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International Journal of Stochastic Analysis
Volume 2013 (2013), Article ID 196372, 14 pages
Research Article

Analysis of Queue-Length Dependent Vacations and P-Limited Service in BMAP/G/1/N Systems: Stationary Distributions and Optimal Control

School of Basic Sciences, Indian Institute of Technology, Samantapuri, Nandan Kanan Road, Bhubaneswar 751 013, India

Received 27 May 2013; Accepted 26 August 2013

Academic Editor: Enzo Orsingher

Copyright © 2013 A. D. Banik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider a finite-buffer single server queueing system with queue-length dependent vacations where arrivals occur according to a batch Markovian arrival process (BMAP). The service discipline is P-limited service, also called E-limited with limit variation (ELV) where the server serves until either the system is emptied or a randomly chosen limit of customers has been served. Depending on the number of customers present in the system, the server will monitor his vacation times. Queue-length distributions at various epochs such as before, arrival, arbitrary and after, departure have been obtained. Several other service disciplines like Bernoulli scheduling, nonexhaustive service, and E-limited service can be treated as special cases of the P-limited service. Finally, the total expected cost function per unit time is considered to determine locally optimal values of or a maximum limit of as the number of customers served during a service period at a minimum cost.