Page 48 - A First Course In Stochastic Models

RENEWAL-REWARD PROCESSES 39

provided that S − s is sufficiently large compared with E(weekly demand). In
practice this is a useful approximation for S−s > α when the weekly demand is not
highly variable and has a squared coefficient of variation between 0.2 and 1 (say).
Another illustration of the importance of the excess variable is given by the
famous waiting-time paradox.

Example 2.1.4 The waiting-time paradox
We have all experienced long waits at a bus stop when buses depart irregularly and
we arrive at the bus stop at random. A theoretical explanation of this phenomenon is
provided by the expression for lim t→∞ E(γ t ). Therefore it is convenient to rewrite
(2.1.8) as
1 2
lim E(γ t ) = (1 + c )µ 1 , (2.1.9)
X
t→∞ 2
where
2
σ (X 1 )
2
c =
X 2
E (X 1 )
is the squared coefficient of variation of the interdeparture times X 1 , X 2 , . . . . The
equivalent expression (2.1.9) follows from (2.1.8) by noting that
µ 2 − µ 2
2 1 µ 2
1 + c = 1 + 2 = 2 . (2.1.10)
X
µ 1 µ 1
The representation (2.1.9) makes clear that

2
< µ 1 if c < 1,
X
lim E(γ t ) = 2
t→∞ > µ 1 if c > 1.
X
Thus the mean waiting time for the next bus depends on the regularity of the bus
service and increases with the coefficient of variation of the interdeparture times. If
2
we arrive at the bus stop at random, then for highly irregular service (c > 1) the
X
mean waiting time for the next bus is even larger than the mean interdeparture time.
This surprising result is sometimes called the waiting-time paradox. A heuristic
explanation is that it is more likely to hit a long interdeparture time than a short
one when arriving at the bus stop at random. To illustrate this, consider the extreme
situation in which the interdeparture time is 0 minutes with probability 9/10 and is
10 minutes with probability 1/10. Then the mean interdeparture time is 1 minute,
but your mean waiting time for the next bus is 5 minutes when you arrive at the
bus stop at random.

2.2 RENEWAL-REWARD PROCESSES

A powerful tool in the analysis of numerous applied probability models is the
renewal-reward model. This model is also very useful for theoretical purposes. In