M/M/1 Non-preemptive Priority Queuing System with Multiple Vacations and Vacation Interruptions

Non-preemptive priority queue system is a type of priority queue where customers with higher priorities cannot interrupt low priority one while being served. High priority consumers will still be at the head of the queue. This article discusses the non-preemptive priority queue system with multiple working vacations, where the vacation can be interrupted. Customers are classified into two classes, namely class I (non-preemptive priority customers) and class II, with exponentially distributed service rates. Customers will still receive services at a slower rate than during normal busy periods when they enter the system while it is on vacation. Suppose other customers are waiting in the queue after completing service on vacation. In that case, the vacation will be interrupted, and the service rate will switch to the busy period service rate. The model's performance measurements are obtained using the complementary variable method and analyzing the state change equation following the birth and death processes to find probability generating function for both classes. The results of the numerical solution show that the expected value number of customers and waiting time of customers in the queue for both class customers will be reduced when the vacation times rate (θ) and the vacation service rate (μ_0 ) increase

Transient Solution of an M/M/1 Retrial Queue with Reneging from Orbit

In this paper, the transient behavior of an M/M/1 retrial queueing model is analyzed where the customers in the orbit possess the reneging behavior. There is no waiting room in the system for the arrivals. If the server is not free when the occurrence of an arrival, the arriving customer moves to the waiting group, known as orbit and retries for his service. If the server is idle when an arrival occurs (either coming from outside the queueing system or from the waiting group), the arrival immediately gets the service and leaves the system. Each individual customer in the orbit, retrying for his service, becomes impatient and starts reneging from the orbit. Here the reneging of customers is due to the long wait in the orbit. Using continued fractions, the transient probabilities of orbit size for this model are derived explicitly. Average and variance of orbit size at time t are also obtained. Further, numerical illustrations of performance measures are done to analyze the effect of parameters

Joint queue length distribution of multi-class, single server queues with preemptive priorities

In this paper we analyze an $M/M/1$ queueing system with an arbitrary number of customer classes, with class-dependent exponential service rates and preemptive priorities between classes. The queuing system can be described by a multi-dimensional Markov process, where the coordinates keep track of the number of customers of each class in the system. Based on matrix-analytic techniques and probabilistic arguments we develop a recursive method for the exact determination of the equilibrium joint queue length distribution. The method is applied to a spare parts logistics problem to illustrate the effect of setting repair priorities on the performance of the system. We conclude by briefly indicating how the method can be extended to an $M/M/1$ queueing system with non-preemptive priorities between customer classes.Comment: 15 pages, 5 figures -- version 3 incorporates minor textual changes and fixes a few math typo

On impatience in M/M/1/N/DWV queue with vacation interruption

In this paper, we establish a cost optimization analysis for an M/M/1/N queuing system with differentiated working vacations, Bernoulli schedule vacation interruption, balking and reneging. Once the system is empty, the server waits a random amount of time before he goes on working vacation during which service is provided with a lower rate. At the instant of the service achievement in the vacation period, if there are customers present in the system, the vacation is interrupted and the server returns to the regular busy period with probability β\u27 or continues the working vacation with probability 1 - β\u27. Whenever the working vacation is ended, the server comes back to the normal busy period. If the system is empty, the server can take another working vacation of shorter duration. In addition, it is supposed that during both busy and working vacation periods, arriving customers may become impatient with individual timers exponentially distributed. Explicit expressions for the steady-state system size probabilities are derived using recursive technique. Further, interesting performance measures are explicitly obtained. Then, we construct a cost model in order to determine the optimal values of service rates, simultaneously, to minimize the total expected cost per unit time by using a quadratic fit search method (QFSM). Finally, numerical illustrations are added to validate the theoretical results