Page 328 - A First Course In Stochastic Models

ALTERNATING RENEWAL PROCESSES 323

Corollary 8.3.2 Suppose that the on-times and off-times have exponential distri-
butions with respective means 1/α and 1/β. Then, for any t 0 > 0 and 0 ≤ x < t 0 ,

lim P {U(t + t 0 ) − U(t) ≤ x}
t→∞
n
∞ n k
β −β(t 0 −x) [β(t 0 − x)] −αx (αx)
= e 1 − e
α + β n! k!
n=0 k=0

∞ n n−1 k
α −β(t 0 −x) [β(t 0 − x)] −αx (αx)
+ e 1 − e . (8.3.3)
α + β n! k!
n=0 k=0
Proof Since lim t→∞ P on (t) = β/(α + β), it follows that
lim P {U(t + t 0 ) − U(t) ≤ x}
t→∞
β
= P {U(t 0 ) ≤ x}
α + β
α t 0 −x −βy ∞ −βy
+ P {U(t 0 − y) ≤ x}βe dy + βe dy} .
α + β 0 t 0 −x
Next it is a matter of algebra to obtain the desired result from (8.3.2).
Exercises 8.4 to 8.8 give results for the alternating renewal process with non-
exponential on- and off-times. The alternating renewal process is particularly useful
in reliability applications. This is illustrated by the next example.

Example 8.3.1 The 1-out-of-2 reliability model with repair
The 1-out-of-2 reliability model deals with a repairable system that has one operat-
ing unit and one cold standby unit as protection against failures. The lifetime of an
operating unit has a general probability distribution function F L (x) having density
f L (x) with mean µ L . If the operating unit fails, it is replaced immediately by the
standby unit if available. The failed unit is sent to a repair facility and immediately
enters repair if the facility is idle. Only one unit can be in repair at a time. The
repair time of a failed unit has a general probability distribution function G R (x)
with mean µ R . It is assumed that µ R << µ L . The operating times and repair times
are mutually independent. The system is down when both units are broken down
and is up otherwise.
We are interested in the probability distribution function

A(x, t 0 ) = lim P {the total uptime in (t, t + t 0 ] is ≤ x}
t→∞
for an interval of length t 0 . In other words, the performance measure is the prob-
ability distribution function of the total amount of time the system is available
during a time interval of given length t 0 when the system has reached statistical