×

Strategic equilibrium versus global optimum for a pair of competing servers. (English) Zbl 1169.90333

Summary: D. Christ and B. Avi-Itzhak [Manage. Sci. 48, No. 6, 813–820 (2002)] analyzed a queueing system with two competing servers who determine their service rates so as to optimize their individual utilities. The system is formulated as a two-person game; Christ and Avi-Itzhak (loc. cit.) proved the existence of a unique Nash equilibrium which is symmetric. In this paper, we explore globally optimal solutions. We prove that the unique Nash equilibrium is generally strictly inferior to a globally optimal solution and that optimal solutions are symmetric and require the servers to adopt service rates that are smaller than those occurring in equilibrium. Furthermore, given a symmetric globally optimal solution, we show how to impose linear penalties on the service rates so that the given optimal solution becomes a unique Nash equilibrium. When service rates are not observable, we show how the same effect is achieved by imposing linear penalties on a corresponding signal.

MSC:

90B22 Queues and service in operations research
60K25 Queueing theory (aspects of probability theory)
68M20 Performance evaluation, queueing, and scheduling in the context of computer systems
91A05 2-person games
91B52 Special types of economic equilibria
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Beckmann, M., McGuire, C. B. and Winsten, C. B. (1956). Studies in the Economics of Transportation . Yale University Press.
[2] Cachon, G. P. and Netessine, S. (2004). Game theoretic applications in supply chain analysis. In Handbook of Quantitative Supply Chain Analysis: Modeling in the eBusiness Era , eds S. D. Wu, D. Simchi-Levi and Z. Shen, Kluwer, Dordrecht, pp. 13–66. · Zbl 1138.91365
[3] Christ, D. and Avi-Itzhak, B. (2002). Strategic equilibrium for a pair of competing servers with convex cost and balking. Manag. Sci. 48 , 813–820. · Zbl 1232.90141 · doi:10.1287/mnsc.48.6.813.191
[4] Golany, B. and Rothblum, U. G. (2006). Inducing coordination in supply chains through linear reward schemes. Naval Res. Logistics 53 , 1–15. · Zbl 1112.90040 · doi:10.1002/nav.20117
[5] Hardin, G. (1968). The tragedy of the commons. Science 162 , 1243–1248.
[6] Hassin, R. and Haviv, M. (2003). To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems . Kluwer, Norwell, MA. · Zbl 1064.60002
[7] Johari, R. and Tsitsiklis, J. N. (2004). Efficiency loss in a network resource allocation game. Math. Operat. Res. 29 , 407–435. · Zbl 1082.90015 · doi:10.1287/moor.1040.0091
[8] Kalai, E., Kamien, M. I. and Rubinovitz, M. (1992). Optimal service speeds in a competitive environment. Manag. Sci. 38 , 1154–1163. · Zbl 0767.90016 · doi:10.1287/mnsc.38.8.1154
[9] Naor, P. (1969). The regulation of queue size by levying tolls. Econometrica 37 , 15–24. · Zbl 0172.21801 · doi:10.2307/1909200
[10] Rapoport, A. and Chammah, A. M. (1965). Prisoner’s Dilemma . University of Michigan Press.
[11] Rothblum, U. G. (2005). Optimality vs. equilibrium: inducing stability by linear rewards and penalties. Unpublished manuscript.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.