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In this paper, we have obtained explicit expressions for the time dependent probabilities of the /1 /M M queue with server vacations under a multiple vacation scheme. The corresponding steady state probabilities have been obtained. We also obtain the time dependent performance measures of the systems.

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We consider an M/G/1 retrial queueing system with two phases of heterogeneous service and a finite number of immediate Bernoulli feedbacks. If an arriving customer finds an idle server, service commences immediately. Otherwise, the blocked customer either joins the infinite waiting room with probability p or leaves the service area and enters the retrial group with complementary probability q. All arriving customers are provided with the same type of service in the first phase. In the second phase, the customer has to choose from one of the several optional services which are available in the system. After having completed both phases of service, the customer is allowed to make an immediate feedback. The feedback service also consists of two phases. In the feedback, the first phase of service is of the same type as in the previous service. However, in the second phase the customer may be permitted to choose an optional service different from one chosen earlier. In this way, the customer is permitted to make a finite number feedbacks. We employ the embedded Markov chain technique to obtain a sufficient condition for the system to attain the steady state. We obtain the probability generating function of the system size and the marginal probability generating functions of the queue size and orbit size. We also obtain the distribution of the server state and some useful performance measures. We also obtain the stochastic decomposition law. We study the asymptotic behaviour under high rate of retrials. Finally, numerical calculations are used to observe system performance.