International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 22 (2017) pp.11965-11969 © Research India Publications. http://www.ripublication.com 11965 Feedback Queue with Services in Different Stations under Reneging and Vacation Policies Sundar Rajan B Research Scholar, Research & Development Centre, Bharathiyar University, Coimbatore, Tamil Nadu - 641014, India. Corresponding author Orcid Id: 0000-0003-3340-9892 Ganesan V Associate Professor (Retd.), Department of Statistics, Periyar E.V.R. College, Tiruchirappalli, Tamil Nadu - 620023, India. Rita S Associate Professor, School of Mathematics, Department of Statistics, Periyar University, Salem, Tamil Nadu - 636011, India. Abstract This paper describes a single server queuing system where the units arrive in batches of variable size under Poisson stream. The server provides services in k stations. The service times in each station follows general distribution. The principles of feedback, vacation and reneging are employed in the system. The steady state probability functions that the server is providing service in any service station and that on vacation are derived. The corresponding steady state probabilities are also obtained. The expected number of units in the queue has been obtained for some special cases. Keywords: Single server, feedback queue, reneging, steady state probability, vacation INTRODUCTION An emerging area of queueing theory is the bulk queue, in which arrival and/or departure can happen in batches either fixed or in variable size. Vital applications of this queueing model can be seen implied in many areas like communication system, computer networks, production industry, logistic sector etc. Queueing models with vacation policies have been studied by many researchers including reneging. In real life, there are some queueing situations when some unit is discouraged by long waits in the queue. The units may be decided to balking or reneging. Balking and reneging have attracted the attention of many researchers and study on queues with behavior of such units has developed extensive amount of literature. Concept of reneging in queueing systems was introduced by Ancker and Gafarian [1] and Daley [2]. Bae et al., [3] have studied the waiting time of M/G/1 queue with impatient customers. Medhi [4] has explained a single server Poisson arrival queue along with a second optional channel. Altman and Yechaili [5] have analyzed customer impatience in queues with server vacation. Choudhury and Medhi [6] have studied balking and reneging in multi-server Markovian queueing systems. Vacation policies in queues were extensively studied previously by Baba [7], Doshi [8], Keilson and Servi [9], Madan et al., [10, 11, 12], Borthakur and Choudhury [13, 14, 15]. Ganesan and Sundar Rajan [16] have studied bulk arrival queue with breakdown analysis. Thangaraj and Vanitha [17] have discussed two-stage heterogeneous service, compulsory vacation and random breakdowns. Ganesan and Sundar Rajan [18] have analyzed a queue with heterogeneous services and random breakdowns. Ayyappan and Sathiya [19] have studied three stage heterogeneous service and server vacation of M x /G/1 feedback queueing model as well as two types of random breakdowns and multiple vacations with restricted admissibility [20]. Monita Baruah et al.,[21] have studied bulk input queue which have second optional service along with reneging during vacation. The present study deals M x /(G1, G2,……Gk)/1 queueing model, where the units arrive in batches of variable size defined by Bailey [22] and once the units enter the initial service process, it must go through k stations of service on first in first out (FIFO) discipline. The service time of this model follows general distribution. The unit follows Bernoulli feedback after completion of k stations of service and if it has not received a quality service, the unit will rejoin at the end of the queue with probability p or leaves forever from the system with probability (1-p). Whenever the system is empty, the server may go on vacation for a random duration. Here, the server adopts multiple vacation policy until at least one batch of arrival present in the system. The vacation periods of the server are distributed according to general distribution. It is assumed that when the server is on vacation the unit may renege from queue and it follows exponential distribution with parameter β.