Gilboa-Freedman, G.

Hassin, R.

Kerner, Y.

English

Pure OAI Repository

 Price of anarchy in the Markovian single server queueCitation for published version (APA):Gilboa-Freedman, G., Hassin, R., & Kerner, Y. (2008). Price of anarchy in the Markovian single server queue.(Report Eurandom; Vol. 2008043). Eurandom.Document status and date:Published: 01/01/2008Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 04. Oct. 2023Price of Anarchy in the Markovian Single ServerQueueGail Gilboa-Freedman,* Refael Hassin∗, Yoav Kerner†AbstractThe Price of Anarchy (PoA) is the ratio of optimal and equilibriumsocial welfare. We consider the most fundamental queueing model in-volving customers’ decisions, as studied by Naor (1969). We show thatthe PoA has an odd behavior as a function of the system’s congestion.In particular, when the ratio of the potential arriving rate and the ser-vice rate is at most 0.98, PoA < 1.5. As the arrival rate comes close tothe service rate, the bound on the PoA increases to 2, and when theratio of the potential arriving rate and the service rate exceeds 1, PoAis unbounded.1 IntroductionCongested systems in general, and queueing systems in particular, may in-volve decision making by individuals that affect not only the welfare of thosewho act but also the welfare of other individuals in the system. Individu-als are motivated then to act strategically in order to maximize their ownwelfare given the strategies adopted by others. The most common solutionconcept in such models is the Nash equilibrium. The equilibrium solution isoften different from the solution which maximizes overall social welfare. Acommon measure to this inefficiency is the Price of Anarchy (PoA), i.e., theratio between the social welfare under optimum and the social welfare underequilibrium1. The PoA has been studied in many settings, see for example[1, 7, 9].This paper is the first to investigate the PoA in the most fundamentalqueueing model that involves customers’ decision making. The equilibriumand socially optimal behavior in this model were analyzed in the seminalpaper of Naor [6]. The model assumes a First-Come First-Served queue,∗Department of Statistics and Operation Research, Tel Aviv University, Israel. Emails:gailgf@post.tau.ac.il, hassin@post.tau.ac.il†EURANDOM, Eindhoven University of Technology, The Netherlands.Email:kerner@eurandom.tue.nl.1When the equilibrium is not unique, the convention is inserting the worse equilibrium,from the society’s point of view.1observable by the decision maker, with a single server, Poisson arrivals, ex-ponential service times, linear waiting costs and fixed rewards from obtainingservice. Balking is associated with zero reward. The equilibrium solution inthis model is very simple since there exists a dominant pure threshold strat-egy. Namely, for some integer n, an arriving customer joins the queue if andonly if the observed queue length upon arrival is shorter than n, and thisstrategy maximizes the individual’s expected welfare no matter what strate-gies are adopted by the others. The optimal behavior is also characterizedby a threshold strategy. Moreover Naor observed that its threshold param-eter is in general smaller than that of the equilibrium strategy. This resultis robust in the sense that it also holds for more general queuing models ,[4], [5] and [8].Following Naor’s notation (see also [2]), we denote the arrival rate byλ , and the service rate by µ . There is a reward of value R the customerobtains upon completing service, and a cost of C per unit of time spentwaiting or in service. The system’s utilization factor is ρ = λµ , and we useνs = RµC to denote the value of service in terms of expected waiting costduring a service duration.2The model’s parameters can be normalized so that the only relevantparameters are ρ and νs. Note that the model doesn’t need to assume ρ < 1for stability, and in fact this number is of no significance in Naor’s results.It comes therefore as a surprise that if we impose no restrictions on νs thenPoA is bounded if and only if ρ ≤ 1. The behavior of PoA as a functionof the utilization factor ρ can be roughly separated into three regions. Asmentioned, for ρ > 1 PoA is unbounded. We derive the exact value of PoAfor ρ < 0.9818, and show that everywhere in this range PoA < 1.48635. Themost interesting behavior of PoA is observed in the range 0.9818 < ρ < 1,where it sharply increases but remains bounded by 2.One can summarize therefore, that since in most real situations ρ < 0.98,the price of anarchy in the single server queue is relatively small, comparedwith other models discussed in the literature.Finally, we mention that the only paper that deals with PoA in a queuingsystem is a study by Haviv and Roughgarden [3], who consider a multi-server queueing system with exponential service times, and show that thePoA is bounded from above by the number of servers used under socialoptimization.2 PoA general behaviorThe equilibrium and optimal thresholds, ne and n∗ respectively, and theirassociated social welfare, have been studied by Naor [6]. It is straightforward2To avoid triviality, νs ≥ 1. Otherwise, an arriving customer would balk even if thesystem is empty.2thatne =⌊RµC⌋= bνsc.Defineg(ν) =ν(1− ρ)− ρ(1− ρν)(1− ρ)2,then n∗ = bν∗c, where ν∗ is the unique solution tog(ν) = νs. (1)Naor showed that n∗ ≤ ne. Moreover, n∗ = ne if and only if ne = 1.The social welfare associated with a threshold n isSn = Rλ1− ρn1− ρn+1− C[ρ1− ρ− (n+ 1)ρn+11− ρn+1].PoA is defined as the ratio of Sn∗ and Sne , i.e.,PoA(ρ, νs) =1−ρn∗1−ρn∗+1 −1νs[11−ρ −(n∗+1)ρn∗1−ρn∗+1]1−ρbνsc1−ρbνsc+1 −1νs[11−ρ −(bνsc+1)ρbνsc1−ρbνsc+1] (2)=pjoin(n∗)− 1νs q(n∗)pjoin(ne)− 1νs q(ne), (3)where q(n) is the expected queue length under a threshold n, and pjoin(n)is the probability that an arriving customer joins the queue.PoA(ρ, νs) is shown in Figures 1 and 2. In the following subsections, wetake a closer look at these graphs.3Figure 1: PoA Vs. ρ and νsFigure 2: PoA Vs. ρ and νs - zoom in43 PoA as a function of νsFor ρ < 1, we observe two local maxima, one of them when νs = 2. It isdemonstrated in Figure 3 for three different values of ρ. In the next sectionwe’ll show that, except for a small range of ρ values near but below 1, theglobal maximum is achieved when νs = 2.Figure 3: PoA(νs) for various values of ρFor ρ ≥ 1 the PoA is monotone increasing when νs → ∞. The asymp-totic behavior is described by the following lemmas:Lemma 3.1. limνs→∞ PoA(1, νs) = 2.Proof. For ρ = 1, the queue length distribution is uniform on 0, 1, . . . , n andin particular, q(n) = n/2 and pjoin = n/(n+ 1).Consider the functionh(x) =xx+ 1− x2νs.A continuous (with respect to x) analysis of this expression yields that it ismaximized by x =√2νs − 1. Thus, Sn∗ = h(n∗) ≤ h(√2νs − 1).Also, by n∗ < νs and concavity of h(x), Sne = h(bνsc) ≥ h(νs).Therefore,PoA(1, νs) ≤√2νs−1√2νs−√2νs−12νsνsνs+1− 12, (4)which tends to 2 when νs →∞.Lemma 3.2. ∀ρ > 1,PoA(ρ, νs)→∞ for νs →∞.5Proof. When ρ > 1 and when νs →∞, by (1)n∗ = logρ νs + o(logρ νs)By (2)PoA(ρ, νs) ∼1−ρlogρ νs1−ρlogρ νs+1− 1νs[11−ρ −(logρ νs+1)ρlogρ νs1−ρlogρ νs+1]1−ρbνsc1−ρbνsc+1 −1νs[11−ρ −(bνsc+1)ρbνsc1−ρbνsc+1] .Since1−ρlogρ νs1−ρlogρ νs+1− 1νs[11−ρ −(logρ νs+1)ρlogρ νs1−ρlogρ νs+1]= 1−νs1−ρνs −1νs[11−ρ −(logρ νs+1)νs1−ρνs]∼1ρ −1νs[11−ρ +logρ νsρ]= νs(1−ρ)−ρ−(1−ρ) logρ νsνs(1−ρ)ρ ∼1ρand1−ρbνsc1−ρbνsc+1 −1νs[11−ρ −(bνsc+1)ρbνsc1−ρbνsc+1]∼ 1ρ −1νs[11−ρ +bνscρ]= νs(1−ρ)−ρ−(1−ρ)bνscνs(1−ρ)ρ =(νs−bνsc)(1−ρ)−ρνs(1−ρ)ρ <1−2ρνs(1−ρ)ρ −−−−→νs→∞0,we conclude thatPoA(ρ, νs) −−−−→νs→∞∞In contrast, ∀ρ < 1, the PoA is monotone decreasing to 1 when νs →∞.Lemma 3.3. ∀ρ < 1, limνs→∞ PoA(ρ, νs) = 1.Proof. When ρ ≤ 1, the solution of Equation (1) satisfieslimνs→∞n∗νs= 1− ρ.Substituting this relation into (2) gives:limνs→∞PoA(ρ, νs) =1−ρνs(1−ρ)1−ρνs(1−ρ)+1 −1r[11−ρ −(νs(1−ρ)+1)ρνs(1−ρ)1−ρνs(1−ρ)+1]1−ρbνsc1−ρbνsc+1 −1νs[11−ρ −(bνsc+1)ρbνsc1−ρbνsc+1] −−−−→νs→∞164 PoA as a function of ρIn this section we consider the upper envelope of the function PoA(ρ, νs) forνs ≥ 1:PoA(ρ) = supνs≥1PoA(ρ, νs).Lemma 4.1. For any ρ > 1, PoA(ρ) is unbounded.Proof. By Lemma 3.2.Remark. Under the limit case, when ρ → ∞, given any threshold n,the number of customers in the system is always n. This is because atany departure instant, an arrival occurs, due to the infinite arrival rate.The social welfare per unit of time is then Rµ − nC. Thus, n∗ = 1. Anexplanation to the latter is that in general, under optimization, customersare allowed to wait to prevent idleness of the server in the future. When thearrival rate is infinite, idleness never occurs. On the other hand, for fixedµ,C and R, ne does not depend on λ and ne remains the same. The PoA isRµ− CRµ− Cne=1− 1νs1− bνscνs=νs − 1νs − bνscνs→∞−−−−→∞Lemma 4.2. PoA(1) = 2.Proof. By (4)PoA(1, νs) ≤√2νs−1√2νs−√2νs−12νsνsνs+1− 12≤ (√2νs − 1)2(νs + 1)νs(νs − 1). (5)For νs = 1, 2, 3, the righthand side is less than 1.5, and for νs > 2 +√3 itsderivative is positive. Therefore, PoA(1, νs) is bounded by the limit whenνs goes to infinity, which is 2.The following Lemmas will be used in the proof of Theorem 4.5.Lemma 4.3. For any ρ < 1, if PoA(ρ, νs) is maximized at νs then νs is aninteger.Proof. We show that PoA is strictly monotone decreasing with νs in therange where ne is fixed, i.e. νs ∈ [n, n + 1). This range is divided to afinite number of intervals, such that in each interval n∗ is also fixed. ThePoA is continuous where n∗ changes because at these values Sn∗ = Sn∗+1.Therefore it is sufficient to show that PoA is strictly monotone decreasingwith νs in each of the intervals, where both n∗ and ne are fixed.7Consider two values ν1s < ν2s in such interval. The derivative of (3) withrespect to νs isPoA′(ρ, νs) =q(n∗)pjoin(ne)− q(ne)pjoin(n∗)ν2s[pjoin(ne)− 1νs q(ne)]2which is negative whenq(n∗)pjoin(ne) < q(ne)pjoin(n∗)or(11− ρ− (n∗ + 1)ρn∗1− ρn∗+1)(1− ρne1− ρne+1)<(11− ρ− (ne + 1)ρne1− ρne+1)(1− ρn∗1− ρn∗+1)This inequality holds when(1− ρne)(1− ρn∗+1)− (n∗ + 1)ρn∗(1− ρne)(1− ρ)−(1− ρn∗)(1− ρne+1) + (ne + 1)ρne (1− ρn∗)(1− ρ) < 0Simplifying the last expression, it is left to show thatneρne(1− ρn∗)− n∗ρn∗(1− ρne ) < 0orρ−n∗(1− ρn∗)n∗− ρ−ne(1− ρne)ne< 0which is true since n∗ < ne andρ−n(1−ρn)n is increasing with n when ρ < 1.We conclude that PoA(ρ, νs) is strictly decreasing with νs ∈ [n, n + 1]. Inparticular this means that the maximum value of PoA in this range is ob-tained at νs = n.Lemma 4.4. For ρ < 1, 1+(n+1)ρn−ρn+1(n+2)1−ρn+1 is a monotone decreasing func-tion of n.Proof. Note that1 + (n+ 1)ρn − ρn+1(n+ 2)1− ρn+1= 1+(n+ 1)(1− ρ)ρn1− ρn+1= 1+n+ 1n∑i=0ρi−n= 1+n+ 1n∑i=0ρ−i.Since ρ < 1,n∑i=0ρ−in+1 is an average of an increasing sequence, and hence it isincreasing with n. Thus, its inverse is decreasing with n.8Theorem 4.5. ∀ρ < 1, PoA(ρ) < 2.Proof. By (2) , we need to prove that ∀ ρ < 1 and r ≥ 1:1− ρn∗1− ρn∗+1− 1νs(11− ρ− (n∗ + 1) ρn∗1− ρn∗+1)≤2[1− ρbνsc1− ρbνsc+1− 1νs(11− ρ− (bνsc+ 1)ρbνsc1− ρbνsc+1)].By Lemma 4.3 it is sufficient to consider νs = bνsc. Since νs is an integerand n∗ < νs, it would be sufficient to prove that for any two integers n < m:1− ρn1− ρn+1− 1m(11− ρ− (n+ 1) ρn1− ρn+1)≤2[1− ρm1− ρm+1− 1m(11− ρ− (m+ 1) ρm1− ρm+1)].Since 1−ρn1−ρn+1 is monotone increasing in n, it is sufficient to prove1m11− ρ+1m(n+ 1)ρn1− ρn+1≤ 1− ρm1− ρm+1+2m(m+ 1)ρm1− ρm+1.Multiplying by m(1− ρ), it is sufficient to prove1 + (n+ 1)ρn − ρn+1(n+ 2)1− ρn+1≤ m+ ρm(m+ 2)1− ρm+1(1− ρ).By Lemma 4.4, the left-hand side is monotone decreasing with n. Thereforeit is sufficient to show that the inequality holds for n = 1:1 + 2ρ− 3ρ21− ρ2<m+ ρm(m+ 2)(1− ρ)1− ρm+1,or0 < 2 + (m− 3)(1 + ρ) + (m+ 1)ρm + ρm(1 + ρ)− (m− 1)ρm+2,which is true by(m+ 1)ρm > (m− 1)ρm+2.The behavior of PoA(ρ) in the range ρ < 1 is as shown in Figure 4. Onesees that except for a small range near ρ = 1, PoA(ρ) is bounded by 1.5.This bound is smaller than the bound in Theorem 4.5. However, when ρbecomes close to 1 the bound sharply increases to 2. To understand thisoddness, we examine the behavior of the PoA for various values of νs.Figure 5 demonstrates that when ρ < 0.98175, PoA(ρ) = PoA(ρ, 2). Incontrast, when ρ ∈ [0.98175, 1], PoA(ρ) is not equal to a single functionPoA(ρ, νs). Instead, there is an infinite number of functions which definethe upper envelope of the PoA in this range.9Figure 4: PoA Vs. ρFigure 5: PoA(ρ) for various values of νsObservation 4.6. If ρ ∈ [0, 0.98175], PoA(ρ, νs) ≤ 1+ρ+ρ21+ρ < 1.48635Proof. We observe that PoA(ρ) = PoA(ρ, 2) when ρ < 0.98175. Substitutingνs = 2 in (2) we havePoA(ρ, νs) ≤ 1+ρ+ρ21+ρ . We also have that PoA is uniformly bounded bythe maximum of 1+ρ+ρ21+ρ over [0, 0.98175] which is 1.48635.10References[1] M.M. Halldorsson, J.Y. Halpern, L.E. Li and V.S. Mirrokni, “On spec-trum sharing games,” Proceedings of the Twenty-Third Annual ACMSymposium on Principles of Distributed Computing 107-114 (2004).[2] R. Hassin and M. Haviv, To Queue or not to Queue: Equilibrium Be-havior in Queueing Systems, Kluwer International Series, 2003.[3] M. Haviv and T. Roughgarden, “The price of anarchy in an exponentialmulti-server,” Operations Research Letters 35 421-426, (2007).[4] N.C. Knudsen, “Bounding the inefficiency of equilibria in nonatomiccongestion games,” Econometrica 47 515-528, (1972).[5] S. A. Lippman and S. Stidham, “Individual versus social optimizationin exponential congested systems,” Operations Research 25 233-247,(1977).[6] P. Naor, “The regulation of queue size by levying tolls,” Econometrica37 15-24, (1969).[7] T. Roughgarden and E. Tardos , “Individual and social optimization ina multiserver queue with a general cost-benefit structure,” Games andEconomic Behavior 47 389-403, (2004).[8] S. Stidham, “Socially and individually optimal control of arrivals to aGI/M/1 queue,” Management Science 47 1598-1610, (1978).[9] A. Vetta, “Nash equilibria in competitive societies, with applicationsto facility location, traffic routing and auctions,” Proceedings of the43rd Annual IEEE Symposium on Foundations of Computer Science,416-425, (2002).11

Price of anarchy in the Markovian single server queue

 Price of anarchy in the Markovian single server queueCitation for published version (APA):Gilboa-Freedman, G., Hassin, R., & Kerner, Y. (2008). Price of anarchy in the Markovian single server queue.(Report Eurandom; Vol. 2008043). Eurandom.Document status and date:Published: 01/01/2008Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 05. Oct. 2023Price of Anarchy in the Markovian Single ServerQueueGail Gilboa-Freedman,* Refael Hassin∗, Yoav Kerner†AbstractThe Price of Anarchy (PoA) is the ratio of optimal and equilibriumsocial welfare. We consider the most fundamental queueing model in-volving customers’ decisions, as studied by Naor (1969). We show thatthe PoA has an odd behavior as a function of the system’s congestion.In particular, when the ratio of the potential arriving rate and the ser-vice rate is at most 0.98, PoA < 1.5. As the arrival rate comes close tothe service rate, the bound on the PoA increases to 2, and when theratio of the potential arriving rate and the service rate exceeds 1, PoAis unbounded.1 IntroductionCongested systems in general, and queueing systems in particular, may in-volve decision making by individuals that affect not only the welfare of thosewho act but also the welfare of other individuals in the system. Individu-als are motivated then to act strategically in order to maximize their ownwelfare given the strategies adopted by others. The most common solutionconcept in such models is the Nash equilibrium. The equilibrium solution isoften different from the solution which maximizes overall social welfare. Acommon measure to this inefficiency is the Price of Anarchy (PoA), i.e., theratio between the social welfare under optimum and the social welfare underequilibrium1. The PoA has been studied in many settings, see for example[1, 7, 9].This paper is the first to investigate the PoA in the most fundamentalqueueing model that involves customers’ decision making. The equilibriumand socially optimal behavior in this model were analyzed in the seminalpaper of Naor [6]. The model assumes a First-Come First-Served queue,∗Department of Statistics and Operation Research, Tel Aviv University, Israel. Emails:gailgf@post.tau.ac.il, hassin@post.tau.ac.il†EURANDOM, Eindhoven University of Technology, The Netherlands.Email:kerner@eurandom.tue.nl.1When the equilibrium is not unique, the convention is inserting the worse equilibrium,from the society’s point of view.1observable by the decision maker, with a single server, Poisson arrivals, ex-ponential service times, linear waiting costs and fixed rewards from obtainingservice. Balking is associated with zero reward. The equilibrium solution inthis model is very simple since there exists a dominant pure threshold strat-egy. Namely, for some integer n, an arriving customer joins the queue if andonly if the observed queue length upon arrival is shorter than n, and thisstrategy maximizes the individual’s expected welfare no matter what strate-gies are adopted by the others. The optimal behavior is also characterizedby a threshold strategy. Moreover Naor observed that its threshold param-eter is in general smaller than that of the equilibrium strategy. This resultis robust in the sense that it also holds for more general queuing models ,[4], [5] and [8].Following Naor’s notation (see also [2]), we denote the arrival rate byλ , and the service rate by µ . There is a reward of value R the customerobtains upon completing service, and a cost of C per unit of time spentwaiting or in service. The system’s utilization factor is ρ = λµ , and we useνs = RµC to denote the value of service in terms of expected waiting costduring a service duration.2The model’s parameters can be normalized so that the only relevantparameters are ρ and νs. Note that the model doesn’t need to assume ρ < 1for stability, and in fact this number is of no significance in Naor’s results.It comes therefore as a surprise that if we impose no restrictions on νs thenPoA is bounded if and only if ρ ≤ 1. The behavior of PoA as a functionof the utilization factor ρ can be roughly separated into three regions. Asmentioned, for ρ > 1 PoA is unbounded. We derive the exact value of PoAfor ρ < 0.9818, and show that everywhere in this range PoA < 1.48635. Themost interesting behavior of PoA is observed in the range 0.9818 < ρ < 1,where it sharply increases but remains bounded by 2.One can summarize therefore, that since in most real situations ρ < 0.98,the price of anarchy in the single server queue is relatively small, comparedwith other models discussed in the literature.Finally, we mention that the only paper that deals with PoA in a queuingsystem is a study by Haviv and Roughgarden [3], who consider a multi-server queueing system with exponential service times, and show that thePoA is bounded from above by the number of servers used under socialoptimization.2 PoA general behaviorThe equilibrium and optimal thresholds, ne and n∗ respectively, and theirassociated social welfare, have been studied by Naor [6]. It is straightforward2To avoid triviality, νs ≥ 1. Otherwise, an arriving customer would balk even if thesystem is empty.2thatne =⌊RµC⌋= bνsc.Defineg(ν) =ν(1− ρ)− ρ(1− ρν)(1− ρ)2,then n∗ = bν∗c, where ν∗ is the unique solution tog(ν) = νs. (1)Naor showed that n∗ ≤ ne. Moreover, n∗ = ne if and only if ne = 1.The social welfare associated with a threshold n isSn = Rλ1− ρn1− ρn+1− C[ρ1− ρ− (n+ 1)ρn+11− ρn+1].PoA is defined as the ratio of Sn∗ and Sne , i.e.,PoA(ρ, νs) =1−ρn∗1−ρn∗+1 −1νs[11−ρ −(n∗+1)ρn∗1−ρn∗+1]1−ρbνsc1−ρbνsc+1 −1νs[11−ρ −(bνsc+1)ρbνsc1−ρbνsc+1] (2)=pjoin(n∗)− 1νs q(n∗)pjoin(ne)− 1νs q(ne), (3)where q(n) is the expected queue length under a threshold n, and pjoin(n)is the probability that an arriving customer joins the queue.PoA(ρ, νs) is shown in Figures 1 and 2. In the following subsections, wetake a closer look at these graphs.3Figure 1: PoA Vs. ρ and νsFigure 2: PoA Vs. ρ and νs - zoom in43 PoA as a function of νsFor ρ < 1, we observe two local maxima, one of them when νs = 2. It isdemonstrated in Figure 3 for three different values of ρ. In the next sectionwe’ll show that, except for a small range of ρ values near but below 1, theglobal maximum is achieved when νs = 2.Figure 3: PoA(νs) for various values of ρFor ρ ≥ 1 the PoA is monotone increasing when νs → ∞. The asymp-totic behavior is described by the following lemmas:Lemma 3.1. limνs→∞ PoA(1, νs) = 2.Proof. For ρ = 1, the queue length distribution is uniform on 0, 1, . . . , n andin particular, q(n) = n/2 and pjoin = n/(n+ 1).Consider the functionh(x) =xx+ 1− x2νs.A continuous (with respect to x) analysis of this expression yields that it ismaximized by x =√2νs − 1. Thus, Sn∗ = h(n∗) ≤ h(√2νs − 1).Also, by n∗ < νs and concavity of h(x), Sne = h(bνsc) ≥ h(νs).Therefore,PoA(1, νs) ≤√2νs−1√2νs−√2νs−12νsνsνs+1− 12, (4)which tends to 2 when νs →∞.Lemma 3.2. ∀ρ > 1,PoA(ρ, νs)→∞ for νs →∞.5Proof. When ρ > 1 and when νs →∞, by (1)n∗ = logρ νs + o(logρ νs)By (2)PoA(ρ, νs) ∼1−ρlogρ νs1−ρlogρ νs+1− 1νs[11−ρ −(logρ νs+1)ρlogρ νs1−ρlogρ νs+1]1−ρbνsc1−ρbνsc+1 −1νs[11−ρ −(bνsc+1)ρbνsc1−ρbνsc+1] .Since1−ρlogρ νs1−ρlogρ νs+1− 1νs[11−ρ −(logρ νs+1)ρlogρ νs1−ρlogρ νs+1]= 1−νs1−ρνs −1νs[11−ρ −(logρ νs+1)νs1−ρνs]∼1ρ −1νs[11−ρ +logρ νsρ]= νs(1−ρ)−ρ−(1−ρ) logρ νsνs(1−ρ)ρ ∼1ρand1−ρbνsc1−ρbνsc+1 −1νs[11−ρ −(bνsc+1)ρbνsc1−ρbνsc+1]∼ 1ρ −1νs[11−ρ +bνscρ]= νs(1−ρ)−ρ−(1−ρ)bνscνs(1−ρ)ρ =(νs−bνsc)(1−ρ)−ρνs(1−ρ)ρ <1−2ρνs(1−ρ)ρ −−−−→νs→∞0,we conclude thatPoA(ρ, νs) −−−−→νs→∞∞In contrast, ∀ρ < 1, the PoA is monotone decreasing to 1 when νs →∞.Lemma 3.3. ∀ρ < 1, limνs→∞ PoA(ρ, νs) = 1.Proof. When ρ ≤ 1, the solution of Equation (1) satisfieslimνs→∞n∗νs= 1− ρ.Substituting this relation into (2) gives:limνs→∞PoA(ρ, νs) =1−ρνs(1−ρ)1−ρνs(1−ρ)+1 −1r[11−ρ −(νs(1−ρ)+1)ρνs(1−ρ)1−ρνs(1−ρ)+1]1−ρbνsc1−ρbνsc+1 −1νs[11−ρ −(bνsc+1)ρbνsc1−ρbνsc+1] −−−−→νs→∞164 PoA as a function of ρIn this section we consider the upper envelope of the function PoA(ρ, νs) forνs ≥ 1:PoA(ρ) = supνs≥1PoA(ρ, νs).Lemma 4.1. For any ρ > 1, PoA(ρ) is unbounded.Proof. By Lemma 3.2.Remark. Under the limit case, when ρ → ∞, given any threshold n,the number of customers in the system is always n. This is because atany departure instant, an arrival occurs, due to the infinite arrival rate.The social welfare per unit of time is then Rµ − nC. Thus, n∗ = 1. Anexplanation to the latter is that in general, under optimization, customersare allowed to wait to prevent idleness of the server in the future. When thearrival rate is infinite, idleness never occurs. On the other hand, for fixedµ,C and R, ne does not depend on λ and ne remains the same. The PoA isRµ− CRµ− Cne=1− 1νs1− bνscνs=νs − 1νs − bνscνs→∞−−−−→∞Lemma 4.2. PoA(1) = 2.Proof. By (4)PoA(1, νs) ≤√2νs−1√2νs−√2νs−12νsνsνs+1− 12≤ (√2νs − 1)2(νs + 1)νs(νs − 1). (5)For νs = 1, 2, 3, the righthand side is less than 1.5, and for νs > 2 +√3 itsderivative is positive. Therefore, PoA(1, νs) is bounded by the limit whenνs goes to infinity, which is 2.The following Lemmas will be used in the proof of Theorem 4.5.Lemma 4.3. For any ρ < 1, if PoA(ρ, νs) is maximized at νs then νs is aninteger.Proof. We show that PoA is strictly monotone decreasing with νs in therange where ne is fixed, i.e. νs ∈ [n, n + 1). This range is divided to afinite number of intervals, such that in each interval n∗ is also fixed. ThePoA is continuous where n∗ changes because at these values Sn∗ = Sn∗+1.Therefore it is sufficient to show that PoA is strictly monotone decreasingwith νs in each of the intervals, where both n∗ and ne are fixed.7Consider two values ν1s < ν2s in such interval. The derivative of (3) withrespect to νs isPoA′(ρ, νs) =q(n∗)pjoin(ne)− q(ne)pjoin(n∗)ν2s[pjoin(ne)− 1νs q(ne)]2which is negative whenq(n∗)pjoin(ne) < q(ne)pjoin(n∗)or(11− ρ− (n∗ + 1)ρn∗1− ρn∗+1)(1− ρne1− ρne+1)<(11− ρ− (ne + 1)ρne1− ρne+1)(1− ρn∗1− ρn∗+1)This inequality holds when(1− ρne)(1− ρn∗+1)− (n∗ + 1)ρn∗(1− ρne)(1− ρ)−(1− ρn∗)(1− ρne+1) + (ne + 1)ρne (1− ρn∗)(1− ρ) < 0Simplifying the last expression, it is left to show thatneρne(1− ρn∗)− n∗ρn∗(1− ρne ) < 0orρ−n∗(1− ρn∗)n∗− ρ−ne(1− ρne)ne< 0which is true since n∗ < ne andρ−n(1−ρn)n is increasing with n when ρ < 1.We conclude that PoA(ρ, νs) is strictly decreasing with νs ∈ [n, n + 1]. Inparticular this means that the maximum value of PoA in this range is ob-tained at νs = n.Lemma 4.4. For ρ < 1, 1+(n+1)ρn−ρn+1(n+2)1−ρn+1 is a monotone decreasing func-tion of n.Proof. Note that1 + (n+ 1)ρn − ρn+1(n+ 2)1− ρn+1= 1+(n+ 1)(1− ρ)ρn1− ρn+1= 1+n+ 1n∑i=0ρi−n= 1+n+ 1n∑i=0ρ−i.Since ρ < 1,n∑i=0ρ−in+1 is an average of an increasing sequence, and hence it isincreasing with n. Thus, its inverse is decreasing with n.8Theorem 4.5. ∀ρ < 1, PoA(ρ) < 2.Proof. By (2) , we need to prove that ∀ ρ < 1 and r ≥ 1:1− ρn∗1− ρn∗+1− 1νs(11− ρ− (n∗ + 1) ρn∗1− ρn∗+1)≤2[1− ρbνsc1− ρbνsc+1− 1νs(11− ρ− (bνsc+ 1)ρbνsc1− ρbνsc+1)].By Lemma 4.3 it is sufficient to consider νs = bνsc. Since νs is an integerand n∗ < νs, it would be sufficient to prove that for any two integers n < m:1− ρn1− ρn+1− 1m(11− ρ− (n+ 1) ρn1− ρn+1)≤2[1− ρm1− ρm+1− 1m(11− ρ− (m+ 1) ρm1− ρm+1)].Since 1−ρn1−ρn+1 is monotone increasing in n, it is sufficient to prove1m11− ρ+1m(n+ 1)ρn1− ρn+1≤ 1− ρm1− ρm+1+2m(m+ 1)ρm1− ρm+1.Multiplying by m(1− ρ), it is sufficient to prove1 + (n+ 1)ρn − ρn+1(n+ 2)1− ρn+1≤ m+ ρm(m+ 2)1− ρm+1(1− ρ).By Lemma 4.4, the left-hand side is monotone decreasing with n. Thereforeit is sufficient to show that the inequality holds for n = 1:1 + 2ρ− 3ρ21− ρ2<m+ ρm(m+ 2)(1− ρ)1− ρm+1,or0 < 2 + (m− 3)(1 + ρ) + (m+ 1)ρm + ρm(1 + ρ)− (m− 1)ρm+2,which is true by(m+ 1)ρm > (m− 1)ρm+2.The behavior of PoA(ρ) in the range ρ < 1 is as shown in Figure 4. Onesees that except for a small range near ρ = 1, PoA(ρ) is bounded by 1.5.This bound is smaller than the bound in Theorem 4.5. However, when ρbecomes close to 1 the bound sharply increases to 2. To understand thisoddness, we examine the behavior of the PoA for various values of νs.Figure 5 demonstrates that when ρ < 0.98175, PoA(ρ) = PoA(ρ, 2). Incontrast, when ρ ∈ [0.98175, 1], PoA(ρ) is not equal to a single functionPoA(ρ, νs). Instead, there is an infinite number of functions which definethe upper envelope of the PoA in this range.9Figure 4: PoA Vs. ρFigure 5: PoA(ρ) for various values of νsObservation 4.6. If ρ ∈ [0, 0.98175], PoA(ρ, νs) ≤ 1+ρ+ρ21+ρ < 1.48635Proof. We observe that PoA(ρ) = PoA(ρ, 2) when ρ < 0.98175. Substitutingνs = 2 in (2) we havePoA(ρ, νs) ≤ 1+ρ+ρ21+ρ . We also have that PoA is uniformly bounded bythe maximum of 1+ρ+ρ21+ρ over [0, 0.98175] which is 1.48635.10References[1] M.M. Halldorsson, J.Y. Halpern, L.E. Li and V.S. Mirrokni, “On spec-trum sharing games,” Proceedings of the Twenty-Third Annual ACMSymposium on Principles of Distributed Computing 107-114 (2004).[2] R. Hassin and M. Haviv, To Queue or not to Queue: Equilibrium Be-havior in Queueing Systems, Kluwer International Series, 2003.[3] M. Haviv and T. Roughgarden, “The price of anarchy in an exponentialmulti-server,” Operations Research Letters 35 421-426, (2007).[4] N.C. Knudsen, “Bounding the inefficiency of equilibria in nonatomiccongestion games,” Econometrica 47 515-528, (1972).[5] S. A. Lippman and S. Stidham, “Individual versus social optimizationin exponential congested systems,” Operations Research 25 233-247,(1977).[6] P. Naor, “The regulation of queue size by levying tolls,” Econometrica37 15-24, (1969).[7] T. Roughgarden and E. Tardos , “Individual and social optimization ina multiserver queue with a general cost-benefit structure,” Games andEconomic Behavior 47 389-403, (2004).[8] S. Stidham, “Socially and individually optimal control of arrivals to aGI/M/1 queue,” Management Science 47 1598-1610, (1978).[9] A. Vetta, “Nash equilibria in competitive societies, with applicationsto facility location, traffic routing and auctions,” Proceedings of the43rd Annual IEEE Symposium on Foundations of Computer Science,416-425, (2002).11

Price of anarchy in the Markovian single server queue

Abstract

Similar works

Full text

Available Versions

Pure OAI Repository

Pure OAI Repository

Repository TU/e