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An algorithm to compute blocking probabilities in multi-rate multi-class multi-resource loss models

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  01 July 2016

Gagan L. Choudhury ,
Kin K. Leung  and
Ward Whitt
Gagan L. Choudhury*
Affiliation:
AT&T Bell Laboratories
Kin K. Leung*
Affiliation:
AT&T Bell Laboratories
Ward Whitt*
Affiliation:
AT&T Bell Laboratories
*
* AT&T Bell Laboratories, Holmdel, NJ 07733-3030, USA.
* AT&T Bell Laboratories, Holmdel, NJ 07733-3030, USA.
** AT&T Bell Laboratories, Murray Hill, NJ 07974-0636, USA.
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Abstract

In this paper we consider a family of product-form loss models, including loss networks (or circuit-switched communication networks) and a class of resource-sharing models. There can be multiple classes of requests for multiple resources. Requests arrive according to independent Poisson processes. The requests can be for multiple units in each resource (the multi-rate case, e.g. several circuits on a trunk). There can be upper-limit and guaranteed-minimum sharing policies as well as the standard complete-sharing policy. If all the requirements of a request cannot be met upon arrival, then the request is blocked and lost. We develop an algorithm for computing the (exact) steady-state blocking probability of each class and other steady state descriptions in these loss models. The algorithm is based on numerically inverting generating functions of the normalization constants. In a previous paper we introduced this approach to product-form models and developed a full algorithm for a class of closed queueing networks. The inversion algorithm promises to be even more useful for loss models than for closed queueing networks because fewer alternative algorithms are available for loss models. Indeed, for many loss models with sharing policies other than traditional complete sharing, our algorithm is the first effective algorithm. Unlike some recursive algorithms, our algorithm has a low storage requirement. To treat the loss models here, we derive the generating functions of the normalization constants and develop a new scaling algorithm especially tailored to the loss models. In general, the computational complexity grows exponentially in the number of resources, but the computation can often be reduced dramatically by exploiting conditional decomposition based on special structure and by appropriately truncating large finite sums. We illustrate our numerical inversion algorithm by applying it to several examples. To validate our algorithm on small models, we also develop a direct algorithm. The direct algorithm itself is of interest, because it tends to be more efficient when the number of resources is large, but the number of request classes is small. Furthermore, it also allows a form of conditional decomposition based on special structure.

Keywords

PRODUCT-FORM MODELSLOSS NETWORKSCIRCUIT-SWITCHED COMMUNICATION NETWORKSRESOURCE-SHARING MODELSCOORDINATE-CONVEX RESOURCE-SHARING POLICIESGENERATING FUNCTIONSNUMERICAL TRANSFORM INVERSIONNORMALIZATION CONSTANTSPARTITION FUNCTIONS

MSC classification

Primary: 60K25: Queueing theory
Secondary: 90B15: Network models, stochastic 90B22: Queues and service 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 27 , Issue 4 , December 1995 , pp. 1104 - 1143
Copyright
Copyright © Applied Probability Trust 1995 

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