Transient Analysis of $$M^{[X_{1}]},M^{[X_{2}]}/G_{1},G_{2}/1$$ M [ X 1 ] , M [ X 2 ] / G 1 , G 2 / 1 Retrial Queueing System with Priority Services, Working Vacations and Vacation Interruption, Emergency Vacation, Negative Arrival and Delayed Repair

Abstract

This paper deals with the analysis of two classes of batch arrivals, one is high priority and other is low priority(retrial) customers with non-preemptive priority service, working vacations, negative arrival, emergency vacation for an unreliable server, which consists of a break-down by negative arrival and delay time to start repair. When there are no customers in the queue and in the orbit at the time of service completion for a positive customers, the server goes for a working vacation. The server works at a lower service rate during working vacation period. If there are no customers in the system at the end of vacation, the server becomes idle and ready for serving the new arrivals with probability p(single working vacation) or it remains on vacation with probability $$q=(1{-}p)$$ (multiple working vacations). Here we assume that customers arrive according to compound Poisson process in which high priority customers are assigned to class one and class two customers are of a low priority type. The priority customers that find the server busy are queued and then served in accordance with FCFS discipline. The arriving low priority customers on finding the server busy are queued in the orbit in accordance with FCFS constant retrial policy. The arrival of negative customer remove the customer being in service and makes the server breakdown and the server fail for a short interval of time. When the server fails it will send to repair. But the repair time do not start immediately, there is a delay time to start the repair process. We consider an emergency vacation which means that during the service time the server suddenly go for a vacation and the interrupted customer waits to getting the remaining service. The retrial time, service time, emergency vacation time, delay time and repair time are all follows general(arbitrary) distribution, working vacation time and negative arrival follows exponential distribution. The time dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results are obtained explicitly. Also some performance measures of our model such as the system state probabilities, reliability indices, the average number of customer in the high priority queue and the low priority in the orbit and the average waiting time are derived. Numerical results are computed and graphical representations are presented.