William J. Stewart

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
When j >= 1, the probability of a departure from S2 in [t, t+dt] is j lambda dt. External arrivals are assumed to be Poisson at rate A. These rates are assigned in the subroutine rate of the tcomm.f source code file.

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

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

```