On The Renewal Input Batch-arrival Queue Under Single And Multiple Working Vacation Policy With Application To EPON Dibyajyoti Guha and A. D. Banik* School of Basic Sciences, Indian Institute of Technology, Samantapuri, Nandan Kanan Road, Bhubaneswar-751013, India, e-mail: dibya.guha@gmail.com, adattabanik@iitbbs.ac.in, banikad@gmail.com Abstract—This paper analyzes renewal input batch arrival queues with multiple and single working vacation policy. We obtain the steady state system-length distributions at pre-arrival and arbitrary epochs. The Laplace-Stieltjes transforms (LST) of the sojourn time distribution for the first customer and an arbitrary customer in an accepted batch have been derived. Also the probability density function of sojourn time distribution for the first customer in an accepted batch has been obtained. The proposed analysis is based on the roots of the characteristics equations formed using the probability generating functions (p.g.f.) of embedded pre-arrival epoch probabilities. Several performance measures such as the mean system-length and mean sojourn times of first and an arbitrary customer in an accepted batch are presented. The suggested queues have potential application in ethernet passive optical network (EPON). Keywords GI/M/I queue, batch arrival, multiple-, single-working vacation, sojourn time analysis, roots 1 INTRODUCTION Priority queueing is an important aspect to improve the delay- throughput characteristics of IEEE 802.11, Bluetooth, router or I/ O subsystems. Vacation queueing provides a good mathematical abstraction for performance evaluation of priority queueing systems. A queueing system in which an arriving customer finds that the server is not available to serve the customers is known as vacation queue. Vacation queueing is classified in two categories based on the status of the server during vacation: i) non working vacation: the server remains inactive during vacation and ii) working vacation: the server serves the customers at a lower rate during vacation. A vast literature survey on vacation queue- ing system have been performed by several researchers (Doshi, 1986; Takagi, 1991; Tian & Zhang, 2006). Vacation queues may further be classified as multiple vacation and single vacation based on the number of vacations taken by the server. In case of multiple vacation queueing system the server keeps on taking vacations until a customer is found to be waiting at the end of a vacation. In case of single vacation system the server resumes service after completing a single vacation. If there is no customer found to be waiting at the end of single vacation, the server remains idle and waits for an arrival of a new customer at normal service mode. For notational purpose, multiple working vacation queueing system and single working vacation queueing system are denoted by MWV and SWV , respectively. An M∕M∕1 multiple working vacation queue in which the inter-arrival duration, service period and vacation period are considered to be independent and exponentially distributed, has been presented (Servi and Finn, 2002). They also proposed that M∕M∕1∕MWV can be deployed in wave length division multi- plexing (WDM) based optical access network in which wave lengths can be reconfigured. On the similar direction, bulk arrival for M∕M∕1 and Geom∕Geom∕1 queue with multiple working vacations have been considered (Xu et al., 2009). An extension of M∕M∕1∕MWV to a GI ∕M∕1∕MWV queueing system where the inter-arrival times follows an independent identically dis- tributed general distribution function has been presented (Baba, 2005). In general, the service process, vacation process and arrival process are mutually independent. Also, the service times during a normal service period, the service times during a working vacation and working vacation times are exponentially distributed with different rates. In the same direction, a GI ∕M∕1∕N multiple working vacation queue with a limited buffer capacity have been analyzed (Banik et al., 2007). An M∕G∕1∕MWV model where the service times during service Received May 2013; Revised April 2014; Accepted July 2014 INFOR, Vol. 51, No. 4, November 2013, pp. 175–191 ISSN 0315-5986|EISSN 1916-0615 175 http://www.utpjournals.press/doi/pdf/10.3138/infor.51.4.175 - Friday, June 03, 2016 4:43:50 PM - IP Address:91.229.229.202