Page 20 - A First Course In Stochastic Models

THE POISSON PROCESS 11

discontinuity points, we find for almost all t > 0 that
p j (t + t) = (1 − λ t)p j (t) + λ tB(t + t)p j (t)

+ λ t{1 − B(t + t)}p j−1 (t) + o( t).

Subtracting p j (t) from p j (t + t), dividing by t and letting t → 0, we find
′
p (t) = −λ(1 − B(t))p 0 (t)
0
′
p (t) = −λ(1 − B(t))p j (t) + λ(1 − B(t))p j−1 (t), j = 1, 2, . . . .
j
Next, by induction on j, it is readily verified that

j
t
λ (1 − B(x)) dx
t (1−B(x)) dx 0
−λ
p j (t) = e 0 , j = 0, 1, . . . .
j!
∞
By a continuity argument this relation holds for all t ≥ 0. Since [1−
0
B(x)] dx = µ, the result (1.1.6) follows. Another proof of (1.1.6) is indicated
in Exercise 1.14.
Example 1.1.3 A stochastic allocation problem
A nationwide courier service has purchased a large number of transport vehicles
for a new service the company is providing. The management has to allocate these
vehicles to a number of regional centres. In total C vehicles have been purchased
and these vehicles must be allocated to F regional centres. The regional centres
operate independently of each other and each regional centre services its own group
of customers. In region i customer orders arrive at the base station according to
a Poisson process with rate λ i for i = 1, . . . , F. Each customer order requires
a separate transport vehicle. A customer order that finds all vehicles occupied
upon arrival is delayed until a vehicle becomes available. The processing time of
a customer order in region i has a lognormal distribution with mean E(S i ) and
standard deviation σ(S i ). The processing time includes the time the vehicle needs
to return to its base station. The management of the company wishes to allocate
the vehicles to the regions in such a way that all regions provide, as nearly as
possible, a uniform level of service to the customers. The service level in a region
is measured as the long-run fraction of time that all vehicles are occupied (it will
be seen in Section 2.4 that the long-run fraction of delayed customer orders is also
given by this service measure).
Let us assume that the parameters are such that each region gets a large number
of vehicles and most of the time is able to directly provide a vehicle for an arriving
customer order. Then the M/G/∞ model can be used as an approximate model
to obtain a satisfactory solution. Let the dimensionless quantity R i denote

R i = λ i E(S i ), i = 1, . . . , F,