We study an (s, S) inventory system with positive service time (for an overview of the work reported so far in inventory with positive service time one may refer to Krishnamoorthy, Lakshmi and Manikandan: A survey on inventory models with positive service time, OPSEARCH, DOI 10.1007/s12597-010-0032-z). This leads to a queue of demands being formed. The process of demand arrival constitutes a Poisson process. The duration of each service is exponentially distributed. Our model is a supply chain where items are added to the inventory through a production process. This starts each time the inventory level goes down to s and continues to be on until inventory level reaches S with the time required to add one unit of the item into the inventory when the production is on, are independent, identically distributed exponential random variables. Further all distributions involved in this paper are assumed to be mutually independent. We assume that no customer joins the queue when the inventory level is 0. This assumption leads us to an explicit product form solution for the steady state probability vector, using a simple approach. This is despite the fact that there is a strong correlation between lead time (the time required to add an item into the inventory) and the number of customers joining the queue during the lead time (except when the inventory level is zero during which time no customer joins the queue). The technique is to combine the steady state probability vector of the classical M/M/1 queue and that of the production inventory system where each service requires negligible time and no backlogs are allowed. Using a similar technique, the expected length of a production cycle is also obtained explicitly. The optimality of the highest inventory level S and the production switching on level s has been studied using a cost function constructed using the steady state system performance measures. Since we have obtained explicit expressions for these measures, analytic expressions have been derived for the optimal values of S and s. To show that our method can be applied to other similar problems, we analyze in detail a variant of the above problem (discussed in Schwarz M, Sauer. C, Daduna, H., Kulik, R and Szekli, R: M/M/1 Queueing systems with inventory, Queueing Systems, 54,55-78, 2006). For that model, we assume that in a production run, production occurs only once in a cycle and the amount produced is sufficient to take the inventory level back to S. A brief discussion on the application of our method to inventory system with lead time for replenishment has also been provided.
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