Binomial Schedule for an M/G/1 Type Queueing System with an Unreliable Server under -Policy

We consider in this paper an M/G/1 type queueing system with the following extensions. First, the server is unreliable and is subject to random breakdowns. Second, the server also implements the well-known -policy. Third, instead of a Bernoulli vacation schedule, the more general notion of binomial schedule with vacations is applied. A cost function with two decision variables is developed. A numerical example shows the effect of the system parameters on the optimal management policy. 1. Introduction Queueing systems where the server uses her/his idle time to perform some secondary job such as maintenance are called systems with server vacations. These systems have received a lot of attention due to their wide applications in different domains such as telecommunications, computer systems, service systems, and production and quality control problems. Survey papers have been written on this subject; the most recent one being that of Ke et al. [1]. Keilson and Servi [2] introduced a class of vacation models called the Bernoulli vacation schedule. When a customer has just been served and other customers are present, the server serves the next customer in line with probability or takes a vacation of random duration with probability . The Bernoulli vacation schedule has been extensively considered. Among the most recent references we cite Kumar et al. [3], Choudhury and Ke [4], Gao and Liu [5], Tao et al. [6, 7], and Wu and Lian [8]. Kella [9] generalized the Bernoulli vacation schedule to a more general scheme according to which the server goes on consecutive vacations with probability if the queue upon her/his return is empty. Ba-Rukab et al. [10] propose another generalization. They argued that since the server may attend different activities while idle, a binomial vacation schedule may be more appropriate than a Bernoulli vacation schedule. In that case, instead of taking just one vacation, the server may take many vacations, for a maximum number of, say, vacations. Yadin and Naor introduced the -policy in which, following an idle period, the server resumes his service only when the number of waiting customers reaches the level . This policy is efficient in that it reduces setup costs. The -policy too has been extensively studied by researches. We refer the reader to the following recent references: Kumar and Jain [11], Lee and Yang [12], Lim et al. [13], and Wei et al. [14]. Another characteristic of servers in a queueing system is that they may break down while providing service. White and Christie [15] were the first to study a queueing system with an