TCOMM --- A Telecommunications Model of Impatient Customers

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

- a service completion at S1 in
*[t, t+dt]*is*mu dt* - a departure due to impatience in
*[t, t+dt]*is*i tau dt*

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.

*********************************************** 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 ***********************************************