Computers & Operations Research 37 (2010) 1181--1190 Contents lists available at ScienceDirect Computers & Operations Research journal homepage: www.elsevier.com/locate/cor A repairable queueing model with two-phase service, start-up times and retrial customers Ioannis Dimitriou ∗ , Christos Langaris Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece ARTICLE INFO ABSTRACT Available online 19 March 2009 Keywords: Poisson arrivals Two-phase service Retrial queue Breakdowns Repairs Start-up time Vacation A repairable queueing model with a two-phase service in succession, provided by a single server, is inves- tigated. Customers arrive in a single ordinary queue and after the completion of the first phase service, either proceed to the second phase or join a retrial box from where they retry, after a random amount of time and independently of the other customers in orbit, to find a position for service in the second phase. Moreover, the server is subject to breakdowns and repairs in both phases, while a start-up time is needed in order to start serving a retrial customer. When the server, upon a service or a repair completion finds no customers waiting to be served, he departs for a single vacation of an arbitrarily distributed length. The arrival process is assumed to be Poisson and all service and repair times are arbitrarily distributed. For such a system the stability conditions and steady state analysis are investigated. Numerical results are finally obtained and used to investigate system performance. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction The main characteristics of the queueing model analysed in this paper are (i) the retrial customers (jobs), (ii) the server breakdowns and repairs, (iii) the two-phase service and (iv) the start-up (system preparation) times. Queueing systems with repeated attempts (retrials) are charac- terized by the feature that an arriving customer who finds the server unavailable, leaves the system, joins a pool of unsatisfied customers, the so-called retrial box, and repeats his demand for service after a random amount of time. Retrial queues have been widely used to model many problems in telephone switching systems, telecommu- nications networks and computer units. For a complete survey on this topic we refer Artalejo [3], Kulkarni and Liang [17], and the books of Falin and Templeton [13], and Artalejo and Gomez-Corral [5]. In most of the queueing literature, the server is assumed to be reliable and always available to customers. However, in practice, we often meet cases where the server may breakdown and has to be repaired. In queueing literature, there have been several works taking into account both retrial phenomenon and server breakdowns with repairs. As related works we mention the papers by Aissani [1], Aissani and Artalejo [2], Kulkarni and Choi [16], Wang et al. [25], Kumar et al. [20]. ∗ Corresponding author. E-mail addresses: dimitriougiannis@yahoo.gr (I. Dimitriou), clagar@cc.uoi.gr (C. Langaris). 0305-0548/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2009.03.003 The assumption of a two-phase service provided by a single server has been proved useful to analyse many practical situations arising in packet transmissions, multimedia communications, central pro- cessors, etc. Such kind of systems have been discussed for the first time by Krishna and Lee [15] and Doshi [12], and more recently have been generalized to include models with vacations, N-policy, etc. (see [6,9,14]). Wang [24], considered a two-phase queueing model with the as- sumptions of breakdowns and repairs, in which he assumed that the second optional service follows an exponential distribution. Kumar et al. [18], Artalejo and Choudhury [4], and Choudhury [7] are the first who imposed the concept of retrial customers in the two-phase models. Kumar et al. [18] generalize their previous work of a single service station [19] by considering now a two-phase service system where an arriving customer who finds the server unavailable joins the retrial box from where only the first customer can retry for ser- vice after an arbitrarily distributed time period while in the work of Choudhury [7] the investigated model includes Bernoulli server va- cations and linear retrial policy. The common feature of the above papers is that there are no server breakdowns, no ordinary queue and all waiting customers join the retrial box. Choudhury and Deka [8], generalize the works by Wang [24] and Artalejo and Choud- hury [4] by considering an M/G/1 retrial queue with second optional service channel which is subject to server breakdowns and repairs. Wang and Li [26] consider a similar model, where only the first re- trial customer can retry for service after an arbitrarily distributed time period.