On a Retrial Production Inventory System with Vacation and Multiple Servers

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In this paper, the model of a production inventory system with heterogeneous servers, the vacation of one of the servers (vacationing server), and retrial customers is considered. The customers reach the system each demanding exactly one unit of the item, according to a Poisson process. Service times and lead time are independent and follow exponential distributions. A single unit of the item is produced at a time according to (s, S) policy. It is assumed that when the inventory level reaches zero or when the orbit is empty or both, one of the servers(server 2) takes multiple, exponentially distributed vacation time. At the end of the vacation, that server immediately takes another vacation, if he/she finds empty stock or empty orbit or both. During the stock out period or server busy period or vacation period, the demands that arrive enter into the orbit of infinite size. If a customer finds both the servers busy or server 1 busy and server 2 in vacation or inventory level is zero, in accordance with Bernoulli trials may wait in the orbit or leave forever. When either the customer in the orbit cannot make an attempt due to the busy servers or inventory level zero, he/she under Bernoulli trials may come back to the orbit or leave the orbit. The rate of retrial customers from orbit is linear. The production process is switched on when the inventory level reaches to s, and similarly when the level of inventory reaches back to S the production process is switched off. The production process releases items that have inter arrival times that are exponentially distributed. Ergodicity condition is obtained and matrix analytic method is used to calculate the steady-state probabilities of the constructed 4D Markov chain. Minimum expected total cost per unit time and many significant system performance measures are obtained. Sensitivity analysis is relevant to check various performance measures that is highly applicable in the realistic situation and it also provides managerial insights.