THE BULK INPUT M~[X]/M/1 QUEUE WITH WORKING VACATIONS

In this paper, we analyze a bulk input M^[X]/M/1 queue with multiple working vacations. A quasi upper triangle transition probability matrix of two-dimensional Markov chain in this model is obtained, and with the matrix analysis method, highly complicated probability generating function（PGF） of the stationary queue length is firstly derived, from which we got the stochastic decomposition result for the stationary queue length which indicates the evident relationship with that of the classical M^[X]/M/1 queue without vacation. It is important that we find the upper and the lower bounds of the stationary waiting time in the Laplace transform order using the properties of the conditional Erlang distribution. Furthermore, we gain the mean queue length and the upper and the lower bounds of the mean waiting time.

Stationary Analysis and Optimal Control Under Multiple Working Vacation Policy in a GI/Ma,b/1 Queue

This paper considers an infinite buffer renewal input queue with multiple working vacation policy wherein customers are served by a single server according to general bulk service （a,b）-rule （1 ≤ a ≤ b）. If the number of waiting customers in the system at a service completion epoch （during a normal busy period） is lower than ＇a＇, then the server starts a vacation. During a vacation if the number of waiting customers reaches the minimum threshold size ＇a＇, then the server starts serving this batch with a lower rate than that of the normal busy period. After completion of a batch service during working vacation, if the server finds less than takes another vacation, otherwise the server rate. The maximum allowed size of a batch a＇ customers accumulated in the system, then the server continues to serve the available batch with that lower in service is ＇b＇. The authors derive both queue-length and system-length distributions at pre-arrival epoch using both embedded Markov chain approach and the roots method. The arbitrary epoch probabilities are obtained using the classical argument based on renewal theory. Several performance measures like average queue and system-length, mean waiting-time, cost and profit optimization are studied and numerically computed.

Transient Behavior of a Single-Server Markovian Queue with Balking and Working Vacation Interruptions

This paper studies the time-dependent analysis of an M/M/1 queueing model with single,multiple working vacation,balking and vacation interruptions.Whenever the system becomes empty,the server commences working vacation.During the working vacation period,if the queue length reaches a positive threshold value‘k’,the working vacation of the server is interrupted and it immediately starts the service in an exhaustive manner.During working vacations,the customers become discouraged due to the slow service and possess balking behavior.The transient system size probabilities of the proposed model are derived explicitly using the method of generating function and continued fraction.The performance indices such as average and variance of system size are also obtained.Further,numerical simulations are presented to analyze the impact of system parameters.