Queue with preemptions and repeat or resumption of preempted service

Here we consider a single server queueing model consisting of two queues-an infinite capacity queue of low priority customers and a finite capacity N of high priority customers. Customers join the system according to a MMAP. If the server is free, at the epoch of an arrival of a customer (low priority/ high priority) can immediately join for service. An (N +1) faces solid figure with the face marked 0 to N, is tossed at the beginning of the service of an ordinary customer. ith face turns up with probability qi (0 ≤ i ≤ d). This decides the maximum number of priority customer(s) allowed to be served during the service of the specified ordinary customer. During the service of a low priority customer pre-emption can take place by the arrival of a high priority customer. Then the preempted customer waits at the head of the low priority queue till either the high priority queue becomes empty or the maximum number of high priority customers permitted to be served, as per the outcome of the toss of solid object, whichever occurs first. The restart/ resumption of pre-empted customer’s service takes place when the high priority queue becomes empty or the maximum number of high priority customer’s service permitted during his effective service is realized. We introduce a threshold random variable which competes with the duration of each pre-emption; if this realizes before completion of preemption then the pre-empted customer has to get its service repeated; otherwise the service is resumed. Here the random variable corresponding to low priority customers service, high priority customers service and threshold random variable are all distinct and independent PH distributed. The system is analysed under stable regime. A few useful measures for system performance are obtained. These help in designing an efficient system. Numerical results are provided to illustrate the system performance.