Priority queueing model with balking and reneging

This investigation deals with two levels, single server preemptive priority queueing model with discouragement behaviour (balking and reneging) of customers. Arrivals to each level are assumed to follow a Poisson process and service times are exponentially distributed. The decision to balk / renege is made on the basis of queue length only. Two specific forms of balking behaviour are considered. The system under consideration is solved by using a finite difference equation approach for solving the governing balance equations of the queueing model, with infinite population of level 1 customer. The steady state probability distribution of the number of customers in the system is obtained

(R1984) Analysis of M^[X1], M^[X2]/G1, G_2^(a,b)/1 Queue with Priority Services, Server Breakdown, Repair, Modified Bernoulli Vacation, Immediate Feedback

In this investigation, the steady state analysis of two individualistic batch arrival queues with immediate feedback, modified Bernoulli vacation and server breakdown are introduced. Two different categories of customers like priority and ordinary are to be considered. This model propose nonpreemptive priority discipline. Ordinary and priority customers arrive as per Poisson processes. The server consistently afford single service for priority customers and the general bulk service for the ordinary customers and the service follows general distribution. The ordinary customers to be served only if the batch size should be greater than or equal to a , else the server should not start service until a customers have accumulated. Meanwhile priority queue is empty; the server becomes idle or go for vacation. If server gets breakdown while the priority customers are being served, they may wait in the head of the queue and get fresh service after repair completion, but in case of ordinary customers they may leave the system. After completion of each priority service, customer may rejoin the system as a feedback customer for receiving regular service because of inappropriate quality of service. Supplementary variable technique and probability generating function are generally used to solve the Laplace transforms of time-dependent probabilities of system states. Finally, some performance measures are evaluated and express the numerical results

Stochastic decomposition in discrete-time queues with generalized vacations and applications

For several specific queueing models with a vacation policy, the stationary system occupancy at the beginning of a rantdom slot is distributed as the sum of two independent random variables. One of these variables is the stationary number of customers in an equivalent queueing system with no vacations. For models in continuous time with Poissonian arrivals, this result is well-known, and referred to as stochastic decomposition, with proof provided by Fuhrmann and Cooper. For models in discrete time, this result received less attention, with no proof available to date. In this paper, we first establish a proof of the decomposition result in discrete time. When compared to the proof in continuous time, conditions for the proof in discrete time are somewhat more general. Second, we explore four different examples: non-preemptive proirity systems, slot-bound priority systems, polling systems, and fiber delay line (FDL) buffer systems. The first two examples are known results from literature that are given here as an illustration. The third is a new example, and the last one (FDL buffer systems) shows new results. It is shown that in some cases the queueing analysis can be considerably simplified using this property

Priority Queueing System with a Single Server Serving Two Queues M[X1],M[X2]/G1,G2/1 with Balking and Optional Server Vacation

In this paper we study a vacation queueing system with a single server simultaneously dealing with an M[x1] /G1/1 and an M[x2] /G2/1 queues. Two classes of units, priority and non-priority, arrive at the system in two independent compound Poisson streams. Under a non-preemptive priority rule, the server provides a general service to the priority and non-priority units. We further assume that the server may take a vacation of random length just after serving the last customer in the priority unit present in the system. If the server is busy or on vacation, an arriving non-priority customer either join the queue with probability b or balks(does not join the queue) with probability (1 - b). The time dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results are obtained explicitly. Also the average number of customer in the priority and the non-priority queue and the average waiting time are derived. Numerical results are computed

Transient Analysis of Two- Class Priority Queuing System

A transient solution of Two-class priority queuing system was discussed in this paper, the first class with high priority and the second one with low priority and the capacity of the system is infinite. Various performance measures are examined. Some numerical analyses are discussed and some special cases are deduced and confirmed the results

Analysis of Two Stage M[X1],M[X2]/G1,G2/1 Retrial G-queue with Discretionary Priority Services, Working Breakdown, Bernoulli Vacation, Preferred and Impatient Units

In this paper, we study M[X1] , M[X2] /G1 ,G2 /1 retrial queueing system with discretionary priority services. There are two stages of service for the ordinary units. During the first stage of service of the ordinary unit, arriving priority units can have an option to interrupt the service, but, in the second stage of service it cannot interrupt. When ordinary units enter the system, they may get the service even if the server is busy with the first stage of service of an ordinary unit or may enter into the orbit or leave the system. Also, the system may breakdown at any point of time when the server is in regular service period. During the breakdown period, the interrupted priority unit will get the fresh service at a slower rate but the ordinary unit can not get the service and the server will go for repair immediately. During the ordinary unit service period, the arrival of negative unit will interrupt the service and it may enter into an orbit or leave the system. After completion of each priority unit’s service, the server goes for a vacation with a certain probability. We allow reneging to happen during repair and vacation periods. Using the supplementary variable technique, the Laplace transforms of time-dependent probabilities of system state are derived. From this, we deduce the steady-state results. Also, the expected number of units in the respective queues and the expected waiting times, are computed. Finally, the numerical results are graphically expressed

Transient Solution of M[X1],M[X2]/G1,G2/1 with Priority Services, Modified Bernoulli Vacation, Bernoulli Feedback, Breakdown, Delaying Repair and Reneging

This paper considers a queuing system which facilitates a single server that serves two classes of units: high priority and low priority units. These two classes of units arrive at the system in two independent compound Poisson processes. It aims to decipher average queue size and average waiting time of the units. Under the pre-emptive priority rule, the server provides a general service to these arriving units. It is further assumed the server may take a vacation after serving the last high priority unit present in the system or at the service completion of each low priority unit present in the system. Otherwise, he may remain in the system. Also, if a high priority unit is not satisfied with the service given it may join the tail of the queue as a feedback unit or leave the system. The server may break down exponentially while serving the units. The repair process of the broken server is not immediate. There is a delay time to start the repair. The delay time to repair and repair time follow general distributions. We consider reneging to occur for the low priority units when the server is unavailable due to breakdown or vacation. We concentrate on deriving the transient solutions by using supplementary variable technique. Further, some special cases are also discussed and numerical examples are presented