Stochastic process fundamentals - Modelling with Generalised Stochastic Petri Nets - page 277

Figure A.5: State transition rate diagram of the single processor model

Alternatively, the 2-state CTMC can be characterized with the graph shown
in Fig. A.5, that is often called the state transition rate diagram of the
CTMC.
The steady-state distribution of the 2-state DTMC can be found by solving
the system of linear equations

= − λ η 1 + µ η 2

0

= λ η 1 − µ η 2 0

1 = η 1 + η 2

Note that the first two equations are linearly dependent, so that one can be
discarded.
The steady-state distribution can immediately be found to be

µ
λ + µ ,

λ
λ + µ

η =

this means for example that at steady-state the probability of the processor
being executing in its private memory is µ

λ + µ .
Coming now to the two-processor system, we name the two processors A
and B, and characterize the workload parameters with the appropriate sub-
scripts. We can observe that the system can be in the same five states that
were already identified in the discrete-time model:

1. both processors are executing in their private memories;

2. processor B is executing in its private memory, and processor A is
accessing the common memory;

3. processor A is executing in its private memory, and processor B is
accessing the common memory;