Stochastic process fundamentals - Modelling with Generalised Stochastic Petri Nets - page 278

4. processor B is waiting for the common memory, and processor A is
accessing the common memory.

5. processor A is waiting for the common memory, and processor B is
accessing the common memory;

The system behaviour can thus be modeled with a five-state continuous-
time process. Again, the assumptions introduced on the system workload
are such that no memory is present in the system, so that the behaviour is
Markovian.
We can thus describe the system operations with a 5-state CTMC with
infinitesimal generator

2 3

− λ A − λ B λ A λ B 0 0

4 5

µ A − µ A − λ B 0 λ B 0

0 − λ A − µ B

Q = µ B 0 λ A

µ A − µ A 0 0 0

0 − µ B 0 µ B 0

Alternatively, the 5-state CTMC can be characterized with the state tran-
sition rate diagram shown in Fig. A.6.
It is interesting to note that now the model description is simpler than it was
in the discrete-time case. This is due to the fact that the use of continuous
random variables for the durations of execution and access times makes the
probability of two simultaneous events go to zero. In the construction of the
continuous-time model thus we just have to describe the events relating to
the individual processor activities, without worrying about the possibility
of two (or more) events happening at the same time.

A.5 Aggregation of States in Markov Chains

Very often Markovian models of real systems comprise quite a large number
of states, so that the numerical evaluation of the steady-state distribution is
computationally very expensive, and in some cases is even infeasible. On the
other hand, the equilibrium solution is often used to derive specific quantities
by performing a weighted summation of steady-state probabilities. In most
of these cases, the weights are not different for each state, so that the same
result could be achieved by solving a reduced problem in which the model
comprises fewer macrostates, each macrostate containing all (or some) of
the states for which the associated performance measure is the same.
In this section we address this problem by giving conditions for the existence
of a compact aggregate Markovian model.