Blocking probability in the multi-service switching networks with multicast traffic Maciej Stasiak, Piotr Zwierzykowski and Mariusz Glqbowski* Abstract- In this paper, a new approximate method of pointto-group blocking probability calculation in switching networks carrying a mixture of different multi-rate unicast and multicast traffic streams is presented. Special attention is paid to the methods for determining the effective availability parameter in the case of multicast connections. The results of analytical calculations are compared with the data of digital simulations of switching networks with unicast and multicast connections. Introduction
Multicast switching has found applications in virtual circuit networks such as B-ISDN (Broadband Integrated Services Digital Network). These applications include many services such & videoconferencing, video-on-demand and distributed data processing. Multicast service in multiservice switching network requires a communication path fkom an incoming link to certain number of outgoing links. Thus, multicast connection induces a communication tree in the switching network, connecting an incoming link to any subset of outgoing directions. For the analysis of multi-service switching systems the socalled multi-rate models [l-21 are mainly used. In these models the system carries call demands having an integer number of BBU (Basic Bandwidth Units). In accordance with a bandwidth discretization [l-21, the BBU is defined as the greatest common divisor of all call demands. In [3-4] the model with multi-rate traffic (full-availability group) has been considered. In [5] the separate l i model (limited-availability group) has been derived. In [6] this model has been used for analysis of switching nodes with multicast connections. In [7-81 the models of limitedavailability group have been used in order to approximate the outgoing groups of links in the switching networks carrying a mixture of multi-rate traftk streams. Switching networks carrying a mixture of different traffic streams were the subject of many analyses ed. [7-121. In [13-141 the methods for blocking probability calculation in single-service switching networks with multicast traffic stream have been proposed. In [15-171 the non-blocking multi-service switching networks for multicast connections have been considered. Solutions associated with multicast connections in blocking multi-service switching networks have not been provided so far. The paper deals with model of multi-service switching network carrying both the mixture of different multi-rate unicast traffic streams and the mixture of different multirate multicast traffic streams. We assumed in the paper that Institute of Electronics a n d Telecommunications Poznan University of Technology ul. Piotorowo 3 A , 60-965 P o z n a n , Poland ?-mail:
[email protected] Mariusz Gtqbowski is supported by the Foundation for Polish Science
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communication trees of multicast connections are executed in the last-stage switches of the switching networks. The proposed model is based on the PGBMT (Point To Group Blocking for Multi-&"el Traffk) method of point-togroup blocking in the multi-service switching network [7]. I. MULTI-RATE MODELS In multi-rate models, resources required for connections of particular classes are the multiplicity of certain value of bandwidth, the so-called BBU (Basic Bandwidth Unit). While constructing multi-rate models for broadband systems, it is assumed that BBU is the greatest common divisor of the equivalent bandwidth of all the call streams offered to the system [l-21. Let us consider a multi-rate system with a capacity of V BBU. The system services A4 independent classes of Poisson traffic streams with the following intensities: AI, Az, ..., AM The class i call requires ti BBU to set up a connection. The holding time for the calls of particular classes has exponential distribution with the parameters: p1, pz, ..., h.Thus, the mean trafXc offered to the system by the class i traffic stream is equal to:
.
ai = jl,/Pt
(1)
A. Basic recurrence relations A multi-dimensional service process occurring in a system with different multi-rate traffic streams can be approximated by the one-dimensional Markov chain characterised by a product form solution. This approximation leads to a very simple recurrent formula for occupancy distribution in the multi-rate system, known as generalised Kaumn-Roberts recursion, [5,9, 181:
Ca,t,cr,(n - t, )p
