This paper is concerned with the analysis of a single server retrial queue with vacation and orbital search. The server is subject to starting failure and repair. At the completion epoch of each service, the server may take a single vacation. After vacation completion, the server searches for the customers in the orbit or remains idle. Retrial times, service times and vacation times are assumed to be arbitrarily distributed. Various performance measures are derived and numerical results are presented.

Batch arrival retrial queue with positive and negative customers is considered. Server provides M types of service. Positive customers arrive in batches according to Poisson process. If the server is idle upon the arrival of a batch, one of the customers in the batch receives any one the types immediately and others join the orbit. The server is subject to two different modes of failure. Mode 1 failure occurs due to the arrival of negative customer and Mode 2 due to random breakdown of the server. In both cases, repair starts after some random amount of time. The server failed under mode 2 continues the interrupted service or waits for the same customer after the repair completion. Generating function technique is employed to obtain joint distributions of the server state and orbit length. Expected system size, expected orbit size, availability of the server and failure frequency of the server are derived. Stochastic decomposition law is also verified.

This paper analyses the steady state behavior of an M/G/1 retrial queueing system with Bernoulli and phase type vacations. Customers arrive one by one at the system in a Poisson stream. At the arrival epoch, if the server is busy then the arriving customer joins the orbit. If the server is free, then the arriving customer starts its service immediately. The service time of a customer is assumed to be general. At each service completion epoch, the server may opt to take a phase 1 vacation with probability p or else with probability 1-p stay in the system for the next service. After the completion of phase 1 vacation the server may take phase 2 vacation with probability q or return back to the system with probability 1-q. The vacation times are assumed to be general. The service times and vacation times are independent of each other. Generating function technique is applied to obtain the system size and orbit size. Numerical examples are provided to illustrate the sensitivity of the performance measures for changes in the parametric of the system. Mathematics Subject Classification 60K25, 90B22

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