Analysis of Tandem Queue with Multi-Server Stages and Group Service at the Second Stage

Abstract

In this paper, we consider a tandem dual queuing system consisting of multi-server stages. Stage 1 is characterized by an infinite buffer, one-by-one service of customers, and an exponential distribution of service times. Stage 2 is characterized by a finite buffer and a phase-type distribution of service times. Service at Stage 2 is provided to groups of customers. The service time of a group depends on the size of the group. The size is restricted by two thresholds. The waiting time of a customer at each stage is limited by a random variable with an exponential distribution, with the parameter depending on the stage. After service at Stage 1, a customer can depart from the system or try to enter Stage 2. If the buffer at this stage is full, the customer is either lost or returns for service at Stage 1. Customer arrivals are described by the versatile Markov arrival process. The system is studied via consideration of a multi-dimensional continuous-time Markov chain. Numerical examples, which highlight the influence of the thresholds on the system performance measures, are presented. The possibility of solving optimization problems is illustrated.