Journal of Probability and Statistics
Volume 2009 (2009), Article ID 716364, 18 pages
Research Article

Computational Procedures for a Class of GI/D/k Systems in Discrete Time

Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB, R3T 5V6, Canada

Received 30 April 2009; Accepted 31 August 2009

Academic Editor: Nikolaos E. Limnios

Copyright © 2009 Md. Mostafizur Rahman and Attahiru Sule Alfa. 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.


A class of discrete time GI/D/k systems is considered for which the interarrival times have finite support and customers are served in first-in first-out (FIFO) order. The system is formulated as a single server queue with new general independent interarrival times and constant service duration by assuming cyclic assignment of customers to the identical servers. Then the queue length is set up as a quasi-birth-death (QBD) type Markov chain. It is shown that this transformed GI/D/1 system has special structures which make the computation of the matrix R simple and efficient, thereby reducing the number of multiplications in each iteration significantly. As a result we were able to keep the computation time very low. Moreover, use of the resulting structural properties makes the computation of the distribution of queue length of the transformed system efficient. The computation of the distribution of waiting time is also shown to be simple by exploiting the special structures.