector γ tar in a distributed anner with only local inforation. In fact, if the target SINR vector is infeasible, the proble of finding the largest set of links that can be supported at the given SINRs is proved to be NP-Coplete [17]. Therefore, we adopt the following heuristic strategies to handle the case SINR γ tar 2566

7 TABLE II DCPC ALGORITHM Initialization: 1 BS obtains b, D,andB for user un ; 2 BS coputes SINR bounds γ ax and γ in ; 3 BS sets γ tar = γax and P() (, P ]; Iteration: 4 BS receives{ SINR feedback γ i and updates its power as: P i =in ( ) } P, γ tar /γi P i 1 ; 5 If ((P i = P for β iterations) & (γ i = γ tar )) reset the target SINR as: γ tar = γ in + α (γ tar γ in ); 6 i = i +1and go to Step 4; when the target SINR vector cannot be achieved by a feasible power assignent due to deep fading channels. i) In the first tie slot, if the DCPC algorith does not converge in a certain nuber of steps, suspend the transission of the video with the largest frae size for soetie and retry the algorith. ii) Adopt the acceleration enhanceent as in the centralized algorith, which is described in Section III-C. iii) If the DCPC algorith does not converge for the reduced (see Line 5 in Table II), further reduce the target = γ in + α (γ tar γ in ). If still no convergence when γ tar =(1+ε) γ in,for <ε 1, γ tar SINR as γ tar all the links whose buffer will not be epty in the next tie slot will pause their transissions. Since the algorith always tries to transit as ore data as possible (i.e., by setting a high target SINR whenever possible), it is highly likely that such links won t have buffer underflow in the following tie slots. iv) If all the above steps fail, the BS suspends its transission and the user freezes the playout precess until the next tie slot. V. SIMULATION RESULTS To evaluate the perforance of the proposed algoriths, we siulate streaing VBR videos in a 7-cell wireless network. We assue the channels within a cell are orthogonal and inter-cell interference is the ajor liiting factor. The channel bandwidth is B w =1MHz. The path gain averages are set to G k = d 4 k, where d k is the physical distance fro BS k to user un. We assue Rayleigh fading channels in all the siulations, where the noralized path gain is exponentially distributed as f(g k ) = exp{ G k / G k } for G k. The distance fro a user to its corresponding BS is uniforly distributed fro 1 to 1 and the inter-cell BS distance is fro 16 to 2. The teperature is T = 29 Kelvin and the equivalent noise bandwidth is also 1 MHz. The peak power constraint is P =1Watt. In each cell, the channel is dedicated to one obile user for VBR video streaing. We assue BS s 1, 4 and 7 are streaing ovie Star Wars, BS s 2 and 5 are streaing NBC News, and the reaining links 3 and 6 are transitting Tokyo Olypics. We use the VBR traces for these videos fro the Video Trace Library hosted at Arizona State University [11] in all the siulations. The playout buffer size is set to be 1.5 ties of the largest frae size in the requested VBR video. A. Centralized Algorith We ipleent the branch-and-bound centralized algorith using MATLAB. We choose ε = 1% for the siulations. Fro the VBR video traces, we derive the cuulative consuption and overflow curves. The centralized algorith coputes the optiized power assignent for the BS s at beginning of each tie slot. In Fig. 4(a), we plot the cuulative consuption, overflow and transission curves for Star Wars transitted on link 1. The top subfigure is for 1, fraes. We also plot the curves fro frae 1,96 to frae 1,98 in the botto subfigure, while frae 1,969 has the largest size aong the 1, fraes. We observe that the cuulative transission curve X 1 (t) is very close to the cuulative overflow curve B 1 (t), indicating that the centralized algorith always ais to axiize the transission rate as allowed by the buffer and power constraints, and the playout buffer is fully utilized for ost of the tie. There is no playout buffer overflow or underflow for the entire range of the ovies. In Fig. 5, we plot the upper and lower bounds for objective function (7) for tie slot 1. This is the hardest tie slot with respect to power control, since all the sessions are transitting I-fraes and all the playout buffers are epty in this tie slot in our siulations. We observe the optiality gap between UB and LB is continuously decreased until the ε =.1 threshold is reached. In other tie slots where the frae sizes are not consistently large and the playout buffers are close to full, it usually takes only a few (e.g., 5 or 6) iterations to reach the optiality gap threshold. We also evaluate the accelerated schee under the sae video and network conditions. The curves for link 1 are plotted in Fig. 4(b). It can be seen that during tie slots 1,963, 1,967, and 1,971, there is no transission on link 1 since the playout buffer is over 8% full. Pausing transission in these tie slots akes it easier for other links to transit large fraes and speeds up the convergence of the algorith, while causing no buffer underflow at link 1. Since usually large fraes rarely occur in the sae tie slot (except for tie slot 1), this is analogous to statistical ultiplexing of VBR videos. We find in the siulation, a link can pause in over 6% of the tie slots with the acceleration heuristic, resulting in significant reduction in coputation tie. B. Distributed Algorith We next exaine the perforance of DCPC. The network and video setups are the sae as those in the centralized algorith siulations. The cuulative overflow, transission, and consuption curves obtained by DCPC are plotted in Fig. 4(c) for Star Wars transitted on link 1. We observe very siilar perforance as in the case of the centralized algorith shown in Fig. 4(a). The cuulative transission curve is again very close to B (t), and there is neither buffer overflow nor underflow during the transission of 1, fraes. To copare the distributed and centralized algoriths, we copute the su of the bit rates of all the links in each tie slot. The acceleration schee is not used for both algoriths in this siulation. The rate sus are plotted in Fig. 6(a) 2567

8 Cuulative bits (kbits) 15 x Frae-tie x 1 4 Cuulative bits (kbits) 15 x Frae-tie x 1 4 Cuulative bits (kbits) 15 x Frae-tie x 1 4 Cuulative bits (kbits) Cuulative consuption curve Cuulative overflow curve 2.34 Cuulative transission curve Frae-tie (a) Centralized algorith Cuulative bits (kbits) Cuulative consuption curve Cuulative overflow curve 2.34 Cuulative transission curve Frae-tie (b) Accelerated centralized algorith Cuulative bits (kbits) Cuulative consuption curve Cuulative overflow curve 2.34 Cuulative transission curve Frae-tie (c) DCPC Fig. 4. The cuulative overflow, transission, and consuption curves when transitting Star Wars at link 1 in the seven-cell network. Su Throughput (bits/sec/hz) UB LB Nuber of Iterations Fig. 5. Convergence of the branch-and-bound algorith in tie slot 1 when all the I-fraes are transitted and all the buffers are epty (i.e., the worst case scenario). fro tie slot 6,8 to 6,84. We observe that the su rates achieved by the centralized algorith and that by the distributed algorith are identical for ost part of this interval. Exaining the rate sus for the entire 1, tie slots, we find that the rate su achieved by the DCPC algorith is within 99% of the corresponding rate su achieved by the centralized algorith in over 97% of the tie slots. The convergence of the distributed DCPC algorith is plotted in Figs. 6(b) and 6(c) for one of the tie slots. The accelerated schee is incorporated with DCPC, such that a link ay pause its transission if its buffer is over 8% full and X (t 1) >D (t). The evolution of the BS transit powers are plotted in Fig. 6(b), where after 23 steps, all the transit powers converges to a value between and P =1 Watt. The converged power vector is P = [.23,.28,.185,.13,.163, ,.188] Watt. The evolution of the bit rates are plotted in Fig. 6(c). It is interesting to see the data rates converge faster than the transit powers in this case. All the data rates reach stable values after a few steps. VI. RELATED WORK Most of the prior work on VBR video streaing consider wired networks, which can be classified according to their traffic odels, i.e., statistical or deterinistic odels. With the forer approach, stochastic odels are developed to capture the burstiness in VBR traffic. In [2], [3], the authors observed the long-range-dependence in VBR video traffic and odeled the autocorrelation with self-siilar processes. This class of work provides valuable insights on the nature of VBR video traffic. The stochastic odels can be incorporated in QoS echaniss for VBR videos, and for traffic synthesizing in siulations [18]. With the deterinistic approach, the piecewise-constantrate transission and transport (PCRTT) ethod was used, aiing to optiize one or ore objectives while preserving continuous video playout. In [4], Liew and Chan proposed bandwidth allocation schees for dynaically sharing a CBR channel aong ultiple VBR video streas, either i) to iniize the total receiver buffer size, or ii) to avoid underflow and overflow for a given playout buffer size. In [5], Salehi et al. considered soothing VBR video over a CBR link and developed an effective algorith to achieve the greatest soothness in rate. In [19], McManus and Ross introduced a dynaic prograing fraework to set PCRTT rates and intervals to optiize different objective functions. These techniques do not directly apply to our proble of VBR over ulticell wireless networks, due to the fundaental difference between wireless and wired CBR links. In two recent papers [2] and [7], the authors studied the ore challenging proble of transitting one VBR video over a given tie-varying (i.e., VBR) wireless channel. In [2], it was shown that the separation between a delay jitter buffer and a decoder buffer is in general suboptial, and several critical paraeters are derived for the syste. In [7], the authors studied the frequency of jitters under both network and video syste constraint and provided a fraework for quantifying the trade-offs aong several syste paraeters. In this paper, we take advantage of power control in wireless networks to adjust the capacity of wireless links based on video frae size inforation, such that we can jointly optiize the transission of ultiple VBR video sessions over ultiple VBR channels. Our approach does not depend on any channel or video traffic 2568

9 Su of bit rates (kbps) Centralized algorith Distributed algorith Power (Watts) Link1 Link2 Link3 Link4 Link5 Link6 Link7 Bit rate (kbps) Link1 Link2 Link3 Link4 Link5 Link6 Link Frae tie (a) Rate sus with the algoriths Steps (b) Convergence of transit powers with DCPC Steps (c) Convergence of bit rates with DCPC Fig. 6. Siulation results with a seven-link network. odels, and can be adopted for CBR video as well. Power control is an iportant proble for interferenceliited wireless networks. Most prior work focuses on axiizing network utility in the fors of SINR or bit rate [8] [1]. In [1], Grandhi, Zander, and Yates presented centralized and distributed power control algoriths for achieving target SINRs in a cellular network. In [8], Chiang studied the proble of joint power control and congestion control, aing to axiize the throughput of TCP-Vegas over an ad hoc network. Gjendesj et al. [9] presented centralized binary power control algoriths for axiizing the su rate over ultiple interfering links. Although laid out the theoretical foundation and developed effective algoriths, these techniques cannot be directly applied for VBR video over ulticell wireless networks with buffer and delay constraints. VII. CONCLUSION We studied downlink power control for VBR video streaing in ulticell wireless networks. The proble forulation considers downlink power control, inter-cell interference, VBR video characteristics, and playout buffer requireents. We developed a centralized algorith that can provide (1-ε)-optial solutions, and a fast distributed algorith that only needs local inforation. The algoriths are evaluated with extensive siulations with VBR video traces and fading channels, and are deonstrated to be effective for streaing VBR videos over ulticell wireless networks. ACKNOWLEDGMENT This work is supported in part by the US National Science Foundation (NSF) under Grants ECCS-82113, NSF-CNS , and NSF-IIP-1322, and through the Wireless Internet Center for Advanced Technology (WICAT) at Auburn University. Any opinions, findings, and conclusions or recoendations expressed in this aterial are those of the author(s) and do not necessarily reflect the views of the NSF. REFERENCES [1] Cisco, Cisco visual networking index: Global obile data traffic forecast update, , Feb. 21, [online] Available: cisco.co. [2] M. W. Garrett and W. Willinger, Analysis, odeling and generation of self-siilar VBR video traffic, ACM SIGCOMM Coput. Coun. Rev., vol. 24, no. 4, pp , [3] J. Beran, R. Sheran, M. Taqqu, and W. Willinger, Long-range dependence in variable-bit-rate video traffic, IEEE Trans. Coun., vol. 43, no. 2/3/4, pp , Feb./Mar./Apr [4] S. Liew and H. Chan, Lossless aggregation: a schee for transitting ultiple stored VBR video streas over a shared counications channel without loss of iage quality, IEEE J. Sel. Areas Coun., vol. 15, no. 6, pp , Aug [5] J. Salehi, Z.-L. Zhang, J. Kurose, and D. Towsley, Supporting stored video: reducing rate variability and end-to-end resource requireents through optial soothing, IEEE/ACM Trans. Networking., vol. 6, no. 4, pp , Aug [6] S. Sen, D. Towsley, Z. Zhang, and J. K. Dey, Optial ulticast soothing of streaing video over the internet, IEEE J. Sel. Areas Coun., vol. 2, no. 7, pp , Sep. 22. [7] G. Liang and B. Liang, Balancing interruption frequency and buffering penalties in VBR video streaing, in Proc. IEEE INFOCOM 7, Anchorage, AK, May 27, pp [8] M. Chiang, Balancing transport and physical layers in wireless ultihop networks: jointly optial congestion control and power control, IEEE J. Sel. Areas Coun., vol. 23, no. 1, pp , Jan. 25. [9] A. Gjendesj, D. Gesbert, G. Oien, and S. Kiani, Binary power control for su rate axiization over ultiple interfering links, IEEE Trans. Wireless Coun., vol. 7, no. 8, pp , Aug. 28. [1] S. Grandhi, J. Zander, and R. Yates, Constrained power control, Int. J. Wireless Personal Coun., vol. 1, no. 4, pp , Apr [11] M. Reisslein, Video trace library, Arizona State University, [online] Available: [12] M. Chen and A. Zakhor, Multiple TFRC connections based rate control for wireless networks, IEEE Trans. Multiedia, vol. 8, no. 5, pp , Oct. 26. [13] J. Lee, R. Mazudar, and N. Shroff, Downlink power allocation for ulti-class wireless systes, IEEE/ACM Trans. Networking, vol. 13, no. 4, pp , Aug. 25. [14] N. Babos, S. C. Chen, and G. J. Pottie, Radio link adission algorith for wireless networks with power control and active link quality protection, in Proc. IEEE INFOCOM 95, Boston, MA, Apr. 1995, pp [15] H. D. Sherali and W. P. Adas, A Reforulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Probles. London, UK: Kluwer Acadeic Publishers, [16] S. Kopella, S. Mao, Y. Hou, and H. Sherali, On path selection and rate allocation for video in wireless esh networks, IEEE/ACM Trans. Netw., vol. 17, no. 1, pp , Feb. 29. [17] M. Andersin, Z. Rosberg, and J. Zander, Gradual reovals in cellular PCS with constrained power control and noise, in Proc. IEEE PIMRC 95, Toronto, Canada, Sept. 1995, pp [18] D. P. Heyan and T. V. Lakshanr, What are the iplications of long-range dependence for VBR-video traffic engineering? IEEE/ACM Trans. Networking, vol. 4, no. 3, pp , June [19] J. M. Mcanus and K. W. Ross, A dynaic prograing ethodology for anaging prerecorded vbr sources in packet-switched networks, in in Proceedings SPIE, Perforance and Control of Network Systes, 1997, pp [2] T. Stockhaer, H. Jenkac, and G. Kuhn, Streaing video over variable-bit-rate wireless channels, IEEE Trans. Multiedia, vol. 6, no. 2, pp , Apr