We studied an M/G/1 queue with multiple vacations and exhaustive service discipline such that the server works with different service times rather than completely stopping service during a vacation.Such a vacation is called a working vacation(WV).Both service times in a vacation and in a service period are generally distributed random variables.We derived the distributions for the queue size and generating function in the steady state.

M/G/1 Queue with Multiple Working Vacations

Transient analysis of a two-heterogeneous servers queue with impatient behavior

This paper presents the analysis of a renewal input multiple working vacations queue with balking, reneging and heterogeneous servers. Whenever the system becomes empty the second server leaves for a working vacation whereas the first server remains idle in the system. During a working vacation the second server provides service at a slower rate rather than completely stopping service. The steady-state probabilities of the model are obtained using supplementary variable and recursive techniques. Various performance measures of the model such as expected system length, expected balking rate, etc., have been discussed. Finally, some numerical results have been presented to show the effect of model parameters on the system performance measures.

Balking and Reneging Multiple Working Vacations Queue with Heterogeneous Servers

We consider an M/M/1 multiple vacation queueing system with two types of server vacations. Type 1 vacation is taken after the server has exhaustively served all the customers in the system, where the number of customers served is at least one. Type 2 vacation is taken when the server returns from a vacation and finds no customer waiting. Each type of vacation can be interrupted when the number of customers in the system reaches two predefined thresholds, where each vacation type has a different threshold. It is assumed that service times and vacation durations are exponentially distributed with different means. We present a steady-state solution of the system under two vacation interruption policies.

M/M/1 Multiple Vacation Queueing Systems With Differentiated Vacations and Vacation Interruptions

We consider a single-server vacation queueing system that operates in the following manner. When the server returns from a vacation, it observes the following rule. If there is at least one customer in the system, the server commences service and serves exhaustively before taking another vacation. If the server finds the system empty, it waits a fixed time c. At the expiration of this time, the server commences another vacation if no customer has arrived; otherwise, it serves exhaustively before commencing another vacation. Analytical results are derived for the mean waiting time in the system. The timeout scheme is shown to be a generalized scheme of which both the single vacation and multiple vacations schemes are special cases, with c=∞ and c=0, respectively. The model is extended to the N-policy vacation queueing system.

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M/G/1 Vacation Queueing Systems with Server Timeout

This paper analyzes an queueing system with two heterogeneous servers, one of which is always available but the other goes on vacation in the absence of customers waiting for service. The vacationing server, however, returns to serve at a low rate as an arrival finds the other server busy. The system is analyzed in the steady state using matrix geometric method. Busy period of the system is analyzed and mean waiting time in the stationary regime computed. Conditional stochastic decomposition of stationary queue length is obtained. An illustrative example is also provided.

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A. Krishnamoorthy

Papers

An M/M/2 Queueing System with Heterogeneous Servers Including One with Working Vacation