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Journal of Applied Mathematics and Stochastic Analysis
Volume 2008 (2008), Article ID 415692, 34 pages
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.


We consider a model for a single link in a circuit-switched network. The link has 𝐢 circuits, and the input consists of offered calls of two types, that we call primary and secondary traffic. Of the 𝐢 links, 𝑅 are reserved for primary traffic. We assume that both traffic types arrive as Poisson arrival streams. Assuming that 𝐢 is large and 𝑅=𝑂(1), the arrival rate of primary traffic is 𝑂(𝐢), while that of secondary traffic is smaller, of the order βˆšπ‘‚(𝐢). 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, βˆšπ‘‚(𝐢). Thus, the primary calls have fast arrivals and fast service, compared to the secondary calls. The loads for both traffic types are comparable (𝑂(𝐢)), and we assume that the system is β€œcritically 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.