Analysis of a discrete-time Geo^X/G/1 repairable queue with multiple vacations, feedback and p-entering discipline

Discrete-time vacation queues have been widely studied over the past decades due to their wide applications in broadband integrated service digital networks (BISDN), asynchronous transfer mode (ATM) and computer telecommunication systems. For a comprehensive review of the main results, methods and applications, readers may refer to the papers [1-7], the books [8-9] and their references. Based on the fact that the sever of a queueing system is subject to unpredictable breakdowns and repairs when serving a customer, some researchers, such as Tang et al.[10], Liu and Gao[11], Lan and Tang[12,13], Kulshrestha et al.[14] and so on, analyzed the reliability of the server in some discrete-time repairable queueing systems. However, existing studies have mainly focused on the discrete-time repairable queues with a constant arrival rate. In fact, the customer arrival rate may have something to do with server states. For example, in a telecommunication network the arrival rate of a message may vary when the server is under a maintenance activity (e.g. virus scan). On the other hand, the feedback is a common phenomenon in real world, e.g., in telecommunication systems the messages that produce errors at the destination need to be sent again. In a call center a user may call again when their problem is not completely solved. Thus the reliability study of discrete-time repairable queues with vacations, feedback and variable arrival rate is not only significant for theoretical investigations but also valuable for practical applications. In this paper, we consider the reliability of the server for a discrete-time Geo^X/G/1 queue with vacations, feedback and p-entering discipline. The arriving batch enters the system with probability p or is lost with probability 1-p during server vacations. The customer who has just been served returns to the queue with probability 1-q for another service or leaves permanently with probability q. The server may break down during service and does not continue to work until it is repaired. Using the z-transform and renewal process theory, the server reliability indices, such as the transient and steady-state unavailability, the expected failure number during (0^+,n^+] and the steady-state failure frequency, are studied. The transient structure of the server reliability indices in this type of repairable queue is characterized. The proposed queueing system can be used to model a network access proxy system (NAPS). In such a system, maintenance activities (vacations), such as virus scanning, are required to keep the proxy server operating properly. After completing one maintenance activity, the proxy server either performs another maintenance activity or begins serving waiting requests. A service request received with errors at the destination may be retransmitted (feedback). In addition, a busy server may stop working when unpredictable events occur, such as network congestion. The service interruption is repaired immediately, and once the interruption is recovered, the server resumes service. Because channel requests, grants, data transmissions, and receptions all occur in fixed time intervals, and because the batch-request arrival rate differs between the server maintenance period and the busy period, this system can be modeled as a discrete-time repairable queue with vacations, feedback, and a variable arrival rate. Moreover, for the NAPS described above, the server reliability indices obtained in Section 3 are useful for analyzing the effects of system parameters on proxy-server performance (see Section 4). The remainder of this paper is organized as follows. Section 2 presents the queueing assumptions and several preliminaries. Section 3 derives the server reliability indices using the z-transform and renewal-point techniques and discusses several special cases. Section 4 provides numerical examples to validate the theoretical results for a network access proxy system. Section 5 concludes the paper.