Abstract:

In this paper, we consider a Markovian $\left({s, S}\right)$ queueing-inventory system in which only partial servers take multiple synchronous vacations when the on-hand inventory level is zero. It is assumed that the vacation time follows an exponential distribution. The customers arrive according to a Poisson process, and the service time of the customers is distributed exponentially. The lead times for the orders are assumed to have independent and identical exponential distributions. Using the theory of quasi-birth-and-death process, the matrix-geometric solution of the steady-state probability is derived. On this basis, the steady-state performance measures and cost function of the system are obtained. Finally, the effect of the parameters on cost function is analyzed by numerical examples, and the optimal inventory policy and the optimal expected cost are also computed.

Key words: queueing-inventory system, vacations of partial servers, (s, S) policy, quasi-birth-and-death process, matrix-geometric solution