———————————————————————-
(b) Formulate a CTMC describing the evolution of the system.
———————————————————————-
However, we can use 5 states with the states being: 0 for no copiers failed, 1 for copier 1
is failed (and copier 2 is working), 2 for copier 2 is failed (and copier 1 is working), (1
,
2) for
both copiers down (failed) with copier 1 having failed first and being repaired, and (2
,
1) for
both copiers down with copier 2 having failed first and being repaired. (Of course, these states
could be relabelled 0, 1, 2, 3 and 4, but we do not do that.)
From the problem specification, it is natural to work with transition rates, where these
transition rates are obtained directly from the originally-specified failure rates and repair rates
(the rates of the exponential random variables).
In Figure 1 we display a
rate diagram
showing the possible transitions with these 5 states together with the appropriate rates. It can
be helpful to construct such rate diagrams as part of the modelling process.
From Figure 1, we see that there are 8 possible transitions.
The 8 possible transitions
should clearly have transition rates
Q
0
,
1
=
γ
1
, Q
0
,
2
=
γ
2
, Q
1
,
0
=
β
1
, Q
1
,
(1
,
2)
=
γ
2
, Q
2
,
0
=
β
2
, Q
2
,
(2
,
1)
=
γ
1
, Q
(1
,
2)
,
2
=
β
1
, Q
(2
,
1)
,
1
=
β
2
.
3