Page 18 - A First Course In Stochastic Models

THE POISSON PROCESS 9

1.1.3 The M/G/∞ Queue ∗
Suppose that customers arrive at a service facility according to a Poisson process
with rate λ. The service facility has an ample number of servers. In other words,
it is assumed that each customer gets immediately assigned a new server upon
arrival. The service times of the customers are independent random variables hav-
ing a common probability distribution with finite mean µ. The service times are
independent of the arrival process. This versatile model is very useful in applica-
tions. An interesting question is: what is the limiting distribution of the number of
busy servers? The surprisingly simple answer to this question is that the limiting
distribution is a Poisson distribution with mean λµ:
(λµ) k
−λµ
lim P (k servers are busy at time t) = e (1.1.6)
t→∞ k!
for k = 0, 1, . . . . This limiting distribution does not require the shape of the
service-time distribution, but uses the service-time distribution only through its
mean µ. This famous insensitivity result is extremely useful for applications.
The M/G/∞ model has applications in various fields. A nice application is the
(S − 1, S) inventory system with back ordering. In this model customers asking
for a certain product arrive according to a Poisson process with rate λ. Each cus-
tomer asks for one unit of the product. The initial on-hand inventory is S. Each
time a customer demand occurs, a replenishment order is placed for exactly one
unit of the product. A customer demand that occurs when the on-hand inventory
is zero also triggers a replenishment order and the demand is back ordered until
a unit becomes available to satisfy the demand. The lead times of the replenish-
ment orders are independent random variables each having the same probability
distribution with mean τ. Some reflections show that this (S − 1, S) inventory sys-
tem can be translated into the M/G/∞ queueing model: identify the outstanding
replenishment orders with customers in service and identify the lead times of the
replenishment orders with the service times. Thus the limiting distribution of the
number of outstanding replenishment orders is a Poisson distribution with mean
λτ. In particular,
S k
−λτ (λτ)
the long-run average on-hand inventory = (S − k) e .
k!
k=0
Returning to the M/G/∞ model, we first give a heuristic argument for (1.1.6)
and next a rigorous proof.

Heuristic derivation

Suppose first that the service times are deterministic and are equal to the constant
D = µ. Fix t with t > D. If each service time is precisely equal to the constant

∗ This section can be skipped at first reading.