THE QUEUE-LENGTH DISTRIBUTION FOR M^x/G/1 QUEUE WITH SINGLE SERVER VACATION

This paper studies the bulk-arrival M-x/G/1 queue with single server vacation. By introducing the server busy period and using the Laplace transform, the recursion expression of the Laplace transform of the transient queue-length distribution is derived. Furthermore, the distribution and stochastic decomposition result of the queue length at a random point in equilibrium are obtained. Especially some results for the single-arrival M/G/1 queue with single server vacation and bulk-arrival M-x/G/1 queue but with no server vacation can be derived directly by the results obtained in this paper.

The Recursive Solution for Geom/G/1(E,SV) Queue with Feedback and Single Server Vacation

Using recursive method, this paper studies the queue size properties at any epoch n＋ in Geom/G/ I（E, SV） queueing model with feedback under LASDA （late arrival system with delayed access） setup. Some new results about the recursive expressions of queue size distribution at different epoch （n＋, n, n-） are obtained. Furthermore the important relations between stationary queue size distribution at different epochs are discovered. The results are different from the relations given in M/G/1 queueing system. The model discussed in this paper can be widely applied in many kinds of communications and computer network.

The Queue Length Distribution for M/G/1 Queue with Delay Single Server Vacation

In this paper we study the transient and equilibrium distributions of the queue length for the M/G/1 queueing system with delay single server vacation.By the server busy period and the Laplace transformation we directly obtain the recursion formula of the L transformation of the transient queue length distribution at any time t , as well as the recursion formula of the equilibrium distribution for calculating conveniently.Furthermore we obtain the stochastic decompositions of the queue length and waiting time in equilibrium.

The Structure of Departure Process and Optimal Control Strategy N^＊ for Geo/G/1 Discrete-Time Queue with Multiple Server Vacations and Min(N, V)-Policy

This paper considers the departure process and the optimal control strategy for a discretetime Geo/G/1 queueing model in which the system operates under the control of multiple server vacations and Min(N, V)-policy. Using the law of total probability decomposition, the renewal theory and the probability generating function technique, the transient and the steady-state probabilities that the server is busy at any epoch n^+ are derived. The authors also obtain the explicit expression of the probability generating function for the expected number of departures occurring in the time interval (0^+, n^+] from any initial state. Meanwhile, the relationship among departure process, server's state process and service renewal process in server busy period is found, which shows the special structure of departure process. Especially, some corresponding results of departure process for special discrete-time queues are directly gained by our results. Furthermore, the approximate expansion for calculating the expected number of departures is presented. In addition, some other important performance measures,including the expected length of server busy period, server's actual vacation period and busy cycle period etc., are analyzed. Finally, some numerical results are provided to determine the optimum value N*for minimizing the system cost under a given cost structure.

ON THE TRANSIENT DEPARTURE PROCESS OF M^x/G/1 QUEUEING SYSTEM WITH SINGLE SERVER VACATION

This paper studies the transient departure process of M^x/G/1 queueing system with single server vacation. We present a simple probability decomposition method to derive the expected number of departures occurring in finite time interval from any initial state and the asymptotic expansion of the expected number. Especially, we derive some more practical results for some special cases.

Effects of Reneging, Server Breakdowns and Vacation on a Batch Arrival Single Server Queueing System with Three Fluctuating Modes of Service

This article examines the effects of reneging, server breakdown and server vacation on the various states of the batch arrivals queueing system with single server providing service to customers in three fluctuating modes. In this queueing system, any batch arrival joins the queue if the server is busy or on vacation or under repair. However, if the server is free, one customer from the arriving batch joins the service immediately while others join the queue. In case of server breakdown, the customer whose service is interrupted returns back to the head of the queue. As soon as the server has is repaired, the server attends to the customer in mode 1. For this queueing system, customers that are impatient due to breakdown and server vacation may renege (leave the queue without getting service). Due to fluctuating modes of service delivery, the system may provide service with complete or reduced efficiency. Consequently, we construct the mathematical model and derive the probability generating functions of the steady state probabilities of several states of the system including the steady state queue size distribution. Further, we discuss some particular cases of the proposed queueing model. We present numerical examples in order to demonstrate the effects of server vacation and reneging on the various states of the system. The study revealed that an increase in reneging and a decrease in server vacation results in a decrease in server utilization and an increase in server’s idle time provided rates of server breakdown and repair completion are constant. In addition, the probability of server vacation, the probability of system is under repair and the probabilities that the server provides service in three fluctuating modes decreases due to an increase in reneging and a decrease in vacation completion rates.

An M/G/1 Queueing System with Delay Server Vacations

In this paper we study the transient property of the queue length for the M/G/1 queueing system with delay server vacations.Using the server busy period and the Laplace transformation we directly obtain the recursion formula of the L transformation of the transient queue length distribution at time t , and also the recursion formula of the equilibrium distribution for calculating conveniently.As will be seen, this paper provides an intuitive and elegant method for studying transient properties of M/G/1 type queueing systems.