 
Summary: Transition probabilities for Inhomogeneous
Continuous Time Markov Chains
Alexandru Mereacre
February 21, 2008
Definition 1 (ICTMC). An inhomogeneous continuoustime Markov chain is
a tuple C = (S, R) where:
S = {1, 2, . . ., n} is a finite set of states.
R(t) = [Ri,j(t)] Rn×n
+ is a timedependent rate matrix, with i, j S and
t 0.
Here the exit rate of a state i S at time t is Ei(t) = jS Ri,j(t).
Example 1. Fig. 1 shows an example of a simple queue with three capacities
and two servers. The arrival process to the queue is a Poisson process with rate
constant and the service rate is a function µ(t) which depends on the global
time of the system. Initially the service rate starts at µmax and decreases linearly
till µmin at t = a. From that moment on, all users are served with constant rate.
An interesting property which can be defined for every ICTMC is the distri
bution of waiting time in a state. Before that, let us first define the notion of a
nonhomogeneous Poisson process:
Definition 2 (Inhomogeneous Poisson process). A stochastic process Z :
