On the Price of Anarchy and the Optimal Routing of Parallel non

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Jonatha Anselmi — Bruno Gaujal

N°

apport de recherche

ISRN INRIA/RR-–-FR+ENG

Févrière 2010

SN 0249-6399

inria-00457603, version 1 - 18 Feb 2010

On the Price of Anarchy and the Optimal Routing of Parallel non-Observable Queues

inria-00457603, version 1 - 18 Feb 2010

On the Price of Anarchy and the Optimal Routing of Parallel non-Observable Queues Jonatha Anselmi , Bruno Gaujal Thème : Calcul distribué et applications à très haute performance Équipes-Projets Mescal

inria-00457603, version 1 - 18 Feb 2010

Rapport de recherche n°  Févrière 2010  21 pages

Abstract:

We consider a network of parallel, non-observable queues and analyze the price of anarchy, an index

measuring the worst-case performance loss of a decentralized system with respect to its centralized counterpart. Our analysis is undertaken from the new point of view where the router has the memory of previous dispatching choices, which signicantly complicates the nature of the problem.

In the limiting regime where the demands

proportionally grow with the network capacity, we provide a tight lower bound on the socially-optimal response time and a tight upper bound on the price of anarchy by means of convex programming. Then, we exploit this result to show, by simulation, that the billiard routing scheme yields a response time which is remarkably close to our lower bound, implying that billiards minimize response time. To study the added-value of non-Bernoulli routers, we introduce the price of forgetting and prove that it is bounded from above by two, which is tight in heavy-trac.

Finally, other structural properties are derived numerically for the price of forgetting.

These

claim that the benet of having memory in the router is independent of the network size and heterogeneity, while monotonically depending on the network load only.

These properties yield simple product-forms well-

approximating the socially-optimal response time.

Key-words:

Price of anarchy, Optimal routing, Parallel queues, Convex programming, Lambert

Centre de recherche INRIA Grenoble – Rhône-Alpes 655, avenue de l’Europe, 38334 Montbonnot Saint Ismier Téléphone : +33 4 76 61 52 00 — Télécopie +33 4 76 61 52 52

W

function

Sur le prix de l'anarchie et le routage optimal des les d'attente non-observables et parallèles

Résumé :

Dans cet article, nous considérons un réseau de les d'attentes en parallèles, non-observables et nous

étudions le prix de l'anarchie, qui mesure la perte de performance pour un système décentralisé par rapport à son analogue centralisé. Notre analyse introduit un nouveau concept, celui de controleur de routage centralisé avec mémoire, qui change la nature du problème. Dans le régime limite avec des demande qui croissent proportionnellement à la taille du réseau, nous donnons une borne inférieure sur le temps de réponse socialement optimal, ce qui fournit une borne supérieure sur le prix de l'anarchie dans ce contexte. Ensuite, nous exploitons ce résultat pour montrer par simulation que le routage par suites billards a un comportement remarquablement proche de la borne inférieure. Pour mettre en évidence l'apport de routage nonBernoulli, nous introduisons le prix de l'oubli qui est la rapport entre la performance d'un routage sans mémoire (Bernoulli) et l'un routage avec mémoire, et nous montrons qu'il est borné par 2, cette borne étant exacte dans le cas fortement chargé. Finalement, d'autres propriétés structurelles sont établies numériquement. Elles mettent en évidence que le

inria-00457603, version 1 - 18 Feb 2010

bénéce obtenu grâce à la mémoire du routeur est indépendant de la taille et de l'hétérogénéité du système mais a une dépendance monotone de la charge globale. Cela donne une forme produit simple approximant le prix de l'anarchie de tels systèmes.

Mots-clés :

Prix de l'anarchie, Routage Optimal, Files d'attentes parallèles

Optimal routing in parallel queues and price of anarchy

3

1 Introduction The price of anarchy [24, 28] is an index measuring the eectiveness of a centralized system with respect to its decentralized counterpart to tradeo, in service networks, among performance, scalability, and reliability. It is dened as the worst-case response-time ratio between the situation where jobs behave selshly to maximize

Nash equilibrium, and the opposite situation where jobs are controlled by a central social optimum or social welfare. While the former identies the equilibrium point for

their own benet, yielding the authority, yielding the

which any unilateral deviation of each job strategy does not lower its delay, the latter represents the optimal strategy for all jobs in a centralized setting.

Therefore, it is the merging of game and queueing theories that

provides, in general, the mathematical framework for analyzing this index. The interest for the price of anarchy in the context of queueing models is currently growing because of its large spectrum of applications:

Network routing, load balancing, peer-to-peer and content delivery networks,

wireless networks, server farms, grid computing clusters, desktop-grid computing, and database systems; see [29, 27, 22, 20, 13, 34, 18, 9, 1, 4, 6, 30] and the references therein. The great majority of these works provide mathematical tools for characterizing and computing the mean response times in both the situations above and try to relate the price of anarchy to the network size in dierent settings.

This lets designers quantitatively

inria-00457603, version 1 - 18 Feb 2010

evaluate the loss of performance when shifting to decentralized solutions (corresponding to Nash equilibria) and subsequently perform a suitable dimensioning of the system.

In [29] it is shown that the price of anarchy is

independent of the network topology as long as the mean jobs arrival rate is less than the mean service rate of the slowest server, and, in this light-load regime, an upper bound is provided. When heterogeneous processorsharing queues are considered, it is shown in [18, 34] that the price of anarchy, in general, linearly scales with the network size, and it can only depend on the heterogeneity degree of the queues provided that these adopt the shortest-remaining-processing-time scheduling discipline [9]. In the case of multiple central authorities, which can be the case of large server farms, the price of anarchy is shown to be lower bounded by the square root of the number of authorities [4]. We observe that a key point common to all the above works is that the central authority, which in the remainder of the paper we refer to as

router, achieves the social optimum in a Bernoulli setting, making its routing decisions

independent each other, i.e., with no memory. In fact, the social optimum is commonly searched among all the possible Bernoulli policies through a non-linear optimization problem. In several cases, this restriction is known to yield tractable formulas for mean response times, but it may play a non-innocuous role in the analysis. In a more general framework, in fact, the optimal jobs inter-arrival times of each queue are not even i.i.d. [17]. Except for special cases, this

dynamic

routing scheme notoriously complicates the nature of the problem so that the

assessment of the routing policy which minimizes the mean response time as well as the analysis of such response time are current open problems in the literature; see, e.g., [17, 5, 10, 14] for the case of parallel and non-observable queues. In this framework, it is also shown in [5] that nding the optimal

1.1

cyclic

policy is NP-complete.

Our Contribution

In this paper, we tackle the problem of analyzing the price of anarchy in open queueing systems of parallel and non-observable queues. We undertake this analysis from a new point of view: In contrast with the existing works above, the key point of our analysis is to consider routers with the memory of previous dispatching choices. This feature of our approach severely complicates the problem of determining the optimal response time achievable by the system, i.e., the social optimum, but we strongly believe that this is necessary for an accurate analysis of nowadays networks. In fact, dispatching schemes with memory, e.g., round-robin, can be easily implemented in network routers with very limited costs. Given the intrinsic intractability of the problem, our analysis is performed in the limiting regime where demands, i.e., jobs arrival rate, proportionally grow with the network size, i.e., the number of queues. This limiting regime is very popular in queueing theory to approximate the behavior of large systems in a tractable manner; see, e.g., [33, 21]. First, we provide a stochastic comparison result providing a tight lower bound on the optimal (mean) response time achievable by the system. This bound is expressed in terms of a convex optimization program which integrates the mean response time of a parallel system of independent

Γ/M/1

queues. New asymptotically-exact bounds on

their response times follows as function of the Lambert W function [11]. Then, we introduce the

price of forgetting, an index measuring the worst-case ratio between the socially-optimal

response time of a memoryless router with respect to its memory counterpart. We prove that it is bounded from above by two being exact in heavy-trac. Thus, the price of anarchy of a router with memory can be understood as the price of anarchy of a Bernoulli router times a correcting factor less than two. When homogeneous queues

RR n°

Anselmi and Gaujal

4

µ1

µ2 λN

Router . . .

µN

Figure 1: Queueing model under investigation. The router dispatches incoming jobs to the queues according to some policy and with no delay. The mean arrival rate is proportional to the number of queues.

are considered, our analysis simplies and an explicit expression is found for the price of anarchy which equals

inria-00457603, version 1 - 18 Feb 2010

the price of forgetting. Here, we prove that it strictly increases in the queues utilization, implying that it is not possible to design a network where the choices of selsh jobs have no impact on performance. This is in contrast with the case of memoryless routers, which make the price of anarchy equal to one for any load [6, 18]. Our analysis also allows us to assess the quality of heuristics for the optimal (non-Bernoulli) routing. exhaustive numerical analysis reveals that a router forwarding jobs to queues according to a

billiard

An

scheme, a

generalization of round-robin, [3, 16, 19], yields a response time which is remarkably close to our lower bound. In other words, we have the two-fold result that billiard routings achieve, in practice, the minimum response time which, in turn, is very-well captured by our bound and approximations. Finally, we give numerical evidence of the fact that the price of forgetting is insensitive to the network size (N ) and heterogeneity, while monotonically depending on the network load (L) only. These structural properties

f (N )F (L) where i) f (N ) is linear and refers to the price F (L), the price of forgetting, The term F (L), explicitely given, is thus interpreted as the

entail that the price of anarchy admits the product-form

of anarchy achieved by a Bernoulli router (which is well-understood [6, 18]), and ii) is increasing in

L

and bounded from above by two.

added-value of a router with memory or, more technically, as the response-time improvement which is obtained when the input process of each queue is Gamma instead of Poisson. This paper is organized as follows.

Section 2 introduces the model under investigation and the necessary

preliminaries. In Section 3, we propose an upper bound on the price of anarchy in terms of a convex program as well as an improved approximation. In Section 4, we dene the price of forgetting which is analyzed to derive qualitative properties on the benet of a router with memory. In Section 5, we show how a non-Bernoulli router should operate to minimize response time, and, in Section 6, we measure its impact on system performance exhibiting structural properties that we interpret. Finally, Section 7 draws the conclusions of this work.

2 Model and Preliminaries We consider a queueing system composed of

N

innite-room queues working in parallel as shown in Figure 2.

Jobs arrive from an external Poissonian source having intensity jobs to one of the

N

queues according to a given

λN

to a router which instantaneously dispatches

policy, i.e., routing rule.

We assume that the router cannot observe the state of the queues, i.e., their current number of jobs. In queue

µ−1 = O(1). i

i,

we assume that jobs require service for an exponentially-distributed amount of time having mean

The service times are i.i.d. and independent of the arrival process and

N.

Initially, all queues are

supposed to be empty (this assumption can be relaxed because the mean performance does not depend on initial states, but it is useful in our proofs for technical reasons). The scheduling discipline of each queue is assumed to be First-Come-First-Served or Processor Sharing. In the remainder of the paper, index to

N,

i

implicitly ranges from

1

if not otherwise specied.

def 1 N The routing policy a = (a , . . . , a ) of jobs into queues is given by the sequences i th an = 1 if the n job is sent to queue i, and is 0 otherwise. By denition, if ain = 1 then a job is routed to a single queue.

(ain )n∈N ∈ {0, 1}, where ajn = 0 for all j 6= i, i.e.,

INRIA

Optimal routing in parallel queues and price of anarchy

5

In contrast with existing works, we analyze the case where the router has memory of which queues previous jobs have been dispatched.

This feature of our model is innovative in the context of evaluating the price of

anarchy, easy to implement in real-world routers and plays a key role in our analysis. Let

def

L = λN/ be the

network load,

i=1

(1)

µi

an index measuring the network utilization.

considered queueing model is stable if and only if We also denote by

PN

R = R(a)

Within the optimal routing policy, the

L < 1.

the mean response time, or sojourn time, of jobs in the system under policy

provided the expectation exists (the dependence of

a

a,

will be reported when necessary). In the remainder of the

paper, we omit the words mean when we refer to response time for simplicity.

2.1

Nash Equilibrium and Social Optimization

Within the model introduced above, we consider two dierent scenarios. In the rst one, selsh jobs choose to join a queue to minimize their response time individually, and we refer to this situation as e.g., [6, 18]. The response time achievable in this scenario is denoted by

RN e .

Nash equilibrium ; see,

In the second one, jobs choose to

inria-00457603, version 1 - 18 Feb 2010

join a queue to minimize the response time of all jobs taking into account choices of previous jobs. We refer to this situation as

social optimization, and the mean response time achievable in this scenario is denoted by RSo .

These scenarios reect the conicting situations where jobs moves in an infrastructure with neither control nor shared information with respect to the case where a centralized object controls the dynamics of the system to maximize the prot of all jobs. Our notion of social optimization diers from the one considered in existing approaches in the sense that we let the router operate with the memory of previous decisions. This means that the set of policies handled by the router is much larger than the set of the Bernoulli ones, because the routing decisions are no more independent each other.

The main consequence of this point is that the optimal jobs inter-arrivals of each queue are not

i.i.d. [17], and, thus, the analysis becomes much more dicult.

In the case of Nash equilibrium, however, we

observe that the optimal policy must be Bernoulli because jobs make their decisions independently of the others (in fact, no shared information is available in a fully-decentralized system before the arrivals of jobs). consequence, existing works apply to our model in this case (see [6] for a formula for

As a

RN e ).

It is clear that both the situations depicted above can be modeled in our queueing system by specifying a suitable policy in the router. We measure the eciency of the social optimization scenario with respect to the Nash equilibrium by means of the price of anarchy

A,

which is dened as follows

def

A = RN e /RSo . Evidently, large values of

(2)

A indicate that the impact of a centralized control drastically improves the performance

of the system, and vice versa. On the other hand, we notice that a centralized control design is less scalable and reliable than a distributed one because the system has a single point of failure. We also stress that the

worst-case

A

represents

delay ineciency of an uncontrolled infrastructure with respect to a controlled one. By denition,

A ≥ 1. RN e , there are currently no exact analyses for the ecient computation of RSo So could be determined, in our settings, by applying introduction). Even though R

In contrast with the analysis of when

N

is generic (see the

Markov decision process algorithms, e.g., [26], the intrinsic computational complexity of this approach makes it feasible only when the number of queues is very limited. In the particular case where of

RSo

N = 2,

an ecient analysis

can be found in [14]. This current diculty prevents the understanding of the problem and motivated the

authors to investigate alternative computational techniques for the approximate analysis of (2).

3 Analysis In this section, we develop an approximation and a lower bound on the socially-optimal response time by means of convex programming. Approximations and bounds on the price of anarchy immediately follow by (2).

RR n°

Anselmi and Gaujal

6

3.1

Bounding the Optimal Response Time

We now introduce a lower bound on the socially optimum response time

i i i Let qn (a1 , . . . , an ) be the amount of work in queue

i

after

n

RSo .

arrivals. We denote by

def

Qin (ai1 , . . . , ain ) = Eqni (ai1 , . . . , ain ),

(3)

the mean work where the expectation is taken over all arrival times and service times. Now, let

i Cesaro limit of Qn , i.e.,

Ri (ai )

m 1 X i i Qn (a1 , . . . , ain ). Ri (a ) = lim m→∞ m n=1 i

be the

(4)

Ri (ai ) is the mean response time of the jobs sent to queue i. P∞ (1 − δ) k=1 δ k−1 aik , that exists since all ain are bounded. By denition of piδ , L of limit points of (p1δ , . . . , pN δ ), when δ → 1, also has a sum equal to one.

Using the PASTA property, e.g., [7],

def 1, let piδ =

0