Multi-Class, Multi-Server Queueing Inventory System with Batch Service

Abstract

In this paper, we consider a queueing inventory system with batch arrival and batch service processes. Customers arrive in batches of sizes 1,2,…,k, according to a marked compound Poisson process. We call a batch of customers as belonging to j when there are j individual customers in that batch. The service facility has waiting rooms for each category of customers and also a room to serve them. Except for category 1, all other customers have finite waiting rooms. The service room has only a limited number of seats. These seats are arranged in such a fashion that customers belonging to category j have groups of seats, each with j seats for j=1,2,…,k. Customers are taken for service according to the availability of seats designated to each category. A category j customer can be taken for service only if j items are available in the inventory. The service time of customers of category j is exponentially distributed with parameter depending on j for j=1,2,…,k. The number of seats available in the service room for each category of customers is restricted to a finite number. The replenishment for items follows the (s,S) policy: fill up to S at the time of replenishment. Lead time follows an exponential distribution. We analyze the system in the equilibrium state. Performance characteristics are evaluated and a number of numerical illustrations are provided.

Keywords

• queueing inventory
• lead time
• marked compound poisson process
• batch arrival
• batch service
• matrix analytic method