~110~ International Journal of Statistics and Applied Mathematics 2018; 3(6): 110-118 ISSN: 2456-1452 Maths 2018; 3(6): 110-118 © 2018 Stats & Maths www.mathsjournal.com Received: 15-09-2018 Accepted: 16-10-2018 Deepa Chauhan Department of Mathematics, Axis Institute of Technology and Management, Kanpur, Uttar Pradesh, India Correspondence Deepa Chauhan Department of Mathematics, Axis Institute of Technology and Management, Kanpur, Uttar Pradesh, India Maximum entropy analysis of unreliable M X / (G 1 , G 2 )/1 queue with Bernoulli vacation schedule Deepa Chauhan Abstract This paper examines the steady state behavior of unreliable server in a batch arrival queue with two phases of heterogeneous service along with Bernoulli schedule. The server provides two kinds of services in succession, the first stage service (FSS) followed by second stage service (SSS). As soon as both services are completed, the server may take a vacation or may continue staying in the system. The server may break down during the service; the failure and repair times are assumed to follow a general distribution. By using the maximum entropy method (MEM), the approximate formulae for the probability distributions of the number of customers in the system have been derived, which are further used to obtain various system performance measures. A comparative analysis between approximate results established and exact results has also been performed. It is noticed that the maximum entropy approach provides reasonably good approximate solutions for practical purpose. Keywords: Maximum entropy, Bernoulli vacation schedule, batch arrival, two phase services, un- reliable server 1. Introduction Information theory provides a constructive criterion for setting up probability distribution on the basis of partial knowledge, called the maximum entropy estimate. It is least biased estimate, possible on the given information. Jain and Singh (2000) [6] employed MEM to analyze the optimal flow control of G/G/c finite capacity queue via diffusion process. Herrero (2002) [2] proposed a direct method to compute the second moment and also for probability of customers being served in a busy period. The methodology of maximum entropy has been used by Guan et al. (2009) [1] to characterize closed form expression for the state and blocking probabilities for threshold based discrete time queue. Wang et al. (2002) [9] analyzed the N-policy M/G/1 queueing system with removable server by using maximum entropy. Jain and Dhakad (2003a) [3] provided the steady state queue size distribution for G/G/1 queue by using maximum entropy approach. Moreover, Jain and Dhakad (2003b) [4] provided the steady state queue size distribution for M x /G/1 queueing system with server breakdowns and general setup times. Entropy maximization and queuing network with priorities and blocking have been analyzed by Kouvatsos and Awan (2003) [8] . Maximum entropy principle is used by Wang et al. (2005) [10] , to derive approximate formulae for the steady state probability distribution of the queue length. Recently, Jain and Jain (2006) [5] obtained approximate results for the queue size distribution for G/G/1 queue with vacation under N-policy. A comparative analysis has been made between the approximate results and exact results for M x /G/1 queueing system with server vacation by Ke and Lin (2006) [7] . A single unreliable server M X /G/1 queueing system with multiple vacations was considered by Wang et al. (2007) [11] . Wang and Huang (2009) [12] analyzed a single removable and unreliable server queue and use maximum entropy approach to develop the approximate formulae for waiting time in the system. In this investigation, we have derived the approximate formulae for the expected system size and expected waiting time for an M x /G/1 queueing model with Bernoulli schedule vacations and repairable server. The organization of the paper is as follows. The model under consideration is described along with assumptions and limiting probabilities and governing