Stochastic process fundamentals - Modelling with Generalised Stochastic Petri Nets - page 276

where SJ (X j is the average sojourn time in state j measured on the process
X(t), and the v ij are the visit ratios relative to the EMC. Thus the steady-
state probability of any state j in a CTMC can be evaluated as the ratio
of the average sojourn time in state j to the sum of the products of the
average sojourn times in all states i in the state space S multiplied by the
visit ratios v ij , i.e., by the average number of times that the process visits
state i between two successive visits to state j. Note that we are considering
a portion of the process that begins with the entrance into state j and ends
just before the next entrance into state j. Call this portion of process a
cycle. The steady-state probability of state j is obtained by dividing the
average amount of time spent in state j during a cycle by the average cycle
duration.

A.4.5 Example

Consider the continuous-time version of the two-processor system that was
considered as an example of DTMC model. We again have two CPUs with
their own private memories, and one common memory that can be accessed
only by one processor at a time.
The two CPUs execute in their private memories for a random time before
issuing a common memory access request. Assume that this random time is
exponentially distributed with parameter λ, so that the average time spent
by each CPU executing in private memory between two common memory
access requests is λ −1 .
The common memory access duration is also assumed to be an exponentially
distributed random variable, with parameter µ, so that the average duration
of the common memory access is µ −1 .
Also in this case, let us initially assume that only one processor is opera-
tional. Again the system can only be in one of two states (the same two
states as in the discrete-time case):

1. the processor is executing in its private memory;

2. the processor is accessing the common memory.

The system behaviour can thus be modeled with a two-state continuous-
time process. Also in this case, 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 2-state CTMC with
infinitesimal generator:

" #

− λ λ
µ − µ

Q =