Page 47 - A First Course In Stochastic Models

38 RENEWAL-REWARD PROCESSES

In Corollary 8.2.4 it will be shown that
µ 2 2 µ 3
lim E(γ t ) = and lim E(γ ) = (2.1.8)
t
t→∞ 2µ 1 t→∞ 3µ 1
k
with µ k = E(X ) for k = 1, 2, 3, provided that the interoccurrence times have a
1
positive density on some interval. An illustration of the usefulness of the concept
of excess variable is provided by the next example.
Example 2.1.3 The average order size in an (s, S) inventory system

Suppose a periodic-review inventory system for which the demands X 1 , X 2 , . . .
for a single product in the successive weeks 1, 2, . . . are independent random
variables having a common probability density f (x) with finite mean α and finite
standard deviation σ. Any demand exceeding the current inventory is backlogged
until inventory becomes available by the arrival of a replenishment order. The
inventory position is reviewed at the beginning of each week and is controlled by
an (s, S) rule with 0 ≤ s < S. Under this control rule, a replenishment order of
size S −x is placed when the review reveals that the inventory level x is below the
reorder point s; otherwise, no ordering is done. We assume instantaneous delivery
of every replenishment order.
We are interested in the average order size. Since the inventory process starts
from scratch each time the inventory position is ordered up to level S, the operating
characteristics can be calculated by using a renewal model in which the weekly
demand sizes X 1 , X 2 , . . . represent the interoccurrence times of renewals. The
number of weeks between two consecutive orderings equals the number of weeks
needed for a cumulative demand larger than S − s. The order size is the sum of
S − s and the undershoot of the reorder point s at the epoch of ordering (see
Figure 2.1.2 in which a renewal occurrence is denoted by an ×). Denote by {N(t)}
the renewal process associated with the weekly demands X 1 , X 2 , . . . . Then the
number of weeks needed for a cumulative demand exceeding S − s is given by
1 + N(S − s). The undershoot of the reorder point s is just the excess life γ S−s of
the renewal process. Hence

E[order size] = S − s + E(γ S−s ).
From (2.1.8) it follows that the average order size can be approximated by

2
σ + α 2
E[order size] ≈ S − s +
2α
g
X 1 X 2 S − s
0 S − s Cumulative demand
Figure 2.1.2 The inventory process modelled as a renewal process