M/G/1/N Queue with Vacation Time and Exhaustive Service Discipline

Abstract

This paper studies an M/G/1 queueing system with a finite waiting room and with server vacation times consisting of periods of time that the server is away from the queue doing additional work. This model has been used in conjunction with a related model to analyze the performance of a processor with a cyclic scheduling algorithm and where, due to finite queueing capacities, losses are a primary concern. Service at the queue is exhaustive, in that a busy period at the queue ends only when the queue is empty. At each termination of a busy period, the server takes an independent vacation. The queue length process is studied using the embedded Markov chain. Using a combination of the supplementary variable and sample biasing techniques, we derive the general queue length distribution of the time continuous process, as well as the blocking probability of the system, due to the finite waiting room in the queue. We also obtain the busy period and waiting time distributions.

Busy period analysis of the level dependent PH/PH/1/K queue

“…Taking the Laplace transforms of the Kolmogorov backward equations of the AMC, we find that (8) where e n ⊗ f n ⊗ s n represents the initial state vector of the AMC, and the matrices A, B, C, and D are given in Appendix A. Multiplying (8) by z l 1 z m 2 z o 3 and summing the result first over all o, then m, and finally l yields that…”

“…However, Rosenlund's final results for the trivariate transform for M/G/1/K and G/M/1/K queues are represented as a fraction of two contour integrals. For more recent works on the busy period analysis of the M/G/1/K queue we refer to [8,16]. Recently, there has been an increased interest in the expected number of losses during the busy period in the M/G/1/K queue with equal arrival and service rate; see, e.g., [1,12,17].…”

A Non-Preemptive Priority Queueing System with a Single Server Serving Two Queues M/G/1 and M/D/1 with Optional Server Vacations Based on Exhaustive Service of the Priority Units

“…Thus their model is a combination of the M/G/1 and M/D/1 queues and the server keeps switching over these two queues depending on the class of units present in the system. For separate references on M/G/1 and M/D/1 queues, the reader is referred to Bhat [10], Levy and Yechiali [11], Kleinrock [12], Cohen [13], Lee [14], Gross and Harris [5], Cox and Miller [16], Tijms [17], Yang and Li [18], Bunday [19] and Madan [20,21]. However, in the present paper, we generalize Madan and Abu-Dayyeh [9] paper by adding a significant assumption to their model that the server may take a vacation of random length but we assume that no vacation is allowed if there is even a single priority unit present in the system.…”

Performance analysis of a finite-buffer discrete-time queue with bulk arrival, bulk service and vacations

“…Lee [23] studies the ÿnite-bu er M=G=1=K +1 queue with multiple vacations with exhaustive services based on the sample biasing technique and the supplementary variable technique. Lee [24] analyzes ÿnite-bu er M=G=1=K + 1 queue with multiple vacations and E-limited service discipline based on the same approach of Ref.…”