CONGESTION control is a key functionality in modern

Transcription

1 IEEE TRANSACTIONS ON INFORMATION TEORY, VOL. X, NO. X, XXXXXXX 2008 On the Connection-Level Stability of Congetion-Controlled Communication Network Xiaojun Lin, Member, IEEE, Ne B. Shroff, Fellow, IEEE, and R. Srikant, Fellow, IEEE Abtract We are intereted in the connection-level tability of a network employing congetion control. In particular, we tudy how the tability region of the network (i.e., the et of offered load for which the number of active uer in the network remain finite) i affected by congetion control. Previou work in the literature typically adopt a time-cale eparation aumption, which aume that, whenever the number of uer in the ytem change, the data rate of the uer are adjuted intantaneouly to the optimal and fair rate allocation. Under thi aumption, it ha been hown that uch rate aignment policie can achieve the larget poible tability region. In thi paper, thi time-cale eparation aumption i removed and it i hown that the larget poible tability region can till be achieved by a large cla of congetion control algorithm. A econd aumption often made in prior work i that the packet of a ource (or uer) are offered to each link along it path intantaneouly, rather than paing through one queue at a time. We how that connection-level tability i again maintained when thi aumption i removed, provided that a back-preure cheduling algorithm i ued jointly with the appropriate congetion controller. Index Term Communication network, congetion control, connection-level dynamic, tability, time-cale eparation. I. INTRODUCTION CONGESTION control i a key functionality in modern communication network. The objective of congetion control i to regulate the data rate at which each uer inject data into the network uch that (a) the network capacity i fully utilized, (b) exceive congetion inide the network i avoided, and (c) ome form of fairne (in term of the amount of ervice that each uer receive) i enured. Since the eminal work by Kelly et al 3, it i clear that thee objective can be mapped to a global optimization problem that maximize the total ytem utility, where different fairne objective can be achieved by appropriately chooing the utility function. Congetion control can then be viewed a a ditributed iterative Manucript received December 8, 2005; revied May 7, 2007 and Augut 30, Xiaojun Lin i with School of Electrical and Computer Engineering, Purdue Univerity, Wet Lafayette, IN ( linx@ecn.purdue.edu). Ne B. Shroff i with Department of Electrical and Computer Engineering and Department of Computer Science and Engineering, The Ohio State Univerity, Columbu, O 4320 ( hroff@ece.ou.edu). R. Srikant i with Coordinated Science Laboratory and Department of Electrical and Computer Engineering, Univerity of Illinoi at Urbana-Champaign, IL 680 ( rrikant@uiuc.edu). Earlier verion of thi paper have been preented at the 42rd Annual Allerton Conference on Communication, Control, and Computing, Monticello, IL, October 2004, and the IEEE Computer Communication Workhop, October Thi work ha been partially upported by the NSF grant CNS , CNS , CNS , CNS , CCF and CCF olution to the afore-mentioned global optimization problem 3 8. Significant advance in the undertanding of congetion control have been made under thi optimization framework. The reult can roughly be categorized into two group. In the firt body of work, it i aumed that the number of uer in the network i fixed and each uer ha infinite data to tranfer. Thi reearch focue on developing ditributed iterative algorithm that converge to a fair rate allocation, which correpond to the olution of the global optimization problem. Variou iue have been addreed in thi body of work, including global convergence of the congetion control algorithm, local tability of the equilibrium rate allocation under feedback delay, the impact of random noie, and the aymptotic behavior of the ytem when the number of uer i large. The econd body of work tudie a network with connection-level dynamic, i.e., when the uer randomly enter and leave the network. When the uer are in ervice, their data rate are dynamically controlled according to the congetion level in the network. A baic quetion in tudying the connection-level dynamic i the tability region of the ytem employing congetion control. ere, by tability, we mean that the number of active uer in the ytem and the queue length at each link in the network remain finite. The tability region of the ytem under a given congetion control algorithm i the et of offered load under which the ytem i table. Thi body of work typically aume that, whenever the number of uer in the ytem change, the data rate of the uer are adjuted intantaneouly to the optimal (and fair) rate allocation computed by the global optimization problem. Thi model eentially aume a time-cale eparation, i.e., the time cale of the arrival and departure of the uer i much lower than that of the dynamic determined by the congetion control algorithm derived in the firt body of work. Under thi time-cale eparation aumption, it ha been hown that the larget poible tability region can be achieved by allocating data rate among the uer according to certain fairne criteria 9 2. Further, it ha been hown in 0 that unfair reource allocation uch a priority cheduling may not achieve the larget poible tability region. Thu, fairne i not merely an aethetic property, but it actually ha a trong global performance implication, i.e., in achieving the larget poible tability region. In thi paper, we tudy the connection-level tability region of congetion controlled communication network without uing the time-cale eparation aumption. Note that the timecale eparation aumption arie from the following wellknown obervation: mot of the traffic in the Internet i due

2 IEEE TRANSACTIONS ON INFORMATION TEORY, VOL. X, NO. X, XXXXXXX to a mall fraction of file which are extremely large. Since thee file only form a mall fraction of the traffic and thee are one for which congetion control can be aumed to reach a teady-tate, it would eem reaonable to aume that thee file arrive, tay in the ytem for a long time and then depart, thu leading to two time-cale behavior. While thi intuition i reaonable, in reality, the ditinction between mall file and large file i not a precie a thi model might ugget and therefore, it i hard to rigorouly jutify the time-cale eparation aumption. We will how that, even when we remove the time-cale eparation aumption, the larget poible tability region can till be achieved by a large cla of congetion control algorithm that are derived from the optimization framework. ence, our reult reinforce the performance benefit of congetion control in a tronger ene. A econd aumption that i uually made in the congetion control literature i that the packet of a ource (or uer) are offered to each link along it path intantaneouly, rather than going through one queue at a time. While we adopt thi aumption in Section 2-4, we will tudy a different model in Section 5 that directly take into account the packet dynamic of going through one queue at a time. We will how that connection-level tability i again maintained when the uer-rate-applied-imultaneouly-to-all-link aumption i removed, provided that a back-preure cheduling algorithm i ued jointly with the appropriate congetion controller. The ret of the paper i tructured a follow. In Section II, we preent the ytem model and review ome related reult in the literature. Our main reult i preented in Section III, where we how that the larget tability region can be achieved even without the time-cale eparation aumption. In Section IV, we extend our main reult to the cae with multi-path routing and the cae with time-varying capacity. In Section V, we extend our main reult to explicitly account for the packet dynamic of going through one queue at a time. We conclude in Section VI. II. TE SYSTEM MODEL AND RELATED RESULTS In thi ection, we decribe our ytem model and review everal related work. We conider a network with L link and S clae of uer. Let E {, 2,..., L} denote the et of link. For now, we aume that the capacity of each link l E i R l, and uer of each cla have one path through the network. (The extenion to the cae with time-varying capacity and multi-path routing are treated in Section IV.) Let l, if the path of uer of cla ue link l, and l 0, otherwie. A. Congetion Control We firt motivate the congetion controller ued in the paper. Note that congetion control i a key functionality in modern communication network to efficiently utilize the network reource while avoiding exceive congetion, and to provide fair rate-allocation to the uer. Without proper congetion control, the network can run into congetion-collape 3, a tate where the ueful throughput of the network drop by order of magnitude due to exceive congetion inide the network. Since the eminal work 3 by Kelly et al, it i clear that congetion control can be modeled a a ditributed olution to the following network utility maximization problem. We firt conider the cae when the number of uer n of each cla i fixed, and each uer ha an infinite backlog to tranfer. Let x denote the rate at which each uer of cla end data into the network, and let U (x ) be the utility received by the uer of cla when it end data at rate x. The utility function U ( ) characterize the atifaction level of a uer of cla when it end data at a certain rate, and a we will oon dicu, it alo correpond to a certain fairne objective. A i typically aumed in the literature, we aume that each uer of cla ha a maximum data-rate limit of M, and the utility function U ( ) i increaing, trictly concave, and twice continuouly differentiable on (0, M 4. Let n n,..., n S and x x,..., x S. Congetion control can then be formulated a the following global optimization problem 3: max x:0x M,,...,S ubject to S n U (x ) () S n l x R l for all l,..., L. ) Fair and Efficient Rate Allocation: The contraint in () enure that the network utilization never exceed the capacity, while the increaing utility function enure that the capacity will be utilized a much a poible. Further, the fairne objective can be achieved by appropriately chooing the utility function 0, 4. For example, utility function of the form U (x ) w log x (2) correpond to weighted proportional fairne, where w,,..., S are the weight. A more general form of the utility function i x β U (x ) w, for ome β > 0 and β. (3) β Maximizing the total ytem utility will correpond to maximizing weighted throughput a β 0, weighted proportional fairne a β, and max-min faine a β. 2) A Ditributed Congetion-Controller and it Convergence: Congetion control can then be viewed a a ditributed iterative olution to the above global optimization problem 3 8. The type of congetion controller that i mot related to thi paper i the o-called dual olution given below 4. We aociate an implicit cot q l with each link l and let q q,..., q L. The following iterative algorithm can olve problem () with an appropriate choice of the tep-ize. Algorithm A: At each time intant t, The data rate of each uer of cla i determined by: x (t) argmax 0x M U (x ) x L q l l (t). (4)

3 IEEE TRANSACTIONS ON INFORMATION TEORY, VOL. X, NO. X, XXXXXXX The implicit cot at each link l i updated by: ( S ) q l (t) q l (t) α l n l x (t) R l, (5) where denote the projection to 0, ) and α l i a poitive tep-ize for each link l. The following propoition wa hown in 4 with lightly different notation. Propoition : Aume that the number of uer in the ytem i fixed. Further, aume that the curvature of U ( ) are bounded away from zero on (0, M, i.e., there exit a poitive number γ for each cla uch that U (x ) γ > 0 for all x (0, M. (6) Let x denote the optimal olution to problem (). Let S S max l n l denote the maximum number of uer uing any link, and let L max L l denote the maximum number of link ued by any uer. If max α l 2 l SL min γ, (7) then Algorithm A converge, i.e., x(t) x a t. ence, when the tep-ize α l are ufficiently mall, the dual olution can olve the problem () a t. B. Connection-Level Stability The dicuion of congetion control in Section II-A ha focued on the tatic etting, i.e., we have aumed that the number of uer of each cla i fixed and each uer ha an infinite amount of data to tranfer. In real network, uer alo exhibit connection-level dynamic, i.e., they randomly enter the network, have only a finite amount of data to tranfer, and leave the network after the data tranfer i completed. When the uer are in ervice, their data rate are aumed to be dynamically controlled according to the congetion controller preented earlier. owever, becaue the number of active uer contantly change, the congetion control algorithm may not be able to converge before the next time intant when a uer enter or leave the ytem. ence, although convergence i an important goal for the tatic etting, we would be intereted in other performance meaure (uch a tability and completion time) when there are connection-level dynamic. In particular, Stability i often a firt-order and important performance meaure to tudy 9 2, 5. ere, by tability, we mean that the number of active uer in the ytem and the queue length at each link in the network remain finite. To be precie, we aume that uer of cla arrive to the network 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 /µ. The load brought by uer of cla i then ρ λ /µ. Let ρ ρ,..., ρ S. Let n (t) denote the number of active uer of cla that till have data to inject into the ytem at time t and let n(t) n (t),..., n S (t). Note that according to thi definition of n(t), once a uer inject all of it data (from the file that it bring) into the network, even though the data may not have reached the detination yet, we aume that the uer will immediately leave the ytem. We aume that uer of the ame cla end data into the network at the ame rate. Let x (t) denote the data injection rate of uer of cla at time t and let x(t) x (t),..., x S (t). The data injection rate x (t) will be determined by the congetion control algorithm. Due to the aumption on Poion arrival and exponential file ize ditribution, the evolution of n(t) will be governed by a Markov proce. It tranition rate are given by: n (t) n (t), with rate λ, n (t) n (t), with rate µ n (t)x (t) if n (t) > 0. (8) The data injected into the network create queue-backlog at the link. We aume that the queue length are updated every time lot of length T. Auming that the data injection rate of each uer i applied intantaneouly over all link that the uer route travere, the evolution of the queue length Q l at link l i then governed by: 2 Q l ((k )T ) Q l ( ) ( S ) (k)t n (t)x (t) T R l. (9) l Let Q(t) Q (t),..., Q L (t). We can now mathematically define the notion of tability that we are intereted in. Note that here the tability of the ytem mut be defined jointly for n(t) and Q(t). To ee thi, conider a rate allocation policy that allocate a contant rate x (t) x for each uer of cla. At an arbitrarily large offered load ρ, if we can chooe x ufficiently large, then the number of active uer n(t) in the ytem can be eaily made table. (Recall that in our definition of n(t), a uer i conidered to have left the ytem once it inject all of it data into the network.) owever, the link inide the network may be everely congeted and the queue length Q(t) may approach infinity. On the other hand, if we chooe x ufficiently mall, then the queue length Q(t) can be eaily made table. owever, each uer will then take a long time to complete ervice, and hence the number of active uer n(t) may approach infinity. Clearly, for the ytem to operate correctly, we hould avoid both of the above two cae. ence, it i neceary to define the notion of tability jointly for n(t), Q(t). We ay that the Markov proce n(t), Q(t) i table-inthe-mean or imply table 6 if lim up t t t 0 S E n (t) L Q l (t) <. (0) Although we do not pecify the unit of thee quantitie, we do aume that their unit are conitent. For example, if the file ize i in bit, and time i in econd, then the unit of λ and /µ are per econd and bit, repectively, and the unit of ρ, x and the link capacity R l are all bit per econd. 2 Thi aumption will be removed in Section V.

4 IEEE TRANSACTIONS ON INFORMATION TEORY, VOL. X, NO. X, XXXXXXX By the Markov inequality, thi implie that, a M, t S L lim up P n (t) Q l (t) > M 0. t t 0 Roughly peaking, the ytem i table if both the number of uer in the ytem and the queue length at each link in the network remain finite. Note that the queue length Q l in our model can be any real poitive number. One could have ued the technique of 2 to dicretize Q l, and modeled the ytem a a countable tate-pace Markov proce. If in addition thi countable tate-pace Markov proce i irreducible, then the above notion of tability alo implie poitive recurrence. owever, in thi paper we do not purue thi approach, and we imply ue tability-in-the-mean to be our notion of tability. We define the tability region Θ of the ytem under a given congetion controller to be the et of offered load ρ uch that the ytem i table for any ρ Θ. We ay that the tability region achieved by a congetion controller i the larget poible when the following hold: for any offered load, if thi congetion controller cannot tabilize the ytem, no other congetion controller can. Trivially, the capacity contraint determine an upper bound 3 on the tability region achieved by any congetion controller, i.e., { } S Θ Θ 0 ρ ρ l R l for all l. () In the ret of the paper, we will be intereted in tudying whether the congetion controller developed through a tatic etting (uch a the one in Section II-A) can alo achieve the larget poible tability region when there are connectionlevel dynamic. A we reviewed related work in Section II-D, there are cae where the congetion controller (or more preciely, the rate allocation algorithm) i not properly deigned, and can lead to a trictly maller tability region. ence, the connection-level tability of a ytem employing congetion control i not a trivial problem. In prior work, the connection tability problem ha typically been tudied under the following time-cale eparation aumption. The Time-Scale Separation Aumption: The data rate x(t) of the uer at each time intant t are adjuted intantaneouly to the optimal rate allocation computed by the global optimization problem () with n n(t). We refer to a congetion controller that allocate data rate according to the above time-cale eparation aumption a the idealized congetion controller. Note that in thi cae, ince the congetion controller chooe the data rate x(t) uch that the aggregate data arrival rate at each link i no greater than the link capacity (ee the contraint in ()), the queue Q l (t) are alway table according to (9). ence, we can model the ytem a a countable tate-pace Markov chain by denoting the number of active uer n(t), a the ytem tate. The tability of the ytem i then equivalent to the poitive recurrence of the Markov chain n(t). 3 Thi upper bound can be etablihed along the line of Lemma 3.3 in 7 or Section III.A of 8, i.e., it can be hown that the Markov proce will not be table if ρ lie outide the et Θ 0. The next propoition from 0 how that the tability region achieved by the idealized congetion controller i indeed the larget poible found on the right hand ide of (). Propoition 2: Under the time-cale eparation aumption, if the utility function are of the form in (2) or (3) for ome β > 0, then for any offered load ρ that reide trictly inide Θ 0, the Markov proce n(t) i poitive recurrent and hence, lim up t t t 0 { S P 0, a M. n (t)>m} Proof of the above reult uing fluid limit and the Foter- Lyapunov theorem can alo be found in 9 and 20, repectively. C. Problem Statement A dicued in the Introduction, time-cale eparation i difficult to verify in real network. When the number of uer in the network change rapidly, the data rate of the uer employing a congetion control algorithm (uch a algorithm A) may never converge. Further, note that the tep-ize condition (7) in Propoition become more tringent a the number of uer in the ytem increae. A the offered load ρ approache the boundary of the tability region Θ 0, the number of uer in the ytem will approach infinity. ence, given a choen et of tep-ize, algorithm A will fail to converge when the offered load i cloe to the boundary of Θ 0. The time-cale eparation aumption will not hold in either of thee two cae. In thi paper, we will tudy the connection-level tability region of congetion controlled communication network without uing the time-cale eparation aumption. In particular, we are intereted in the following quetion: will the cla of congetion control algorithm introduced in Section II-A be able to achieve the larget tability region Θ 0, even when the time-cale eparation aumption i removed? D. Related Work Stability i an important ubject for many tochatic ytem. The tability region for arbitrary network ha been tudied in 7, 2. The original model in 7, 2 do not have congetion control or connection-level dynamic. Intead, they aume that packet arrive to the ytem according to a given tochatic proce, and hence the packet injection rate cannot be controlled. The role of the network i to route thee packet through the ytem, and, if the link capacity can alo be controlled, to determine the optimal link tranmiion pattern. Optimal control cheme are developed in 7, 2 that achieve the larget tability region, which i in a imilar form a () for the cae when the link capacity i fixed. It would be intructive to map the reult in 7, 2 to the model in Section II-B. We can either take x (t) a being infinitely large, in which cae when a uer arrive to the ytem the entire file i injected immediately into the network; or, we can take x (t) to be at a fixed value. In both cae, the packet arrival proce i completely determined once the value of x (t) i fixed. ence, we can apply the control cheme and

5 IEEE TRANSACTIONS ON INFORMATION TEORY, VOL. X, NO. X, XXXXXXX the reult from 7, 2. In particular, ince the average packet arrival rate by uer of cla i exactly equal to the offered load ρ when x (t) i a contant, if we apply the control cheme in 7, 2, the reult there imply that the ytem will be table a long a ρ fall trictly inide the et Θ 0 given by (). Note that when the rate x (t) are fixed, we only need to conider the tability of the queue length Q(t), becaue at any fixed x (t) the number of active uer n(t) i alway finite by tandard reult from M/M/ / queue. owever, chooing a fixed x (t) mean that the ytem doe not have the capability to dynamically control the data rate baed on the current congetion level. ence, we will refer to the above tability reult (that i directly derived from 7, 2) a tability without congetion control. itorically, the Internet without proper congetion control ha uffered from congetion collape 3, a tate where the ueful throughput of the network drop by order of magnitude due to exceive congetion inide the ytem. ence, mot modern communication network do employ ome form of congetion control mechanim. In thi paper, we are intereted in the tability of the ytem when congetion control algorithm are employed. In order to differentiate with the notion of tability without congetion control 7, 2, we refer to the tability of a ytem employing congetion control a connection-level tability. Note that connection-level tability cannot be inferred from the reult of 7, 2. Specifically, in 7, 2, although the control policy itelf doe not require any knowledge of the tatitic of the packet arrival proce, the tability reult doe require certain aumption on it correlation tructure. When one employ a dynamic congetion controller, it i unclear whether thee aumption hold for the tochatic proce with which uer inject data into the network. ence, we cannot ue the tability reult of 7, 2. The author in 9 2, 5 tudy the connection-level tability problem under the time-cale eparation aumption that, at each time-intant t, the data injection rate x(t) are choen a a function of the current number of active uer n(t). In particular, the data injection rate x (t) may be choen to maximize the total ytem utility a in (), where n i replaced by the intantaneou number of uer n (t) in the ytem 0, 2. Alternatively, the rate x(t) can be choen according to other fairne and optimality criteria (e.g., according to max-min fairne 9). Under thi timecale eparation aumption, it ha been pointed out that the introduction of certain congetion-control mechanim may prevent link capacitie from being fully utilized, which might reduce the tability region of the ytem 9. In fact, example have been provided in 0 that, under certain unfair rateallocation mechanim (e.g., thoe uing priority-cheduling), the connection-level tability region of the ytem i indeed trictly maller than the upper bound Θ 0 given in (). Therefore, the quetion of connection-level tability i not a trivial one. On the poitive ide, it ha been hown that certain type of fair congetion controller can indeed achieve the larget tability region (e.g., in Propoition 2), when timecale eparation i aumed. Thu, the fairne objective of congetion control i not merely an aethetic property, but it actually ha a trong global performance implication, i.e., in achieving the larget poible tability region. III. CONNECTION-LEVEL STABILITY WITOUT TIME-SCALE SEPARATION In thi ection, we tudy the connection-level tability of ytem employing congetion control, without the aforementioned time-cale eparation aumption. We firt decribe ome more detail on the dynamic of the ytem. We aume that time i divided into lot of length T. We ue the following congetion controller imilar to Algorithm A in Section II-A. We till aign an implicit cot q l with each link l. The implicit cot q l correpond to the congetion level at link l (and we will oon ee that q l i imply a calar multiple of the real queue length at link l). We aume that the implicit cot are updated only at the end of each time lot. owever, uer may arrive and depart at any point within a time lot. Let q( ) denote the implicit cot at time lot k. At any time t, the data rate of uer of cla i given by /β w x (t) x ( ) min, M, (2) L q l l ( ) for t < (k )T, where β > 0. Note that Equation (2) i the olution to (4) when the utility function i of the form (2) or (3). (We ue the convention β when the utility function i of the form (2).) At the end of each time lot k, the implicit cot are updated by q l ((k )T ) q l ( ) ( S ) (k)t α l n (t)x ( ) T R l. (3) l Note that both equation are cloely related to Algorithm A. owever, unlike the cae in Propoition 2 where the rate allocation x(t) i determined by the olution to (), which by itelf require Algorithm A to converge intantaneouly at each time, now the rate allocation x(t) i determined by the current implicit cot. ence, we have removed the time-cale eparation aumption. Remark: By comparing (3) with (9), we oberve that the implicit cot q l (t) i imply a caled verion of the queue length Q l (t), i.e., q l (t) α l Q l (t) auming q l (0) Q l (0) 0. ence, in the equel we will focu on the tability of the Markov proce n( ), q( ) becaue it i equivalent to the tability of n( ), Q( ) a defined in (0). The following propoition how that, even when the timecale eparation aumption i removed, the above congetion control algorithm can till achieve the larget poible tability region. Propoition 3: Aume that the utility function are either of the form in (2) (in which cae we ue the convention that β ), or of the form in (3) for ome β >, and that the data rate of the uer are controlled by (2) and (3). Let S S max l l denote the maximum number of clae

6 IEEE TRANSACTIONS ON INFORMATION TEORY, VOL. X, NO. X, XXXXXXX uing any link, and let L max L l denote the maximum number of link ued by any cla, If max l α l 2 β min T S L 6 w ρ M β, β, (4) then for any offered load ρ that reide trictly inide Θ 0, the ytem decribed by the Markov proce n( ), q( ) i table. Remark: Since Θ 0 in () i the upper bound on the tability region achieved by any congetion controller, Propoition 3 implie that the congetion controller in (2) and (3) can achieve the larget poible tability region. Before we tate the proof for Propoition 3, we would like to highlight the difference between the reult in Propoition 2 and 3. Firt, no time-cale eparation aumption i required in Propoition 3. ence, we do not require the data rate of the uer to be choen at each time according to the olution to (), which by itelf require an iterative procedure. Second, a tep-ize rule that i independent of the intantaneou number of uer in the ytem i provided in (4) (note the difference between S and S). Given our dicuion at the beginning of thi ection, it i quite urpriing that we do not need to reduce the tep-ize even when the offered load i cloe to the boundary of the tability region. In fact, ince the et Θ 0 i bounded, the feaible value of ρ i alo upperbounded by max ρ Θ0 ρ. ence, the tep-ize can be choen independently of the offered load. The tep-ize rule (4) i alo dependent on M, which i the maximum data rate of uer belonging to cla. Thi dependence i not urpriing. Recall that Equation (2) i the olution to (4) when the utility function U ( ) i of the form (2) or (3). We thu have, U (x ) β w x β ence, the minimum curvature of U ( ) i γ βw M β Let ñ ρ /M, which can be interpreted a the average number of uer of cla in a (fictitiou) M/M/ / ytem, where each uer of cla i erved at it maximum data rate M. The tep-ize condition (4) then become max l α l 2 β T S L 6β min γ, ñ which i comparable to (7). owever, note that ñ i quite different from En (t), the average number of uer of cla in the real ytem. Again, in our model without time-cale eparation, ince ñ i alway bounded, the tep-ize can be choen independently of the offered load. Sketch of the Proof of Propoition 3: The precie proof i quite technical. ence, here we will ue a heuritic fluid-model argument to illutrate the main idea of the proof and relegate the full proof to the Appendix. Thi fluid-model argument will be ued again in Section IV and V to treat the extenion of Propoition 3. We will look at the following fluid model that.. approximate the dynamic of the original ytem in (8), (2) and (3): d n (t) λ µ n (t)x (t), (5) w x (t) L q l l (t) /β, (6) d q l(t) (7) ( S ) α l n l (t)x (t) R l, when S n l (t)x (t) R l or q l > 0, 0, otherwie. Roughly peaking, thi fluid model become a good approximation of the original ytem when the quantitie n(t) and q(t) are large, in which cae the effect of the randomne in (8) and the jump in the dicrete-time update (3) are dominated by the overall trend that i decribed by the continuou-time differential equation (5)-(7). We conider the following Lyapunov function V( n, q) V n ( n) V q ( q), (8) where V n ( n) V q ( q) 2 β L S K w n β µ ρ β K l2 (q l ) 2 α l, and the poitive parameter K and K l2 will be determined later on. (Recall that the implicit cot q l i equal to α l multiplied by the queue length Q l. ence, in order to how tability we can focu on q l intead of Q l.) The rationale for chooing uch a Lyapunov function i a follow: the firt term V n ( n) i the Lyapunov function ued in 0 to prove connection-level tability with the time-cale eparation aumption. The econd term V q ( q) i a natural Lyapunov function which can be ued to prove the tability of (7) if the arrival rate at the link are fixed. Thu, we hope that ome linear combination of thee two term would erve a the Lyapunov function to etablih the connection-level tability of the fluid model ytem (5)-(7). Note that, according to (5), we have, ence, d nβ (t) (β )n β (t) d n (t) (β )n β (t)(λ µ n (t)x (t)). dv n ( n(t)) S K w n β (t) µ ρ β (λ µ n (t)x (t)) S K w n β (t) (ρ n (t)x (t)). (9) ρ β

7 IEEE TRANSACTIONS ON INFORMATION TEORY, VOL. X, NO. X, XXXXXXX From the aumption that ρ reide trictly inide Θ 0, there exit an ɛ > 0 uch that S ( 2ɛ) ρ l R l, for all l. (20) Adding and ubtracting ɛ S K w n β (t), we have (dropping the variable t for eae of expoition), dv n S S K w n β K w n β S K w n β ( ɛ)ρ ρ β n x (2) S K ( ɛ) β w ( ɛ)ρ n x (A) (22) S K w n β ρ ( S L ) K ( ɛ) β q l l ( ɛ)ρ x β n x (A) (23) where in (22) we have denoted S ( ɛ) β (A) K w x β nβ ρ β ( ɛ)ρ n x, (24) and in (23) we have ued Equation (6). Further, according to (7), we have q l (t) d S ql (t) α l q l (t) n l (t)x (t) R l under both condition in Equation (7). ence, dv q ( q(t)) L S K l2 q l (t) l n (t)x (t) R l. (25) Combining (23) and (25), we can compute the derivative of V( n(t), q(t)) with repect to t a (again we drop the variable t for eae of expoition): dv S K w n β ρ ( S L ) K ( ɛ) β q l l ( ɛ)ρ n x L S K l2 q l n l x R l (A) S K w n β L S q l K ( ɛ) β ρ l S K ( ɛ) β n l x L S K l2 q l n l x R l (A), (26) where in the lat tep we have interchanged the order of the ummation over l and over. Note that by contruction (24), (A) S ( ɛ) β ρ β n β x β K w x β ρ β ( ɛ)ρ n x 0. (27) Further, if we chooe K /( ɛ) β and K l2, then L S q l K ( ɛ) β n l x L S K l2 q l n l x and they cancel each other in (26). We thu have, dv (28) S K w n β ρ L S q ( l ɛ) ρ l R l (A) (29) ɛ S K w n β ɛ L q l S l ρ (A), (30) where in (30) we have ued Inequality (20). Thi provide the negative drift 22 neceary to etablih the tability of the ytem governed by (5)-(7). For the original ytem governed by (8), (2) and (3), the above fluid-model argument may erve a a plauible explanation for it connection-level tability. Note that if we could etablih that the fluid model (5)-(7) i indeed a fluid limit of the original ytem, we could have ued the reult of 23 to how the tability of the original ytem. owever, there are ome technical difficultie in puruing thi approach, namely, in etablihing that the fluid model ytem (5)-(7) i indeed the fluid limit of the original ytem under ome limiting regime. Typically, we need ome form of uniform convergence over compact interval to contruct uch a fluid limit 23, which uually require that the rate of the tatetranition in (8) and the rate of change in the update (3) are bounded. Unfortunately, the tate-tranition rate in (8) and the rate of change in (3) can both be arbitrarily large when n (t)x (t) i large, or, in other word, when n(t) i large and q(t) i mall. Thi unboundedne become an obtacle for contructing the appropriate fluid limit. (Note that thi problem doe not arie when we ue the time-cale eparation aumption 0, ince in that cae there i no q(t), and

8 IEEE TRANSACTIONS ON INFORMATION TEORY, VOL. X, NO. X, XXXXXXX n (t)x (t) i upper-bounded by the capacity of the larget link in the network.) There are two approache to reolving thi difficulty. The firt approach would be to impoe an additional contraint on n (t)x (t), i.e., one can aume that the aggregate data rate of uer of each cla i bounded by a number ˆM. Thu, the data rate of each cla- uer i governed by the following equation, which replaced (2): /β w ˆM x (t) min,. (3) L n q l l (t) In 2, we have ued thi approach to etablih the connectionlevel tability for all β > 0. Further, the tepize rule in (4) i alo not needed if the above additional contraint i impoed. The weakne of thi approach i that equation (3) require ome additional congetion control mechanim to hare the bandwih between n flow when the upper bound ˆM i met. An alternative approach i to earch for a Lyapunov function for the original ytem directly. In fact, the fluid-model argument ha already dicloed to u the Lyapunov function (8) and the main idea to obtain the negative drift. The key tep i in (28) where the term containing n (t)x (t) cancel each other. In the Appendix, we will etablih the negative drift of the Lyapunov function (8) for the original ytem directly. Reader will find that the line of argument there cloely reemble that of the fluid-model argument in thi ection. The main difference in the full proof in the Appendix i that we will carefully bound the additional term due to the randomne and the jump in the dicrete-time update. The weakne of thi econd approach, however, i that it cannot be ued to etablih the connection-level tability for β <. We briefly outline the main difference in the full technical proof. Lemma 5 in Appendix A can be compared to Equation (23). Note that when we compute the derivative of V n ( n(t)) in Equation (23), we have ignored all econd-order variation. owever, to compute the difference of V n ( n((k )T )) V n ( n( )) in Lemma 5, the econdorder term mut be carefully accounted for. Further, the effect of M (the upper bound on the data injection rate) wa ignored in our heuritic fluid-model argument, and mut alo be accounted for. In Lemma 5, we will how that the term (A) in (27) i non-poitive and i on the order of (t)x (t), and the effect of all econd-order change can be bounded by O(n β (t)x (t)). ence, the effect of the econd-order change can be bounded by (A) /2 plu a contant. Further, we will how that the effect of M can alo be bounded by a contant. ence, the net effect of thee additional term plu the term (A) i bounded by (A)/2, which i non-poitive and i on the order of n β (t)x (t)/2. When β, ince x (t) M, the net effect (A)/2 can be further upper bounded by n 2 (t)x 2 (t)/(2m ) time a contant 4. Then, in Appendix C, we how Lemma 6, which n β 4 It i in thi tep that we need the condition β. can be compared to Equation (25). We again how that thoe econd-order variation ignored in our heuritic fluid-model argument i on the order of O(n 2 (t)x 2 (t)). Note that thi econd-order term can thu be dominated by (A)/2 with appropriate choice of the tepize. Finally, in Appendix D, the tep are comparable to Equation (30), when the term containing n (t)x (t) cancel each other (reader can compare Equation (28) with Equation (74)), and all econd-order term can be bounded by a contant provided that the tepize condition (4) hold. We thu etablih the negative drift for proving the tability of the original ytem. IV. EXTENSIONS TO SYSTEMS WIT MULTIPAT ROUTING AND TIME-VARYING CAPACITY In thi ection, we extend our main reult to ytem with multipath routing and time-varying capacity. For eae of expoition, we will mainly ue the type of heuritic fluid-model argument a in Section III to illutrate thee reult. owever, rigorou proof of thee reult can be readily obtained by following the technique in the Appendix. A. Multipath Routing We now aume that each uer of cla can ue θ() alternate path through the network. Let j l if path j of cla ue link l, j l 0 otherwie. Let x j denote the data rate on path j of each cla- uer. Let x x,..., x,θ(). The exitence of multiple alternate path affect our network model in a number of way. Firtly, for connection-level dynamic, the tranition rate of the number of active uer are now given by: n (t) n (t), with rate λ, n (t) n (t), with rate µ n (t) θ() x j(t) if n (t) > 0. (32) Secondly, to determine an upper bound on the tability region of the ytem, note that for any offered load ρ that the ytem can tabilize, there mut exit ρ j, j,..., θ(),,..., S, uch that θ() ρ j ρ, for all and θ() S jρ l j R l, for all l. (33) ence, an upper bound on the tability region i the et Θ 0 given by: Θ 0 { ρ (33) can be atified}. (34) Finally, the congetion controller i alo affected by multipath. We conider the following multipath congetion control algorithm: For ome poitive contant c,,..., S, and non-negative contant y j, j,..., θ(),,..., S,

9 IEEE TRANSACTIONS ON INFORMATION TEORY, VOL. X, NO. X, XXXXXXX At each time lot k, the data rate of uer of cla i given by x (t) x ( ) (35) θ() argmax x P θ() U ( x j ) 0, xjm c 2 θ() (x j y j ) 2 θ() x j L jq l l ( ), for t < (k )T, where M i again the maximum data rate of each uer of cla. At the end of time lot k, the implicit cot are updated by: q l ((k )T ) q l ( ) θ() S α l l j (k)t n (t)x j ( ) T R l). (36) Remark: When the number of uer n in the ytem i fixed, the update in (35) and (36) olve the following optimization problem 24: max x 0 ubject to S n θ() U ( x j ) c 2 θ() (x j y j ) 2 θ() S jn l x j R l for all l θ() x j M for all. (37) We can imagine that when c i mall, the above problem approximate the multipath utility maximization problem given below 24: max x 0 ubject to S θ() n U ( x j ) θ() S jn l x j R l for all l θ() x j M for all, which i comparable to the ingle-path utility-maximization problem (). The contant c and y j in the approximate problem (37) are ued to alleviate the ocillation problem in ytem with multipath routing 24. Thi ocillation problem will occur when c 0, becaue then the maximization in (35) will only ue the path with the mallet cot to tranmit data. (The cot q j of a path j of cla i the um of the implicit cot of all link on the path, i.e., q j L j l ql.) ence, when the cot of two path are cloe to each other, a mall perturbation on the implicit cot will trigger the entire load offered by cla to be witched to another path. When c i i poitive, the maximizer of (35) become continuou with repect to the implicit cot, and uch ocillation will not occur any more. We now provide a heuritic fluid-model argument imilar to Section III that the ytem governed by (32), (35) and (36) will achieve the larget poible tability region Θ 0 given in (34). Again, we aume that the utility function i of the form (2) or (3). We can write down the following fluid model that approximate the dynamic of the original ytem: θ() d n (t) λ µ n (t) x j (t), x (t) argmax x 0 d q l(t) ( S α l θ() U ( θ() x j ) c 2 θ() x j S θ() when or q l > 0, 0, otherwie. θ() (x j y j ) 2 L jq l l (t), ) j l n (t)x j (t) R l We will ue the following Lyapunov function where and V n ( n) β V( n, q) V n ( n) V q ( q), S K w n β µ ρ β V q ( q) L l j n (t)x j (t) R l (q l ) 2 2α l. θ() S c y j n µ, Then, uing the ame type of fluid-model argument a in Section III, we have, dv n ( n(t)) S K w n β θ() (t) ρ n (t) x j (t) ρ β θ() θ() S c y j ρ n (t) x j (t). Thi equation i comparable to (9). From the tability condition, if ρ lie trictly inide Θ 0, there exit an ɛ > 0 uch that ( 2ɛ) ρ Θ 0. Thi implie that there exit ρ j uch that ρ θ() ρ j for all and ( 2ɛ) θ() S jρ l j R l for all l.

10 IEEE TRANSACTIONS ON INFORMATION TEORY, VOL. X, NO. X, XXXXXXX Adding and ubtracting ɛ S K w n β a in Section III, we have (dropping the variable t for eae of expoition), dv n S K w n β ρ S K w n β θ() ( ɛ)ρ n ɛ S c ρ β θ() y j x j θ() ( ɛ)ρ n x j θ() S c y j ρ (38) S K w n β ρ S K ( ɛ) β θ() w ( ( θ() x ɛ)ρ n x j j) β θ() θ() S ( ɛ)ρ n ɛ c y j x j θ() S c y j ρ (A), (39) where S ( ɛ) (A) {K β w ( θ() x j) nβ β ρ β θ() ( ɛ)ρ n x j 0. (40) Uing ρ θ() ρ j, we have S K ( ɛ) β θ() w ( ( θ() x ɛ)ρ n x j j) β θ() S K ( ɛ) β w ( θ() h x h) ( ɛ)ρ β j n x j and S c θ() y j θ() ( ɛ)ρ n θ() S c y j ( ɛ)ρ j n x j x j θ() S c y j ( ɛ)ρ h n x h. h j ence, adding and ubtracting S n x j, we have, where dv n ɛ F 0 ( ɛ) S θ() c x j (ɛ)ρ j K w n β θ() S K ( ɛ) β w ρ ( θ() h x h) β c (x j y j ) ( ɛ)ρ j n x j θ() S c x j ( ɛ)ρ j n x j F 0 (4) S θ() c θ() S y j ρ h ɛ c y j ρ (A). h j Note that Equation (4) i comparable to Equation (22). Since x j M for all, j, we have θ() S c x j ( ɛ)ρ j n x j F, (42) where F ( ɛ) S θ() c M ρ j. Further, auming that the utility function U ( ) i of the form (2) or (3), ince x olve (35), there mut exit ome δ j 0, j,..., θ() and δ 0 0 uch that w ( θ() h x h) c (x j y j )δ j δ 0 q j, for all, j, β (43) where q j L j l ql. ere, δ j, j,..., θ() are the Lagrange multipler for the contraint x j 0, and δ 0 i the Lagrange multiplier for the contraint θ() x j M. Further, δ j x j 0 and δ 0 ( θ() x j M ) 0 due to the complementary lackne condition. ence, if we let K /( ɛ) β, we have θ() S K ( ɛ) β w ( θ() h x h) c (x j y j ) β ( ɛ)ρ j n x j θ() S (q j δ j δ 0 )( ɛ)ρ j n x j θ() S q j ( ɛ)ρ j n x j L q l θ() S ( ɛ)δ 0 ρ j (uing δ j x j 0) ( ɛ) θ() S jρ l j θ() S jn l x j θ() S ( ɛ)δ 0 ρ j. (44)

11 IEEE TRANSACTIONS ON INFORMATION TEORY, VOL. X, NO. X, XXXXXXX 2008 It i eay to how that δ 0 i bounded for all. Specifically, note that when δ 0 > 0, we mut have θ() x j M. Summing (43) over all x j > 0, and noting that δ j 0 when x j > 0, we have, for all, J w M β c (M j:x j>0 y j ) J δ 0 j:x j>0 q j, where J θ() i the cardinality of the et {j : x j > 0}. Since q j 0, δ 0 i therefore bounded by δ 0 w M β θ() c y j /J F, where F w θ() c M β y j. Finally, ubtituting (42) and (44) into (4), we have, dv n ɛ S K w n β L q l ρ ( ɛ) θ() S l jρ j θ() S jn l x j F 0 F F 2, where F 2 ( ɛ) S θ() F ρ j. Thi equation i comparable to (23). Now, uing the argument in Section III again (ee Equation (25)), we have, dv q L q l θ() S jn l x j R l. The term containing n x j in the expreion of dvn cancel each other. We thu have, dv ɛ S K w n β ρ L q l ( ɛ) F 0 F F 2 S K w n β L ɛ ρ ɛ F 0 F F 2. and dvq θ() S jρ l j R l q l S θ() jρ l j Thi thu provide the negative drift for tability. Similar to the dicuion at the end of Section III, the full proof that take into account the econd-order variation can be obtained along the line of proof in the Appendix. Specifically, all econd-order variation (due to randomne and dicrete-time update) can be bounded by (A) /2, where the quantity (A) i given by (40) and i non-poitive. B. Time-Varying Capacity Our main reult can alo be readily extended to the cae with time-varying capacity. Time-varying capacity arie naturally in wirele network. When there i channel fading, the propagation gain between the tranmitter and the receiver varie over time. ence, even when the tranmiion power i fixed, the capacity at each link i till time-varying 25. Further, even if there i no fading in a wirele network, it i uually preferable to activate different et of link alternately 7, 2, 26, 27. Otherwie, if all link are activated at the ame time, they may create o much mutual interference uch that none of the link can carry data at an acceptable rate. Through the above dicuion on wirele network, we oberve two factor that lead to time-varying capacity. The firt factor i the change in the environment (i.e., channel fading). The econd factor i that different configuration (i.e., activation pattern of link) can be choen at different time even if the outide environment i identical. We now decribe a general model that capture both of thee two factor. We ue κ to denote the tate of the environment. In the cae of wirele network, the tate κ ummarize the current et of propagation gain between tranmitter and receiver. We aume that time i again divided into lot of length T, and the tate of the environment i fixed within each time lot. Let κ( ) denote the tate of the environment at time lot k. We aume that, at each time lot, the tate κ( ) i choen independently and identically from a et K, following the ditribution π κ, κ K. Next, let R R,..., R l denote the vector of link capacitie on all link. We aume that, when the tate of the environment i κ, the vector R can be choen within a et R κ. In other word, thi et R κ model the poible configuration of the link capacitie at tate κ. For technical reaon, we aume that R κ i cloed and bounded for each κ K. Conider the model of Section II-A, where n uer of cla,..., S are haring the network capacity. Each uer of cla may till travere multiple link, where routing i defined by l (for implicity we aume ingle-path routing here). The goal of congetion control i again to maximize the total ytem utility, ubject to the contraint that the aggregate rate allocated to the uer are no greater than the link ervice capacity upported by chooing appropriate chedule R(t). With time-varying capacity, the congetion controller mut be deigned jointly with the chedule R(t) for each time lot In the literature, the following congetion controller ha often been ued to achieve thi goal. Auming that the rate-aignment and the cheduling deciion are alo updated at the beginning of every time-lot of length T. The ratecontrol component of the congetion controller i in fact nearly identical to the one in Section III. We till aociate an implicit cot q l for each link l. At each time lot k, the data rate of

12 IEEE TRANSACTIONS ON INFORMATION TEORY, VOL. X, NO. X, XXXXXXX uer of cla i given by /β w x (t) x ( ) min, M, L q l l ( ) (45) for t < (k )T, (ue β when the utility function are of the form (2)). At the end of each time lot, the implicit cot are updated by ( S q l ((k )T ) q l ( ) α l n l ( )x ( ) R l ( ) ). (46) (Note that the capacity of each link R l (t) i now time-varying.) The cheduling policy i choen a follow: at each time lot k, the chedule R( ) i choen to be the vector R R κ( ) that maximize L q l ( )R l, i.e., R(t) R( L ) argmax q l ( )R l, (47) R R κ( ) for t < (k )T. Thi cheduling policy i in fact identical to the cheduling policie in 7, 2, 27. It can be hown that, when the number of uer n of each cla i fixed, and the poitive tepize α l are ufficiently mall, the above joint congetion-control and cheduling algorithm will converge to a neighborhood of the rate-allocation that maximize the total ytem utility 28, 29. Next, conider the model with connection-level uerdynamic (a in Section II-B), where uer of cla arrive according to a Poion proce with rate λ, that each uer bring with it a file for tranfer whoe ize i exponentially ditributed with mean /µ. We ak the quetion again: without the time-cale eparation aumption, doe the above congetion controller achieve the larget poible tability region? We firt anwer the following quetion: what i the larget poible tability region for uch ytem with time-varying capacity. Conider the following ytem which i generated from the original ytem. For each node i that can act a the ource node of a certain cla, we divide node i into two node i A and i B. The new ytem ha the ame uer arrival proce a the original ytem. owever, in the new ytem uer of cla arrive at node i A firt. Further, whenever a uer of cla arrive to the new ytem, the entire file i injected into node i A, and the uer leave the ytem immediately afterward. Node i A i connected to node i B with a link with infinite capacity, and node i B will be connected to the ret of the network according to the original network topology. We now argue that if the original network with congetion control can be tabilized, then thi new ytem can alo be tabilized. Specifically, the congetion control mechanim imply regulate the rate between each pair of node i A and node i B. ence, the tability region of the original ytem with congetion control mut be no larger than the tability region of the new ytem where we do not impoe any retriction on whether or not to ue congetion control. Now, let u focu on the new ytem: the tatitic of the data arrival proce at each ource node i A i known, and the average data injection rate by cla- uer i equal to ρ. ence, we can now ue ome of the tability reult from 7, 2. Let { S Θ 0 ρ ρ l } π κ Co(R κ ). (48) κ K It ha been hown in 7, 2 that no control cheme can tabilize the new ytem if the average rate ρ of the data injection proce i outide the et Θ 0. Therefore, we can conclude that Θ 0 i alo an upper bound on the tability region of our original ytem. On the other hand, although 7, 2 provide control cheme to tabilize the new ytem for any offered load ρ that lie trictly inide Θ 0, a we dicued in Section II-D, thee reult correpond to the cae with no congetion control. ence, we cannot ue them to deduce the connection-level tability of the original ytem when congetion control i enforced. We next how that the congetion controller in (45)-(47) indeed achieve the larget poible tability region Θ 0, without the time-cale eparation aumption. We firt pecify ome more detail of the ytem dynamic. The congetion control and cheduling deciion are till aumed to be updated every time-lot of length T according to (45) and (47). Thu, the value of x (t) and R(t) are equal to x ( ) and R( ), repectively, when t < (k )T. owever, uer may enter or leave the ytem within a time-lot (according to the tranition rate in (8)). Therefore, we change the implicit-cot update at the end of the time-lot to q l ((k )T ) q l ( ) α l ( S l (k)t n (t)x ( ) T R l ( ) ). (49) Note that q l ( ) can again be viewed a a calar-multiple of the real queue length Q l ( ) at link l. We now provide a heuritic fluid-model argument imilar to Section III that the above joint congetion-control and cheduling policy achieve the larget poible tability region Θ 0. We ue the Lyapunov function V(, ) in (8). Following the heuritic fluid-model argument in Section III, we can how that, given n(t), q(t) and κ(t), dv( n(t), q(t)) S K w n β (t) ρ L S q l (t) ( ɛ) ρ l R l (t) (A). Note that thi equation i comparable to Equation (29), where the only change i in the time-varying capacity R l (t). The quantity (A) i again given by (24) and i non-poitive. Taking