Journal of Applied Mathematics and Stochastic Analysis
Volume 2008 (2008), Article ID 415692, 34 pages
doi:10.1155/2008/415692
Research Article
Asymptotic Analysis of a Loss Model with Trunk Reservation I: Trunks Reserved for Fast Traffic
1Alcatel-Lucent Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974, USA
2Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL 60607, USA
Received 16 January 2007; Accepted 21 November 2007
Academic Editor: Benjamin Melamed
Copyright © 2008 John A. Morrison and Charles Knessl. 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.
Abstract
We consider a model for a single link in a circuit-switched
network. The link has C circuits, and the input consists
of offered calls of two types, that we call primary and secondary traffic. Of the C links, R are reserved for primary traffic. We
assume that both traffic types arrive as Poisson arrival streams.
Assuming that C is large and R=O(1), the arrival rate of primary traffic is O(C), while that of secondary traffic is smaller, of the order O(C). The holding times of the primary calls are
assumed to be exponentially distributed with unit mean. Those
of the secondary calls are exponentially distributed with a large
mean, that is, O(C). Thus, the primary calls have fast arrivals
and fast service, compared to the secondary calls. The loads for
both traffic types are comparable
(O(C)), and we assume that the system is “critically loaded”; that is, the system's capacity is approximately
equal to the total load. We analyze asymptotically
the steady state probability that n1 (resp.,
n2) circuits are occupied
by primary (resp., secondary) calls. In particular, we obtain
two-term asymptotic approximations to the blocking probabilities
for both traffic types.