[CST-2] More CSM: 1996 p8 q3 and p9 q3

[Tim Harris]

> What is that "neccessary condition for stability of [an M/M/4/4] system"?

The system will always reach stability: although clearly if the arrivals
rate is very high then it will almost always be in the "four calls in
progress" state.

> 1996 P9 Q3 was walked through in the second exaples class, which I missed.

The solution notes from that are on the web from

     http://www.cl.cam.ac.uk/Teaching/2000/CompSysMod/

The transitions go 0->2, 1->3, 2->4, etc because each arrivals brings
'two units' of work (one for the first stage of the server and one for
the second stage).  In the other direction transitions go 4->3, 3->2,
2->1, 1->0.

You might find it clearer with some alternative kind of labels on the
states, e.g. if the two stages of the server are termed A and B then
you could have states

   0_I  (0 queueing, servers idle)
   0_A  (0 queueing, server A busy)
   0_B  (0 queueing, server B busy)
   1_A  (1 queueing, server A busy)
   1_B  (1 queueing, server B busy)
   etc

so an arrival from e.g. 0_I would go to O_A (first customer arrives and
enters service at stage A without queueing), 2_A would go to 3_A, from
2_B to 3_B etc.  Departures would be modelled by e.g. 3_A to 3_B and
then 3_B to 2_A.

Tim