Selfish Traffic Allocation for Server Fars Artur Czua Departent of Coputer Science New Jersey Institute of Technology czua@cis.nit.edu Piotr Krysta Max-Planck-Institut für Inforatik Saarrücken, Gerany krysta@pi-s.pg.de Berthold Vöcking Max-Planck-Institut für Inforatik Saarrücken, Gerany voecking@pi-s.pg.de ABSTRACT We investigate the price of selfish routing in non-cooperative networks in ters of the coordination and icriteria ratios in the recently introduced gae theoretic network odel of Koutsoupias and Papadiitriou. We present the first thorough study of this odel for general, onotone failies of cost functions and for cost functions fro Queueing Theory. Our ain results can e suarized as follows. We give a precise characterization of cost functions having a ounded/unounded coordination ratio. For exaple, cost functions that descrie the expected delay in queueing systes have an unounded coordination ratio. We show that an unounded coordination ratio iplies additionally an extreely high perforance degradation under icriteria easures. We deonstrate that the price of selfish routing can e as high as a andwidth degradation y a factor that is linear in the network size. We separate the gae theoretic integral allocation odel fro the fractional flow odel y deonstrating that even a very sall, in fact negligile, aount of integrality can lead to a draatic perforance degradation. We unify recent results on selfish routing under different oectives y showing that an unounded coordination ratio under the in-ax oective iplies an unounded coordination ratio under the average-cost or total-latency oective and vice versa. Our special focus lies on cost functions descriing the ehavior of We servers that can open only a liited nuer of TCP connections. In particular, we copare the perforance of queueing systes that serve all incoing requests with servers that reect requests in case of overload. Supported in part y NSF grant CCR-0105701, y SBR grant No. 421090, y DFG grant Vo889/1-1, and y the IST progra of the EU under contract nuer IST-1999-14186 ALCOM-FT. Perission to ake digital or hard copies of all or part of this work for personal or classroo use is granted without fee provided that copies are not ade or distriuted for profit or coercial advantage and that copies ear this notice and the full citation on the first page. To copy otherwise, to repulish, to post on servers or to redistriute to lists, requires prior specific perission and/or a fee. STOC 02, May 19 21, 2002, Montreal, Queec, Canada. Copyright 2002 ACM 1-58113-495-9/02/0005...$5.00. Fro the result presented in this paper we conclude that queuing systes without reection cannot give any reasonale guarantee on the expected delay of requests under selfish routing even when the inected load is far away fro the capacity of the syste. In contrast, We server fars that are allowed to reect requests can guarantee a high quality of service for every individual request strea even under relatively high inection rates. 1. INTRODUCTION In large-scale counication networks, like the Internet, it is usually ipossile to gloally anage network traffic. In the asence of gloal control it is therefore a reasonale assuption in traffic odeling that network users follow the ost rational approach, that is, they ehave selfishly to optiize their own individual welfare. This otivates the analysis of network traffic using odels fro Gae Theory in which each player is aware of the situation facing all other players and tries to iniize its cost. Under these assuptions, the routing process should arrive into a socalled Nash equiliriu in which no network user has an incentive to change its strategy. It is well known and easy to see that Nash equiliria do not always optiize the overall perforance of the syste. Therefore, in order to understand the phenoenon of non-cooperative systes Koutsoupias and Papadiitriou [14] initiated investigations of the coordination ratio, which is the ratio etween the worst possile Nash equiliriu and the social i.e., overall optiu. In other words, this analysis seeks the price of uncoordinated selfish decisions the price of anarchy. Koutsoupias and Papadiitriou [14, 19] proposed to investigate the coordination ratio for routing proles in which a set of agents is sending traffic along a set of parallel links with linear cost functions. In this paper, we generalize the odel of Koutsoupias and Papadiitriou towards ore realistic cost functions. Our ain focus is on a specific exaple in which parallel links are the natural choice: we investigate the effects of selfish ehavior on a We server far. Suppose soe copanies aintain a set of servers distriuted all over the world and offer content providers to store data for the. Such servers could store, e.g., large pictures and other eedded files, since this kind of data akes up ost of the load. The request streas that would norally go to the servers of the content provider ust now e redirected to these new We servers. Clearly, this defines a load alancing prole in which streas ust e apped to the servers such that a high quality of service can e guaranteed for every strea. Iportant aspects that have to e taken into account is that different streas ight have different characteristics, e.g., caused y different file lengths. For practical studies that investigate the reasons and ipacts of this variaility in traffic see, e.g., [2, 4, 5, 22, 23].
In nowadays server fars the apping of data streas to servers is typically done y a centralized or distriuted algorith that is under control of the provider of the server far. We can iagine, however, that such a service can e offered in a copletely different way without gloal control. For exaple, each strea of requests is anaged y a selfish agent e.g., the content provider that decides to which server the strea is directed. In this case, every agent would ai to iniize its own cost, e.g., the expected latency experienced y the requests in the strea or the fraction of requests that are reected. In this paper, we present the first thorough study of coordination and icriteria ratios under ore realistic cost functions. In previous works on traffic analysis in networks, it has een typically assued that every data strea is copletely descried y a single rate. In soe of our investigations we will ake this assuption, too. In order to incorporate the variaility of traffic, however, we will additionally study selfish routing under ore coplex cost functions that take into account different session length distriutions. In fact, we will consider general distriutions for session lengths. We further distinguish etween hoogeneous traffic in which all streas have the sae session length distriution and heterogeneous traffic in which different streas ight have different session length distriutions. 1.1 Definition of the Routing Prole The routing prole descried aove can e forally defined as an assignent prole with n data streas and servers or parallel links. The set of streas is denoted y [n] = {1,..., n} and the set of servers is denoted y [] = {1,..., }. The data streas shall e apped to the servers such that a cost function descriing, e.g., waiting or service ties is iniized. We ai at coparing the assignent otained y selfish agents with a inax optial assignent. A server far is a set of servers, all using the sae policy to serve requests. Different servers, however, ay have different andwidths. Let denote the andwidth of server. Data streas are infinite sequences of requests for service to the server far. These sequences are assued to e of a stochastic nature. For siplicity, we ake a standard assuption that requests are issued y a large nuer of independent users and hence they arrive with Poisson distriution. Let r i denote the inection rate of data strea i. For the lengths of the sessions, however, we allow general proaility distriutions. In particular, we assue the session length of strea i is deterined y an aritrary proaility distriution D i. We define the weight of strea i to e λ i = r i E[session length wrt. D i ]. We distinguish etween fractional and integral assignents of data streas to the servers. In an integral assignent, every strea ust e assigned to exactly one server. The apping is descried y an assignent atrix X = x i i [n], [], where x i is an indicator variale with x i = 1 if strea i is assigned to server and 0 otherwise. In a fractional assignent the variales x i can take aritrary real values fro [0, 1], suect to the constraint [] x i = 1, for every i [n]. The cost occurring at the servers under soe fixed assignent is defined y failies of cost functions F B = {f B}, where B denotes the doain of possile andwidth values and f descries the cost function for servers with andwidth B. Typically, we will assue B = R >0 or B = N >0, ut occasionally we will study finite doains of andwidth. For exaple, a collection of identical servers with soe specified andwidth is forally descried y a faily of cost functions F B with B = {}. The load of a server under an assignent X is defined y w = n i=1 λ i x i and the cost of server is defined y C = f x 1,..., x n. Unless otherwise stated, we consider the routing prole with respect to the in-ax oective. That is, we assue an optial assignent iniizes the axiu cost over all servers: opt = in X ax [] f x 1,..., x n. We note here, that in the definition of opt the iniization is over all atrices X that are either integral in the case of integral assignents or fractional in the case of fractional assignents. This distinction will e clear fro the context. 1.2 Preliinaries in Gae Theory Integral assignents and Nash equiliria. We assue the decision aout the assignent of a data strea i [n] to a server is perfored y an agent i who uses certain strategy to assign its data strea. Gae Theory distinguishes etween ixed and pure strategies. The set of pure strategies for agent i [n] is [], that is, a pure strategy aps every strea to exactly one server and hence can e descried y an integral assignent atrix X. A ixed strategy is defined to e a proaility distriution over pure strategies. In particular, the proaility that agent i aps its strea to server is denoted y p i. Oserve that under these assuptions the load w and the cost C of server are rando variales wrt. the proailities p i, i [n]. Let l denote the expected cost on server, that is, l = E[C ]. For a strea i, let us define the expected cost of strea i on server y c i = E[C x i = 1]. Recall that our oective is to iniize the axiu cost over all servers. Therefore, we define the social cost of a ixed assignent y C = E[ ax [] C ]. If selfish players ai to iniize their individual cost, then the resulting possile ixed assignent is in Nash equiliriu, that is, p i > 0 iplies c i cq i, for every i [n] and, q []. In other words, a Nash equiliriu is characterized y the property that there is no incentive for any task to change its strategy. Coordination and icriteria ratios. The coordination ratio for a fixed set of servers and streas is defined y ax C, where the opt axiu is over all Nash equiliria. Thus, the coordination ratio specifies how any ties the cost can increase due to selfish ehavior. The coordination ratio R over a faily of cost functions F B is defined to e the axiu coordination ratio over all possile sets of streas and servers with cost functions fro F B. Typically R is descried y an asyptotic function in. In our study, we will identify several instances of cost functions for which R is unounded. In this case, we will investigate icriteria characteristics of the syste. Let opt Γ denote the value of an optial solution over pure strategies assuing that all the inection rates r i are increased y a factor of Γ. Then, the icriteria ratio R is defined to e the sallest Γ satisfying C opt Γ over all Nash equiliria. In other words, the icriteria ratio descries how any ties the inected rates the aount of traffic in the syste ust e decreased so that the worst-case cost in Nash equiliriu cannot exceed the optial cost for the original rates. Fractional assignents and selfish flow. The otivation to consider fractional assignents is to assue that every strea consists of infinitely any units each carrying an infinitesial and thus negligile aount of flow traffic. Each such a unit ehaves in a
selfish way. Intuitively, we expect each unit to e assigned selfishly to a server proising iniu cost, taking into account the ehavior of other units of flow. Assuing infinitesial sall units of flow, we coe to the following fractional variant of the integral assignent odel. This fractional odel has een frequently considered in the literature, see, e.g., [8, 24, 25, 26, 27]. The fractional odel does not distinguish etween ixed and pure strategies. There are several equivalent ways to define a Nash equiliriu in this odel. We use the characterization of Wardrop [29], see also [24]. A fractional assignent is in Nash equiliriu if x i > 0 iplies C C q, for every i [n] and, q []. The coordination ratio is defined analogously to the integral assignent odel, and the coordination ratio over a faily of cost functions is denoted y R. 1.3 Previous Research The gae theoretical integral assignent odel for server fars or parallel links as descried aove, has een introduced y Koutsoupias and Papadiitriou [14]. Their focus is on the integral assignent odel and linear cost functions, that is, functions of the for f x 1,..., x n = n i=1 λ ix i /. Koutsoupias and Papadiitriou give first results for the coordination ratio in this odel e.g., tight ounds on the coordination ratio for two links. These results have een later largely extended in [3, 13, 16], and in particular, tight ounds in this odel are estalished y Czua and Vöcking in [3]. We are not aware of any results for non-linear cost functions in this odel. Roughgarden and Tardos [24] study the cost of selfish routing in a general network odel, where the streas ay e required to e routed through a network fro given sources to given destinations. They focus ainly on the fractional flow odel aking the assuption that each strea consists of infinitely any units, each of which ehaves in a selfish way. The traffic network paraeter to e iniized is the weighted average-cost over all streas instead of the in-ax cost over all servers. We will discuss this oective function in a ore detail in Section 2.5. They showed that when the cost functions of the edges are linear then the aver- α 2 α age cost in a Nash equiliriu is at ost 4 ties the average cost 3 in an optial routing. For aritrary nondecreasing and continuous cost functions, they show the existence of Nash equiliria whose total cost ay e aritrarily larger than the average cost in an optial routing. On the other hand, they give a icriteria result that the average cost in Nash equiliriu is upper ounded y the average cost in an optial routing for twice the aount of flow. The sae oective function has een investigated recently y Friedan [8], who studied how the aount of flow influences the icriteria ratio under cost functions fro Queueing Theory. Besides the selfish flow, Roughgarden and Tardos [24] consider also integral assignents. They give an exaple of a network with an unounded icriteria ratio. They present also a sufficient condition under which the icriteria ratio is ounded y a certain function: they prove that the icriteria ratio is upper ounded y if data streas are so sall that adding any strea to any server increases the total cost at ost y a factor of α, where α 2. This condition is restricted to pure assignents only. Furtherore, they reark that a useful application of this condition requires failies of cost functions with f 0 > γ, for any > 0 and a fixed γ > 0. 1.4 Outline Our new results are presented in the following two sections. In Section 2, we will focus on general, onotone cost functions. In Section 3, we will consider failies of cost functions fro Queueing Theory. Section 4 contains conclusions. In order to have space for a coprehensive discussion of our results, the technical part containing the proofs is oved to the Appendix. Missing proofs of Theores 9 and 11 will appear in the full version. 2. MONOTONE FAMILIES OF COST FUNCTIONS A cost function f is called siple if it depends only on the inected load, that is, if the cost of a server is a function of the su of the weights apped to the server ut does not depend on other characteristics like, e.g., the session length distriution. This is a typical assuption in previous works. A siple cost function is called onotone if it is non-negative, continuous, and nondecreasing. For an ordered set B, a faily of siple cost functions F B is called onotone if i f is onotone for every B and ii the cost functions are non-increasing in, i.e., f λ f λ, for every λ 0 and 0 <. 2.1 Fractional assignents Our first result is that all onotone cost functions ehave very well under fractional assignents. Recall that R denotes the coordination ratio for fractional assignents. OBSERVATION 1. For every server far whose servers are descried y onotone cost functions, R = 1. This result follows alost directly fro the definition of Nash equiliria specifying that all servers with positive flow at Nash equiliriu ust have sae cost, which under onotone cost functions iplies that all Nash equiliria have the sae social cost C = opt see, e.g., [24, Lea 2.5]. The oservation separates coordination ratios for the in-ax oective investigated y Koutsoupias and Papadiitriou [14] fro ratios for the averagecost oective y Roughgarden and Tardos [24] and Friedan [8] see also Section 2.5. In sharp contrast to Oservation 1, the results in [8, 14] show that there exist instances of fractional flow on parallel links in which the average-cost coordination ratio is unounded [24]. This shows that the average-cost and in-ax oective can differ aritrarily under general, onotone cost functions. In the context that we consider here, that of server fars, the in-ax oective sees to e the natural choice as it guarantees fairness and efficiency siultaneously. 2.2 Integral assignents Now let us consider the integral assignent odel. We say a coordination ratio R over a faily of cost functions F B is ounded if for every and every server far with servers with cost functions fro F B there exists Γ > 0 such that for every set of streas the value of the worst-case Nash equiliriu is at ost Γ opt. Oserve that Γ ight depend on. Thus, ounded eans ounded y a function in. Otherwise, the coordination ratio is unounded. Our first result is an exact characterization of those onotone failies of cost functions for which the coordination ratio is ounded. THEOREM 2. The coordination ratio R over a onotone faily F B of cost functions is ounded if and only if α 1 B λ > 0 f 2 λ α f λ. Notice that this characterization of ounded vs. unounded coordination ratios can e applied also to server fars with identical servers. Recall that such fars are descried y failies of cost functions consisting only of a single cost function.
Clearly, for every faily of onotone cost functions one can identify a iniu α R 1 { } fulfilling the conditions specified in the theore. A natural question is, how does the coordination ratio depend on this iniu α? In fact, our analysis see the proof of Theore 2 and Lea 14 shows that the coordination ratio is at ost Olog α. Oserve, if the faily of cost functions F B is assued to e fixed, then α is a constant or infinity. Thus, we can conclude that for every fixed faily of cost functions the coordination ratio is either unounded or it is polynoially ounded in the nuer of servers,. Let us illustrate the power of the aove theore y investigating soe exaples. First, we consider failies over polynoial cost functions, i.e., functions of the for k r=0 a r λ r, for a fixed k 0. For these failies we can pick α = a k 2 k to conclude that here the coordination ratio is ounded. In contrast, there is no such α for exponential cost functions, i.e., cost functions for which an additive increase in the load leads to a ultiplicative increase in the cost. COROLLARY 3. The coordination ratio R for server fars with polynoial cost functions is ounded, whereas the coordination ratio for server fars of possily identical servers with exponential cost functions is unounded. In the next sections we shall discuss several other, practically otivated exaples of failies of cost functions with unounded coordination ratios ased on well-known forulas fro Queueing Theory. We want to point out that unounded coordination ratios are not only a special phenoenon of cost functions having a pole or an unounded first derivative. Later in the paper, we will see a practical exaple of a faily of cost functions ased on the Erlang loss forula that has an unounded coordination ratio although the functions in this faily as well as their first derivatives are ounded aove y a sall constant, naely one. 2.3 Integral assignents with negligile weights If we copare the results in Oservation 1 and Theore 2 then we coe to the conclusion that integrality can lead to a draatic perforance degradation. As entioned efore, the fractional flow odel is assued to e a siplification that ais to odel the situation in which each strea carries only a negligile fraction of the total load [24]. Therefore, let us investigate the relationship etween fractional flow and integral assignents of streas with tiny weights ore closely. For this purpose we define the notion of ɛ-sall streas. For a server far with identical servers we have the following siple definition. Strea i is called ɛ-sall if λ i ɛ λ, [] that is, the strea has at ost an ɛ -fraction of the overall weight. In the case of servers with different andwidths, we use the following, slightly ore technical definition. Let us fix a server far and a set of streas with positive weights. Let opt denote the iniu axiu cost over all fractional assignents. For a strea i [n], define the scaled strea i to e a strea with rate r i = r i /ɛ and session length distriution D i = D i. Oserve that this iplies that the weight of the scaled strea is λ i = λ i /ɛ. Then strea i is called ɛ-sall if, for every [], the cost of server is at ost opt when this server gets assigned the scaled strea i and no other strea. Now, we define R ɛ to e the worst-case coordination ratio under the restriction that all streas are ɛ-sall. THEOREM 4. Given any onotone faily of cost functions F B. For every ɛ > 0, the coordination ratio R ɛ over ɛ-sall streas is ounded if and only if the coordination ratio R is ounded. A otivation for considering fractional flow instead of integral assignents is that these two odels are soeties assued to e essentially equivalent, see, e.g., Reark 2.3 in [24]. Theore 4 disproves this equivalence for general cost functions. It iplies, that there are cost functions with li ɛ 0 R ɛ = R = and R = 1. Moreover, the instances proving the characterization of unounded coordination ratios use only pure strategies. Hence, even pure assignents with negligile weights are different fro fractional flow. 2.4 Integral assignents under icriteria easures It is not surprising that selfish routing can lead to a draatic cost increase when the cost function has an -pole. In principle, icriteria easures can e uch ore inforative as they in soe sense filter out the extree ehavior of such cost functions at the pole. The following theore, however, shows that an unounded coordination ratio R iplies a very poor worst-case ehavior under icriteria easures as well. Recall that R denotes the icriteria ratio over integral assignents, that is, R specifies y how uch the inected rates ust e increased to ensure that the worst-case cost in Nash equiliriu will not exceed the optial cost for the increased rates. THEOREM 5. Consider a server far with servers with cost functions drawn fro a onotone faily F B. If the coordination ratio R is unounded, then the icriteria ratio R has value at least, even if all streas are restricted to e ɛ-sall. The exaple proving this ad ratio is a server far of identical servers. In fact, for the case of identical servers one can easily show that a icriteria ratio of is the worst possile. This is ecause a Nash equiliriu cannot e worse than apping all streas to the sae server, and the cost of this extreely unalanced solution is ounded aove y an optial assignent for an instance with all weights lown up y a factor of. 2.5 Min-ax versus average-cost oective functions Besides the in-ax oective function investigated aove, we study also the average-cost or total latency oective function that has een investigated y Roughgarden and Tardos [24] see also, e.g., [6, 28] for related results. This oective function ais at iniizing the expected weighted average cost over all streas. Forally, the cost under this oective function is defined y C ave = 1 E[w C ], λ [] and the social optiu is defined y opt ave = in 1 C f x X λ 1,..., x n, [] where λ = i [n] λ i is the total inected weight and the iniu is taken over all integral assignent atrices. These definitions are equivalent to the respective definitions in the Roughgarden-Tardos odel [24] when noralizing λ to one. We can consider various coordination ratios for the average-cost oective function siilarly as for the in-ax oective function. These average-cost coordination ratios are defined in the sae way as for the in-ax odel considered efore, the only difference is that now one copares the average cost in Nash equiliriu with the average-cost optiu.
THEOREM 6. In the case of integral assignents, the averagecost oective leads to exactly the sae characterizations for coordination and icriteria ratios as those given in the Theores 2, 4 and 5 for the in-ax oective. We want to point out that this equivalence is non-trivial. In general, the ehavior under the two oective functions can e quite different. For exaple, as descried in the discussion elow Oservation 1, in case of fractional assignents, the coordination ratios can e copletely opposite under average-cost and in-ax oective functions. 3. COST FUNCTIONS FROM QUEUEING THEORY A typical exaple of a onotone faily of cost functions that is derived fro the forula for the expected syste tie delay on an M/M/1 server with inection rate λ and service rate, naely 1. Already Koutsoupias and Papadiitriou in their seinal work [14] ask for the price of selfish routing under cost func- in{,λ} tions of this for. Our characterization of ounded and unounded coordination ratios given in Theore 2 iediately iplies that the integral coordination ratios for this faily of functions are unounded, which answers the open question fro [14]. Of course, this is only one particular exaple for cost functions fro Queueing Theory. Selfish Routing under siilar functions have een widely studied, e.g., in [11, 12, 15, 18, 19, 20]. We want to have a closer look at such functions in various server far odels. We distinguish two general kinds of servers in server fars. A parallel server has ultiple service channels. Each service channel can serve a session independently fro other channels. The nuer of channels on server corresponds to its andwidth and all channels serve requests with the sae service rate one. Thus, the tie a channel needs to serve a session is equal to the length of the session. Recall that the session lengths in strea i are deterined y a proaility distriution D i. For exaple, the nuer of channels ay correspond to the nuer of TCP connections that are allowed to e opened siultaneously on a We server. In contrast, a sequential server has only one service channel. This channel works at service rate. Thus, the tie a channel needs to serve a session is equal to the length of the session divided y. We will consider oth fars of parallel servers and fars of sequential servers. Another iportant aspect that leads to different cost functions is what happens in case of overload. Again, we distinguish two extree variants, naely, queueing and reection. In the queuing odel, every server aintains an FCFS queue in which it inserts requests when all availale channels are in use y other requests. When a service channel finishes a session and the queue is nonepty the server iediately starts serving the next request in the queue. All channels on the sae server share the sae queue. In this odel, the oective is to iniize the axiu expected delay. In the reection odel, locked requests are reected and disappear fro the syste. A natural oective in this odel is to iniize the nuer of reected requests and hence the cost function should descrie the fraction of reected requests. Using the standard notation fro Queueing Theory, a server with k channels that queues requests in case of overload corresponds to a so-called M/D/k/ or short M/D/k process, where D corresponds to the service tie session length distriution of the inected request strea. When requests are reected then the server is descried y a so-called M/D/k/k process. 3.1 Queueing systes without reection In order to avoid discussions aout what is exactly the right queueing odel e.g., M/M/1, M/D/1, M/G/1, M/G/c,... and what is exactly the right cost to consider e.g., expected waiting tie or expected syste tie, let us introduce a generic concept of onotone queueing functions. A onotone cost function is called a onotone queueing function if it satisfies li λ f λ =. Oserve that the onotonicity iplies f λ =, for every λ. This assuption is otivated y the fact that the expected waiting tie as well as the expected syste tie in every queueing process without reection goes to infinity when the inection rate approaches the service rate or andwidth of the server. Clearly, an iediate consequence of the -pole is that the paraeter α introduced in Theore 2 is. Thus, y Oservation 1 and Theores 2 and 5, we otain the following corollary. COROLLARY 7. For every faily F R>0 of onotone queueing functions, R = 1, R =, and R, even under the restriction that all streas are ɛ-sall. The proof for this negative result uses a Nash equiliriu in which all streas are identical and the total inected load i [n] λ i is less than the andwidth of a single server. Thus, selfish routing can lead to a catastrophic perforance degradation even under icriteria easures in extreely lightly loaded cases. Recall that the instances proving the unounded coordination ratio R are constructed using only pure strategies. However, the ad instances for the icriteria ratio R that we have seen until now use ixed strategies. This otivates us to investigate whether the randoness introduced due to the choice of ixed strategies is the only source of troules under icriteria easures. The following theore deonstrates that icriteria ratios can also e poor in case of pure strategies only. THEOREM 8. Let F R>0 e any faily of onotone queueing functions. Suppose is the nuer of servers and there exists > such that f 1 < f1, where f1, f F R>0. Then, the icriteria ratio over pure strategies is at least. 2 in{,λ} 1 in{,λ} This theore needs soe explanation. For exaple, the cost λ function for M/M/1 waiting tie is and the cost function for M/M/1 syste tie is. If we assue the cost function for waiting tie then the theore iplies a icriteria ratio over pure strategies of Ω 1/3. Siilarly, for syste tie the icriteria ratio is Ω 1/2. In oth cases, the total inected load in the exaple that gives these ad results is very sall. We investigate this closer, and show that the Ω 1/3 ound for M/M/1 waiting tie is essentially tight, proving the following theore. THEOREM 9. Let us consider the integral allocation odel and cost function on a server eing the M/M/1 waiting tie. The icriteria ratio with pure strategies, R, in this odel has value at ost O 1/3 log, where is the nuer of servers. Suarizing, even for pure strategies and under a sall total inection rate, the slowdown due to the lack of coordination can e draatic. An alternative icriteria easure. For the faily of onotone queueing functions there is another interesting icriteria easure. It is a very natural question to ask y how uch one has to decrease the andwidths of the servers such that an optial assignent under the decreased andwidths is at least as expensive as a
Nash equiliriu for the original syste. Let R w denote the corresponding worst-case icriteria ratio. It turns out that for ost functions fro Queueing Theory, the effect of changing the andwidths is larger than the effect of changing the inection rate. In fact, ost of these functions show superlinear scaling, i.e., f λ 1 α f /αλ/α, for every λ [0, and α 1. Applying this property, we can deterine the icriteria andwidth ratio exactly. THEOREM 10. For every faily F B of onotone queueing functions with super-linear scaling, R w =, where is the nuer of servers. We want to ephasize that this theore gives tight results, e.g., for expected waiting tie or syste tie in the queueing systes M/M/1, M/D/1, M/G/1, M/M/c or M/G/1. 3.2 Queueing under heterogeneous traffic Until now we iplicitly assued hoogeneous traffic, i.e., all streas have the sae general session length distriution. However, several practical studies show that Internet traffic is far away fro eing hoogeneous, see, e.g., [2, 5, 22]. Following these studies, one has to take into account different session lengths distriutions. The Pollaczek-Khinchin P-K forula see, e.g., [10] descries expected waiting tie in M/G/1 queues, that is, the expected delay of requests on sequential servers under heterogeneous traffic with aritrary service tie distriutions. We can use this forula and transfor it into a faily of cost functions depending only on two paraeters, naely, the weight λ and the variance V, of the coined streas inected into server. Let us descrie this in a ore detail. Suppose every strea i is characterized y two weights λ i and V i corresponding to the expected load i.e., the nuer of ytes requested per unit of tie and the variance of the load. Then, the P-K cost function faily F R>0 can e defined as follows: n i=1 f x 1,..., x n = V i x i n i=1 λ. i x i The rearkale fact here is that oth paraeters, the expected load and the variance, can e aggregated independently in a siple linear fashion. That is, the expected load inected into the server is λ = n i=1 λ i x i and the variance of this load is V = n i=1 V i x i. Oserve that if we assue λ i = V i then we are ack in the hoogeneous odel with identical session length distriution and we otain a onotone queueing function with only one paraeter, λ. Consequently R = and R for the P-K cost function faily. In the fractional flow odel, however, we will coe to different results. Recall that R = 1 under hoogeneous traffic. THEOREM 11. The coordination ratio R for the P-K cost function faily under heterogeneous traffic is unounded. If the ratio etween the andwidth of the fastest and slowest server is restricted to e at ost S then R = S. We conclude that the optiality of fractional flow in Nash equiliriu is a special property of hoogeneous traffic on parallel links, and hence one ust take into account the heterogeneous nature of We traffic when studying the price of selfish routing in the Internet. 3.3 Servers with parallel channels and reection Until now, we assued that all requests are served, regardless of how long they have to wait for service. In practice, however, We servers reect requests when they are overloaded. For siplicity, let us assue that a server reects requests whenever all service channels are occupied and then these requests disappear fro the syste. In this case, the fraction of reected requests is copletely independent of the service tie distriution. In other words, there is no difference etween hoogeneous and heterogeneous traffic under this service odel. In fact, the fraction of reected requests can e derived fro the Erlang loss forula, see, e.g., [8, 9]. We otain the following cost function faily F N>0 for servers that can open up to channels siultaneously: f x 1,..., x n = λ /! k=0 λk /k! with λ = n λ i x i. On the first glance, the faily of Erlang loss functions akes an innocent ipression. Indeed, these functions are continuous, convex, onotonically increasing in λ and f λ 1/, for every λ 0. Hence R = 1. Nevertheless, the following corollary shows that the coordination ratio R for integral assignents is unounded. COROLLARY 12. For the faily F N>0 of Erlang loss cost functions, R = and R. The corollary follows fro Theores 2 and 5, as α =. This can e seen as follows. Let us consider the faily of functions F N>0 with F x = f x, i.e., the Erlang loss functions in ters of relative load. We oserve that for every x 0, li F x = ax { 0, x 1 x Thus, the liit of these functions ehaves in a very extree way at x = 1, where the cost suddenly increases y an unounded factor. This iplies that α =. In contrast to the onotone queueing functions fro Section 3.1, however, the source of the troules is not an -pole, ut the rapid increase fro cost f 1 ɛ exp ɛ 2 to cost f 1 + ɛ ɛ, or in other words, the rapid increase fro tiny to sall cost. Hence, one ight hope that the asolute cost of selfish routing under the Erlang loss cost function faily is sall. In fact, this is confired y the following theore. THEOREM 13. Let δ satisfy δ 2/ log 2. Consider a server far of servers with andwidths 1 and cost functions fro the faily F N>0 of Erlang loss cost functions. Suppose i [n] λ i 1 6e [] and ax i [n] λ i. Then, 3δ log 2 any Nash equiliriu has social cost at ost δ+1 + 2 /4. log Hence, if the total inected load is at ost a constant fraction of the total andwidth and every strea has not too large weight, 1 that is, streas are O -sall, then the fraction of reected requests is at ost δ+1 + 2 /4, assuing constant δ. Under the sae conditions, an optial assignent would reect a fraction of 2 Θ 1 packets. Taking into account that typical We servers can open several hundred TCP connections siultaneously, so that can e assued to e quite large, we conclude that the cost of selfish routing is very sall in asolute ters, even though the coordination ratio coparing this cost with the optial cost is unounded. 4. CONCLUSIONS In this paper we present the first thorough theoretical study of the price of selfish routing in server fars for general cost function. In }. i=1
our investigations we paid special attention on cost functions fro Queueing Theory. Our results have soe iportant algorithic consequences in these odels. They show that the choice of the queueing discipline should take into account the possile perforance degradation due to selfish and uncoordinated ehavior of network users. We have shown that the coordination ratio for queueing systes without reection is unounded. The sae is true for server fars that reect requests in case of overload. However, there is a fundaental difference etween these two kinds of queueing policies. Because of the infinity pole, the delay under selfish routing in the queuing systes without reection is in general unounded. In fact, we have explicitly shown that the selfish routing in such queueing systes can lead to an aritrary large delay even when the total inected load can potentially e served y a single server. In contrast, the fraction of reected requests under selfish routing can e ounded aove y a function that is exponentially sall in the nuer of TCP connections that can e opened siultaneously. We conclude that server fars that serve all requests, regardless of how long requests have to wait, cannot give any reasonale guarantee on the quality of service when selfish agents anage the traffic. However, if requests are allowed to e reected, then it is possile to guarantee a high quality of service for every individual request strea. Thus, the typical practice of reecting requests in case of overload is a necessary condition to ensure efficient service under gae theoretic easures. 5. REFERENCES [1] M. Beckann, C. B. McGuire, and C. B. Winston. Studies in the Econoiices of Transportation. Yale University Press, 1956. [2] M. E. Crovella and A. Bestavros. Self-siilarity in World Wide We traffic: Evidence and possile causes. IEEE/ACM Transactions on Networking, 56:835 846, 1997. [3] A. Czua and B. Vöcking. Tight ounds for worst-case equiliria. In Proc. 13th ACM-SIAM SODA, 2002. [4] F. Douglis, A. Feldann, B. Krishnaurthy, and J. Mogul. Rate of change and other etrics: a live study of the World Wide We. In Proc. USENIX Syposiu on Internet Technologies and Systes, pp. 147 158, Deceer 1997. [5] A. Feldann, A. C. Gilert, P. Huang, and W. Willinger. Dynaics of IP traffic: A study of the role of variaility and the ipact of control. In Proc. ACM SIGCOMM 99, pp. 301 313, 1999. [6] M. Frank. The Braess paradox. Matheatical Prograing Study, 20:283 302, 1981. [7] P. Franken, D. Konig, U. Arndt, and V. Schidt. Queues and Point Processes. Wiley, Chichester, 1982. [8] E. Friedan. A generic analysis of selfish routing. Manuscript, 2001. [9] D. Gross and C. M. Harris. Queueing Theory. Third Edition John Wiley & Sons, New York, NY, 1998. [10] L. Kleinrock. Queueing Systes. Volue I: Theory. John Wiley & Sons, New York, NY, 1975. [11] Y. A. Korilis, A. A. Lazar, and A. Orda. Capacity allocation under noncooperative routing. In IEEE Transactions on Autoatic Control, 423:309 325, 1997. [12] Y. A. Korilis, A. A. Lazar, and A. Orda. Avoiding the Braess paradox in noncooperative networks. In Journal of Applied Proaility, 361:211 212, 1999. [13] E. Koutsoupias, M. Mavronicolas, and P. Spirakis. Personal counication, 2001. [14] E. Koutsoupias and C. H. Papadiitriou. Worst-case equiliria. In Proc. 16th STACS, pp. 404 413, 1999. [15] A. A. Lazar, A. Orda, and D. E. Penderakis. Virtual path andwidth allocation in ultiuser networks. In IEEE/ACM Transactions on Networking, 5:861 871, 1997. [16] M. Mavronicolas and P. Spirakis. The price of selfish routing. In Proc. 33rd ACM STOC, pp. 510 519, 2001. [17] R. Motwani and P. Raghavan. Randoized Algoriths. Caridge University Press, New York, NY, 1995. [18] A. Orda, R. Ro, and N. Shikin. Copetitive routing in ulti-user counication networks. In IEEE/ACM Transactions on Networking, 1:510 521, 1993. [19] C. H. Papadiitriou. Algoriths, gaes, and the Internet. In Proc. 33rd ACM STOC, pp. 749 753, 2001. [20] C. H. Papadiitriou. Gae theory and atheatical econoics: A theoretical coputer scientist s introduction Tutorial. In Proc. 42th IEEE FOCS, pp. 4 8, 2001. [21] C. H. Papadiitriou and M. Yannakakis. On coplexity as ounded rationality. In Proc. 26th ACM STOC, pp. 726 733, 1994. [22] K. Park, G. Ki, and M. E. Crovella. On the relationship etween file sizes, transport protocols, and self-siilar network traffic. In Proc. IEEE International Conference on Network Protocols, pp. 171 180, 1996. [23] V. Paxson and S. Floyd. Wide area traffic: The failure of Poisson odeling. IEEE/ACM Transactions on Networking, 3:226 244, 1995. [24] T. Roughgarden and É. Tardos. How ad is selfish routing? In Proc. 41st IEEE FOCS, pp. 93 102, 2000. [25] T. Roughgarden. Stackelerg scheduling strategies. In Proc. 33rd ACM STOC, pp. 104 113, 2001. [26] T. Roughgarden. Designing networks for selfish users is hard. In Proc. 42nd IEEE FOCS, pp. 472 481, 2001. [27] T. Roughgarden. How unfair is optial routing? In Proc. 13th ACM-SIAM SODA, 2002. [28] R. Steinerg and W. I. Zangwill. The prevalence of Braess paradox. Transportation Science, 173:301 318, 1983. [29] J. G. Wardrop. Soe theoretical aspects of road traffic research. In Proc. Institution of Civil Engineers, Part II, volue 1, pp. 325 362, 1952. APPENDIX A. PROOF OF THEOREMS 2 AND 4 We prove Theores 2 and 4 in a single proof using two leas. The first lea proves sufficient conditions for a ounded coordination ratio, while the second lea proves necessary conditions. LEMMA 14. Suppose we are given a server far with servers having cost functions fro a fixed, onotone faily F B satisfying α 1 B λ > 0 f 2 λ αf λ. Then, for every set of streas the worst-case cost over all Nash equiliria is upper ounded y opt O1. PROOF. Fix an aritrary allocation in Nash equiliriu. Let C denote the cost of this allocation. We will show that C α log opt = O1 opt. First, we oserve that f s λ α log s f λ, for every s 1. Let X := i [n] λ i denote the total inected load. Let x denote the load on server 1 with the iggest andwidth
under an optial fractional allocation, i.e., an allocation with iniu axiu cost over all servers assuing that streas can e split aritrarily. Without loss of generality, we assue that server 1 has the axiu load over all servers, and hence X x. Recall that since server 1 is the server with the iggest andwidth and each function f F B is nondecreasing in λ and nonincreasing in, there exists an optial fractional allocation with axiu load for server 1. In this way, f 1 X f 1 x α log f 1 x α log opt. Let M denote the set of servers [] with i [n] p i > 0. Pick any server in M. Let i [n] denote a strea with p i > 0. Then, the Nash equiliriu property guarantees that c i c1 i. Hence, for every M, ] E [C ] E [C x i = 1 = c i c1 i f 1 X α log opt. which copletes the proof of Lea 14. Now, we prove a sufficient condition for an unounded coordination ratio. Oserve that the negation of the property gives the sufficient condition for a ounded coordination ratio as specified in Theore 2. LEMMA 15. Let ɛ > 0 e chosen aritrarily. Suppose we are given a server far with only two identical servers, each with the sae onotone cost function f satisfying α 1 λ > 0 f2 λ > α fλ. Then, for every Γ 1, there exists a pure Nash equiliriu over ɛ-sall streas with cost C > Γ opt. PROOF. First, let us show that the aove property of the function f iplies that α 1 β > 0 λ > 0 f1 + β λ > α fλ. 1 Indeed, if we consider the negations of the two stateents then we oserve that α 1, β > 0 λ > 0 f1 + β λ α fλ iplies that α 1 λ > 0 f2 λ α fλ, e.g., we can set α = α 1/ log1+β. Now, consider a server far with two identical servers, each with the sae onotone cost function f satisfying condition 1. For the purpose of contradiction, assue there exists Γ with C Γ opt for every Nash equiliriu over streas of axiu weight ɛ. Therefore, y 1, there exists λ > 0 with f1 + ɛ/10 λ > Γ fλ. Using this assuption, we will define a Nash equiliriu over streas of axiu weight ɛ > 0 with cost C > Γ opt. First, let us consider streas of identical weight w [λɛ/2, λɛ] and assign the to the servers so that the cost on each server is exactly fλ. Let τ e the nuer of streas per server in this allocation. Now, let us slightly change the instance y taking two streas, one fro each server, and reak each of the into two saller streas, one of weight 3 w, the other of weight 2 w. It 5 5 is easy to see that the optial allocation for this instance has cost opt = fλ. Let us consider a different allocations of the streas to the servers. We assign τ 1 streas of weight w and two streas of weight 3 w to the first server and the reaining streas to the second server. In this way, the first server has cost fλ+ 1 w whereas 5 5 the second server has cost fλ 1 w. This allocation defines a 5 Nash equiliriu, ecause the streas have iniu weight 2 w 5 and therefore there is no incentive for any of the to change its strategy. The cost of this Nash equiliriu, however, is C = f λ + w f λ + λ ɛ > Γ fλ = Γ opt. 5 10 Clearly this contradicts our initial assuption that C Γ opt for any Nash equiliriu over streas of axiu weight ɛ. This copletes the proof of Lea 15. Theores 2 and 4 follow iediately fro Leas 14 and 15. B. PROOF OF THEOREM 5 Let n e the nuer of streas and let α = n 1. Assue, that n is such that ɛ. Since ratio R is unounded, Theore 2 n iplies that there exist f F B and λ > 0 with f 2λ > αf λ. Furtherore, for [] \ M, we have E [C ] = f 0 opt α log opt. As a consequence, Assue that we have identical servers, each with andwidth = [], and a set of n identical data streas, each having [ ] weight of λ i = λ i [n], where Γ > 0 will e specified later. Γn C = E ax [] ax ] α log opt = O1 opt, [] Define the proailities p i as p i = 1, for each i, []. These proailities define a Nash equiliriu, since all the expected costs c i have the sae value. Let us fix a server. The proaility that all the data streas are assigned to server is Π i p i = n. In this case, the cost on server is f λ Γ, since we have n streas, each of weight λ, and so Γn their total weight is λ n = λ. Therefore, with proaility Γn Γ n, the cost on a particular server is at least f λ Γ, and thus also ax C f λ Γ, over all []. Additionally, these events corresponding to different servers are pairwise disoint. Using these oservations, we can estiate C = E [ax C ] as [ ] λ E ax C f n = f λ Γ. Γ n 1 We want to show that if C opt Γ then Γ. We proceed y contradiction and let us assue there is Γ such that C opt Γ and Γ <. Then, we otain f λ Γ λ opt n 1 Γ f Γn n Γ = f λ, where the last inequality follows y oserving that the value of opt Γ is at ost the value of a solution in which we assign n we can assue that n is a positive integer data streas to each server, after lowing up each data strea y Γ. Then, we otain λ f n 1 f λ. 2 Γ By our assuption, α 1 λ > 0f 2λ > αf λ. This, y the arguent fro the proof of Lea 15, yields α 1, β > 0 λ > 0 f 1 + βλ > αf λ. Now, we plug α = n 1, 1 + β =, to otain a contradiction with inequality 2. Γ Since all the servers are identical, a data strea of weight λ i is ɛ-sall if λ i ɛ λ. Since in our case λ i = λ for all Γn i [n], the last condition is equivalent to λ ɛ λ, which gives Γn Γ ɛ. The last inequality is true y the choice of n, and so our n data streas are ɛ-sall. C. PROOF OF THEOREM 6 Fix a onotone faily F B = {f B} of cost functions. First let us show that the coordination ratio under average-cost oective, denoted y Σ, for this faily is ounded if the coordination
ratio R is ounded. Clearly, if R is ounded then we otain fro Theore 2 α 1 B λ > 0 f 2λ αf λ. We will show that Σ is ounded provided this property is given. Fix any Nash equiliriu with proailities p i, i [n], []. We have to give an upper ound on the ratio etween the expected total latency given y the proailities p i, on one hand, and the optial total latency, on the other hand. Let w = w 1,..., w denote a vector of rando variales with w descriing the inected load of server. Let w denote a corresponding load vector of an optial allocation that iniizes the total latency. We have to show that there exists Γ 1 such that E w f w Γ w f w. [] [] Define X = [] w = [] w. Then f 1 X corresponds to the cost that is otained y assigning all the load to the fastest server. Using the sae arguents as in the proof of Lea 14, we otain E [ f w ] f 1 X, for every [], such that E w f w X f 1 X [] [] 2 α 1+log X X f 1 2 α 1+log [] w f w. The second inequality follows fro our assuptions on the faily of cost functions. The third inequality needs soe ore explanation. Oserve that there exists [] with w X/. Applying onotonicity of our cost function faily, we otain X f 1 X w f w, which yields the inequality. As a consequence, Σ 2 α 1+log, that is, Σ is ounded. It reains to show that an unounded coordination ratio R iplies an unounded ratio Σ. If R is unounded then the faily of cost functions satisfies α 1 B λ > 0 f 2λ > αf λ. In Lea 15, we descried an instance with two identical servers and a set of ɛ-sall streas such that oth servers have identical load in an optial allocation ut there is a Nash equiliriu in which one of the servers receives ore load than the other server. Using the aove property we show that if one server has load at least 1 + ɛ/10 ties average load then the cost on this server can deviate y an aritrary large factor fro the optial cost, which then proves that R is unounded. A straightforward adaption of these arguents show also that if the aove condition is fulfilled then Σ is unounded. The proof of Theore 5 in the average-cost case is asically the sae as in the in-ax case, since the values of the two oectives are the sae in the lower ound instance. This copletes the proof of Theore 6. D. PROOF OF THEOREM 8 We prove the following lea that directly iplies Theore 8. LEMMA 16. Let F R>0 e a faily of onotone queueing functions. Assue we are given + 2 servers with andwidths 1 = 1 = 2 =... = +1 +2 =, such that f F R>0, for each = 1, 2,..., + 2. Assue, oreover, that there exists an ε > 0 such that functions f 1 and f fulfill f 1 1 1 + ε f 1, 3 where Γ = and ε/2 > 1. Then, in this syste, C > opt 2 Γ, where C is the axiu value over all Nash equiliria assuing only pure strategies. In particular, R. 2 PROOF. We define the following instance of the prole. Let the servers 1, 2,..., e called slow. We also have a fast server with +2 = and an additional server with andwidth +1 =. Assue that each slow server holds one sall data strea with 1 weight and let the fast server have one large data strea with weight ε/2. The additional server holds no data strea. In the instance we have ust defined, each slow server has cost after lowing the streas up y Γ f 1 1 2 and the fast server has cost f εγ 2 2. Let costε = ax { f 1/Γ 1/2 Γ, f /Γ /2 Γ ε/2 }, and oserve that y the definition of onotone queueing functions, costε is finite even if ε 0, and oviously opt Γ costε. Let us define a Nash equiliriu for our instance. We assign to the fast server a total aount of 1 1 + ε of sall streas. Oserve that we have enough sall streas to achieve this, since their total size is equal to > 1 1 notice that if ε is not sall enough, then we can use a saller ε still satisfying 3 and ε/2 > 1 guaranteed y the onotonicity of function f. One can also easily show that the large strea ust e assigned to the additional server + 1. We clai we have defined a Nash equiliriu. First, none of the sall streas would go fro the fast server to a slow server, + ε f1 1. Also, none since y 3 we have f 1 1 of the sall streas would go fro the fast server to the additional server, since the reaining space on the additional server is ε/2 and it can e ade aritrarily sall. Finally, the large strea cannot go fro the additional server to any slow server since ε/2 > 1 nor to the fast server since it would exceed the capacity on the fast server. Notice that it is possile that a sall reaining data strea, if any, can go fro a slow server to the fast server. But then our construction works as well. The cost on the additional server in this Nash equiliriu is f / ε/2 and so y the properties of onotone queue- ing functions, li ε 0 f / ε/2 =. This iplies that there exists an ε > 0, such that C is aritrarily large, and in particular larger than cost0. By onotonicity, we have cost0 costε. Thus we otain opt Γ costε cost0 < C. E. PROOF OF THEOREM 10 Let OP T Γ denote the optiu solution in a syste where andwidths of all servers are slowed down y a factor of Γ. LEMMA 17. Fix an aritrary onotone queueing function f = f. Consider a server far of identical servers with cost function f. Then for every ɛ > 0 and Γ < there exists a Nash equiliriu over ɛ-sall streas such that C > OP T Γ. PROOF. Assue we have identical servers, each with andwidth =, and a set of n identical data streas, each of weight, where n is s.t. ɛ, and δ [0, will e specified n, for each i, []. Since all the expected costs c i have the sae value, p i s define a Nash equiliriu. Fix a server []. The proaility that all data streas are assigned to server is Π i p i = n. In this case, the cost on server λ i = δ n later. Let p i = 1 is f δ, since the total weight of the streas is δ n = δ. n