A single server finite population queueing model with compulsory server vacation and with fixed batch service has been considered. For this model the system steady state probabilities are obtained. Some performance measures are calculated and numerical examples are also given.

An M/M/1 queueing system with second optional service and unreliable server is studied. We consider that the server works at different rate rather than being idle during the vacation period. The customers arrive to the system according to Poisson process with state dependent rates depending upon the server’s status. All customers demand the first essential service whereas only some of them demand the second optional service. A customer either may leave the system after the first essential service with probability (1-r) or at the completion of the first essential service go for second optional service with probability r (0£r£1). The server may breaks down according to Poisson process during the busy and working vacation duration. Both service times in vacation and in service period are exponentially distributed. The matrix geometric technique is used for the analysis of the concerned queueing system. The sensitive analysis is also performed to examine the variation of the system performance characteristics with various input parameters.

This study is concerned with the performance modeling of a fault tolerant system consisting of operating units supported by a combination of warm and cold spares. The on-line as well as warm standby units are subject to failures and are send for the repair to a repair facility having single repairman which is prone to failure. If the failed unit is not detected, the system enters into an unsafe state from which it is cleared by the reboot and recovery action. The server is allowed to go for vacation if there is no failed unit present in the system. Markov model is developed to obtain the transient probabilities associated with the system states. Runge–Kutta method is used to evaluate the system state probabilities and queueing measures. To explore the sensitivity and cost associated with the system, numerical simulation is conducted.

This paper studies the operating characteristics of an MX/Hk/1 queueing system under multiple vacation policy. It is assumed that the server goes for vacation as soon as the system becomes empty. When he returns from a vacation and there is one or more customers waiting in the queue, he serves these customers until the system becomes empty again, otherwise goes for another vacation. The breakdown and repair times of the server are assumed to follow a negative exponential distribution. By using a generating function, we derive various performance indices. The approximate formulas for the probability distribution of the waiting time of the customers in the system by using the maximum entropy principle (MEP) are obtained. This approach is accurate enough for practical purposes and is a useful method for solving complex queueing systems. The sensitivity analysis is carried out by taking a numerical illustration.

A single server provides service to all arriving customers with service time following general distribution. After every service completion the server has the option to leave for phase one vacation of random length with probability p or continue to stay in the system with probability 1-p. As soon as the completion of phase one vacation, the server may take phase two vacation with probability q or to remain in the system with probability 1-q, after phase two vacation again the server has the option to take phase three vacation with probability r or to remain in the system with probability 1-r. The vacation times are assumed to be general. The server is interrupted at random and the duration of attending interruption follows exponential distribution. Also we assume, the customer whose service is interrupted goes back to the head of the queue where the arrivals are Poisson. The time dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been obtained explicitly. Also the mean number of customers in the queue and system and the waiting time in the queue and system are also derived. Particular cases and numerical results are discussed.