The model illustrated has been used to determine the effect
of impatient telephone customers on a computerized telephone exchange.
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]
Station S1 represents a node dedicated to a special processing task and required by all customers. A customer is prepared to wait for a certain period of time to get a reply. If at the end of that period, the reply has not arrived, the customer may either give up and leave the network or else wait awhile before trying again. Customers at station S1 are processed by a single server according to a processor sharing discipline. Each customer possesses a limited amount of patience which is defined as an upper bound on its service duration; when its patience is exhausted, the customer simply gives up processing. This impatient customer may simply quit the network (with a fixed probability, 1-h); otherwise it joins an infinite server station S2 where it remains for a certain period, called the thinking time, and then re-joins station S1 for another attempt.
A state of the network may be described by the pair (i, j) where i and j are the number of customers in stations S1 and S2 respectively. When i >= 1, the probability of
To obtain a finite Markov chain, we assume that K1 is the maximum
number of customers permitted in station S1 and K2 the maximum
permitted in S2. Customers arriving to a full station are lost.
In the dataset corresponding to this model,
tcomm_in, the values of K1 and
K2 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 K1, K2, n and nz for the 20 datasets:
K1 K2 n nz
5 110 666 3,091
10 110 1,221 5,851
15 110 1,771 8,611
20 110 2,331 11,371
10 220 2,431 11,681
15 220 3,536 17,191
20 220 4,641 22,701
30 220 6,851 33,721
10 330 3,641 17,511
15 330 5,296 25,771
20 330 6,951 34,031
30 330 10,261 50,551
10 440 4,851 23,341
15 440 7,056 34,351
20 440 9,261 45,361
30 440 13,671 67,381
10 550 6,061 29,171
15 550 8,816 42,931
20 550 11,571 56,691
30 550 17,081 84,211
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