Operations Research Letters 36 (2008) 696–699 Contents lists available at ScienceDirect Operations Research Letters journal homepage: www.elsevier.com/locate/orl Equilibrium balking strategies in the observable single-server queue with breakdowns and repairs Antonis Economou ∗ , Spyridoula Kanta University of Athens, Greece article info Article history: Received 24 April 2008 Accepted 27 June 2008 Available online 18 July 2008 Keywords: Queueing Balking Unreliable server Equilibrium strategies Partial information abstract We consider the Markovian single-server queue that alternates between on and off periods. Upon arriving, the customers observe the queue length and decide whether to join or balk. We derive equilibrium threshold balking strategies in two cases, according to the information for the server’s state. © 2008 Elsevier B.V. All rights reserved. 1. Introduction During the last few decades, there has been an emerging interest in the economic analysis of queueing systems. These ideas go back at least to the pioneering works of Naor [8] and Edelson and Hildebrand [2]. In such an economic analysis, the customers are allowed to take their own decisions and therefore the system can be modeled as a game among the customers. The fundamental problem is to identify the Nash equilibria. For an introduction to this area see [5]. A significant part of the literature in the game theoretic analysis of queueing systems is devoted to two-dimensional Markov models (see e.g. [3,4,1]). In such models, the state of the system is represented by two variables that correspond to the number of customers and some additional information. The analysis of systems with general service times is very involved (see e.g. the analysis of the M/G/1 queue in [7,6]). Hence, the analysis of two- dimensional systems is limited to the Markovian case in almost all of the relevant papers. In the present paper we consider the Markovian single-server queue with an unreliable server. The customers observe the queue length and then decide whether to join or balk. We consider separately two information cases and we identify the (Nash) equilibrium balking strategies. ∗ Corresponding address: University of Athens, Department of Mathematics, Section of Statistics and Operations Research, Panepistemioupolis, Athens 15784, Greece. E-mail addresses: aeconom@math.uoa.gr (A. Economou), spkanta@math.uoa.gr (S. Kanta). The paper is organized as follows. In Section 2, we describe the model and the decision structure. In Section 3, we identify the equilibrium strategies. Finally, in Section 4, we discuss some qualitative implications and we point to possible generalizations. 2. The model We consider the single-server queue with an infinite waiting room, subject to a Poisson arrival process with rate λ. We assume that the service times are exponentially distributed with rate μ. The server alternates between on and off periods that are also exponentially distributed at rates ζ and θ respectively. We represent the state of the system at time t by a pair (N (t ), I (t )), where N (t ) and I (t ) denote the number of customers and the state of the server (0: off, 1: on) respectively. The process {(N (t ), I (t )) : t ≥ 0} is a continuous time Markov chain with non- zero transition rates given by q (n,i)(n+1,i) = λ, n = 0, 1, 2,... and i = 0, 1 q (n,1)(n−1,1) = μ, n = 1, 2, 3,... q (n,0)(n,1) = θ, n = 0, 1, 2,... q (n,1)(n,0) = ζ, n = 0, 1, 2, .... We assume that the customers are allowed to decide whether to join or balk upon their arrival. Every customer receives a reward of R units for completing service. Moreover, there exists a waiting cost of C units per time unit that is continuously accumulated from the time that she/he arrives at the system till the time she/he leaves after being served. Customers want to maximize their expected net benefit. Their decisions are taken only at their arrival instants and they are irrevocable in the sense that retrials of balking customers and reneging of entering customers are not allowed. 0167-6377/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2008.06.006