Analysis of a queuing process with balking, reneging and pre-emptive priority

Waiting lines or queues are omnipresent. Irregularities in arrival pattern or service mechanism of the system generate waiting lines. Since irregularities of the system depend upon chance, so feasibility of probability theory was realised in the study of queues. Cobham (1954), Kesten and Runnenberg (1957), Stephan (1958), White and Christie (1958), Cox and Smith (1961), studied the queuing models in the steady-state having Poisson arrivals, exponential service times and priority queue discipline. Vikram (1996) formulated the queuing model where a customer may leave the system with or without getting service and pre-emptive priority is the queue discipline for service.

We construct the queuing model having balking and reneging where the customers arrive in Poisson stream, service time distribution is negative exponential and queue discipline for service is pre-emptive priority. It has been considered here that the customers may leave the system with or without getting service and may not join the queue due to large number of customers already present there. We write its difference-differential equations and steady-state equations. The Steady-state solutions are found by iterative method. The performance of the system is projected by calculating the mean queue length for infinite waiting space.