Page 15 - A First Course In Stochastic Models

6 THE POISSON PROCESS AND RELATED PROCESSES

The mathematical symbol o(h) is the generic notation for any function f (h) with
the property that lim h→0 f (h)/h = 0, that is, o(h) is some unspecified term that
2
is negligibly small compared to h itself as h → 0. For example, f (h) = h is an
o(h)-function. Using the expansion of e −h , it readily follows from (1.1.4) that

(C) The probability of one arrival occurring in a time interval of length t is
λ t + o( t) for t → 0.
(D) The probability of two or more arrivals occurring in a time interval of length
t is o( t) for t → 0.

The property (D) states that the probability of two or more arrivals in a very small
time interval of length t is negligibly small compared to t itself as t → 0.
The Poisson process could alternatively be defined by taking (A), (B), (C) and
(D) as postulates. This alternative definition proves to be useful in the analysis of
continuous-time Markov chains in Chapter 4. Also, the alternative definition of the
Poisson process has the advantage that it can be generalized to an arrival process
with time-dependent arrival rate.

1.1.2 Merging and Splitting of Poisson Processes
Many applications involve the merging of independent Poisson processes or the
splitting of events of a Poisson process in different categories. The next theorem
shows that these situations again lead to Poisson processes.

Theorem 1.1.3 (a) Suppose that {N 1 (t), t ≥ 0} and {N 2 (t), t ≥ 0} are indepen-
dent Poisson processes with respective rates λ 1 and λ 2 , where the process {N i (t)}
corresponds to type i arrivals. Let N(t) = N 1 (t) + N 2 (t), t ≥ 0. Then the merged
process {N(t), t ≥ 0} is a Poisson process with rate λ = λ 1 + λ 2 . Denoting by Z k
the interarrival time between the (k − 1)th and kth arrival in the merged process
and letting I k = i if the kth arrival in the merged process is a type i arrival, then
for any k = 1, 2, . . . ,

λ i
P {I k = i | Z k = t} = , i = 1, 2, (1.1.5)
λ 1 + λ 2
independently of t.
(b) Let {N(t), t ≥ 0} be a Poisson process with rate λ. Suppose that each arrival
of the process is classified as being a type 1 arrival or type 2 arrival with respective
probabilities p 1 and p 2 , independently of all other arrivals. Let N i (t) be the number
of type i arrivals up to time t. Then {N 1 (t)} and {N 2 (t)} are two independent Poisson
processes having respective rates λp 1 and λp 2 .

Proof We give only a sketch of the proof using the properties (A), (B), (C)
and (D).