Unreliable Batch Arrival Retrial Queue with Negative Arrivals, Multi-Types Of Heterogeneous Service, Feedback and Randomized J Vacations

A batch arrival retrial queue with two types of arrival - positive and negative is considered. Positive customers arrive in batches according to Poisson process. Server provides M types of service. If the server is idle upon the arrival of a batch, one of the customers in the batch receives any one of the types immediately and others join the orbit. Otherwise, all the customers in the batch join the orbit. After completion of service, the unsatisfied customers join the orbit as feedback customers. Whenever the orbit becomes empty the server takes a vacation of random length. At each vacation completion epoch, if the system is empty, the server leaves for another vacation of same length or remains idle in the system. This pattern continues until the server finds at least one customer in the orbit or the number of vacations reaches J. At the end of J th vacation, even if the orbit is empty the server remains in the system for new arrival. The arrival of negative customer brings the server down and makes the interrupted customer to leave the system. Using supplementary variable technique, various performance measures are derived. Stochastic decomposition property is established and particular cases are deduced.