A Two-Server Queueing System with Processing of Service Items by a Server

Abstract

We consider a two-server (S1 and S2) queueing system in which the customers arrive according to Markovian arrival process. Each customer is to be provided with a processed item (inventory) at the end of his service. S1 provides service alone, whereas S2 provides service and also processes the items required to serve the customers. The maximum number of processed item permitted is L. The processing time follows phase type distribution. When the inventory level hits L, S2 starts serving customers if any waiting; else stays idle. S1 is dedicated to service only. Service is rendered only if there are processed items. Also, when a customer arrives to the system when both servers are idle, S1 provides him service and S2 continuously remains idle even if it has completed the processing of L items. The duration of service time given by both servers follows phase type distributions of same order, but S1 provides service at a slower rate than S2. If the inventory level drops to a predetermined level s due to a service completion by S2, then he starts processing items. If the inventory level drops to level s due to a service completion by S1, then the customer served by S2 is shifted to S1 to provide him the residual service; S2 starts processing items. The arrival process is independent of the inventory processing and service process. The long run behavior of the system is analyzed under condition for stability. We derive some important distributions associated with the model. Numerical investigation of the optimal values of L and s is provided.