Abstract
We attempt to derive the steady-state distribution of theM/M/cqueueing-inventory system with positive service time. First we analyze the case ofc=2servers which are assumed to be homogeneous and that the service time follows exponential distribution. The inventory replenishment follows the(s,Q)policy. We obtain a product form solution of the steady-state distribution under the assumption that customers do not join the system when the inventory level is zero. An average system cost function is constructed and the optimal pair(s,Q)and the corresponding expected minimum cost are computed. As in the case ofM/M/cretrial queue withc≥3, we conjecture thatM/M/cforc≥3, queueing-inventory problems, do not have analytical solution. So we proceed to analyze such cases using algorithmic approach. We derive an explicit expression for the stability condition of the system. Conditional distribution of the inventory level, conditioned on the number of customers in the system, and conditional distribution of the number of customers, conditioned on the inventory level, are derived. The distribution of two consecutivestostransitions of the inventory level (i.e., the first return time tos) is computed. We also obtain several system performance measures.
Highlights
The notion of inventory with positive service time was first introduced by Sigman and Simchi-Levi [1]
In this paper we studied multiserver queueing-inventory system with positive service time
First we considered two server queueing-inventory systems, where the steady-state distributions are obtained in product form
Summary
The notion of inventory with positive service time was first introduced by Sigman and Simchi-Levi [1]. A recent contribution of interest to inventory with positive service time involving a random environment is by Krenzler and Daduna [15] wherein a stochastic decomposition of the system is established They prove a necessary and sufficient condition for a product form steady-state distribution of the joint queueing-environment process to exist. A still more recent paper by Krenzler and Daduna [12] investigates inventory with positive service time in a random environment embedded in a Markov chain They provide a counter example to show that the steadystate distribution of an M/G/1/∞ system with (s, S) policy and lost sales need not have a product form. With a bit of algebra, this simplifies to λ < μ[2−βζ0/γμ(1−ζ0)]
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