Page 21 - A First Course In Stochastic Models

12 THE POISSON PROCESS AND RELATED PROCESSES

that is, R i is the average amount of work that is offered per time unit in region i.
Denoting by c i the number of vehicles to be assigned to region i, we take c i of
the form

c i ≈ R i + k R i , i = 1, . . . , F,
for an appropriate constant k. By using this square-root rule, each region will
provide nearly the same service level to its customers. To explain this, we use for
each region the M/G/∞ model to approximate the probability that all vehicles in
the region are occupied at an arbitrary point of time. It follows from (1.1.6) that
for region i this probability is approximated by

∞ k
R
i
e −R i
k!
k=c i
when c i vehicles are assigned to region i. The Poisson distribution with mean R
can be approximated by a normal distribution with mean R and standard deviation
√
R when R is large enough. Thus we use the approximation
∞ k
R i c i − R i
e −R i ≈ 1 − √ , i = 1, . . . , F,
k! R i
k=c i
where (x) is the standard normal distribution function. By requiring that

c 1 − R 1 c F − R F
√ ≈ · · · ≈ √ ,
R 1 R F
we find the square-root formula for c i . The constant k in this formula must be
chosen such that
F

c i = C.
i=1
Together this requirement and the square-root formula give
F

C − R i
i=1
k ≈ .
F

R i
i=1
This value of k is the guideline for determining the allocation (c 1 , . . . , c F ) so that
each region, as nearly as possible, provides a uniform service level. To illustrate
this, consider the numerical data:
c = 250, F = 5, λ 1 = 5, λ 2 = 10, λ 3 = 10, λ 4 = 50, λ 5 = 37.5,
E(S 1 ) = 2, E(S 2 ) = 2.5, E(S 3 ) = 3.5, E(S 4 ) = 1, E(S 5 ) = 2,

σ(S 1 ) = 1.5, σ(S 2 ) = 2, σ(S 3 ) = 3, σ(S 4 ) = 1, σ(S 5 ) = 2.7.