A Multistage Queueing Model with Priority for Customers Become Fit

Abstract

We consider a multistage single server queueing model with one infinite queue and two finite buffers. The primary customer on arrival joins an infinite queue. The head of the queue finding an idle server, enter into Stage I of service. A customer who found fit, pass on to Stage II with probability p, \(0\le p\le 1\) and those who found unfit pass on to Buffer I with complimentary probability \(1-p\). The customer from Buffer I after an exponential duration of time move to Buffer II. A customer in Buffer II who seems unfit again move to Buffer I. Buffer I and Buffer II individually and collectively should not exceed capacity M. On every service completion epoch, the head of Buffer II is selected for service. We also assume customer reneges from both infinite queue and from Buffer II within an exponential duration of time intervals. Customers arrive according to a Markovian arrival process (MAP). The service time follows phase type distribution. Stability condition of the system is established. Steady-state system size distribution is obtained. Some performance measures of the system are evaluated.