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Inventory control with two switch-over levels for a class of finite capacity M/G/1 queueing systems with variable arrival and service rate

This paper deals with inventory control in a class of finite capacity, M/G/1 queueing systems with the inventory being the virtual waiting time. It is assumed that at each point of time the system is one of two possible stages 1 and 2 where at any moment the system can be switched from one stage to another. The rate of the arrival process and the service rate depend on the stage of the system. The cost structure imposed on the model includes both a holding cost at rate h,(x) when the system is in stage i and the inventory is x, overflow costs and fixed switch-over costs. The functions hi(x)(i = 1, 2) are assumed to be bounded functions having only a finite number of discontinuities. The rule for controlling the inventory is specified by two switch-over levels y, and Y2 and prescribes to switch the system from stage 1 to stage 2 when the inventory exceeds level y, and to switch the system from stage 2 to stage 1 when the inventory has been decreased to the level Y2* Using an embedding approach, we will derive a formula for the long-run average expected costs per unit time of this policy. By an appropriate choice of the cost parameters, we may obtain various operating characteristics for the system amongst which the stationary distribution of the inventory and the average number of switch-overs per unit time. The above control problem includes as special cases a variety of problems previously studied in the literature (e.g. [1]-[4]) and provides thus a unifying treatment of these problems.