Analysis Of Single Server Queueing System With Batch Service Under Multiple Vacations With Loss And Feedback

Consider a single server queueing system with foxed batch service under multiple vacations with loss and feedback in which the arrival rate ? follows a Poisson process and the service time follows an exponential distribution with parameter ?. Assume that the system initially contain k customers when the server enters the system and starts the service in batch. The concept of feedback is incorporated in this model (i.e) after completion of the service, if this batch of customers dissatisfied then this batch may join the queue with probability q and with probability (1-q) leaves the system. This q is called a feedback probability. After completion of the service if he finds more than k customers in the queue then the first k customers will be taken for service and service will be given as a batch of size k and if he finds less than k customers in the queue then he leaves for a multiple vacation of exponential length ?. The impatient behaviour of customer is also studied in this model (i.e) the arriving customer may join the queue with probability p when the server is busy or in vacation. This probability p is called loss probability. 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 analytical solution for mean number of customers and variance of the system. Numerical studies have been done for analysis of mean and variance for various values of ?, µ, ?, p, q and k and also various particular cases of this model have been discussed. Keywords : Single Server , Batch Service, Loss and Feedback,  Multiple vacations, Steady state distribution

Analysis of Single Server Fixed Batch Service Queueing System under Multiple Vacation with Catastrophe

Consider a single server fixed batch service queueing system under multiple vacation with a possibility of catastrophe in which the arrival rate ? follows a Poisson process and the service time follows an exponential distribution with parameter ?. Further we assume that the catastrophe occur at the rate of ? which follows a Poisson process and the length of time the server in vacation follows an exponential distribution with parameter ?.  Assume that the system initially contains k customers when the server enters in to the system and starts the service immediately in a 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 ?. 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. We are analyzing the possibility of catastrophe that is whenever a catastrophe occurs in the system, all the customers who are in the system will be completely destroyed and system becomes an empty and server goes for a multiple vacation. This model is completely solved by constructing the generating function  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 analytical solution for mean number of customers and variance of the system. Numerical studies have been done for analysis of mean and variance of number of customers in the system for various values of ?, µ, ? and k and also various particular cases of this model have been discussed. Keywords: Single server queue , Fixed batch service , Catastrophe, Multiple vacation, Steady state distributio

M/M/1 RETRIAL QUEUEING SYSTEM WITH VACATION INTERRUPTIONS UNDER PRE-EMPTIVE PRIORITY SERVICE

Consider a single server retrial queueing system with pre-emptive priority service and vacation interruptions in which customers arrive in a Poisson process with arrival rate λ1 for low priority customers and λ2 for high priority customers. Further it is assume that the service times follow an exponential distribution with parameters μ1 and μ2 for low and high priority customers respectively. The retrial is introduced for low priority customers only. The server goes for vacation after exhaustively completing the service to both types of customers.  The vacation rate follows an exponential distribution with parameter α. The concept of vacation interruption is used in this paper that is the server comes from the vacation into normal working condition without completing his vacation period subject to some conditions. Let k be the maximum number of waiting spaces for high priority customers in front of the service station. The high priority 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 and probability of server in vacation for various values of λ1 , λ2 , μ1 , μ2, α and σ  in elaborate manner and also various particular cases of  this model have been discussed