The Impact of Imperfect Scheduling on Cross-Layer Rate. Control in Multihop Wireless Networks

Transcription

1 The mpact of mperfect Scheduling on Cro-Layer Rate Control in Multihop Wirele Network Xiaojun Lin and Ne B. Shroff Center for Wirele Sytem and Application (CWSA) School of Electrical and Computer Engineering, Purdue Univerity Wet Lafayette, N 47907, U.S.A. {linx, hroff}@ecn.purdue.edu Abtract n thi paper, we tudy cro-layer deign for rate control in multihop wirele network. n our previou work, we have developed an optimal cro-layered rate control cheme that jointly compute both the rate allocation and the tabilizing chedule that control the reource at the underlying layer. However, the cheduling component in thi optimal cro-layered rate control cheme ha to olve a complex global optimization problem at each time, and hence i too computationally expenive for online implementation. n thi paper, we tudy how the performance of cro-layer rate control will be impacted if the network can only ue an imperfect (and potentially ditributed) cheduling component that i eaier to implement. We tudy both the cae when the number of uer in the ytem i fixed and the cae with dynamic arrival and departure of the uer, and we etablih deirable reult on the performance bound of cro-layered rate control with imperfect Thi work ha been partially upported by the NSF grant AN and the ndiana 21t Century Center for Wirele Communication and Networking. 1

2 cheduling. Compared with a layered approach that doe not deign rate control and cheduling together, our cro-layered approach ha provably better performance bound, and uually ubtantially outperform the layered approach. The inight drawn from our analye alo enable u to deign a fully ditributed cro-layered rate control and cheduling algorithm for a retrictive interference model. Keyword: Cro-layer deign, rate control, multihop wirele network, tability, imperfect cheduling, mathematical programming/optimization, tochatic procee/queueing theory. 1 ntroduction Cro-layer deign i becoming increaingly important for improving the performance of multihop wirele network (ee, e.g., 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and the reference therein). By imultaneouly optimizing the control acro multiple layer of the network, cro-layer deign can ubtantially increae the network capacity, reduce interference and power conumption. n thi paper, we tudy the iue involved in the cro-layer deign of multihop wirele network that employ rate control 8, 9, 10. Rate control (or congetion control) i a key functionality in modern communication network to avoid congetion and to enure fairne among the uer. Although rate control ha been tudied extenively for wireline network (ee 11 for a good urvey), thee reult cannot be applied directly to multihop wirele network. n wireline network, the capacity region (i.e., the et of feaible data rate) i of a imple form, i.e., the um of the data rate at each link hould be le than the link capacity, which i known and fixed. n multihop wirele network, the capacity of each radio link depend on the ignal and interference level, and thu depend on the power and tranmiion chedule at other link. Hence, the capacity region i uually of a complex form that critically depend on the way in which reource at the underlying phyical and MAC layer are cheduled. One poible way to addre thi difficulty i to chooe a rate region within the capacity region, which ha a impler et of contraint imilar to that of wireline network, and compute the rate allocation within 2

3 thi impler rate region 12, 13, 14. Thi approach eentially attempt to make rate control obliviou of the dynamic of the underlying layer. Hence, we will refer to thi approach a the layered approach to rate control. However, it require prior knowledge of the capacity region in order to chooe uch a rate region. For many network etting, even uch a rate region i difficult to find. Further, becaue the rate region reduce the et of feaible rate that rate control can utilize, the layered approach reult in a conervative rate allocation. On the other hand, the cro-layered approach to rate control can allocate data rate without requiring precie prior knowledge of the capacity region. Here, by the cro-layered approach to rate control, we mean that the network jointly optimize both the data rate of the uer and the reource allocation at the underlying layer, which include modulation, coding, power aignment and link chedule, etc. (For the ret of the paper, we will ue the term cheduling to refer to the joint allocation of thee reource at layer under rate control.) n our previou work 8, we have preented an optimal cro-layered rate control cheme and we have hown that our cheme can fully utilize the capacity of the network, maintain fairne, and improve the quality of ervice to the uer. However, the cheduling component in the optimal cro-layered rate control cheme of 8 require olving at each iteration a global optimization problem that i uually quite difficult. n ome cae, the optimization problem doe not even have a polynomial-time olution. n thi work, our objective i to develop a framework for cro-layered rate control that i uitable for online (and potentially ditributed) implementation. The complexity of the cheduling component ha become the main obtacle to developing uch a olution. To overcome thi difficulty, in thi paper we take a different approach. We accept the poibility that only uboptimal olution to the cheduling problem may be computable, which we will refer to a imperfect chedule. ntead, we will tudy the impact of imperfect cheduling on the optimality of cro-layered rate control. n thi paper, we have tudied thi impact for a large cla of imperfect cheduling policie, both for the cae when the number of uer in the 3

4 ytem i fixed, and for the cae when uer dynamically arrive and leave the network. When the number of uer in the ytem i fixed, we are able to obtain ome deirable, but weak, reult on the fairne and convergence propertie of cro-layered rate control with imperfect cheduling. Surpriingly, we are able to obtain far tronger reult on the performance of the ytem when we conider dynamic arrival and departure of the uer. Our numerical reult ugget that, in many network configuration, cro-layered rate control with imperfect cheduling can perform comparably to that with perfect cheduling, while ignificantly reducing the computation overhead of the cheduling component. Further, we find that our cro-layered approach can ubtantially outperform the layered approach. Finally, the inight drawn from our analyi allow u to develop a fully ditributed rate control and cheduling cheme in a more retrictive network etting. The ret of the paper i tructured a follow. The ytem model i preented in Section 2. We review reult with perfect cheduling in Section 3, and tudy the impact of imperfect cheduling in Section 4 and 5. n Section 6, we preent a fully ditributed cro-layered rate control algorithm. Simulation reult are preented in Section 7, and the concluion i given in Section 8. 2 The Sytem Model We conider a multihop wirele network with N node. Let L denote the et of node pair (i, j) (i.e., link) uch that direct tranmiion from node i to node j i allowed. The link are aumed to be directional. Due to the hared nature of the wirele media, the data rate r ij of a link (i, j) depend not only on it own modulation/coding cheme and power aignment P ij, but alo on the interference due to the power aignment on other link. Let P = P ij, (i, j) L denote the vector of global power aignment and let r = r ij, (i, j) L denote the vector of data rate. We aume that r = u( P ), i.e., the data rate are completely determined by the global 4

5 power aignment. The function u( ) i called the rate-power function of the ytem. Note that the global power aignment P and the rate-power function u( ) ummarize the cro-layer control capability of the network at both the phyical layer and the MAC layer. Preciely, the global power aignment determine the Signal-to-nterference-Ratio (SR) at each link. Given the SR, each link can chooe appropriate modulation and coding cheme to achieve the data rate pecified by u( P ). Finally, the network can chedule different et of link to be active (and to ue different power aignment) at different time to achieve maximum capacity 3. There may be contraint on the feaible power aignment. For example, if each node ha a total power contraint P i,max, then j:(i,j) L P ij P i,max. Let Π denote the et of feaible power aignment, and let R = {u( P ), P Π}. We aume that Co(R), the convex hull of R, i cloed and bounded. We aume that time i divided into lot and the power aignment vector P (t) i fixed during each time lot t. We will refer to r(t) = u( P (t)) a the chedule at time lot t. n the ret of the paper, it i uually more convenient to index the link numerically (e.g., link 1, 2,..., L) rather than a node-pair (e.g., link (i, j)). The power aignment vector and the rate vector hould then be written a P = P 1,..., P L and r = r 1,..., r L, repectively. There are S uer and each uer = 1,..., S ha one path through the network. Let H = H l denote the routing matrix, i.e., H l = 1, if the path of uer ue link l, and H l = 0, otherwie. Let x be the rate with which uer inject data into the network. Each uer i aociated with a utility function U (x ), which reflect the level of atifaction of uer when it data rate i x. A i typically aumed in the rate control literature, we aume that each uer ha a maximum data rate M and the utility function U ( ) i trictly concave, non-decreaing and twice continuouly differentiable on (0, M. Although we have not conidered channel variation, e.g., due to fading, our main reult may be generalized to thoe cae. Extenion to the cae with multipath routing are alo poible (ee 8). 5

6 3 Cro-Layer Rate Control with Perfect Scheduling n thi ection, we review the optimal cro-layered rate control cheme that we preented in 8. We firt define the capacity region of the ytem. We ay that a ytem i table if the queue length at all link remain finite. We ay that a uer rate vector x = x 1,..., x i feaible if there exit a cheduling policy that can tabilize the ytem under uer rate x. We define the capacity region to be the et of feaible rate x. t ha been hown in 3, 4, 6 that the optimal capacity region Λ i a convex et and i given by { } Λ = x Hx l Co(R). (1) where S H l x can be interpreted a the total data rate on link l. The convex hull operator Co( ) i due to a tandard time-averaging argument 3, 4, 6. Λ i optimal in the ene that no vector x outide Λ i feaible for any cheduling policy. n 8, we have formulated and olved the following optimal cro-layered rate control problem. The Cro-Layered Rate Control Problem: Find the uer rate vector x in Λ that maximize the total ytem utility, i.e., max 0x M ubject to and U (x ) (2) Hx l r l for all l L (3) r l Co(R). Find the aociated cheduling policy that tabilize the ytem. There are two element in thi cro-layer control problem. One i to determine the rate with which uer inject data into the network. The other i to determine when and at what rate each link in the network hould tranmit. Maximizing the total ytem utility a in (2) ha been hown to be equivalent to ome fairne objective when the utility function are appropriately 6

7 choen 15. For example, utility function of the form U (x ) = w log x (4) correpond to weighted proportional fairne, where w, = 1,..., S are the weight. A more general form of utility function i x 1 β U (x ) = w, β > 0. (5) 1 β Maximizing the total utility will correpond to maximizing weighted throughput a β 0, weighted proportional fairne a β 1, minimizing weighted potential delay a β 2, and max-min faine a β. We now take a duality approach to olve problem (2). We aociate a Lagrange multiplier q l for each contraint in (3). The Lagrangian i then: = = L( x, r, q) U (x ) q l Hx l r l U (x ) Hq l l x + q l r l. The objective function of the dual of problem (2) i then: D( q) = max L( x, r, q) 0x M,,...,S, r Co(R) = B ( q) + V ( q), where and B ( q) = max 0x M U (x ) Hq l l x, (6) V ( q) = max r Co(R) 7 q l r l. (7)

8 Further, becaue the objective function in (7) i a linear function of r, the optimal point mut lie in the et R, i.e., V ( q) = max r R q l r l = max r=u( P ), P Π q l r l. (8) The dual approach thu reult in an elegant decompoition of the original problem. Given the Lagrange multiplier q l, the rate control problem B ( q) and the cheduling problem V ( q) are decompoed. The Lagrange multiplier q l can be interpreted a the implicit cot at link l. Each uer olve it own utility maximization problem B ( q) independently a if the price for uer i L Hq l l. The cheduling problem V ( q) alo compute the power aignment P and the chedule r = u( P ) baed on the implicit cot. Note that V ( q) alo appear a the optimal cheduling policy in 3, 6. The dual problem of (2) i then min D( q). (9) q 0 The dual objective function D( q) i convex. We can how that it ubgradient i given by, ( ) D = H l q x l r l. where x = x and r = r l olve (6) and (8), repectively. We can then ue the ubgradient method to olve the dual problem 16. The olution to the optimal cro-layered rate control problem can be ummarized a follow: The Optimal Cro-Layered Rate Control Algorithm: At each iteration t: The data rate of the uer are determined by x (t) = argmax 0x M U (x ) Hq l l (t)x. (10) The chedule i determined by r(t) = argmax r R q l (t)r l = 8 argmax r=u( P ), P Π q l (t)r l. (11)

9 The implicit cot (i.e., Lagrange multiplier) are updated by ( + q l (t + 1) = q l (t) + α l Hx l (t) r l (t)). (12) The following propoition i given in 8. Propoition 1 a) There i no duality gap, i.e., the minimal value of (9) coincide with the optimal value of (2). b) Let Φ be the et of q that minimize D( q). For any q Φ, let x olve (10), then x i the unique optimal olution x of (2). c) Aume that α l = hαl 0. Let q A = L (q l ) 2 α 0 l and d( q, Φ) = min p Φ q p A. For any ɛ > 0, there exit ome h 0 > 0 uch that, for any h h 0 and any initial implicit cot q(0), there exit a time T 0 uch that for all t T 0, d( q(t), Φ) < ɛ and x(t) x < ɛ. Propoition 1 i a conequence of Theorem 2.3 in 16, p26. The detail of the proof i in Appendix A. t how that, when the tepize α l are mall, the uer rate x(t) will converge within a mall neighborhood of the optimal rate allocation x. The optimal cro-layered rate control algorithm (10)-(12) not only compute the optimal rate allocation, but alo generate the tabilizing cheduling policy by olving (11) at each time lot t. n fact, let Q l denote the queue ize at link l. Then Q l evolve approximately a : ( + Q l (t + 1) Q l (t) + Hx l (t) r l (t)). (13) Comparing (13) with (12), we can ee that Q l (t) q l (t)/α l. From here we can infer that Q l (t) i bounded. An alternate formulation of Propoition 1 i a follow: if the tepize are time varying and they are choen uch that α l (t) = h t α 0 l, h t 0 a t and + t=1 h t = +, then d( q(t), Φ) 0 and x(t) x a t. Note, (13) i an approximation becaue not all link are active at the ame time. Hence, data injected to the network by each uer at time t may take everal time lot to reach downtream link. 9

10 Propoition 2 f the tepize α l are ufficiently mall, then uing the chedule determined by olving (11) at each time lot, we have, up Q l (t) < + for all l L. t We give the proof in Appendix B. Combining Propoition 1 and 2, we conclude that, by chooing the tepize α l ufficiently mall, we can obtain uer rate allocation x a cloe to x a we want, and we can obtain the joint tabilizing cheduling policy at the ame time. Remark: The duality approach that we ued here (and in 8) hare ome imilaritie to the approach in 1, 9, 10. However, there are alo ome major difference. The network model in 1 and 10 aume a retrictive et of rate-power function. They either aume that the data rate at each link i a concave function of it own power aignment, or aume a pecial form of rate-power function that are concave after a change of variable. n thi paper, we impoe no uch retriction. Further, a conequence of the aumption in 10 i that, at their optimal olution, all link will be tranmitting at the ame time. n the more general network model of thi paper, it uually require different et of link to tranmit at different time in order to achieve optimality. n 9, the author propoe a column generation approach for olving (2). Thi approach appear to be more uitable for offline computation a it require olving a equence of approximate problem to (2), each of which require an iterative olution by itelf. n contrat, in thi paper we are more intereted in olution uitable for on-line implementation. Finally, thee previou work have not addreed the joint tabilizing cheduling policy a we did in Propoition 2. 4 The mpact of mperfect Scheduling on Cro-Layered Rate Control: The Static Cae n thi paper, we are intereted in developing cro-layered rate control olution that are uitable for online implementation. The main difficulty in implementing the optimal olution of Section 3 10

11 i the complexity of the cheduling component. Depending on the rate-power function u( ), the cheduling problem (11) i uually a difficult global optimization problem. n ome cae, thi optimization problem doe not even have a polynomial-time olution. Hence, olving (11) exactly at every time lot i too time-conuming. A dicued in the ntroduction, in thi paper, we take a different approach from that of finding optimal rate allocation. We will only compute uboptimal olution to the cheduling problem (11), which we will refer to a imperfect chedule. We will intead tudy how imperfect cheduling impact the optimality of cro-layered rate control. Our objective i to find ome imperfect cheduling policie that are eay to implement and that, when properly deigned with rate control, reult in good overall performance. We will particularly be intereted in the following cla of imperfect cheduling policie: mperfect Scheduling Policy S γ : Fix γ (0, 1. At each time lot t, compute a chedule r(t) R that atifie: r l (t)q l (t) γ max r R r l q l (t). (14) With an imperfect cheduling policy S γ, the dynamic of cro-layered rate control are ummarized by the following et of equation: x (t) = argmax 0x M U (x ) Hq l l (t)x, (15) r(t) T q(t) γ max r T q(t), r(t) Co(R), (16) r R q l (t + 1) = q l (t) + α l ( H l x (t) r l (t)) +. (17) The parameter γ in (14) can be viewed a a tuning parameter indicating the degree of preciion of the imperfect chedule. The complexity of finding a chedule r(t) atifying (14) uually decreae a γ i reduced. When γ = 1, the dynamic (15)-(17) reduce to the cae with perfect cheduling (a in Section 3). Let x,0 denote the olution to the original optimal cro-layered rate control problem (2). The olution to the following problem turn out to be a good reference point for tudying the dynamic (15)-(17) when γ < 1: 11

12 The γ-reduced Problem: max 0x M ubject to U (x ) (18) x γλ. Let x,γ denote the olution to the γ-reduced problem. The choice of γλ in the contraint of the γ-reduced problem i motivated by the following propoition, which how that an imperfect cheduling policy S γ at mot reduce the capacity region by a factor of γ. The proof i given in Appendix C. Propoition 3 f the uer rate x lie trictly inide γλ, then any imperfect cheduling policy S γ can tabilize the ytem. Motivated by Propoition 3, we would expect that the rate allocation computed by the dynamic (15)-(17) will be no wore than x,γ. However, thi aertion i not quite true. A we will ee oon, the interaction between cro-layered rate control and imperfect cheduling i much more complicated. A the data rate of the uer are reacting to the ame implicit cot a the cheduling component i, there i a poibility that the ytem get tuck into local ub-optimal area. We will contruct example where, for a ubet of the uer, their data rate determined by the dynamic (15)-(17) can be much maller than the correponding rate allocation computed by the γ-reduced problem. Nonethele, we will be able to how certain weak but deirable reult on the fairne and convergence propertie of cro-layer rate control with imperfect cheduling. 4.1 Dominance We begin our analyi by tudying whether the rate allocation computed by the dynamic (15)- (17) will dominate x,γ. (Note: a vector x 1,..., x S dominate another vector y 1,..., y S if x i y i for all i = 1,..., S.) t i eay to check that, if we let r(t) = γ r 0 (t), (19) 12

13 the dynamic (15)-(17) will olve the γ-reduced problem. Hence, we can ue (19) a a pecial cae of the imperfect cheduling policy S γ, and tudy firt whether the rate allocation x,0 of the original problem (2) dominate that of the γ-reduced problem (18). The following propoition how that uch dominance hold if the utility function i logarithmic. (Recall that logarithmic utility function are of the form U (x ) = w log x for all uer, where w i the weight for uer. n thi cae, the rate allocation computed by the original problem (2) i weighted proportionally fair 15.) Propoition 4 Aume that the utility function i logarithmic. Let x,0 be the olution to the original problem (2). Then the olution to the γ-reduced problem i x,γ = γ x,0. Proof: n the γ-reduced problem (18), do a change of variable x = x/γ. Uing the fact that U (x ) = w log x + w log γ, one can how that the γ-reduced problem become equivalent to the original problem (2). Hence, x,γ = γ x,0. Q.E.D. However, a hown in the following example, if the utility function i not logarithmic, dominance will not hold in general. Example 1: Conider the following wireline network (note that a wireline network can be viewed a a pecial cae of our network model where the capacity of each link i fixed). There are two link, whoe capacitie are 2 and 7, repectively. There are three uer. The firt uer ue both link, the econd uer ue only the firt link, and the third uer ue only the econd link. Their utility function are U 1 (x) = log x + 6x 13

14 U 2 (x) = log x U 3 (x) = 36 log x. The γ-reduced problem i then max x 1,x 2,x 3 0 ubject to (log x 1 + 6x 1 ) + log x log x 3 x 1 + x 2 2γ x 1 + x 3 7γ. When γ = 1, the olution i x,0 = T. When γ = 0.95, the olution become x,γ = T. Note that the rate of the econd uer increae a γ i reduced. Thi example how that x,0 doe not dominate x,γ in general. 4.2 A Weak Fairne Property For the ret of the paper, we will focu on logarithmic utility function, although mot of the reult that follow can alo be extended to utility function of other form (a in (5)). Note that even though x,0 dominate x,γ when the utility function i logarithmic (a hown in Propoition 4), it doe not imply that the rate allocation computed by the cro-layered rate control algorithm with an arbitrary imperfect cheduling policy S γ will dominate x,γ. n the following propoition, we characterize the likely rate allocation under imperfect cheduling provided that the dynamic (15)-(17) converge. The proof i given in Appendix D. Propoition 5 Aume that the utility function i logarithmic (i.e., of the form in (4)). f the dynamic (15)-(17) converge, i.e., x(t) x, and q(t) q a t, then x, Λ and w x,γ x, w. (20) Propoition 5 can be generalized to other form of utility function (a in (5)). Thi reult can be viewed a a weak fairne property. t how that, if the dynamic (15)-(17) converge, the 14

15 The Optimal Capacity Region Λ x *, Rate Allocation γ of the γ reduced problem D( q ) D γ( q ) Lower Bound γλ Σ w x *, γ x *, w = Σ q *,0 q *,γ Φ γ Figure 1: The weak fairne property (left) and the et Φ γ (right). rate allocation of the uer will lie in a trip defined by (20) (ee Fig. 1). Hence, the rate of each uer i unlikely to be too unfair compared to x,γ. n particular, if w = 1 for all, then by (20), x, will be no maller than x,γ /S. On the negative ide, the rate of ome uer can till be ubtantially maller than their rate computed by the γ-reduced problem, which indicate that cro-layered rate control with imperfect cheduling may indeed get tuck into local ub-optimal region. 4.3 Convergence We next tudy the quetion whether the dynamic (15)-(17) converge in the firt place. Uing a duality approach analogou to that in Section 3, we can define the dual of the γ-reduced problem a D γ ( q) = B ( q) + γv ( q), where B ( q) and V ( q) are till defined a in (6) and (7), repectively. Note that both D( q) and D γ ( q) are convex function and D( q) D γ ( q). Let q,0 denote a minimizer of D( q) and q,γ denote a minimizer of D γ ( q). Further, let Φ γ = { q : D γ ( q) D( q,0 )}. Propoition 6 Aume that α l = hα 0 l. Let q A = L (q l ) 2. For any ɛ > 0, there exit ome α 0 l h 0 > 0 uch that, for any h h 0 and any initial implicit cot q(0), there exit a time T 0 uch 15

16 that for all t T 0, q(t) q,0 A < max p Φ γ p q,0 A + ɛ. The proof i provided in Appendix E. Propoition 6 how that, if the tepize α l are ufficiently mall, the dynamic (15)-(17) will eventually enter a neighborhood of the et Φ γ. Note that both q,0 and q,γ belong to the et Φ γ (ee Fig. 1). Hence, in a weak ene, the dynamic of the ytem are moving in the right direction. However, in general the et Φ γ i quite large and doe not provide much further inight on the eventual rate allocation. We next preent two example illutrating the poible behavior of the dynamic. Example 2: We will firt how that, for any vector q and x, that atify x, = q l, w H l q l, H l x, for all, x, Λ, and > γ max r Co(R) q l, r l, (21) there exit an imperfect cheduling policy S γ uch that the dynamic (15)-(17) converge to q and x,. Note that the above et of condition implie (20). n fact, ince Hx l,γ, l L γco(r), we have, w = = x, q l, w x,γ. x, H l q l, = H l x,γ = q l, x,γ H l x, H l q l, We now how how a uitable imperfect cheduling policy S γ can be contructed. t i eay to verify that x, i the olution to the following optimization problem and q 16 i the correponding

17 Lagrange multiplier. max x 0 ubject to w log x Hx l H l x,. (22) Hence, if we let r l (t) = H l x, for all l and all t, (23) then, uing a tandard gradient decent argument for the dual problem of (22), we can how that the dynamic (15)-(17) will converge to q and x, a t. t remain to be verified whether the chedule in (23) belong to the cla of imperfect cheduling policie S γ. To ee thi, note that if we pick the initial implicit cot vector q(0) to be ufficiently cloe to q, then q(t) q for all t. Hence, q l (t)r l (t) > γ max r Co(R) q l, q l, r l γ max r Co(R) H l x, q l (t)r l, i.e., the chedule in (23) indeed belong to S γ if the initial implicit cot vector q(0) i ufficiently cloe to q. Example 3: We next give another example in which the dynamic (15)-(17) never converge to any point. Conider the following imple wireline network with two uer, each of which ue one different link. The capacity i c for both link. The olution to the γ-reduced problem i imply x,γ 1 = x,γ 2 = γc. Aume that the vector q any time t, define and x, atify the condition in (21) of Example 2. At (t) = q(t) q. 17

18 radiu 2ε radiu ε q * Figure 2: The direction of the update of the implicit cot Let ɛ be a mall poitive number. We now ue the following cheduling policy: x 1 x 2 T ɛ r(t) = r 1 r 2 T = (t), if (t) ɛ, (t) x 1 x 2 T ɛ (t), if ɛ (t) 2ɛ, (t) c, otherwie. With thi choice of the chedule r(t), the update of the implicit cot q(t) will be around a circle when q(t) q ɛ, and it will be toward q when ɛ < q(t) q 2ɛ (ee Fig. 2). Provided that the initial q(0) atifie q(0) q 2ɛ and the tepize are ufficiently mall, the dynamic (15)-(17) will eventually follow the circle q(t) q = ɛ, and hence will never converge. We can verify a in Example 2 that the chedule r(t) doe belong to the cla S γ when the tepize and ɛ are ufficiently mall. To conclude thi ection, we have tudied the impact of imperfect cheduling on the dynamic of cro-layered rate control when the number of uer in the ytem i fixed. We have preented everal example that illutrate the difficulty in characterizing the dynamic preciely. We have hown that the ytem may not even converge in the firt place, or, it may converge to any rate allocation within a fairly large et that doe not poe any deirable dominance property. Thee example indicate that the interaction between cro-layered rate control and imperfect cheduling are quite complicated, and the ytem may indeed get tuck into local ub-optimal 18

19 region. Nonethele, we do how two deirable, but weak, reult on the fairne and convergence propertie of the ytem. n Propoition 6, we are able to how that the dynamic (15)-(17) appear to move in the right direction globally. n Propoition 5, we how that thoe local ub-optimal region are probably not too bad. n the next ection, we will turn to the cae when uer dynamically arrive and depart the network, and urpriingly, we will be able to how far tronger reult on the performance of the ytem there. 5 Stability Region of Cro-Layered Rate Control n thi ection, we turn to the cae when the number of uer in the ytem i itelf a tochatic proce. We will tudy how imperfect cheduling impact the tability region of the ytem employing cro-layer rate control. Here, by tability, we mean that the number of uer in the ytem and the queue length at all link in the network remain finite. The tability region of the ytem i the et of offered load under which the ytem i table. Previou work for wireline network have hown that, by allocating data rate to the uer according to ome fairne criteria, the larget poible tability region can be achieved 15. Thi reult i important a it tell u that fairne i not jut an aethetic property, but it actually ha a trong global performance implication, i.e., in achieving the larget poible tability region. n thi ection, we will how that imilar but tronger reult can be hown for our cro-layered rate control cheme with imperfect cheduling. To be precie, intead of uing the notation for uer, we now ue to denote a cla of uer with the ame utility function and the ame path. We aume that uer of cla arrive according to a Poion proce with rate λ and that each uer bring with it a file for tranfer whoe ize i exponentially ditributed with mean 1/µ. The load brought by uer of cla i then ρ = λ /µ. Let ρ = ρ 1,..., ρ S. Let n (t) denote the number of uer of cla that are in the ytem at time t, and let n(t) = n 1 (t),..., n S (t). We aume that the rate allocation for uer of the ame cla are identical. Let x (t) denote the rate of each uer of cla at time t. 19

20 n the rate aignment model that follow, the evolution of n(t) will be governed by a Markov proce. t tranition rate are given by: n (t) n (t) + 1, with rate λ, n (t) n (t) 1, with rate µ x (t)n (t) if n (t) > 0. A in 17, We ay that the above ytem i table if lim up t 1 t t 0 P 1{ S P n L (t)+ dt 0, q l (t)>m} a M. Thi mean that the fraction of time that the amount of unfinihed work in the ytem exceed a certain level M can be made arbitrary mall a M. The tability region Θ of the ytem under a given rate control and cheduling policy i the et of offered load ρ uch that the ytem i table. We next decribe the rate aignment and implicit cot update policy. We aume that time i divided into lot of length T, and the chedule and implicit cot are only updated at the end of each time lot. However, uer may arrive and depart in the middle of a time lot. Let q( ) denote the implicit cot at time lot k. The data rate of the uer are determined by the current implicit cot a in (10). For implicity, we aume that the utility function i logarithmic (the reult can be readily generalized to utility function of other form in (5)). Further, let M denote the maximum data rate for uer of cla. The rate of each uer of cla i then given by { } w x (t) = x ( ) = min L Hl q l ( ), M (24) for t < (k + 1)T. The chedule r( ) at time lot k i computed according to an imperfect cheduling policy S γ baed on the current implicit cot q( ). Finally, at the end of each time lot, the implicit cot are updated a ( q l ((k + 1)T ) = q l ( ) + α l H l 20 n (t)x ( )dt r l ( )T ) +. (25)

21 The following propoition how that, uing the above cro-layered rate control algorithm with imperfect cheduling policy S γ, the tability region of the ytem i no maller than γλ. Propoition 7 f where S = max l L max α l 1 w min, (26) l L T S L 4ρ M H l i the maximum number of clae uing any link, and L = max H l i the maximum number of link ued by any cla, then for any offered load ρ that reide trictly inide γλ, the ytem decribed by the Markov proce n( ), q( ) i table. Several remark are in order: Firtly, Propoition 7 how that, when imperfect chedule are ued, the tability region of the ytem employing cro-layer rate control i no wore than the capacity region hown in Propoition 3 (and ued by the γ-reduced problem). Thi reult i intereting (and omewhat urpriing) given the fact that, when the number of uer in the ytem i fixed, the dynamic of cro-layered rate control with imperfect cheduling can form loop or get tuck into local ub-optimal region. Nonethele, Propoition 7 how that thee potential local ub-optimum are inconequential when the arrival and departure of the uer are taken into account. Secondly, we do not need the rate of any uer to converge. Previou reult on the tability region of rate control typically adopt a time-cale eparation aumption 15, which aume that the rate allocation x(t) perfectly olve (2) at each time intant t. Such an approach i of little value for the model in thi paper becaue the dynamic (15)-(17) with imperfect cheduling do not even converge in the firt place! Further, the time-cale eparation aumption i rarely realitic in practice: a the number of uer in the ytem i contantly changing, the rate allocation may never have the time to converge. n Propoition 7, we etablih the tability region of the ytem without requiring uch a time-cale eparation aumption. Thi reult i of independent value. For the pecial cae when γ = 1, it can be viewed a a tronger verion of previou reult in the literature (including thoe for wireline network, e.g., Theorem 1 in 15). 21

22 Finally, a imple tepize rule i provided in (26). Note that when the number of uer in the ytem i fixed, we typically require the tepize to be driven to zero for convergence to occur (ee Propoition 1). However, in (26) the tepize can be choen bounded away from zero. n fact, a the et γλ i bounded, the tepize can be choen independently of the offered load. The implicity in the tepize rule i another benefit we obtain by tudying the dynamic arrival and departure of the uer. 5.1 The Main dea of the Proof of Propoition 7 We now ketch the main idea of the proof for Propoition 7 o that the reader can gain ome inight on the dynamic of the ytem. Define the following Lyapunov function, V( n, q) = V n ( n) + V q ( q), where V n ( n) = S w n 2 2λ, and V q ( q) = L (q l ) 2 2α l. We hall how below that V( n, q) ha a negative drift. A a crude firt-order approximation, aume that uer arrive and depart only at the end of each time lot. Thu, n (t) = n ( ) during the k-th time lot. We can how that (ee Appendix F for the detail), EV n ( n((k + 1)T )) V n ( n( )) n( ), q( ) w T ρ n ( )x ( ) + E 1 (k), x ( ) where E 1 (k) i an error term that i roughly on the order of ρ n ( )x ( ). Since the rate allocation i determined by (24), we have (ignoring the maximum data rate M ), EV n ( n((k + 1)T )) V n ( n( )) n( ), q( ) T Hq l l ( ) ρ n ( )x ( ) +E 1 (k). (27) We can alo how that EV q ( q((k + 1)T ) V q ( q( )) n( ), q( ) 22

23 T q l ( ) Hn l ( )x ( ) r l ( ) +E 2 (k), (28) S 2 where E 2 (k) i an error term that i roughly on the order of Hn l ( )x ( ) r l ( ). Hence, by adding (27) and (28), and by changing the order of the ummation, we have EV( n((k + 1)T ), q((k + 1)T )) T V( n( ), q( )) n( ), q( ) q l ( ) Hρ l r l ( ) +E 1 (k) + E 2 (k). (29) By aumption, ρ lie trictly inide γλ. Hence, there exit ome ɛ > 0 uch that (1 + ɛ) Hρ l γco(r). By the definition of the imperfect cheduling policy S γ, q l ( )r l ( ) (1 + ɛ) q l ( ) Hρ l. Subtituting into (29), we have, EV( n((k + 1)T ), q((k + 1)T )) V( n( ), q( )) n( ), q( ) (30) T ɛ q l ( ) Hρ l + E 1 (k) + E 2 (k). Thi how that V(, ) would drift toward zero when q( ) i large and when the error term E 1 (k) and E 2 (k) are bounded. We would then apply Theorem 2 of 17 to etablih the tability of the ytem. To complete the proof, however, we have to addre everal difficultie: 23

24 n order to apply Theorem 2 of 17, a tronger negative drift i required. ntead of (30), we need, EV( n((k + 1)T ), q((k + 1)T )) V( n( ), q( )) n( ), q( ) ɛ ( n( ) + q( ) ) + E 0 for ome poitive contant ɛ and E 0. Further, in order to apply Theorem 2 of 17, the error term E 1 (k) and E 2 (k) have to be bounded, which i not true in (30) ince they both can become large a n ( ) increae. Finally, uer could arrive and depart at any time (not only at the end of a time lot). The complete proof that addree thee difficultie i given in Appendix F. We now give two example howing how efficient cro-layer rate control cheme can be contructed by applying Propoition 7 to different network etting. 5.2 The Node Excluive nterference Model Propoition 7 i mot ueful when an imperfect chedule that atifie (14) can be eaily computed for ome reaonable value of γ. Thi i the cae under the following node excluive interference model. The Node Excluive nterference Model: The data rate of each link i fixed at c l. Each node can only end to or receive from one other node at any time. Thi interference model ha been ued in earlier tudie of rate control in multihop wirele network 12, 13. Under thi model, the perfect chedule (according to (11)) at each time lot correpond to the Maximum Weighted Matching (MWM), where the weight of each link 24

25 i q l c l. (A matching i a ubet of the link uch that no two link hare the ame node. The weight of a matching i the total weight over all link belonging to the matching. A maximumweighted-matching (MWM) i the matching with the maximum weight.) An O(N 3 )-complexity algorithm for MWM can be found in 18, where N i the number of node. On the other hand, the following much impler Greedy Maximal Matching (GMM) algorithm can be ued to compute an imperfect chedule with γ = 1/2. Start from an empty chedule. From all poible link l L, pick the link with the larget q l c l. Add thi link to the chedule. Remove all link that are incident with either the ending node or the receiving node of link l. Pick the link with the larget q l c l from the remaining link, and add to the chedule. Continue until there are no link left. The GMM algorithm ha only O(L log L)-complexity (where L i the number of link), and i much eaier to implement than MWM. Uing the technique in Theorem 10 of 19, we can how that the weight of the chedule computed by the GMM algorithm i at leat 1/2 of the weight of the maximum-weighted-matching. According to Propoition 7, the tability region will be at leat Λ/2 uing our cro-layered rate control cheme with the GMM cheduling policy. For the node-excluive interference model, a layered approach to rate control i alo poible, which conider eparately the dynamic of rate control and cheduling 12, 13. t ha been hown that the optimal capacity region Λ in the node-excluive interference model i bounded by 2Ψ 3 0 Λ Ψ 0, where Ψ 0 = x 1 c l:b(l)=i or l e(l)=i Hx l 1 for all i. and b(l) and e(l) are the ending node and the receiving node, repectively, of link l. The layered approach then chooe the lower bound 2 3 Ψ 0 a the rate region for computing the rate allocation 12, 13. On the other hand, when an imperfect GMM cheduling policy i ued, the capacity region can be reduced by half in the wort cae (according to Propoition 3). Hence, the layered approach then need to ue Ψ 0 /3( Λ/2) a the rate region. Note that for the layered approach with GMM cheduling, Ψ 0 /3 i an upper bound for it tability region, which i maller than the lower bound of the tability region of the correponding cro-layered approach (which i Λ/2 25

26 according to Propoition 7). Hence, due to it conervative nature, the layered approach alway uffer from wort cae inefficiencie. n Section 7, we will ue imulation to how that our cro-layered rate control cheme can in practice ubtantially outperform the layered approach. 5.3 General nterference Model Under general interference model, it may till be time-conuming to compute a chedule that atifie (14) for a given value of γ. We now ue Propoition 7 to develop a cheduling policy that can cut down the frequency of uch computation, and hence effectively reduce the computation overhead. Thi idea i motivated by the obervation that implicit cot, being updated by (17), cannot change abruptly. Hence, there i a high chance that a chedule computed earlier can be reued in ubequent time-lot. To ee thi, aume that we know a chedule r 0 that atifie (14) for an inefficiency factor γ 0 > γ when the implicit cot vector i q 0, i.e., r 0 l q l 0 γ 0 max r R r l q0. l (31) Let the implicit cot vector at the current time lot be q, and let r denote the correponding (but unknown) perfect chedule. We can normalize q 0 and q to be of unit length ince the correponding chedule will remain the ame. We have, where r max l q l rl = (q l q0)r l l + i the maximum rate of link l. Hence, if q0r l l q0r l l 0 q l q0 l + rl max +, γ 0 q l rl 0 γ q l q0 l + rl max + q0r l l 0, γ 0 we can till ue r 0 a the imperfect chedule for q. Thi approach i even more powerful when the network can remember multiple chedule from the pat. Aume that the chedule r 1, r 2,..., r K 26

27 correpond to q 1, q 2,..., q K, repectively, and each pair atifie (31). Then, a long a max k=1,..,k q l rl k (32) min γ k=1,...,k q l q l k + r max l + qk l rk l γ 0, we do not need to compute a new chedule. ntead, we can ue the chedule that maximize the left hand ide of (32). By Propoition 7, the tability region of the ytem uing the above cheduling policy i no maller than γλ. n Section 7, we will ue imulation to how that uch a imple policy can perform very well in practice. 6 A Fully Ditributed Cro-Layered Rate Control and Scheduling Algorithm Propoition 7 open a new avenue for tudying cro-layer deign for rate control in multihop wirele network. ntead of retricting our attention to the rate allocation at each naphot of the ytem (a we did in Section 4 where the reult tend to be weaker), we can now tudy the entire time horizon by focuing on the tability region of uch a cro-layer-deigned ytem. Motivated by Propoition 7, we now preent a fully ditributed cro-layered rate control and cheduling algorithm for the node-excluive interference model in Section 5.2. (n contrat, the GMM algorithm in Section 5.2 till require centralized implementation.) Thi new algorithm can be hown to achieve a tability region no maller than Λ/2. The new algorithm ue Maximal Matching (MM) to compute the chedule at each time 20. A maximal matching i a matching uch that no more link can be added without violating the node-excluive interference contraint. To be precie, let q ij denote the implicit cot at link (i, j). (For convenience, in thi ection we will index a link by a node pair (i, j).) A maximal matching M i a ubet of L uch that q ij 1 for all (i, j) M, and, for each (i, j) L, one of 27

28 the following hold: q ij < 1, or (33) (i, k) M for ome link (i, k) L, or (k, i) M for ome link (k, i) L, or (j, h) M for ome link (j, h) L, or (h, j) M for ome link (h, j) L. Note that a maximal matching can be computed in a ditributed fahion a follow. When a link (i, j) i added to the matching, we ay that both node i and node j are matched. For each node i, if it ha already been matched, no further action i required. Otherwie, node i can it neighboring node. f there exit a neighboring node j uch that node j ha not been matched, node i end a matching requet to node j. t i poible that a matching requet conflict with other matching requet. n thi cae, the node involved in the conflict can ue ome randomization and local coordination to pick any non-conflicting ubet of the matching requet. For thoe node whoe matching requet are declined, they can repeat the above procedure until every node in the network i either matched or ha no neighbor that are not matched. Let Q i = q ij + q ji j:(i,j) L j:(j,i) L denote the total cot of the link that either tart from, or end at node i. Our new cro-layered rate control and cheduling algorithm then proceed a follow. The Ditributed Cro-Layered Rate Control Algorithm: At each time lot, (k + 1)T ): A maximal matching M( ) i computed baed on the implicit cot q( ). The data rate of each uer of cla i determined by x (t) = x ( ) 28

29 w = max 2 (i,j) L Hij Q i( )+Q j, M ( ) c ij (34) where c ij i the capacity of link (i, j), and H ij cla ue link (i, j); and H ij = 0, otherwie. i defined a H l, i.e., H ij = 1, if uer of The implicit cot are updated by: q ij ((k + 1)T ) = q ij ( ) + α ( H ij ) + n (t)x ( ) dt T 1 {(i,j) M( )}. (35) c ij Thi new cro-layered rate control and cheduling algorithm i imilar to the algorithm of Section 4 and 5 in many apect: A uer react to congetion by reducing it data rate when the implicit cot along it path increae. The implicit cot at each link (i, j) i updated baed on the difference between the offered load and the chedule of the link. However, there i a critical difference. When the maximal matching i computed, we do not care about the precie value of the implicit cot (ee (33)). Hence, the maximal matching typically doe not atify the requirement of the imperfect cheduling policy S γ, and Propoition 7 doe not apply either. Further, the rate control part (34) i alo different from that in the earlier ection. t ha been choen pecifically for the maximal matching cheduling policy. Nonethele, uing imilar technique a in Section 5, we can how the following reult on the tability region of the ytem. The detail are given in Appendix G. Propoition 8 f the tepize α i ufficiently mall, then for any offered load ρ that reide trictly inide Λ/2, the ytem with the above ditributed cro-layered rate control algorithm i table. 29

30 Cla 3 0 Cla 0 Cla 1 1 L1 L2 L0 2 L3 L4 L6 3 Cla 2 L5 4 L7 L8 5 Node Poition 0 (0.0, 0.0) 1 (0.0, 2.0) 2 (1.0, 1.0) 3 (2.2, 0.0) 4 (2.5, 2.0) 5 (3.5, 1.0) Cla 4 Figure 3: The Network Topology 7 Numerical Reult We now ue imulation to verify the reult in thi paper. We ue the network in Fig. 3. There are 5 clae of uer, whoe path are hown in Fig. 3. Their utility function are all given by U (x ) = log x. We firt ue the following interference model. The path lo G(i, j) from a node i to a node j i given by G(i, j) = d 4 ij where d ij i the ditance from node i to node j (the poition of the node are alo given in Fig. 3). We aume that the data rate r ij at link (i, j) L i proportional to the SR, i.e., G(i, j)p ij r ij = W N 0 + (k,h) L,(k,h) (i,j) G(k, j)p, k,h where N 0 i the background noie and W i the bandwidth of the ytem. Thi aumption i uitable for CDMA ytem with a moderate proceing gain 6. Each node i ha a power contraint P i,max, i.e., the power allocation mut atify j:(i,j) L P ij P i,max for all i. We firt imulate the cae when there i one uer for each cla. The left figure in Fig. 4 how the evolution of the data rate for all five uer when the network compute the perfect chedule according to (11) at every time lot. We have choen W = 10, N 0 = 1.0, P i,max = 1.0 for all node i and α l = 0.1 for all link l. Note that the cheduling ubproblem (11) for thi interference model i a complex non-convex global optimization problem. n 8, we have given an O(2 N ) algorithm for olving the perfect chedule, where N i the number of node. Executing uch an algorithm at every time-lot i extremely time-conuming. We then imulate the imperfect cheduling policy outlined in Section 5.3 for general interference 30

31 Rate Uer 0 Uer 1 Uer 2 Uer 3 Uer 4 Rate Uer 0 Uer 1 Uer 2 Uer 3 Uer teration teration Figure 4: The evolution of the data rate for all uer with perfect cheduling (left) and with imperfect cheduling (right, γ = 0.5). model. Such an imperfect cheduling policy attempt to reue chedule that have already been computed in the pat. n our imulation, we have choen γ 0 = 1.0 in (31), i.e., each of thee pat chedule are perfect chedule. The computational complexity could have been further reduced if we had choen γ 0 < 1. However, we leave thi for future work. ntead, in thi paper we focu on how the imperfect cheduling policy can reduce the number of time that new perfect chedule have to be computed. The ytem that we imulate can tore at mot 10 pat chedule. f there are already 10 pat chedule and a new perfect chedule i computed, the new chedule will replace the old one that ha the mallet weighted-um L q l r l. n the right figure of Fig. 4, we how the evolution of the data rate when γ = 0.5. Note that the rate allocation eventually converge to value cloe to that with perfect cheduling. We alo record the number of time that perfect chedule are computed. When γ = 0.5, perfect chedule are computed in only 7 iteration among the entire 2000 iteration of the imulation, and mot of thee perfect chedule are computed at the initial tage of the imulation. We have imulated other value of γ and find imilar reult. n fact, by jut reducing γ from 1.0 to 0.9, the number of time that perfect chedule have to be computed i reduced to 34 (over 2000 iteration of imulation). Thee reult indicate that our cro-layered rate control cheme with the imperfect cheduling policy in Section 5.3 can ubtantially reduce the computation overhead and till maintain good performance. 31

32 Average Number of Uer γ=1.0 γ=0.9 γ= ρ Average Number of Uer Joint MWM Layered MWM Joint GMM Layered GMM Joint MM ρ Figure 5: The average number of uer in the ytem veru load. Figure 6: The average number of uer in the ytem veru load: the node-excluive interference model We then imulate the cae when there are dynamic arrival and departure of the uer a in Section 5. Uer of each cla arrive to the network according to a Poion proce with rate λ. Each uer bring with it a file to tranfer whoe ize i exponentially ditributed with mean 1/µ = 100 unit. We vary the arrival rate λ (and hence the load ρ = λ/µ) and record in Fig. 5 the average number of uer in the ytem at any time for different choice of γ. Given γ, the average number of uer in the ytem will increae to infinity a the offered load ρ approache a certain limit. Thi limit can then be viewed a the capacity of the ytem. From Fig. 5, we oberve that the capacity of the ytem i not ignificantly affected when γ i reduced from 1.0 to 0.5. On the other hand, the number of time-lot that new perfect chedule have to be computed i reduced to le than 1% of the total number of time-lot when γ = 0.9, and to le than 0.05% when γ = 0.5. Thee reult confirm again the effectivene of our cro-layered rate control cheme with the imperfect cheduling policy in Section 5.3, in reducing the computation overhead and achieving good overall performance. We next turn to the node-excluive interference model in Section 5.2, where we can draw a comparion with the layered approach to rate control 12, 13. We till ue the network topology in Fig. 3. The capacity of each link i now fixed at 10 unit. Due to pace contraint, we only report the reult for the cae when there are dynamic arrival and departure of the uer. Fig. 6 demontrate the average number of uer in the ytem veru load with different rate control 32