Page 26 - A First Course In Stochastic Models

THE POISSON PROCESS 17

∞ n
−λT (λT )
= E(T − S 1 + T − S 2 + · · · + T − S n | N(T ) = n)e
n!
n=1
∞ n
−λT (λT )
= E(T − U (1) + T − U (2) + · · · + T − U (n) )e .
n!
n=1
This gives
∞ n
−λT (λT )
E(total waiting time up to time T ) = E(nT − (U 1 + · · · + U n ))e
n!
n=1
∞ n
T −λT (λT ) T
= nT − n e = λT,
2 n! 2
n=1
which proves the desired result.
The result (1.1.9) is simple but very useful. It is sometimes used in a somewhat
different form that can be described as follows. Messages arrive at a communication
channel according to a Poisson process with rate λ. The messages are stored in
a buffer with ample capacity. A holding cost at rate h > 0 per unit of time is
incurred for each message in the buffer. Then, by (1.1.9),
h 2
E(holding costs incurred up to time T ) = λT . (1.1.10)
2

Clustering of Poisson arrival epochs

Theorem 1.1.5 expresses that Poisson arrival epochs occur completely randomly
in time. This is in agreement with the lack of memory of the exponential density
λe −λx of the interarrival times. This density is largest at x = 0 and decreases as x
increases. Thus short interarrival times are relatively frequent. This suggests that
the Poisson arrival epochs show a tendency to cluster. Indeed this is confirmed by
simulation experiments. Clustering of points in Poisson processes is of interest in
many applications, including risk analysis and telecommunication. It is therefore
important to have a formula for the probability that a given time interval of length
T contains some time window of length w in which n or more Poisson events
occur. An exact expression for this probability is difficult to give, but a simple and
excellent approximation is provided by
λw

1 − P (n − 1, λw) exp [− 1 − λ(T − w)p(n − 1, λw)],
n
k
where p(k, λw) = e −λw (λw) /k! and P (n, λw) = n k=0 p(k, λw). The approxi-
mation is called Alm’s approximation; see Glaz and Balakrishnan (1999). To illus-
trate the clustering phenomenon, consider the following example. In the first five
months of the year 2000, trams hit and killed seven people in Amsterdam, each