MAP/PH(1), PH(2)/2 finite retrial inventory system with service facility, multiple vacations for servers

In this paper, we consider a retrial (s, S) inventory system with multiple server vacations for two heterogeneous servers. We have assumed that the customers arrive according to a Markovian arrival process and two parallel servers who provide heterogeneous phase type services to customers. The lead times for the orders are assumed to have independent and identical exponential distributions. The vacation times of both servers are assumed to be independent and identically distributed exponential random variables. The arriving customer who finds both servers are busy or both servers are on vacation, joins an orbit of finite size. These orbiting customers retry for their demand after a random time, which is assumed to be exponential distribution. Some important performance measures are obtained and the optimality of an expected total cost rate is shown through numerical illustration.