Page 17 - A First Course In Stochastic Models

8 THE POISSON PROCESS AND RELATED PROCESSES

P {N 1 (t) = k, N 2 (t) = m}
∞

= P {N 1 (t) = k, N 2 (t) = m | N(t) = n}P {N(t) = n}
n=0
= P {N 1 (t) = k, N 2 (t) = m | N(t) = k + m}P {N(t) = k + m}
k+m
k + m k m −λt (λt)
= p p e
1 2
k (k + m)!
(λp 1 t) k −λp 2 t (λp 2 t) m
−λp 1 t
= e e ,
k! m!
showing that P {N 1 (t) = k, N 2 (t) = m} = P {N 1 (t) = k}P {N 2 (t) = m}.

The remarkable result (1.1.5) states that the next arrival is of type i with proba-
bility λ i /(λ 1 +λ 2 ) regardless of how long it takes until the next arrival. This result
is characteristic for competing Poisson processes which are independent of each
other. As an illustration, suppose that long-term parkers and short-term parkers
arrive at a parking lot according to independent Poisson processes with respective
rates λ 1 and λ 2 . Then the merged arrival process of parkers is a Poisson process
with rate λ 1 + λ 2 and the probability that a newly arriving parker is a long-term
parker equals λ 1 /(λ 1 + λ 2 ).

Example 1.1.2 A stock problem with substitutable products

A store has a leftover stock of Q 1 units of product 1 and Q 2 units of product 2.
Both products are taken out of production. Customers asking for product 1 arrive
according to a Poisson process with rate λ 1 . Independently of this process, cus-
tomers asking for product 2 arrive according to a Poisson process with rate λ 2 .
Each customer asks for one unit of the concerning product. The two products serve
as substitute for each other, that is, a customer asking for a product that is sold
out is satisfied with the other product when still in stock. What is the probability
distribution of the time until both products are sold out? What is the probability
that product 1 is sold out before product 2?
To answer the first question, observe that both products are sold out as soon as
Q 1 + Q 2 demands have occurred. The aggregated demand process is a Poisson
process with rate λ 1 + λ 2 . Hence the time until both products are sold out has an
Erlang (Q 1 + Q 2 , λ 1 + λ 2 ) distribution. To answer the second question, observe
that product 1 is sold out before product 2 only if the first Q 1 +Q 2 −1 aggregated
demands have no more than Q 2 − 1 demands for product 2. Hence, by (1.1.5), the
desired probability is given by

k
Q 2 −1 Q 1 +Q 2 −1−k
Q 1 + Q 2 − 1 λ 2 λ 1
.
k λ 1 + λ 2 λ 1 + λ 2
k=0