# William J. Stewart

## A Multi-Class, Finite Buffer, Priority System

The model illustrated below has been used to model ATM networks. It has been discussed in several papers including:
Numerical Methods in Markov Chain Modelling. B. Philippe, Y. Saad and W.J. Stewart, Operations Research, Vol. 40, No. 6, pp. 1156-1179, 1992. [Postscript copy available]

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This model has applicability to telecommunications modelling. It consists of a single service center at which two identical servers provide service to two different classes of customer. The service rates may differ for each class, but both are exponentially distributed. The arrival processes of the two classes are not exponential. It has often been observed that teletraffic is rather bursty in nature and to take this into effect, hyper-exponential interarrival times with large coefficients of variance have been associated with these arrival processes.

Class one customers are assumed to have a high priority. An arriving class one customer is inserted in the queue in front of all class two customers. An idle server will only serve a class two customer if there are no class one customers waiting. However, once a server begins to provide service to a class two customer, it will continue to serve that customer even if a class one customers arrives and is forced to wait; in other words, the service is non-preemptive.

The effect of the limited capacity buffer is to restrict the number of customers that can enter the system. Class two customers that arrive to a full buffer are lost. If the buffer is full and contains both class one and class two customers, an arriving class one customer will displace a class two customer. This class two customer is therefore lost. A class one customer that arrives to a system that is full of class one customers is lost.

For class 1 customers, the rates of the two phases in the arrival process are taken to be 2.0 and 3.0 respectively and the probability of taking the first of these is set at 0.25. For class 2 customers, the corresponding parameter values are set at 2.0, 3.0 and 0.5. Class 1 customers have a service time distribution that is exponential with rate 1.0; for those of class 2, it is also exponential but at rate 1.5. These values are set in the qnatm subroutine of the qnatm.f source code file.

In the dataset corresponding to this model, qnatm_in, the maximum number of customers the buffer can hold, maxbuffer, along with the size of the matrix generated and the number of nonzeros in the matrix, are shown in the table below.

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Values of Buffersize, n and nz for the 7 datasets:

maxbuffer           n             nz

10             740          4,820
16           1,940         12,824
32           7,956         53,176
50          19,620        131,620
64          32,276        216,824
100          79,220        533,120
128         130,068        875,896

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