On a queueing inventory system with lost sales, PH vacation and random order size (s, S) policy

Abstract We propose a queueing-inventory system (QIS) that involves a single server who places a random order quantity ’R’ when stock is depleted completely and immediately takes a vacation. This variable R follows a discrete probability distribution, while the server’s vacation duration follows a phase-type (PH) distribution. Customers arriving during stock-out and vacation periods are turned away, resulting in lost sales. Inter-arrival times, service time, and lead times are all assumed as mutually independent and exponentially distributed random variables. Each customer is served a single unit of the product. Our research primarily examines the stability condition and the stationary distribution of the three-dimensional process encompassing queue length, inventory status, and server states. The stability condition is derived as a function of arrival and service rates. Additionally, we develop a cost model to determine the economic order quantity (EOQ) and provide numerical illustrations to trace EOQ values under various order policies and PH type vacations.