Simplification of Network Dynamics in Large Systems

Transcription

1 Simplifiation of Network Dynamis in Large Systems Xiaojun Lin and Ness B. Shroff Shool of Eletrial and Computer Engineering Purdue University, West Lafayette, IN 47906, U.S.A. {linx, Abstrat In this paper we show that signifiant simpliity an be exploited for priing-based ontrol of large networks. We first onsider a general loss network with Poisson arrivals and arbitrary holding time distributions. In dynami priing shemes, the network provider an harge different pries to the user aording to the urrent utilization level of the network and also other fators. We show that, when the system beomes large, the performane (in terms of expeted revenue) of an appropriately hosen stati priing sheme, whose prie is independent of the urrent network utilization, will approah that of the optimal dynami priing sheme. Further, we show that under ertain onditions, this stati prie is independent of the route that the flows take. We then extend the result to the ase of dynami routing, and show that the performane of an appropriately hosen stati priing sheme with bifuration probability determined by average parameters an also approah that of the optimal dynami routing sheme when the system is large. These results deepen our understanding of priing-based network ontrol. In partiular, they provide us with the insight that an appropriate priing strategy based on the average network onditions (hene, slowly hanging) ould approah optimality when the system is large. We also briefly disuss how these results an be exploited to develop pratial algorithms for network ontrol. Introdution In this work, we use priing as the mehanism of ontrolling a network to ahieve ertain performane objetives. The performane objetives an be modeled by some revenue- or utility-funtions. Suh a framework has reeived signifiant interest in the literature (e.g., see [3, 4, 5, 6, 7] and the referenes therein) wherein prie provides a good ontrol signal beause it arries monetary inentives. The network an use the urrent prie of a resoure as a feedbak signal to oere the users into modifying their ations (e.g., hanging the rate or route). This work has been partially supported by the National Siene Foundation through the NSF award ANI , and the Indiana 2st Century Researh and Tehnology Award Earlier versions of this paper have appeared in the Proeedings of 200 Tyrrhenian International Workshop on Digital Communiations [] and ACM/IEEE IWQoS 02 [2].

2 In [8], Pashalidis and Tsitsiklis have shown that the performane (in terms of expeted revenue or welfare) of an appropriately hosen stati priing sheme approahes the performane of the optimal dynami priing sheme when the number of users and the network apaity beome very large. Note that a dynami priing sheme, is one where the network provider an harge different pries to the user aording to the varying levels of ongestion in the network, while a stati priing sheme is one where the prie only depends on the average levels of ongestion in the network (and is hene invariant to the instantaneous levels of ongestion). The result is obtained under the assumption of Poisson flow arrivals, exponential flow holding times, and a single resoure (single node). This elegant result is an example of the type of simpliity that one an obtain when the system beomes large. In this paper, we find that simple stati network ontrol an also approah the optimal dynami network ontrol under more general assumptions and a variety of other network problems. For simpliity of exposition, we struture the paper as follows: We first extend the result of [8] from the single-link ase to a general loss network with arbitrary holding time distributions. Independently of our work (first reported in []), Pashalidis and Liu have also extended the work in [8] from a single-link ase to a Markovian loss network [9]. Our ontributions in this paper are three-fold. Firstly, we generalize the results of [8] and [9] to non-exponential holding time distributions. Note that while the assumption of Poisson arrivals for flows in the network is usually onsidered reasonable, the assumption of exponential holding time distribution is not. For example, muh of the traffi generated on the Internet is expeted to our from large file transfers whih do not onform to exponential modeling. By weakening the exponential servie time assumption we an extend our results to more realisti systems. We show that a stati priing sheme is still asymptotially optimal, and that the orret stati prie depends on the servie time distribution only through its mean. A nie observation that stems from this result is that under ertain onditions, the stati prie depends only on the prie elastiity of the user, and not on the speifi route or distane. This indiates, for example, that the flat priing sheme used in the domesti long distane telephone servie in the U.S. may be an eonomially good priing mehanism. Seondly, we investigate whether more sophistiated shemes an improve network performane (e.g., shemes that have prior knowledge of the duration of individual flows, shemes that predit future ongestion levels, et.). We find that the performane gains using suh shemes beome inreasingly marginal as the system size grows. Thirdly, we study two types of saling for modeling large networks. One is the original saling that was studied in [8] and [9], whih requires that the number of users between eah soure-destination pair is saled proportionally with the apaity of the network. In this paper, we also study a new type of saling suitable for the ase when the apaity of the network is large but the number of users between eah soure-destination pair is small. Hene, our new saling may be more appropriate for modeling ertain Internet senarios where the topology is omplex and routing is diverse. We show that appropriate stati shemes are asymptotially optimal under both saling models. 2

3 We then weaken the assumption on fixed routing and study a dynami routing model where flows an hoose among several alternative routes based on the urrent network ongestion level. In this more general model, when the system is large, we show that the invariane result still holds, i.e., there still exists a stati priing sheme whose performane an approah that of the optimal dynami sheme. Our work is also related to the work in [0, ], and the referenes therein. However, in their work, the prie is set a priori, and the fous is on how to admit and route eah flow. Our work (as well as [8, 9]) expliitly models the users prie-elastiity, and onsiders the optimality of the priing shemes. The impat of large systems has also been studied for flow ontrol problems in [2, 3], however, these works fous on the single-link ase with a fixed number of flows. In networks of today and in the future, the apaity will be very large, and the network will be able to support a large number of users. The work reported in this paper demonstrates under general assumptions and different network problem settings that, when a network is large, signifiant simpliity an be exploited for priing based network ontrol. Our result also shows the importane of the average network ondition when the system is large, sine the parameters of the stati shemes are determined by average onditions rather than instantaneous onditions. These results will help us develop more effiient and realisti algorithms for ontrolling large networks. In pratie, over a long enough period of time, the network ondition is usually non-stationary, and even the average network statistis ould vary. Hene, in real networks, when we say a stati sheme, it does not neessarily mean that the prie is fixed over the entire time. Rather, the stati sheme should be interpreted as pries being stati over the time period for whih the network statistis do not hange (whih is typially still fairly long in real networks) and the average network ondition should be interpreted as the average of the dominant network ondition over suh a time period. Over longer time sales, the pries should still adapt to the hanges in the dominant network ondition, although relatively slowly. 2 Priing in a General Multi-lass Loss Network 2. Model The basi model that we onsider in this setion is that of a multi-lass loss network with Poisson arrivals and arbitrary servie time distributions. There are L links in the network. Eah link l {,..., L} has apaity R l. There are I lasses of users. We assume that flows generated by users from eah lass have a fixed route through the network. The routes are haraterized by a matrix {Ci l, i =,..., I, l =,..., L}, where Ci l = if the route of lass i traverses link l, and Cl i = 0 otherwise. Let n = {n, n 2,..., n I } denote the state of the system, where n i is the number of flows of lass i urrently in the network. We assume that eah flow of lass i requires a fixed amount of bandwidth r i. Flows of lass i arrive to the network aording to a Poisson proess with rate λ i (u i ). The rate 3

4 λ i (u i ) is a funtion of the prie u i harged to users of lass i. Here u i is defined as the prie per unit time of onnetion. Therefore λ i (u i ) an be viewed as the demand funtion and it represents the prie-elastiity of lass i. We assume that for eah lass i, there is a maximal prie u max,i suh that λ i (u i ) = 0 when u i u max,i. Therefore by setting a high enough prie u i the network an prevent users of lass i from entering the network. We also assume that λ i (u i ) is a ontinuous and stritly dereasing funtion for u i (0, u max,i ). One admitted, a flow of lass i will hold r i amount of resoure in the network and pay a ost of u i per unit time, until it ompletes servie, where u i is the prie set by the network at the time of the flow arrival. The servie times are i.i.d. with mean /. The servie time distribution is general. (By assuming that the demand funtion λ i (u i ), the mean holding time / and the apaity R l are fixed, we have assumed that the network ondition is stationary. We will omment at the end of this setion how the results of this paper should be interpreted when this assumption does not hold.) The bandwidth requirement determines the set of feasible states Ω = { n : n i r i Ci l Rl i= l}. A flow will be bloked if the system beomes infeasible after aommodating it. Other than this feasibility onstraint, the network provider an harge a different prie to eah flow, and by doing so, the network provider strives to maximize the revenue olleted from the users. The way in whih prie is determined an range from the simplest stati priing shemes to more ompliated dynami priing shemes. In a dynami priing sheme, the prie at time t an depend on many fators at the moment t, suh as the urrent ongestion level of the network, et. On the other hand, in a stati priing sheme, the prie is fixed over all time t, and does not depend on these fators. Intuitively, the more fators a priing sheme an be based on, the more information it an exploit, and hene the higher the performane (i.e., revenue) it an ahieve. The dynami priing sheme that we study in this setion is more sophistiated than the one in [8]. Firstly, we allow the network provider to exploit the knowledge of the immediate past history of states up to length d. Note that when the exponential holding time assumption is removed, the system is no longer Markovian. There will typially be orrelations between the past and the future given the urrent state. In order to ahieve a higher revenue, the network provider an potentially take advantage of this orrelation, i.e., it an use the past to predit the future, and use suh predition to determine the prie. Seondly, we allow the network provider to exploit prior knowledge of the parameters of the inoming flows. In partiular, the network provider knows the holding time of the inoming flows, and an harge a different prie aordingly. In order to ahieve a higher revenue, the network provider an thus use priing to ontrol the omposition of flows entering the network, for example, short flows may be favored under ertain network onditions, while long flows are favored under others. We assume that the prie-elastiity of flows is independent of their holding times. For onveniene of exposition, we restrit ourselves to the ase when the range of the servie time an 4

5 be partitioned into a series of disjoint segments, and the prie is the same for flows that are from the same lass and whose servie times fall into the same segment. Note that the range of the servie time is a subset of the set of non-negative real numbers R +. Let {a k }, k =, 2,... be an inreasing series of positive numbers, i.e., 0 < a < a 2 <... and let a 0 = 0. Therefore R + = k= [a k, a k ). The servie time of a flow is a non-negative real number and hene must fall into one of the segments. We assume that at any time t, for all flows of lass i whose servie times T i fall into segment [a k, a k ), we harge the same prie u ik (t), i.e., we do not are about the exat value of T i as long as T i [a k, a k ). The dynami priing sheme an thus be written as u i (t, T i ) = u ik (t) = g ik ( n(s), s [t d, t]), () for T i [a k, a k ), where n(s), s [t d, t] reflets the immediate past history of length d, T i is the holding time of the inoming flow of lass i, and g ik are funtions from Ω [ d,0] to the set of real numbers R. By inorporating the past history in the funtions g ik, we an study the effet of predition on the performane of the dynami priing sheme without speifying the details of how to predit. Let g = {g ik, i =,..., I, k =, 2,...}. The system under suh a dynami priing sheme an be shown to be stationary and ergodi under very general onditions. One suh ondition is stated in the following proposition. Readers an refer to the Appendix for the proof. Proposition Assume that for all lasses i, the arrival rates λ i (u) are bounded from above by some onstant λ 0. Further, for eah lass i, the servie times are i.i.d. with finite mean and independent of the servie times of other lasses and all arrivals. If the prie is only dependent on the urrent state of the system, or a finite amount of past history (i.e., predition based on past history), and/or the parameters of the inoming flows, then the stohasti proess n(t) (i.e. the system state) is asymptotially stationary and the stationary version is ergodi. We are now ready to define the performane objetive funtion. For eah lass i, let T ik = E {T i T i [a k, a k )} be the mean servie time for flows of lass i whose servie time T i falls into segment [a k, a k ). The expetation is taken with respet to the servie time distribution of lass i. Let p ik = P{T i [a k, a k )} be the probability that the servie time T i of an inoming flow of lass i falls into segment [a k, a k ). We an deompose the original arrivals of eah lass into a spetrum of substreams. Substream k of lass i has servie time in [a k, a k ). Its arrival is thus Poisson with rate λ i (u)p ik, sine we assume that the prie-elastiity of flows is independent of T i. For any dynami priing sheme g, the expeted revenue ahieved per unit time is given by [ ] ζ lim ζ ζ E λ i (u ik (t))u ik (t) T ik p ik dt 0 i= k= 5

6 = i= k= [ E λ i (u ik (t))u ik (t) T ik p ik ], where the expetation is taken with respet to the steady state distribution. The limit on the left hand side as the time ζ exists and equals to the right hand side due to stationarity and ergodiity. Note that the right hand side is independent of t (from stationarity). Therefore, the performane of the optimal dynami poliy is J max g i= k= [ E λ i (u ik (t))u ik (t) T ik p ik ]. Finally, we onstrut the performane objetive for the stati priing sheme. In a stati priing sheme, the prie for eah lass is fixed, i.e., it does not depend on the urrent state of the network, nor does it depend on the individual holding time of the flow. Let u i be the stati prie for lass i. Let u = [u,..., u I ]. Under this stati priing sheme u, the expeted revenue per unit time is: J 0 = i= λ i (u i )u i ( P loss,i [ u]), where P loss,i [ u] is the bloking probability for lass i. Therefore the performane of the optimal stati poliy is J s max u i= λ i (u i )u i ( P loss,i [ u]). When the exponential holding time assumption is removed, we an no longer use the MDP approah as in [8] to find the optimal dynami priing sheme. We will instead study the behavior of the optimal dynami priing sheme by bounding its performane. Using these bounds, we will show that, when the network is large, an appropriately hosen stati priing sheme an ahieve almost the same performane as that of the optimal dynami sheme. By definition, the performane of any stati priing sheme beomes a lower bound for the performane of the optimal dynami priing sheme. An upper bound is presented next. 2.2 An Upper Bound The upper bound is of a similar form to that in [8]. Let λ max,i = λ i (0) be the maximal value of λ i. For onveniene, we write u i as a funtion of λ i. Let F i (λ i ) = λ i u i (λ i ), λ i [0, λ max,i ]. Further, let J ub be the optimal value of the following nonlinear programming problem: max λ i,i=,...,i subjet to F i (λ i ) (2) i= i= λ i r i C l i R l for all l, 6

7 where /, r i are the mean holding time and the bandwidth requirement, respetively, for flows from lass i, C l i is the routing matrix and Rl is the apaity of link l. Proposition 2 If the funtion F i is onave in (0, λ max,i ) for all i, then J J ub. Proof: Consider an optimal dynami priing poliy. Let n ik (t) be the number of flows of substream k of lass i in the system at time t, hene n i (t) = n ik (t). Let λ ik (t) = λ i (u ik (t)). From Little s Law, we have k= E[n ik (t)] = E[λ ik (t)p ik ] T ik. The expetation is taken with respet to the steady state distribution. Now let Note that k= p ik Tik = /, therefore λ i = k= λ i = k= E[λ ik (t)]p ik Tik k= E[λ ik (t)]p ik Tik =. p iktik E[n ik (t)] = E[n i (t)]. k= At any time t, n i (t)r i Ci l Rl i= for all l. Therefore i= Sine the funtions F i are onave, we have λ i r i C l i R l for all l. J ub F i (λ i ) µ i= i ) F i (E[λ ik (t)] p iktik i= k= i= k= [ ] E F i (λ ik (t)) p iktik = J, by Jensen s inequality. Q.E.D. The upper bound has a very simple and intuitive form. Its objetive funtion an be viewed as an approximation of the average revenue without taking into aount bloking, while the onstraint is 7

8 simply to keep the load at all links to be no greater than (where the load at a link l is defined by λ i r i Ci l ). As we will see later, when the network is large, the bloking probability of all lasses R l i= will approah zero as long as the network load is not greater than. This implies that the upper bound losely estimates the performane of the optimal dynami priing sheme, when the network is large. The maximizer of the upper bound (2) indues a set of optimal pries u ub i = u i (λ i ). It is interesting to note that although the dynami priing sheme an use predition and exploit prior knowledge of the parameters of the inoming flows, the upper bound (2) and its indued optimal pries are indifferent to these additional mehanisms. Remark: The onavity assumption on F i is essential in the proof of this proposition and the result that follows. A linear λ i (u i ), as used in many appliations, guarantees that F i is onave. We will briefly address the ase of non-onave F i in the onlusion. 2.3 Asymptoti Optimality of the Stati Sheme We are interested in the performane of the optimal dynami sheme and that of the simple stati sheme in large systems, that is, when the apaity and the number of users in the network is very large. We will show that, as the network grows large, the relative differene between the revenue of the optimal dynami sheme, the revenue of the optimal stati sheme, and the upper bound, will approah zero. We will first onsider the following saling (S), whih is also used in [8]. A different saling will be studied in Setion 2.6. (S) Let be a saling fator. We onsider a series of systems saled by. All systems have the same topology. The saled system has apaity R l, = R l at eah link l, and the arrivals of eah lass i has rate λ i (u) = λ i(u). Let J,, Js and Jub be the optimal dynami revenue, optimal stati revenue, and the upper bound, respetively, for the -saled system. We are interested in the performane differene of the dynami priing shemes and the stati priing shemes when. We first note that, under the saling (S), the normalized upper bound Jub / is fixed over all, sine J ub is obtained by maximizing I λ i u i (λ i )/, subjet to the onstraints i= λ i r i Ci l/ R l, for all l. Therefore the optimal prie indued by the upper bound is also independent of. Let P loss,i denote the bloking probability of lass i in the -saled system. The following lemma illustrates the behavior of P loss,i as under the saling (S). Reall that the load at a link l is defined by λ i R l r i Ci l. i= Lemma 3 Let λ i be the arrival rate of flows from lass i and let / be the mean holding time. Under the assumptions of Poisson arrivals and general holding time distributions, if the load at eah resoure i= 8

9 is less than or equal to, i.e., i= λ i r i C l i R l for all l, then under saling (S), as, the bloking probability P loss,i of eah lass i goes to 0, and the speed of onvergene is at least /, that is, there exists some positive onstant D i suh that lim sup P loss,i D i. Further, when the load of all the links that lass i traverses is stritly less than, the speed of onvergene is exponential, i.e., where lim sup log P loss,i max l:c l i = inf w>0 Λ l(w) < 0, Λ l (w) = j= λ j (e r jw )C l j wr l. The proof of this lemma an be found in the Appendix. We will use this lemma to show the following main result: Proposition 4 If the funtion F i is onave in (0, λ max,i ) for all i, then under the saling (S), lim J s = lim J, = lim J ub = J ub. Instead of proving Proposition 4, we will prove the following stronger result that the stati pries indued by the upper bound are in fat asymptotially optimal. Proposition 5 Let u ub i be the stati prie indued by the upper bound, i.e., let λ i be the maximizer of the upper bound (2), then let u ub i = u i (λ i ). Let J s be the revenue for the -saled system under this stati prie. If the funtion F i is onave in (0, λ max,i ) for all i, then under the saling (S), lim J s = lim J ub = J ub. Proof: Sine J s J s J, J ub = J ub, in order to prove the above two propositions, we only need to show that lim J s / = J ub. For every stati prie u = [u,...u I ] falling into the onstraint of J ub, i.e., i= λ i (u i )r i C l i R l for all l, (3) let J0 denote the revenue under this stati prie. Sine (3) guarantees that the ondition of Lemma 3 is met, we have P loss,i [ u] 0, as. Therefore J0 lim = lim = i= i= λ i (u i )u i ( P loss,i [ u]) λ i (u i )u i. (4) 9

10 If we take the stati prie u ub i indued by the upper bound as our stati prie, then inequality (3) is satisfied, J0 = J s, and the right hand side of (4) is exatly the upper bound. Therefore, and Propositions 4 and 5 then follow. lim J s = lim J0 = J ub, Q.E.D. Proposition 2 and 4 are parallel to Theorem 6 and 7, respetively, in [8]. In [8], they are shown under the assumption of a single link and exponential holding time distribution. Proposition 2 and 4 tell us that extending the results of [8] from a single link to a network of links and from exponential holding time distributions to arbitrary holding time distributions does not hange the invariane result. In other words, there still exists stati priing shemes whose performane an approah that of the optimal dynami priing sheme when the system is large. Further, even though the dynami priing sheme an use predition and exploit prior knowledge of the parameters of the inoming flows, the upper bound (2) turns out to be indifferent to these additional mehanisms. Therefore, these extra mehanisms only have a minimal effet on the long term revenue when the system is large. To see how fast the gap between the performane of the optimal dynami sheme and that of the stati shemes dereases as, note that J, J s J, J ub J s J ub J ub J 0 ( uub ) J ub = i= λ i (u ub i i= )u ub i λ i (u ub i P loss,i [ uub ] max P loss,i )u ub i [ uub ]. i Therefore, the speed of onvergene of Propositions 4 and 5 an be determined by the P loss,i [ uub ] that has the slowest speed of onvergene to zero. If at the prie u ub i indued by the upper bound, the load of all links is stritly less than, then the onvergene of Propositions 4 and 5 is exponential: lim sup log J, Js J, lim sup log J ub J s J ub max l inf Λ l(w) < 0. w>0 If the load of some links is equal to, then the onvergene is /, i.e., there exists a positive onstant D suh that J, J lim sup s J J, lim sup ub J s Jub D. Our result is different from [8] and [9] in the following aspets: Firstly, we remove the assumption on the exponential servie time distributions. Seondly, we haraterize two different regions for the speed with whih the performane of the optimal stati sheme approahes that of the optimal dynami sheme. When the load on some links is equal to, the onvergene is /. When the load of all links is stritly less than, the onvergene is exponential. In [9], only the latter region is haraterized. Thirdly, and more importantly, we show in Proposition 5 that the stati prie indued by the upper bound is by itself asymptotially optimal. Hene, this result permits us to use the prie indued by the 0

11 L Class L4 Class 2 L3 Class 4 L2 Class 3 L5 Figure : The network topology upper bound diretly in the stati shemes. In [8] and [9], although the authors show the asymptoti optimality of the optimal stati sheme, it is not lear whether the prie indued by the upper bound an serve as a viable alternative, beause with this prie the load at some links ould be equal to, and the onvergene in this region is not haraterized in their work. 2.4 Distane Neutral Priing When pries are ongestion-dependent, two lasses that traverse different routes will typially be harged different pries, even if they have the same prie-elastiity funtion. A lass that traverses a longer distane, or one that traverses a more ongested route, will likely be harged a higher prie. However, if the following ondition is satisfied, this distane-dependene an be eliminated when we adopt an asymptotially optimal stati priing sheme: We say a network has no signifiant onstraint of resoures if the unonstrained maximizer of F i (λ i ) satisfies the onstraint in (2). If there is no i= signifiant onstraint of resoures, there exists an asymptotially optimal stati sheme whose pries depend only on the prie-elastiity of eah lass, and are independent of their routes. To see this, we go bak to the formulation of the upper bound (2). If the unonstrained maximizer of F i (λ i ) satisfies the onstraint, then it is also the maximizer of the onstrained problem. In this ase i= we an use the pries indued by the upper bound as the asymptotially optimal stati pries, and the stati pries depend only on the funtion F i, whih is determined by the prie elastiity of the users. This result suggests that the use of flat priing, as in inter-state long distane telephone servie in the United States, an also be eonomially near-optimal under appropriate onditions. Assuming that there is no signifiant onstraint of resoures in the domesti telephone network in the U.S., and all onsumers have the same prie-elastiity, then our result indiates that a flat (i.e., independent of both time and distane) priing sheme will suffie, given that the apaity of the network is very large. 2.5 Numerial Results Here we report a few numerial results. Consider the network in Fig.. There are 4 lasses of flows. Their routes are shown in the figure. Their arrivals are Poisson. The funtion λ i (u) for eah lass i is

12 Table : Traffi and prie parameters of 4 lasses Class Class 2 Class 3 Class 4 λ max,i u max,i Servie Rate Bandwidth r i 2 2 Table 2: Solution of the upper bound (2) when the apaity of Link 3 is 5 bandwidth units. The upper bound is J ub = 27.5 Class Class 2 Class 3 Class 4 u i λ i (u i ) λ i (u i )/ of the form λ i (u) = ( [λ max,i u )] +, u max,i i.e., λ i (0) = λ max,i and λ i (u max,i ) = 0 for some onstants λ max,i and u max,i. The prie elastiity is then The funtion F i is thus λ i (u) λ i (u) = /u max,i u/u max,i, for 0 < u < u max,i. F i (λ i ) = λ i ( λ i λ max,i )u max,i, whih is onave in (0, λ max,i ). The holding time is exponential with mean /. The parameters λ max,i, u max,i, servie rates, and bandwidth requirement r i for eah lass are shown in Table. First, we onsider a base system where the 5 links have apaity 0, 0, 5, 5, and 5 respetively. The solution of the upper bound (2) is shown in Table 2. The upper bound is J ub = We then use simulations to verify how tight this upper bound is and how lose the performane of the stati priing poliy an approah this upper bound when the system is large. We use the prie indued by the upper bound alulated above as our stati prie. We first simulate the ase when the holding time distributions are exponential. We simulate -saled versions of the base network where ranges from to 000. For eah saled system, we simulate the stati priing sheme, and report the revenue generated. In Fig. 2 we show the normalized revenue J 0 / as a funtion of. As we an see, when the system grows large, the differene in performane between the stati priing sheme and the upper bound dereases. Although we do not know what the optimal dynami sheme 2

13 Table 3: Solution of the upper bound when the apaity of Link 3 is 5 bandwidth units. The upper bound is J ub = 37.5 Class Class 2 Class 3 Class 4 u i λ i (u i ) λ i (u i )/ Jo/ : saling Jo/ :saling Figure 2: The stati priing poliy ompared with the upper bound: when the apaity of link 3 is 5 bandwidth units. The dotted line is the upper bound. Figure 3: The stati priing poliy ompared with the upper bound: when the apaity of link 3 is 5 bandwidth units. The dotted line is the upper bound. 3

14 Jo/ : saling Figure 4: The stati priing poliy ompared with the upper bound: when the apaity of link 3 is 5 bandwidth units and the servie time distribution is Pareto. is, its normalized revenue J / must lie somewhere between that of the stati sheme and the upper bound. Therefore the differene in performane between the stati priing sheme and the optimal dynami sheme is further redued. For example, when = 0, whih orresponds to the ase when the link apaity an aommodate around 00 flows, the performane gap between the stati poliy and the upper bound is less than 7%. The gap dereases as /. We now hange the apaity of link 3 from 5 bandwidth units to 5 bandwidth units. The solution of the upper bound is shown in Table 3. The upper bound is J ub = The simulation result (Fig. 3) onfirms again that the performane of the stati poliy approahes the upper bound when the system is large. At = 0, the performane gap between the stati poliy and the upper bound is around 0%. The latter example also demonstrates distane-neutral priing. For example, lasses and 2, and lasses 3 and 4 have different routes but have the same prie (and prie-elastiity). Readers an verify that, in this example, if we lift the onstraints in (2), and solve the upper bound again, we will get the same result. Therefore in the latter example, there is no signifiant onstraint of resoures, and the optimal prie will only depend on the prie elastiity of eah lass and not on the speifi route. Sine lass has the same prie elastiity as lass 2, its prie is also the same as that of lass 2, even though it traverses a longer route through the network. We also simulate the ase when the holding time distribution is deterministi. The result is the same as that of the exponential holding time distribution. The simulation result with heavy tail holding time distribution also shows the same trend exept that the sample path onvergene (i.e., onvergene in time) beomes very slow, espeially when the system is large. For example, Fig. 4 is obtained when the holding time distribution is Pareto, i.e., the umulative distribution funtion is /x a, with a =.5. We use the same set of parameters as the onstrained ase above, and let the Pareto distribution have the same mean as that of the exponential distribution. Note that this distribution has finite mean but infinite variane. This demonstrates that our result is indeed invariant of the holding time distribution. 4

15 2.6 Saling with Topologial Changes In Setion 2.3, we have studied a large network under the saling (S), where the network topology is fixed and we inrease both the demand funtion λ i (u i ) and the apaity R l proportionally. Hene, when the network is large, the number of users of eah lass i (i.e., between eah soure-destination pair) is also assumed to be large. This saling is suitable for a large apaity network with a simple topology, e.g., network bakbones. In this setion, we will onsider large networks where the number of users over eah soure-destination pair is muh smaller than the apaity of the network. Large networks of this type arise naturally when the network topology beomes inreasingly omplex, for instane, when we inlude the aess links into the topology. To illustrate a real world example, note that even though the links between Purdue University, Indiana and Columbia University, New York ould have large apaities, the number of users ommuniating between these two institutions at any time is usually quite low. The high-apaity aess link that onnets Purdue University to the Internet is to aommodate the large aggregate traffi between Purdue University and all destinations on the Internet, while the amount of traffi to any single destination is muh smaller. The question we attempt to answer in this setion is: will similar simpliity results as in Setion 2.3 hold in this type of large networks? We will first present a different saling model that allows the topology of the network to beome more omplex as we sale its apaity. We will then show that, under some reasonable assumptions, the performane of appropriately hosen stati shemes will still approah that of the optimal dynami shemes, as long as the the apaity of the network is large. We onsider a series of networks indexed by the saling fator. We use L(), I(), and C() to denote the number of links, the number of lasses, and the routing matrix, respetively, in the -th network. Note that in the saling (S) in Setion 2.3, these quantities are assumed to be fixed for all. In this setion, in order to aommodate topologial hanges, we allow them to vary with. We still use λ i (u) and Rl, to denote the demand funtion for lass i and the apaity of link l, respetively, in the -th system. However, unlike the ase for the saling (S), λ i (u) and Rl, do not need to follow the linear saling rule. The sale of the -th network is defined as S() = min l=,...,l() R l, max r. i:ci l= i We will onsider the saling (S2) that satisfies the following set of assumptions: (A) S(), as. (B) For all networks, the maximum number of links on any route is bounded from above by a number M. The first assumption simply states that, with large, the apaity of eah link in the network will be large ompared to the bandwidth requirement of eah flow that goes through the link. The seond 5

16 assumption limits the maximum number of hops eah flow an traverse. This is a reasonable assumption: for example, in TCP/IP, the TTL (time-to-live) field in the IP header oupies only 8 bits. effetively put an upper bound of 255 on the number of hops a flow an traverse within a network. Real Internet topology exhibits the small world phenomenon [4], where the average number of hops of a soure-destination pair is typially small. Any series of networks under saling (S) trivially satisfy saling (S2). Finally, let λ max,i be the maximal value of λ i (u). Let u i (λ i) be the inverse funtion of λ i (u i). We assume that: (C) The funtion Fi (λ i) = λ i u i (λ i) is onave in (0, λ max,i ) for all and i. Let J,, Js, and Jub be the optimal dynami revenue, the optimal stati revenue, and the upper bound, respetively, for the -th network. By Proposition 2, under Assumption C, the optimal dynami revenue J, is no greater than Jub for all. In general, J ub does not follow the simple linear rule as in saling (S). Hene, Proposition 4 will not hold under saling (S2). Instead, we an show the following: Proposition 6 Under Saling (S2) and Assumption C, as, the relative differene among the optimal dynami revenue J,, the optimal stati revenue Js, and the upper bound Jub will onverges to zero. That is, Jub lim J s k Jub Jub = lim J, k Jub The proof is in the Appendix. Proposition 6 tells us that no matter how omplex the topology of the network is, as long as the apaity of the network is large, the performane of the optimal stati sheme will be lose to that of the optimal dynami sheme. We defer relevant numerial results until Setion 3. The definition of S() allows us to apply the result to ertain networks with heterogeneous apaity. The network ould have both large-apaity links and small-apaity links, and both large-bandwidth flows and small-bandwidth flows. As long as the apaity of eah link is large ompared to the bandwidth requirement of the flows that traverse the link, stati shemes will suffie. On the other hand, there may exist network senarios where S() is not large, whih means that the apaity of some links an only aommodate a small number of flows that go through them. Proposition 6 will not hold in suh senarios. It is easy to hek that Proposition 4 and Proposition 6 are equivalent under saling (S). There is yet a differene between the results we an obtain under the two different types of saling. Under saling (S), we an show that the prie indued by the upper bound suffies to be the near-optimal stati prie (see Proposition 5). Suh onlusion annot be drawn under saling (S2). The diffiulty is that, under saling (S2), we annot show that the bloking probability will go to zero as when the onstraints in the upper bound (2) are satisfied with equality. In spite of this tehnial diffiulty, we expet that in most ases the prie indued by the upper bound would still suffie under saling (S2). We an argue as follows using the familiar independent bloking assumption underlying the Erlang Fixed 6 = 0. This

17 Point approximation [5], i.e., we assume that bloking ours independently at eah link. Under the independent bloking assumption, the bloking probability B l at eah link l an be alulated as if the traffi offered to the link l omprises independent Poisson streams with arrival rates that are thinned by other links in the network. At the prie indued by the upper bound, the offered load at any link l after thinning will be no greater than the offered load before thinning, i.e., λ i i r i, whih is no greater than the apaity R l of the link l. Applying the tehnique in the proof of Lemma 3 to a single link, we an show that, if the independent bloking assumption holds, the bloking probability at eah link will go to zero as S(), and the onvergene is uniform over all links. Sine the number of hops eah route an traverse is upper bounded by M, we an then infer that the bloking probability of all lasses will go to zero uniformly as S(). Then, using an argument similar to that of Proposition 4, we an infer that the prie indued by the upper bound will suffie. The validity of the above argument relies on the independent bloking assumptions. A rigorous haraterization of this onvergene is an interesting problem for future work. We have shown under two different types of saling that the performane of an appropriately hosen stati priing sheme will approah that of the optimal dynami priing sheme when the apaity of the network is large. We onlude this setion with some disussion on the assumption we have made. In our model, we assume that the demand funtion λ i (u i ), the mean servie time / and the apaity R l are fixed. Hene, we have assumed that the network ondition is stationary. However, in pratie the network ondition is usually non-stationary and even the average network statistis an hange over time. If we assume that the hanges of these average network onditions are at a muh slower time sale than that of the flow arrivals and departures, we an still use the result in this setion with the following interpretation: the stati sheme should be interpreted as pries being fixed over a time period for whih the network statistis do not hange (whih is typially a fairly long time in real networks) and the average network ondition should be interpreted as the average of the dominant network ondition over suh a time period. Similarly, the invariane result regarding predition should also be interpreted under this ontext: if the dynami priing sheme uses a predition window (i.e., d in Equation ()) that is smaller than the time sale of the hanges of the average network ondition, then the dynami sheme will not signifiantly outperform the stati priing sheme. Over the longer time sale, the pries should still adapt to the hanges in the dominant network ondition and predition on the hanges of the average network statistis will help. 3 Dynami Routing We next onsider a system with dynami routing. Many results in the QoS routing literature fous on finding the best route for eah individual flow based on the instantaneous network onditions. When these QoS routing algorithms are used in a dynami routing setting, the network is typially required to first ollet link information (suh as available bandwidth, delay, et.) on a regular basis. Then, 7

18 when a request for a new flow arrives, the QoS routing algorithms are invoked to find a route that an aommodate the flow. When there are multiple routes that an satisfy the request, ertain heuristis are used to pik one of the routes. However, suh greedy shemes may be sub-optimal system wide, beause a greedy seletion may result in an unfavorable onfiguration suh that more future flows are bloked. Further, an obstale to the implementation of these dynami shemes is that it onsumes a signifiant amount of resoures to propagate link states throughout the network. Propagation delay and stale information will also degrade the performane of the dynami routing shemes. In this setion, we will formulate a dynami routing problem that diretly optimizes the total system revenue. Although our model is simplified, it reveals important insight on the performane tradeoff among different dynami routing shemes. We will establish an upper bound on the performane of the dynami shemes, and show that the performane of an appropriate hosen stati priing sheme, whih selets routes based on some pre-determined probabilities, an approah the performane of the optimal dynami sheme when the system is large. The stati sheme only requires some average parameters. It onsumes less ommuniation and omputation resoures, and is insensitive to network delay. Thus the stati sheme is an attrative alternative for ontrol of routing in large networks. The network model is the same as in the last setion, exept that now a user of lass i has θ(i) alternative routes that are represented by the matrix {Hij l } suh that Hl ij =, if route j of lass i uses resoure l and Hij l = 0, otherwise. The dynami shemes we onsider have the following idealized properties: the routes of existing flows an be hanged during their onnetion; and the traffi of a given flow an be transmitted on multiple routes at the same time. Thus our model aptures the paket-level dynami routing apability in the urrent Internet. These idealized apabilities allow the dynami shemes to pak more flows into the system. Yet, we will show that an appropriately hosen stati routing sheme will have omparable performane to the optimal dynami sheme. Let n i be the number of flows of lass i urrently in the network. Consider the k-th flow of lass i, k =,..., n i. Let Pij k denote the proportion of traffi of flow k assigned to route j, j =,...θ(i). Then, state n = {n,..., n I } is feasible if and only if There exists Pij k θ(i) suh that and I θ(i) i= j= r i H l ij n i k= j= Pij k =, i, k, P k ij Rl for all l. The set of feasible states is Ω = { n suh that (5) is satisfied}. A dynami sheme an harge pries based on the urrent state of the network, or a finite amount of past history, i.e., predition based on past history. (For simpliity we onsider priing shemes that are insensitive to the individual holding times.) (5) An inoming flow will be admitted if the resulting state is in Ω. One the flow is admitted, its route (i.e., Pij k ) is assigned based on (5), involving (in an idealized dynami sheme) possible rearrangement of routes of all existing flows. We assume that suh rearrangement an be arried out instantaneously. Thus a dynami priing sheme an be modeled by 8

19 u i (t) = g i ( n(s), s [t d, t]), where g i is a funtion from Ω [ d,0] to R. Let g = {g,..., g I }. The performane objetive is again the expeted revenue per unit time generated by the inoming flows admitted into the system. The performane of the optimal dynami routing sheme is given by: J max g E{ λ i (u i (t))u i (t) } (6) i= subjet to (5). The expetation is taken with respet to the steady state distribution. Note that (6) is independent of t beause of stationarity and ergodiity. The set of dynami shemes we have desribed may require omplex apabilities (e.g., rearrangements of routes and transmitting traffi of a single flow over multiple routes) and hene may not be suitable for atual implementation. We make lear here that we do not advoate implementing suh shemes but instead advoate implementing stati shemes. In fat, we will show that, as the system sales, our stati sheme will approah the performane of the optimal idealized dynami sheme. The stati shemes do not require the afore-mentioned omplex apabilities and ould be an attrative alternative for network routing. Let u i = u i (λ i ) and F i (λ i ) = u i (λ i )λ i. Analogous to Proposition 2, we an derive the following upper bound on the optimal revenue in (6). The proof is a natural extension of that of Proposition 2 and are available online in [6]. Proposition 7 If the funtion F i is onave in (0, λ max,i ) for all i, then J J ub, where J ub is defined as the solution for the following optimization problem: J ub max λ ij subjet to θ(i) F i ( λ ij ) (7) i= θ(i) i= j= j= λ ij H l ijr i R l l. We next onstrut our stati routing poliy with probablisti routing as follows: The network harges a stati prie to all inoming flows, and the inoming flows are direted to alternative routes based on predetermined probabilities. Note that the stati poliy does not have the idealized apabilities presribed for the dynami shemes, i.e., all traffi of a flow has to follow the same path, and rearrangement of routes of existing flows is not allowed. Let {u s i, P ij s } denote suh a stati poliy, where us i is the prie for lass i, and Pij s is the bifuration probability that an inoming flow from lass i is direted to route j. Then the optimal stati poliy an be found by solving: J s max u s i,p ij s θ(i), P Pij s = i= j= j= θ(i) λ i (u s i )u s i Pij s [ P Loss,ij ], (8) 9

20 where P Loss,ij is the bloking probability experiened by users of lass i routed to j. We onsider a speial stati poliy derived from the solution of the upper bound in Proposition 7. If θ(i) ij is the maximal solution to the upper bound, we let us i = u i( λ ub ij ), and P s ij = λub ij. The revenue θ(i) λ ub with this stati poliy differs from the upper bound only by the term ( P Loss,ij ), and this revenue will be less than J s. However, under saling (S), we an show that, as, P loss,ij 0. Therefore, we have our invariane result (stated next). Proposition 8 In the dynami routing model, if the funtion F i is onave in (0, λ max,i ) for all i, then under the saling (S), j= lim J s/ = lim J, / = lim J ub / = J ub. P j= λ ub ij Proof: Analogous to that of Proposition 4. Details are available online at [6]. Q.E.D. A similar result under saling (S2) an be stated analogous to Proposition 6. When the routing is fixed, by replaing λ ij with λ i, and Hij l with Cl i, we reover Propositions 2 and 4 from the results in this Setion. When there are multiple available routes, the upper bound in Proposition 7 is typially larger than that of Proposition 2. Therefore one an indeed improve revenue by employing dynami routing. However, Proposition 8 shows that, when the system is large, most of the performane gain an also be obtained by simpler stati shemes that routes inoming flows based on pre-determined probabilities. Further, what we learn is that for large systems the apability to rearrange routes and to transmit traffi of a single flow on multiple routes does not lead to signifiant performane gains. Not only an the stati shemes be asymptotially optimal, they also have a very simple struture. Their parameters are determined by average onditions rather than instantaneous onditions. Colleting average information introdues less ommuniation and proessing overhead, and it is also insensitive to network delay. Hene the stati shemes are muh easier to implement in pratie. The asymptotially optimal stati sheme also reveals the marosopi struture of the optimal dynami routing sheme. For example, the stati prie u s i shows the preferene of some lasses than the others, and the stati bifuration probability Pij s reveals the preferene on ertain routes than the other. While a greedy routing sheme tries to aommodate eah individual flow, the optimal stati sheme may reveal that one should indeed prevent some flows from entering the network, or prevent some routes from being used. For our future work we plan to study effiient distributed algorithms to derive these optimal stati parameters. We use the following examples to illustrate the results in this setion. We first onsider a triangular network (Fig. 5). There are three lasses of flows, AB, BC, CA. There are two possible routes for eah lass of alls, i.e., a diret one-link path (route ), and an indiret two-link path (route 2). Eah all 20