A simple analysis of system characteristics in the batch service queue with infinite-buffer and Markovian service process using the roots method: GI/C-MSP^(a,b)/ 1 / ∞

¹ Department of Mathematics and Computer Science, Royal Military College of Canada, P.O. Box 17000, STN Forces, Kingston Ont., Canada K7K 7B4.
chaudhry-ml@rmc.ca
² School of Basic Sciences, Indian Institute of Technology, Samantapuri, Nandan Kanan Road, 751 013 Bhubaneswar, India.
banikad@gmail.com; souvikghosh589@gmail.com
³ CEMAT and Department of Mathematics, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal.
apacheco@math.tecnico.ulisboa.pt

Received: 9 September 2014
Accepted: 17 September 2015

Abstract

We consider an infinite-buffer single-server queue with renewal input and Markovian service process where customers are served in batches according to a general bulk service rule. Queue-length distributions at epochs of pre-arrival, arbitrary and post-departure have been obtained along with some important performance measures such as mean queue lengths and mean waiting times in both the system as well as the queue. We also obtain the steady-state service batch size distributions as well as system-length distributions. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of queue-length distribution at a pre-arrival epoch. Also, we provide analytical and numerical comparison between the roots method used in this paper and the matrix geometric method in terms of computational complexities and required computation time to evaluate pre-arrival epoch probabilities for both the methods. Later, we have established heavy- and light-traffic approximations as well as an approximation for the tail probabilities at pre-arrival epoch based on one root of the characteristic equation. Numerical results for some cases have been presented to show the effect of model parameters on the performance measures.