Page 64 - A First Course In Stochastic Models

POISSON ARRIVALS SEE TIME AVERAGES 55

In the same way as in Example 2.4.1, we define the random variables

1 if the system is out of stock at time t,
I (t) =
0 otherwise.
and

1 if the system is out of stock when the nth demand occurs,
I n =
0 otherwise.
The continuous-time process {I (t)} and the discrete-time process {I n } are both
regenerative. The regeneration epochs are the demand epochs at which the stock
on hand drops to zero. Why? Let us say that a cycle starts each time the stock on
hand drops to zero. The system is out of stock during the time elapsed from the
beginning of a cycle until the next inventory replenishment. This amount of time
is exponentially distributed with mean 1/µ. The expected amount of time it takes
to go from stock level Q to 0 equals Q/λ. Hence, with probability 1,

the long-run fraction of time the system is out of stock
1/µ
= . (2.4.4)
1/µ + Q/λ
To find the fraction of demand that is lost, note that the expected amount of demand
lost in one cycle equals λ × E(amount of time the system is out of stock during
one cycle) = λ/µ. Hence, with probability 1,

the long-run fraction of demand that is lost
λ/µ
= . (2.4.5)
λ/µ + Q
Together (2.4.4) and (2.4.5) lead to this remarkable result:
the long-run fraction of customers finding the system out of stock
= the long-run fraction of time the system is out of stock. (2.4.6)

The relations (2.4.3) and (2.4.6) are particular instances of the property ‘Poisson
arrivals see time averages’. Roughly stated, this property expresses that in statistical
equilibrium the distribution of the state of the system just prior to an arrival epoch
is the same as the distribution of the state of the system at an arbitrary epoch
when arrivals occur according to a Poisson process. An intuitive explanation of
the property ‘Poisson arrivals see time averages’ is that Poisson arrivals occur
completely randomly in time; cf. Theorem 1.1.5.
Next we discuss the property of ‘Poisson arrivals see time averages’ in a broader
context. For ease of presentation we use the terminology of Poisson arrivals. How-
ever, the results below also apply to Poisson processes in other contexts. For some