This paper deals with a two server (s,S) inventory system with positive service time, positive lead time, retrial of customers and negative arrivals. In this system, arrival of customers form a Poisson process, lead time and service time are exponentially distributed. The system starts with S units of inventory on hand. Each arriving customer is served a single unit of the item by any one of the servers. When the inventory level reaches s, an order is placed for (S-s) units. If the inventory level is zero or both servers are busy, then the arriving customer goes to orbit and becomes a source of repeated calls. Assume that the capacity of the orbit is infinite. The negative arrival plays an important role in this paper and it controls the congestion in the orbit by removing one customer from the orbit and further it is assumed that it removes the customer from the orbit only if inventory level is zero or both servers are busy. It is also assumed that the access from orbit to the service facility is governed by the classical retrial policy. This model is solved by using Direct Truncation Method. Numerical and graphical studies have been done for analysis of mean number of customers in the orbit, average inventory level, Truncation level, mean number of busy servers and system performance measures. A suitable cost function is defined.
Consider a Multi server Retrial queueing system with vacation policies in which arrival rate follows a Poisson distribution with parameter λ and service time follows an exponential distribution with parameter μ. Let c be the number of servers in the system. Two type of vacation policies have been introduced in this paper namely exhaustive service type vacation and Bernoulli vacation. If any one of the server is free at the time of a primary call arrival, the arriving call begins to be served immediately by one of the idle servers and customer leaves the system after service completion. Otherwise, if c servers are busy or c servers are in vacation then the arriving customer goes to orbit and becomes a source of repeated calls. The pool of sources of repeated calls may be viewed as a sort of queue. Every such source produces a Poisson process of repeated calls with intensity σ. If an incoming repeated call finds any one of the servers is free, it is served and leaves the system after service, while the source which produced this repeated call disappears. The access from the orbit to the service facility follows the classical retrial policy. This model is solved by using Direct Truncation Method. Numerical study have been done for Analysis of Mean number of customers in the orbit, Mean number of busy servers, Mean number of servers in vacation, Truncation level and various system measures.
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