Analysis of a Queueing-Inventory System with Synchronous Vacation of Multiple Servers

Abstract

This paper considers a queueing-inventory system with synchronous vacation of
multiple servers. The stocks are replenished by an (s, S) policy. When the inventory is
empty, all servers synchronize multiple vacations, and the vacation time follows an exponential
distribution. In this paper, we first establish a three-dimensional Markov process
for the number of customers, the inventory level and the status of the servers in the system.
Then, we give the solution of the steady-state probability distribution of the system by using
the matrix-geometric solution method, and derive some important performance measures. In
order to deal with the case of larger or super larger dimension of the state space, an approximate
method to calculate the steady-state probability distribution of the system is developed.
We further develop a total average cost function. Finally, we investigate the effect of system
parameters on the optimal number of servers, the optimal inventory policy and the optimal
average cost function through numerical illustration.