International Journal of Engineering Research and Development e-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.com Volume 13, Issue 9 (September 2017), PP.01-11 1 Analysis of a Queueing Model with Balking for Buffer Sharing in Atm * Yogendra Kumar Sharma 1 ,G.C.Sharma 2 ,M. Jain 3 1 Department of Mathematics, Seth Padam Chand Institute of Management, Dr. B. R. A. University, Agra (INDIA) 2 Department of Mathematics, Institute of Basic Science, Dr. B. R. A. University, Agra (India) 3 Department of Mathematics, Indian Institute of Technology, Roorkee, (India) Corresponding Auther: *Yogendra Kumar Sharma ABSTRACT:- In this paper, we study a Markovian queueing model with balking for buffer sharing in ATM. It is assumed that server provides service with priorities termed as first, second and third. As soon as system accumulates L+1, M+1 customers third and second priority customers balk with balking function b n . We have obtained the steady state probabilities for the system by recursive method. Some performance measures have also been obtained. A numerical example is also provided. Keywords:- Markovian Queueing Model, Balking, Buffer sharing, Asynchronous Transfer Mode (ATM). I. INTRODUCTION In many queueing systems the arriving customers may not join the queue due to impatience if the number of customers is greater than the predetermined threshold value, that is, the customers may balk if system size is greater than a threshold value. Clearly balking of customers is a loss to the system and needs attention to improve the same. The improvement is possible only when full analysis of the system is available. Keeping this factor such queueing systems have been studied by many researchers in queueing literature. Ancker and Gafarian (1963) investigated some queueing problems with balking and reneging. Doshi and Jangerman (1986) studied an M/G/1 queue with class dependent balking; they obtained some performance measures for the system. A related concept may be seen in asynchronous transfer mode (ATM) where loss priority control concerned with reducing the loss probability of customers in buffer sharing scheme. Bae and Suda (1991) provided a survey of traffic control scheme in protocols in ATM networks. Abou and Hariri (1992) analysed the M/M/C/N queueing system with balking and reneging. They also obtained some performance measures and numerical results for the system. Suri et al. (1994) evaluated space priority strategies in ATM network. Lee and Ahn (1998) analysed a queueing model for optimal control of partial buffer sharing in ATM. They gave an algorithm to obtain the steady state probabilities and performance measures. In this paper we model M 1 , M 2 , M 3 /G/1/N+1 queue with balking for buffer sharing in ATM. We assume that first, second and third priority customers arrive according to poison with rates 1 , 2 and 3 respectively. The service times are identical for all three types and generally distributed. As soon as the number of customers becomes L+1 and M+1, the third and second priority customers balk with balking function b n . If the number of customers in the system is M+1 only first and second priority customers are allowed to join the queue. Since the buffer size is N, the number of customers in the system can not exceed N+1 including one in service. II. NOTATIONS 1 Arrival rate of first priority customers. 2 Arrival rate of second priority customers. 3 Arrival rate of third priority customers. Total customers arrival rate = 1 + 2 + 3 L,M Thresholds N Buffer capacity S(x) Service time distribution function s(x) Service time probability function S*(θ ) Laplace transform of s(x) E(s) Mean service time P Lost1 Lost probability of first priority customers