Journal of Applied Mathematics
Volume 2012 (2012), Article ID 123808, 14 pages
Research Article

Analysis of the M/M/N/N Queue with Two Types of Arrival Process: Applications to Future Mobile Radio Systems

1Department of Electrical and Computer Engineering, University of Canterbury, Christchurch 8140, New Zealand
2School of Engineering and Computer Science, Victoria University of Wellington, Wellington 6011, New Zealand
3Telecom New Zealand, Wellington 6140, New Zealand

Received 14 July 2011; Revised 18 September 2011; Accepted 19 September 2011

Academic Editor: Yuri Sotskov

Copyright © 2012 Peter J. Smith et al. 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.


The queueing system considered is essentially a M/M/N/N queue where two types of users compete for the 𝑁 resources. The users may have different arrival and service rates and are denoted as primary or secondary users. The primary users have priority access to the resources, and three levels of priority are considered: perfect priority, partial priority, and no priority. This system models the recently developed cognitive radio concept, a methodology that has been proposed for future mobile radio systems. In this context, the primary users have certain rights to use the resources, whereas the secondary users must make opportunistic use of the resources without impacting too much on the performance of the primary users. For all priority settings, the mean number of primary and secondary users is derived as are the blocking probabilities for both users. When no priority is given to the primary user, the system collapses to a truncated form of two independent M/M/∞ queues. The product form solution for this special case is known, and, here, these results are given in a novel, compact form. In the case of nonzero priority, the dropping probability for the secondary users is also derived.