M/M/1 Retrial Queueing System with Negative Arrival under Pre-emptive Priority Service

摘要

Consider a single server retrial queueing system with negative arrival under pre-emptive priority service in which three types of customers arrive in a Poisson process with arrival rate λ_1 for low priority customers and λ_2 for high priority customers and λ_3 for negative arrival. Low and high priority customers are identified as primary calls. Further assume that the service times follow an exponential distribution with parameters μ_1 and μ_2 for low and high priority customers. The retrial and negative arrivals are introduced for low priority customers only. Gelenbe (1991) has introduced a new class of queueing processes in which customers are either Positive or Negative. Positive means a regular customer who is treated in the usual way by a server. Negative customers have the effect of deleting some customer in the queue. In the simplest version, a negative arrival removes an ordinary positive customer or a random batch of positive customers according to some strategy. It is noted that the existence of a flow of negative arrivals provides a control mechanism to control excessive congestion at the retrial group and also assume that the negative customers only act when the server is busy. Let K be the maximum number of waiting spaces for high priority customers in front of the service station. The high priorities customers will be governed by the Pre-emptive priority service.We assume that the access from orbit to the service facility is governed by the classical retrial policy. This model is solved by using Matrix geometric Technique.Numerical study have been done for Analysis of Mean number of low priority customers in the orbit (MNCO), Mean number of high priority customers in the queue (MPQL), Truncation level (OCUT), Probability of server free and Probabilities of server busy with low and high priority customers for various values of λ_1, λ_2, λ_3, μ_1 μ_2, δ and k in elaborate manner and also various v particular cases of this model have been discussed.