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Consider a single server retrial queueing system with negative arrival under non-pre-emptive priority service in which three types of customers arrive in a poisson process with arrival rate λ1 for low priority customers and λ2 for high priority customers and λ3 for negative arrival. Low and high priority customers are identified as primary calls. The service times follow an exponential distribution with parameters μ1 and μ2 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 mechanismto 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 maximumnumber of waiting spaces for high priority customers in front of the service station. The high priorities customers will be governed by the Non-preemptive priority service. The access from the 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 λ1 , λ2 , λ3 , μ1 , μ2 ,σ and k in elaborate manner and also various particular cases of this model have been discussed.

Consider a single server fixed batch service queueing system under multiple vacation with gated service in which the arrival rate λ follows a Poisson process and the service time follows an exponential distribution with parameter μ. Assume that the system initially contain k customers when the server enters into the system and starts the service immediately in batch of size k. After completion of a service, if he finds less than k customers in the queue, then the server goes for a multiple vacation of length α. If there are more than k customers in the queue then the first k customers will be selected from the queue and service will be given as a batch. Gated type service policy is adopted in this model that is once the server starts service for a batch of k customers, no customers will be allowed to enter into the queue. Every time a service is finished, and there are less than k customers in the queue, the server leaves for a vacation of length α. This model is completely solved by constructing the generating function and Rouche's theorem is applied and we have derived the closed form solutions for probability of number of customers in the queue during the server busy and in vacation. Further we are providing the closed form solutions for mean number of customers in the queue, variance and various system performance measures of the system. Numerical studies have been done for analysis of system measures for various values of λ, µ, α and k.

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