Page 22 - A First Course In Stochastic Models

THE POISSON PROCESS 13

Then the estimate for k is 1.8450. Substituting this value into the square-root
formula for c i , we find c 1 ≈ 15.83, c 2 ≈ 34.23, c 3 ≈ 45.92, c 4 ≈ 63.05 and
c 5 ≈ 90.98. This suggests the allocation
(c , c , c , c , c ) = (16, 34, 46, 63, 91).
∗
∗
∗
∗
∗
3
2
1
4
5
Note that in determining this allocation we have used the distributions of the
processing times only through their first moments. The actual value of the long-run
fraction of time during which all vehicles are occupied in region i depends (to
a slight degree) on the probability distribution of the processing time S i . Using
simulation, we find the values 0.056, 0.058, 0.050, 0.051 and 0.050 for the service
level in the respective regions 1, 2, 3, 4 and 5.
The M/G/∞ queue also has applications in the analysis of inventory systems.
Example 1.1.4 A two-echelon inventory system with repairable items
Consider a two-echelon inventory system consisting of a central depot and a num-
ber N of regional bases that operate independently of each other. Failed items
arrive at the base level and are either repaired at the base or at the central depot,
depending on the complexity of the repair. More specifically, failed items arrive
at the bases 1, . . . , N according to independent Poisson processes with respective
rates λ 1 , . . . , λ N . A failed item at base j can be repaired at the base with probabil-
ity r j ; otherwise the item must be repaired at the depot. The average repair time of
an item is µ j at base j and µ 0 at the depot. It takes an average time of τ j to ship
an item from base j to the depot and back. The base immediately replaces a failed
item from base stock if available; otherwise the replacement of the failed item is
back ordered until an item becomes available at the base. If a failed item from base
j arrives at the depot for repair, the depot immediately sends a replacement item to
the base j from depot stock if available; otherwise the replacement is back ordered
until a repaired item becomes available at the depot. In the two-echelon system
a total of J spare parts are available. The goal is to spread these parts over the
bases and the depot in order to minimize the total average number of back orders
outstanding at the bases. This repairable-item inventory model has applications in
the military, among others.
An approximate analysis of this inventory system can be given by using the
M/G/∞ queueing model. Let (S 0 , S 1 , . . . , S N ) be a given design for which S 0
spare parts have been assigned to the depot and S j spare parts to base j for
j = 1, . . . , N such that S 0 + S 1 + · · · + S N = J. At the depot, failed items arrive
according to a Poisson process with rate
N

λ 0 = λ j (1 − r j ).
j=1
Each failed item arriving at the depot immediately goes to repair. The failed items
arriving at the depot can be thought of as customers arriving at a queueing system