On the Stability Region of Congestion Control

Transcription

1 On the Stability Region of Congetion Control Xiaojun Lin and Ne B. Shroff School of Electrical and Computer Engineering Purdue Univerity, Wet Lafayette, IN Abtract It i well known that congetion control can be viewed a a ditributed iterative algorithm olving a global optimization problem that maximize the total ytem utility. In thi paper, we tudy the tability region of a network employing congetion control algorithm derived from uch an optimization framework. 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 rate allocation computed by the global optimization problem. Under thi aumption, it ha been hown that uch rate allocation policie can achieve the larget poible tability region. However, thi time-cale eparation aumption, although technically convenient, rarely hold in practice. In thi paper, we remove thi time-cale eparation aumption and how that the larget poible tability region can till be achieved by a large cla of congetion control algorithm derived from the optimization framework. Our reult provide new inight on the performance implication of congetion control and on the choice of the parameter of the congetion controller. 1

2 1 Introduction Congetion control (or rate control) i a key functionality in modern communication network. The objective of congetion control are two-fold: to utilize a much the available capacity of the network a poible without cauing evere congetion within the network, and to enure ome form of fairne among the uer. Since the eminal work by Kelly 1, it i clear that both of 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 olution to the global optimization problem 1, 2, 3, 4. Significant advance in the undertanding of congetion control have been made under thi optimization framework for congetion control (ee 5 for a good urvey). The reult can be roughly 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 the 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 (in the ene of Lyapunov) of the equilibrium rate allocation, the impact of feedback delay and 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 random dynamic arrival and departure of the uer. Thi reearch tudie the tability region of the ytem employing congetion control. Here, by tability, we mean that the number of 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 2

3 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. It ha been hown that, for a large cla of utility function and fairne objective, the larget poible tability region can be achieved by allocating data rate fairly according to thi timecale eparation aumption 6, 7, 8, 9. Thi reult i important a it tell u that 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. However, for a large network like today Internet, with the continual arrival and departure of the uer, the number of uer in the ytem change contantly. There will rarely be an extended period of time when the number of uer in the ytem i fixed. Hence, the iterative congetion control algorithm in the firt body of work may never have the chance to converge to an optimal (and fair) rate allocation. Therefore, the time-cale eparation aumption ued in the econd body of work, albeit technically convenient, rarely hold in practice. In thi paper, we tudy the tability region of congetion control without requiring uch a timecale 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. Hence, our reult reinforce the performance benefit of congetion control in a tronger ene than previou work. The ret of the paper i tructured a follow. In Section 2, we preent the ytem model and review ome relevant reult in the literature. Our main reult i preented in Section 3, and the proof i given in Section 4. Then we conclude. 2 The Sytem Model and Related Reult In thi ection, we decribe our ytem model and review certain related work. We conider a network with L link and S clae of uer. The capacity of each link l i R l. Uer of each 3

4 cla have one path through the network. Let H l = 1, if the path of uer of cla ue link l, and H l = 0, otherwie. 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 2. Let n, = 1,..., S denote the number of uer of cla that are in the ytem. Let n = n 1,..., n S and x = x 1,..., x S. Congetion control can then be formulated a the following global optimization problem 1: max x:0x M,,...,S ubject to n U (x ) (1) Hn l x R l for all l = 1,..., L. 2.1 Fairne It ha been well known that fairne objective can be achieved by appropriately chooing the utility function 7. For example, utility function of the form U (x ) = w log x (2) correpond to weighted proportional fairne, where w, = 1,..., S are the weight. A more general form of the utility function i x 1 β U (x ) = w, for ome β > 0 and β 1. (3) 1 β Maximizing the total ytem utility will correpond to maximizing weighted throughput a β 0, weighted proportional fairne a β 1, and max-min faine a β. 4

5 2.2 Convergence We firt aume that n, the number of uer in the ytem, i fixed and each uer ha an infinite backlog to tranfer. We aociate an implicit cot q l with each link l and let q = q 1,..., q L. The following iterative algorithm, commonly referred to a the dual olution in the congetion control literature, can olve problem (1) 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 The implicit cot at each link l i updated by: q l (t + 1) = L Hq l l (t). (4) q l (t) + α l ( H l n 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 2 with lightly different notation. Propoition 1 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 (1). Let S = max l number of uer uing any link, and let L = max ued by any uer. If Hn l denote the maximum L H l denote the maximum number of link max α l 2 l SL min then Algorithm A converge, i.e., x(t) x a t. 5 γ, (7)

6 2.3 Stability Region We now turn to the cae when the number of uer in the ytem change dynamically. In thi cae, we will tudy the tability region of ytem. Here, by tability, we mean that the number of 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 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 i identical. Let x (t) denote the rate of uer of cla at time t and let x(t) = x 1 (t),..., x S (t). In the rate aignment model that follow, the evolution of n(t) will be governed by a Markov proce. It 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. We ay that the above ytem i table 10 if lim up t 1 t P t 0 1{ S P dt 0, a M. n L (t)+ q l (t)>m} The tability region Θ of the ytem under a given congetion control algorithm i the et of offered load ρ uch that the ytem i table for any ρ Θ. Pat work on the tability region of congetion control typically adopt 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 (1) with n = n(t). 6

7 We refer to a congetion controller that allocate data rate according to the above time-cale eparation aumption a the perfect congetion controller. 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. Note that the capacity contraint determine an upper bound on the tability region achieved by any congetion controller, i.e. { } Θ Θ 0 ρ Hρ l R l for all l. (8) The next propoition from 7 how that the tability region achieved by the perfect congetion controller i indeed the larget poible found on the right hand ide of (8). 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 1 t P dt 0, a M. t 0 1{ S n (t)>m} 3 Stability Region of Congetion Control Without the Time-Scale Separation Aumption A dicued earlier in the Introduction, the time-cale eparation aumption rarely hold in reality. In typical network, uer arrive and depart contantly. Hence, the data rate of the uer employing a congetion control algorithm uch a algorithm A may never be able to converge. Further, note that the tep-ize condition (7) in Propoition 1 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. Hence, 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 thi cae either. 7

8 In thi ection, we will preent a new reult on the tability region of congetion control without thi time-cale eparation aumption. We firt decribe ome more detail of the dynamic of the ytem. We aume that time i divided into lot of length T, and that the implicit cot at the link are updated only 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. Unlike the cae in Propoition 2, we now let the rate allocation x(t) be determined by the current implicit cot. We aume that the utility function i of the form (2) or (3). Then, by olving (4), the data rate of uer of cla i given by w x (t) = x ( ) = min L Hq l l ( ) 1/β, M, for t < (k + 1)T (9) (ue β = 1 when the utility function are of the form (2)). At the end of each time lot, the implicit cot are updated by ( q l ((k + 1)T ) = q l ( ) + α l H l n (t)x ( )dt T R l ) +. (10) The following propoition how that, even when the time-cale eparation aumption i removed, the above congetion control algorithm can till achieve the larget poible tability region. The proof i given in Section 4. Propoition 3 Aume that the utility function are of the form in (2) or (3) for ome β > 1, and that the data rate of the uer are controlled by (9). Let S = maxl denote the maximum number of clae uing any link, and let L = max number of link ued by any cla, If L H l H l denote the maximum max α l 1 l T S L 2 β 1 16 min w ρ M β (11) (ue β = 1 if the utility function are of the form in (2)), then for any offered load ρ that reide trictly inide Θ 0, the ytem decribed by the Markov proce n( ), q( ) i table. 8

9 Several remark are in order: Firtly, no time-cale eparation aumption i required in Propoition 3. Hence, we do not require that the data rate of the uer converge. Secondly, a tep-ize rule that i independent of the intantaneou number of uer in the ytem i provided in (11) (note the difference between S and S). Given our dicuion at the beginning of thi ection, it i quite remarkable 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 tep-ize can be choen independently of the offered load. The tep-ize rule (11) i dependent on M, the maximum data rate of uer belonging to cla. Thi dependence i not urpriing. Since the utility function are of the form in (2) or (3), we have, Hence, the minimum curvature of U ( ) i U (x ) = β w x β+1. γ = βw M β+1. 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 tepize condition (11) then become max α l 1 l T S L 2 β 1 16β which i comparable to (7). However, note that ñ i quite different from En (t), the average number of uer of cla in the real ytem. Again, a ñ i alway bounded, the tep-ize can be choen independently of the offered load. min γ ñ, 4 Proof of Propoition 3 Define V( n, q) = V n ( n) + V q ( q), 9

10 where V n ( n) = 1 (1 + ɛ) β w n β+1, V (1 + β)µ ρ β q ( q) = L (q l ) 2 2α l, and ɛ i a poitive contant in (0, 1 to be choen later. We hall how that V(, ) i a Lyapunov function of the ytem. We begin with a few lemma. The firt lemma bound the change in V n ( ). Lemma 4 EV n ( n((k + 1)T ) V n ( n( )) n( ), q( ) ɛ E 0 () En β (t) n( ), q( )dt + L Hq l l ( ) (1 + ɛ)ρ T +E 1, 2 β 1 w 8(1 + ɛ) ρ M β t= En 2 (t)x 2 (t) n( ), q( )dt En (t)x (t) n( ), q( )dt (12) where E 0 () and E 1 are finite poitive contant. Proof: Over a mall time interval δt, we have E n β+1 (t + δt) n β+1 (t) n(t), q(t) = (n (t) + 1) β+1 n β+1 (t)λ δt + (n (t) 1) β+1 n β+1 (t)µ n (t)x (t)δt + o(δt). By the Mean-Value Theorem, (n + n) β+1 n β+1 = (β + 1)n β n + for ome ν (0, 1). Hence, letting n = ±1, we have E n β+1 (t + δt) n β+1 (t) n(t), q(t) β(β + 1) (n + ν n) β 1 ( n) 2 2 (β + 1)n β (t)λ δt µ n (t)x (t)δt +2 β 2 β(β + 1)n β 1 (t) λ δt + µ n (t)x (t)δt + N 1 ()δt + o(δt) 10

11 for ome poitive contant N 1 (). We then have, where = EV n ( n(t + δt)) V n ( n(t)) n(t), q(t) δt 1 { w n β (t) λ (1 + ɛ) β µ ρ β µ n (t)x (t) + β2β 2 w n β 1 (t) µ ρ β 1 { w n β (t) ρ (1 + ɛ) β n (t)x (t) ɛ = (1 + ɛ) β ɛ ρ β + β2β 2 w n β 1 ρ β (t) w n β (t) ρ β 1 + β2β 2 w n β 1 ρ β N 0 ()n β (t) + } λ + µ n (t)x (t) + N 1 () + o(1) } ρ + n (t)x (t) + N 1 () + o(1) + 1 (1 + ɛ) β { w n β (t) (1 + ɛ)ρ n (t)x (t) ρ β } (t) ρ + n (t)x (t) + N 1 () + o(1) (13) { L Hq l l (t) (1 + ɛ)ρ n (t)x (t) w L + x β (t) Hq l l (t) (1 + ɛ)ρ n (t)x (t) (14) n β (t) +w ((1 + ɛ)ρ ) 1 (1 + ɛ)ρ β x β n (t)x (t) (15) (t) + β2β 2 w n β 1 (t) ρ (1 + ɛ) β + n (t)x (t) (16) + N 1() (1 + ɛ) β ρ β } + o(1), N 0 () = 1 (1 + ɛ) β w ρ β 1 We hall bound the three term (14-16). By (9), { L w x β (t) = max Hq l l (t), w M β Hence, the term (14) can be bounded by w L x β (t) Hq l l (t) (1 + ɛ)ρ n (t)x (t) 11. }.

12 w L x β (t) Hq l l (t) (1 + ɛ)ρ + w L Hq (t) l l (1 + ɛ)ρ M β N 2 () (1 + ɛ)w ρ. (17) M β Let (A) and (B) denote the term (15) and (16), repectively. Note that n β (t) (A) = w ((1 + ɛ)ρ ) 1 (1 + ɛ)ρ β x β n (t)x (t) (t) (1 + ɛ)ρ n (t)x (t) ((1 + ɛ)ρ ) β n β (t)x β (t) = w ((1 + ɛ)ρ ) β x β (t) 0. (18) If n (t)x (t) 2(1 + ɛ)ρ, then (1 + ɛ)ρ n (t)x (t) ((1 + ɛ)ρ ) β n β (t)x β (t) n (t)x (t) 2 β 1 n β 2 2 β (t)x β (t). (19) Hence, (A) 2β 1 w n β+1 x (t) 2 β+1 ((1 + ɛ)ρ ), β and Since (B) β2β 1 w n β (t)x (t). (1 + ɛ) β β2 β 1 n β (t) 2β 1 2 for ome poitive contant N 3 (), we have, where nβ+1 β+2 ρ β (t) + N 3 () (B) (A) 2 + w N 3 ()x (t) ((1 + ɛ)ρ ) β (A) 2 + N 4(), N 4 () = w N 3 ()M ((1 + ɛ)ρ ) β. 12

13 On the other hand, if n (t)x (t) < 2(1 + ɛ)ρ 4ρ, then (B) 5β2β 2 w n β 1 (t) (1 + ɛ) β ρ β 1 = N 5 ()n β 1 (t), where In both cae, N 5 () = 5β2β 2 (1 + ɛ) β w ρ β 1. Subtituting (17) and (20) back to (14-16), we have, (B) (A) 2 + N 5()n β 1 (t) + N 4 (). (20) EV n ( n(t + δt)) V n ( n(t)) n(t), q(t) δt ɛn0 ()n β (t) N 5 ()n β 1 (t) + + L Hq l l (t) (1 + ɛ)ρ n (t)x (t) (A) 2 + N 1 () + N 2 () + N 4 () + o(1). (21) We hall ue (18) and (19) again to bound (A)/2. Since x (t) M, if n (t)x (t) 2(1 + ɛ)ρ, we have, (A) 2 2 β 1 n β+1 (t)x β+1 (t) w 2 β+2 ((1 + ɛ)ρ ) β M β w 2 β 1 2 β+2 2 β 1 (1 + ɛ)ρ n 2 (t)x 2 (t) w 2 β 1 8(1 + ɛ) n 2 (t)x 2 (t). ρ M β M β (22) On the other hand, if n (t)x (t) < 2(1 + ɛ)ρ, we till have (A)/2 0. Hence, in both cae, (A) 2 w 2 β 1 n 2 (t)x 2 (t) + N 8(1 + ɛ) ρ M β 6 (), (23) 13

14 where Further, note that for ome poitive contant N 7 (). N 6 () = w 2 β 1 8(1 + ɛ) N 5 ()n β 1 (2(1 + ɛ)ρ ) 2. ρ M β (t) ɛn 0() n β (t) + N 7 () (24) 2 Subtituting (23) and (24) into (21), and integrating over, (k + 1)T, the reult (12) then follow with E 0 () = N 0 ()/2 and E 1 = T N 1 () + N 2 () + N 4 () + N 6 () + N 7 (). Q.E.D. The next lemma bound the change in V q ( ). For implicity, we ue the following matrix notation. Let A denote the L L diagonal matrix whoe l-th diagonal element i α l. Let H denote the L S matrix whoe (l, )-element i H l. Let R = R 1,..., R l tr, where tr denote the tranpoe. Further let X (t) = n (t)x (t) and let X(t) = X 1 (t),..., X S (t) tr. Then V q ( q) = q tr A 1 q, 2 and the update on the implicit cot (10) can be written a ( Lemma 5 q((k + 1)T ) = q( ) + A H X(t)dt RT ) +. (25) EV q ( q((k + 1)T ) V q ( q( )) n( ), q( ) q tr ( ) H +T α max S L E X(t) n( ), q( )dt RT (k+1)t En 2 (t)x 2 (t) n( ), q( )dt + E 2, (26) where α max = max l α l, L and S are defined a in Propoition 3, and E2 i a finite poitive contant. 14

15 Proof: By (25), V q ( q((k + 1)T ) V q ( q( )) q tr ( ) H + 1 H 2 q tr ( ) H + H X(t)dt RT tr X(t)dt RT A H X(t)dt RT tr X(t)dt A H X(t)dt RT X(t)dt + T 2 R tr AR. For the econd term, we have, tr (k+1)t (k+1)t H X(t)dt A H X(t)dt L 2 (k+1)t = α l H l n (t)x (t)dt L α l H ( ) 2 (k+1)t l H l n (t)x (t)dt ( S L ) 2 (k+1)t α l H l n (t)x (t)dt T S L n 2 (t)x 2 (t)dt = T S α l H l (k+1)t L n 2 (t)x 2 (t)dt α l H l T α max S L (k+1)t n 2 (t)x 2 (t)dt. (27) Letting E 2 = T 2 R tr AR, the reult (26) then follow. Q.E.D. Proof of Propoition 3 : Adding (12) to (26), and noting that { L } (k+1)t Hq l l ( ) En (t)x (t) n( ), q( )dt 15

16 L = q l ( ) = q tr ( ) H H l En (t)x (t) n( ), q( )dt E X(t) n( ), q( )dt, we have, EV( n((k + 1)T ), q((k + 1)T )) V( n( ), q( )) n( ), q( ) ɛ E 0 () En β (t) n( ), q( )dt + L Hq l l ( ) (1 + ɛ)ρ T q tr ( ) RT 2 β 1 w 8(1 + ɛ) ρ M β +E 3, T α max S L En 2 (t)x 2 (t) n( ), q( )dt (28) where E 3 = E 1 + E 2. If (11) i atified, then the product term in (28) i negative. Hence, by ome rearrangement of the order of the ummation, we have, EV( n((k + 1)T ), q((k + 1)T )) V( n( ), q( )) n( ), q( ) ɛ E 0 () En β (t) n( ), q( )dt + T q tr ( ) (1 + ɛ)h ρ R + E 3. By aumption, ρ lie trictly inide Θ 0. Hence, there exit ome ɛ (0, 1 uch that (1+2ɛ)H ρ R. Ue thi value of ɛ in the definition of V(, ). we then have, EV( n((k + 1)T ), q((k + 1)T )) V( n( ), q( )) n( ), q( ) ɛ E 0 () En β (t) n( ), q( )dt ɛt q tr ( )H ρ + E 3 ɛ n β ( ) + L q l ( ) + E 3 for ome ɛ > 0. By Theorem 2 of 10, the reult then follow. Q.E.D. 16

17 Remark: Thi proof will not work for β < 1, in which cae the relationhip (22) will fail to hold. (We need (22) to cancel the econd term in (26) of the change in V q ( ).) We have not been able to either prove or diprove our reult for β < 1. We could have reorted to the fluid limit technique of 11. However, the difficulty in applying the technique of 11 i that the fluid limit of our ytem i not well defined whenever L Hq l l (t) = 0 for ome cla, which alo correpond to the cae when the econd term in (26) i large. We will leave the cae β < 1 for future work. 5 Concluion In thi paper, we have tudied the tability region of a network employing congetion control algorithm derived from an optimization framework. We have removed the time-cale eparation aumption typical in other related work, and etablihed that the larget poible tability region can be achieved by a large cla of congetion control algorithm (i.e., the o-called dual olution ) derived from the optimization framework. Our reult provide new inight on the performance implication of congetion control, and on the choice of the parameter of the congetion controller. Several direction for future work are poible. Firtly, it would be intereting to ee whether our main reult hold for the o-called primal olution in the literature 1. Secondly, we have aumed a Markovian model in thi paper. We plan to extend our reult to more general uer arrival and departure procee. Our reult can alo be extended to other form of utility function 9. Thirdly, we plan to tudy the impact of feedback delay. We expect that our main reult (Propoition 3) would hold even in the preence of feedback delay, provided that the tepize are appropriately choen. Finally, the extenion to the cae with multipath routing would alo be intereting. 17

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