Page 329 - A First Course In Stochastic Models

324 ADVANCED RENEWAL THEORY

equilibrium. An approximate analysis will be given. The analysis is based on the
following ideas:

1. Compute the means of the up- and down-periods.

2. Approximate the stochastic process of the up- and down-periods by an alter-
nating renewal process in which both the up-periods and the down-periods are
independent, exponential random variables and the up-periods are independent
of the down-periods.

In view of the assumption µ R << µ L , the occurrence of a system failure is a rare
event. This justifies the approximate step of assuming an exponential distribution
for the up-period; see also the discussion on rare events at the end of Section 2.2.
A similar justification for approximating the distribution of the downtime by an
exponential distribution cannot be given. However, in view of the fact that the
uptime dominates the downtime, it is reasonable to expect that the distributional
form of the downtime has only a minor effect on the accuracy of the approximation.
The process alternates between the up-state and the down-state. With the possible
exception of the first up-period, the up-periods start when a unit is put into operation
while the other unit enters repair. The system regenerates itself at the beginning
of those up-periods. We assume that epoch 0 is such a regeneration epoch. Let the
random variables τ up and τ down denote the lengths of an up-period and a down-
period. Denote by the sequences {L i } and {R i } the successive operating times and
the successive repair times. Then

N

E(τ up ) = E L i ,
i=1
where N = min{n ≥ 1 | R n > L n }. The event {N = n} is independent of
L n+1 , L n+2 , . . . for any n ≥ 1. Thus, by Wald’s equation, E(τ up ) = E(N)µ L . Let

q = P {R > L}

where the random variables L and R denote the operating time and the repair time
of a unit. Since P {N = n} = (1 − q) n−1 q for n ≥ 1, we find

µ L
E(τ up ) = .
q
By conditioning on the lifetime, we have

∞
q = {1 − G R (x)}f L (x) dx.
0
To find E(τ down ), note that E(τ down ) = E(R − L | R > L). Using the formula