Page 154 - A First Course In Stochastic Models

THE FLOW RATE EQUATION METHOD 147

and
1 if the unloader is available at time t,

X 2 (t) =
0 if the unloader is in repair at time t.
Since the underlying distributions are exponential, the process {(X 1 (t), X 2 (t))} is
a continuous-time Markov chain. This process has the state space

I = {(i, 0) | i = 1, 2, . . . } ∪ {(i, 1) | i = 0, 1, . . . }.
The next step is to determine the infinitesimal transition rates of the process. Putting
for abbreviation X(t) = (X 1 (t), X 2 (t)), we have

P {X(t + t) = (i, 1) | X(t) = (i, 0)}
= P {the running repair is finished in (t, t + t) and
no arrival occurs in (t, t + t)}
= β t (1 − λ t) + o ( t) = β t + o( t)
for t → 0. This gives

q (i,0)(i,1) = β for i = 1, 2, . . . .
Alternatively, q (i,0)(i,1) could have been obtained by noting that the sojourn time in
state (i, 0) is exponentially distributed with parameter β + λ and noting that with
probability β/(β + λ) the running repair time is finished before an arrival occurs.
Also,

P {X(t + t) = (i + 1, 0)|X(t) = (i, 0)}
= P {an arrival occurs in (t, t + t) and the running repair time
is not finished in (t, t + t)}
= λ t(1 − β t) + o( t) = λ t + o( t)

for t → 0. This gives
q (i,0)(i+1,0) = λ for i ≥ 1.

Similarly, we find

q (i,1)(i,0) = δ, q (i,1)(i+1,1) = λ and q (i,1)(i−1,1) = µ for i ≥ 1.
The state transitions and transition rates are summarized in Figure 4.1.2.

4.2 THE FLOW RATE EQUATION METHOD

This section discusses the flow rate equation method for obtaining the equilibrium
distribution of a continuous-time Markov chain. The emphasis is to give insight