International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8 Issue-9, July, 2019 2636 Published By: Blue Eyes Intelligence Engineering & Sciences Publication Retrieval Number: I8985078919/19©BEIESP DOI:10.35940/ijitee.I8985.078919 Abstract— This paper deals with an M/M/1 queueing system with customer balking and reneging. Balking and reneging of the customers are assumed to occur due to non-availability of the server during vacation and breakdown periods. Steady state probabilities for both the single and multiple vacation scenarios are obtained by employing probability generating functions. We evaluate the explicit expressions for various performance measures of the queueing system. Key words:— impatient customers, balking, reneging, vacation, server breakdowns. 1. INTRODUCTION In real life, it can be observed that most queueing systems are with server vacations, breakdowns, delayed repairs and impatient customers. Some of the common examples include those in manufacturing systems, designing of local area communication networks and data communication networks. In general, the customers are impatient. In our fast life we often see that the customers anticipating service need to form a queue. Going through impatience due to the phenomenon of joining queue, the customers may not join the queue or even if they join, may quit from the queue without before served. The primary aim of this paper is to explore the steady state behavior of the M/M/1 queue with impatient customers under single and multiple vacation policies. The customer impatience with vacation has become the essential feature for the queuing models which are analyzed by the many authors in the past. For the queueing models with balking, reneging during different vacation states the reader may refer to [4 and 5] Dequan Yue, Wuyi Yue, Xiuju Li [1] considered queuing system with impatient customers and multiple vacations under two-phase service. He derived the Probability generating functions were derived for different states of the server focusing the number of customers present in the system. Besides, the closed-form expressions were derived for various performance measures. These measures included the mean system sizes for various states of the server, the average rate of balking, the average rate of reneging and the average rate of loss. Sherif I. Ammar [2] found the transient behavior of an M/M/1 queue with impatient customers and multiple vacations. He obtained the explicit expressions for the mean and variance of the system size in terms of the modified Bessel functions. Revised Manuscript Received on July 10, 2019. Ch. Swathi, Vigana’s Foundation for Science Technology & Research (Deemed to be university), Vadlamudi, Guntur dt, Andhra Pradesh, India. V. Vasanta Kumar, Vigana’s Foundation for Science Technology & Research (Deemed to be university), Vadlamudi, Guntur dt, Andhra Pradesh, India. Hanumantha Rao, Vigana’s Foundation for Science Technology & Research (Deemed to be university), Vadlamudi, Guntur dt, Andhra Pradesh, India. A recent study by S. Hanumantha Rao, V. Vasanta Kumar et.al [3] has focused on the impatient behavior of customers in a two-phase M/M/1 queuing system with server breakdown and delayed repair. They derived the probability generating function of the queue length distribution in steady state, the mean system size and the average rate of loss. In this paper, we consider M/M/1 queuing system with customer balking and reneging during the server vacations or breakdown periods. The current research is presented as per the following sections: The second section presents the mathematical model. The model is considered having multiple vacation policy and server breakdown. The equilibrium analysis of the system states is performed in the third section. Using this, the probability generating functions of the steady state probabilities are obtained. In section 4, the closed form expressions for some performance measures are derived. These measures include mean system size, average rate of customers served and balked. We considered the system under the multiple vacations. In fifth and sixth we describe the mathematical model and derived the steady state probabilities for the system with breakdowns under single vacation policy and obtained the various closed form results respectively. In section seven and eight conclusions and references are given respectively. 2. MULTIPLE VACATION POLICY In the current research M/M/1 queue is considered. Here the server takes multiple vacations. Whenever the server has no customers to serve, the server will enter vacation mode. After the vacation period it returns to the system to resume its services. If the queue line is headed up with the customers for service, it starts its busy period and continues the services until the queue line impoverish. On the other hand, if there are no customers in the queue line by the time the server returns from vacation, the server takes another vacation. Model description The assumptions and notations of the system model in the present research are presented below. a) Customers will arrive into the queue individually with the Poisson arrival rate  . Also, the service times are negative exponentially distributed with rate . M/M/1 Queueing system with Customer Balking and Reneging Ch. Swathi, V. Vasanta Kumar, S. Hanumantha Rao