Note on a Binomial Schedule for an MX/G/1 Queueing System with an Unreliable Server

We consider in this paper a batch arrival queueing system with an unreliable server. If the queue is empty at a service completion, then the server becomes inactive and begins an idle period. However, if the queue is not empty, then the server will take at most vacation before serving the next customer. A linear cost structure is developed for the system and the optimal value of is obtained. 1. Introduction As mentioned by Hlynka [1], “the use of queueing theory in the analysis and management of computer and telecommunication systems has by now a lengthy history, with many important contributions in the design and analysis of such systems. There appear to be no signs of this letting up opportunities abound for queueing to aid in developments in wireless networking, cloud computing, social networking, and many other modern application areas.” The part of queueing theory concerned with the optimization of a queueing system is called the optimal design and control of queues. For a comprehensive survey of the research on this topic, see the book by Stidham [2] and the survey by Tadj and Choudhury [3]. The current paper presents yet another model that combines many known features to accommodate the increasingly complex computer networks and telecommunication systems. Meaningful and systematic scheduling requires sophisticated models to allocate the sometimes scarce available resources. Since most of these traits are standard and can be found in many papers, our review of the literature will make do with citations of surveys where the reader can find the relevant references. The first feature of our model is the batch arrival process. Batch arrival models have been extensively used in an uncountable number of papers. A topical textbook is that of Chaudhry and Templeton [4]. The second feature of the model considered in this paper is that of an unreliable server. Indeed, a server, such as a machine, may break down while providing service. The service of the customer being served is then interrupted and cannot resume until the server is repaired. This is a very realistic assumption that models real-life situations. For a review of the research on this topic, the reader is referred to the broad survey of Tadj et al. [5]. Finally, the last characteristic of the model under consideration is that of a binomial vacation schedule. For a review of vacation queueing systems, see the comprehensive surveys of Doshi [6, 7]. In this class of queueing models, we find the models with Bernoulli vacation schedule where if the queue is empty at a service completion, then the