The paper addresses monotonicity properties of the single server retrial queue with no waiting room and server subject to active breakdowns. The obtained results allow us to place in a prominent position the insensitive bounds for the stationary distribution of the embedded Markov chain related to the model in the study. Numerical illustrations are provided to support the results. 1. Introduction Queueing systems with repeated attempts have been widely used to model many problems in telecommunication and computer systems [1–3]. The essential feature of a retrial queue is that arriving customers who find all servers busy are obliged to abandon the service area and join a retrial group, called orbit, in order to try their luck again after some random time. For a detailed review of the main results and the literature on this topic the reader is referred to the monographs [4, 5]. In recent years, there has been an increasing interest in the investigation of the retrial phenomenon in cellular mobile network, see [6–10] and the references therein, and in many other telecommunication systems including starlike local area networks [11], wavelength-routed optical networks [12], circuit-switched systems with hybrid fiber-coax architecture [13], and wireless sensor networks [14]. On the other hand, in most of the queueing literature, the server is assumed to be always available, although this assumption is evidently unrealistic. In fact, queueing systems with server breakdowns are very common in communication systems and manufacturing systems, the machine may break down due to the machine or job related problems. This results in a period of unavailable time until the servers are repaired. Such a system with repairable server has been studied as a queueing model and a reliability model by many authors. Aissani [15] studies the influence of the reliability of the communication line on the distribution of the number of customers in the retrial queues. A generalization of the well-known Pollaczek-Khinchin formula is given for this case. Aissani [16] considers a retrial queue with redundancy and unreliable server. Dudin [17] treats a problem similar to [16], and the problem of redundancy and related control problem are also discussed. Djellab [18] studies a system with breakdowns in heavy traffic. Kumar et al. [19] consider an retrial queue with feedback and starting failure, which occurs in the startup period and its repair can be interpreted as a warm up period (the server is unavailable to customers). Retrial queues with a server subject to breakdowns and repairs
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