Page 23 - A First Course In Stochastic Models

14 THE POISSON PROCESS AND RELATED PROCESSES

with infinitely many servers. Hence the limiting distribution of the number of items
in repair at the depot at an arbitrary point of time is a Poisson distribution with
mean λ 0 µ 0 . The available stock at the depot is positive only if less than S 0 items
are in repair at the depot. Why? Hence a delay occurs for the replacement of a
failed item arriving at the depot only if S 0 or more items are in repair upon arrival
of the item. Define now

W 0 = the long-run average amount of time a failed item at the depot
waits before a replacement is shipped,
L 0 = the long-run average number of failed items at the depot
waiting for the shipment of a replacement.

A simple relation exists between L 0 and W 0 . On average λ 0 failed items arrive at
the depot per time unit and on average a failed item at the depot waits W 0 time
units before a replacement is shipped. Thus the average number of failed items at
the depot waiting for the shipment of a replacement equals λ 0 W 0 . This heuristic
argument shows that

L 0 = λ 0 W 0 .

This relation is a special case of Little’s formula to be discussed in Section 2.3.
The relation W 0 = L 0 /λ 0 leads to an explicit formula for W 0 , since L 0 is given by

∞ k
(λ 0 µ 0 )
L 0 = (k − S 0 )e −λ 0 µ 0 .
k!
k=S 0
Armed with an explicit expression for W 0 , we are able to give a formula for the
long-run average number of back orders outstanding at the bases. For each base j
the failed items arriving at base j can be thought of as customers entering service
in a queueing system with infinitely many servers. Here the service time should be
defined as the repair time in case of repair at the base and otherwise as the time
until receipt of a replacement from the depot. Thus the average service time of a
customer at base j is given by

β j = r j µ j + (1 − r j )(τ j + W 0 ), j = 1, . . . , N.

The situation at base j can only be modelled approximately as an M/G/∞ queue.
The reason is that the arrival process of failed items interferes with the replacement
times at the depot so that there is some dependency between the service times at
base j. Assuming that this dependency is not substantial, we nevertheless use the
M/G/∞ queue as an approximating model and approximate the limiting distri-
bution of the number of items in service at base j by a Poisson distribution with