Product form solutions of some queueing inventory problems in random environment

Abstract

The public is often attracted by the offers given by the shopkeepers. These offers do make a crucial impact in the company's annual turnover. On a close watch it can be observed that these offers do change from time to time according to the seasonal variations. Here we investigate a single commodity Queueing-Inventory with 'offer' in random environment. The 'offer' can be anything given to the customers as incentives, who purchase the item. There are 'n' distinct environments. The offers in distinct environments are different. In some environments there may not be any offer. Here an environment dependent inventory control policy $ (s_i ,S) $ for $ 1\leq i \leq n $ is followed. Lead time, service and replenishment of the inventories are independent exponentially distributed random variables. It has been observed that when certain restrictions are imposed on the system, it admits asymptotic product form solution. Two such variants are discussed here. Long run behaviour of the system is analysed under the assumption of system stability. An optimization problem is constructed to compute the values of $ s_i $ and $ S $ that minimize the average system cost. The effect of environment in the optimum values of control variables, is also investigated.