  We consider a change detection problem in which the arrival rate of a Poisson
process changes suddenly at some unknown and unobservable disorder time. It is
assumed that the prior distribution of the disorder time is known. The
objective is to detect the disorder time with an online detection rule (a
stopping time) in a way that balances the frequency of false alarm and
detection delay. So far in the study of this problem, the prior distribution of
the disorder time is taken to be exponential distribution for analytical
tractability. Here, we will take the prior distribution to be a phase-type
distribution, which is the distribution of the absorption time of a continuous
time Markov chain with a finite state space. We find an optimal stopping rule
for this general case and give a numerical algorithm that calculates the
parameters of $\eps$-optimal strategies for any $\eps>0$. We illustrate our
findings on two examples.
