1 Continuous Time Processes - IPM

1.2 Inter-arrival Times and Poisson Processes
Poisson processes are perhaps the most basic examples of continuous time
Markov chains. In this subsection we establish their basic properties. To construct
a Poisson process we consider a sequence W1, W2, . . . of iid exponential
random variables with parameter λ. Wj’s are called inter-arrival times. Set
T1 = W1, T2 = W◦ + W1 and Tn = Tn−1 + Wn. Tj’s are called arrival times.
Now define the Poisson process Nt with parameter λ as
Nt = max{n | W1 + W2 + · · · + Wn ≤ t} (1.2.1)
Intuitively we can think of certain events taking place and every time the
event occurs the counter Nt is incremented by 1. We assume N◦ = 0 and
the times between consecutive events, i.e., Wj’s, being iid exponentials with
the same parameter λ. Thus Nt is the number of events that have taken
place until time t. The validity of the Markov property follows from the
construction of Nt and the exponential nature of the inter-arrival times, so
that the Poisson process is a continuous time Markov chain.
The arrival and inter-arrival times can be recovered from Nt by
Tn = sup{t | Nt ≤ n − 1}, (1.2.2)
and Wn = Tn − Tn−1. One can similarly construct other counting processes 2
by considering sequences of independent random variables W1, W2, . . . and
defining Tn and Nt just as above. The assumption that Wj’s are exponential
is necessary and sufficient for the resulting process to be Markov. What
makes Poisson processes special among Markov counting processes is that
the inter-arrival times have the same exponential law. The case where Wj’s
are not necessarily exponential (but iid) is very important and will be treated
in connection with renewal theory later.
The underlying probability space Ω for a Poisson process is the space
non-decreasing right continuous step function functions such that at each
point of discontinuity a ∈ R+
ϕ(a) − lim ϕ(t) = 1,
t→a− reflecting the fact that from state n only transition to state n + 1 is possible.
2 By a counting process we mean a continuous time process Nt taking values in Z+ such
that the only possible transition is from state n to state n + 1.
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