International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 06 | June 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 2004 Two-Class Priority Queueing System with Restricted Number of Priority Customers and Catastrophes A.M.K. Tarabia 1 , A.H. El-Baz 2 , A.M. Elghafod 3 1 Professor of Mathematical Statistics, Dept. of Mathematics, Damietta Faculty of Science, New Damietta, Egypt. 2 Assistant Professor of Statistics, Dept. of Mathematics, Damietta Faculty of Science, New Damietta, Egypt . 3 M.Sc. School Basic Sciences – Department of Mathematical Sciences –Statistics Division - New Damietta, Egypt. ---------------------------------------------------------------------***---------------------------------------------------------------------- Abstract - In this paper, we analyze a two-class single-server preemptive priority queueing system with restricted number of priority customers and catastrophes. The arrivals per class follow the Poisson process with exponentially distributed service times. Customers are served on a first-come, first-served basis within their queue. Explicit expressions for the mean queue length and the joint distribution are derived in the steady-state case for the number of high and low priority customers in the system. The analysis is based on the generating function technique. Also, we study the impact of catastrophe on the system and other performance measures. Key Words: Queueing, Queue length, Preemptive resume, Generating function, Catastrophe 1. INTRODUCTION This Priority queues occur in many aspects of daily life, especially in cases where preferential treatment is given to certain types of individuals, for example. Telecommunications field. Priority mechanisms are an invaluable scheduling method that allows messages from different categories to receive different quality of service. For this reason, the priority queue has received considerable attention in the literature. Cobham [1] was the first to consider the non-preemptive priority of the queue was considered with Poisson inputs and exponential retention time and discussed them for single and multi-channel cases. Holley [2] simplified Cobham’s work. Barry [3] and Stephan [4] studied the problem of preemptive priority system problem with two priorities, Poisson inputs and exponential retention time. White and Christie [5] and Heathcote [6] studied the same problem for two channels and used the generating function method to describe the steady-state and transient queue length distributions, respectively. The main difficulty is to extend these results to multiple categories with general service time allocations in processing Laplace transformations that characterize the dynamics of the system. An extensive study of two priority plans can be cited in Miller [7] where he obtained several results in this model using the functions of generating and transformations of Laplace. A different approach to converting Laguerre to this model has also been applied, see for example Keilson and Sumita [8] and Keilson and Nunn [9]. Miller [10] considered exponential single server priority queues with two classes of customers. He Obtain repeated calculation formulas for steady distributions using Neuts [11] theory of matrix- geometric invariant probability vectors. Shack and Larson [12] are treated with a multi-server, non-preemptive, Multi- stationary queueing system with Poisson arrivals and exponential servers. They assumed a limited discipline in the priority queue and derived many performance measures. Franti [13,14] developed algorithms for a dynamic priority queue to compute performance measures such as the distribution of queue length and moments. Nishida [15] gave an approximation analysis for a heterogeneous multiprocessor system with preemptive and non-preemptive priority discipline and also presented the numerical comparison between simulation and approximation analysis. Recently, Bitran and Caldentey [16] analyzed a two- class single queueing system that features state dependent arrivals and preemptive priority service discipline. For this system, they computed the mean balance queue length and studied the effective service time and other first passage time quantities for both classes. More recently, Drekic and Woolford [17] analyzed a two class, single-server preemptive priority queueing model with low priority balking customers. They considered arrivals to each class are assumed to follow a Poisson with exponentioally distributed service times and obtained the steady state joint distribution of the number of high and low priority customers in the system. Choa and Zheng [20] considered an immigration birth and death process with total catastrophes and studied its transient as well as equilibrium behavior Kirshna Kumar et al. [23,24] obtained transient solution for analytically using continued fractions for the system size in an M/M/1 queueing system with catastrophes, server failures and nonzero repair time , the steady state probability of the system size, and obtained transient solution for the system size in the M/M/1 queueing model with the possibility of catastrophes at the service station in the direct way. Dharmaraja and Rakesh