The MAP/PH/N/∞ Queueing-Inventory System With Demands From a Random Environment

Abstract

This study investigates a multi-server queueing-inventory system in a random environment. The system has an unlimited waiting space and an inventory capacity of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula> units. The customer arrives at the system according to the Markovian arrival process. Whenever all the servers are busy, an arriving customer either goes to an unlimited waiting space or leaves the system. And if there is a free server with positive inventory, then the arriving customer gets service immediately. Inventories are filled under the instantaneous replenishment policy. We consider that the service time follows a phase-type distribution, and we assess the joint probability distribution of the inventory level and the number of customers in the steady-state case. On top of that, we extract the sojourn time distributions of arbitrary customers using the Laplace-Stieltjes transform. Finally, a few numerical examples are provided to illustrate our mathematical model.