Page 33 - A First Course In Stochastic Models

24 THE POISSON PROCESS AND RELATED PROCESSES

Another illustration of the usefulness of the non-stationary Poisson process is
provided by the following example.

Example 1.3.2 Replacement with minimal repair
A machine has a stochastic lifetime with a continuous distribution. The machine is
replaced by a new one at fixed times T , 2T, . . . , whereas a minimal repair is done at
each failure occurring between two planned replacements. A minimal repair returns
the machine into the condition it was in just before the failure. It is assumed that
each minimal repair takes a negligible time. What is the probability distribution of
the total number of minimal repairs between two planned replacements?
Let F(x) and f (x) denote the probability distribution function and the probability
density of the lifetime of the machine. Also, let r(t) = f (t)/[1 − F(t)] denote the
failure rate function of the machine. It is assumed that f (x) is continuous. Then
the answer to the above question is

P {there are k minimal repairs between two planned replacements}
[M(T )] k
= e −M(T ) , k = 0, 1, . . . ,
k!
T
where M(T ) = r(t) dt. This result follows directly from Theorem 1.3.1 by not-
0
ing that the process counting the number of minimal repairs between two planned
replacements satisfies the properties (a), (b), (c) and (d) of Definition 1.3.1. Use
the fact that the probability of a failure of the machine in a small time interval
(t, t + t] is equal to r(t) t + o( t), as shown in Appendix B.

1.4 MARKOV MODULATED BATCH
POISSON PROCESSES ∗

The Markov modulated batch Poisson process generalizes the compound Pois-
son process by allowing for correlated interarrival times. This process is used
extensively in the analysis of teletraffic models (a special case is the compos-
ite model of independent on-off sources multiplexed together). A so-called phase
process underlies the arrival process, where the evolution of the phase process
occurs isolated from the arrivals. The phase process can only assume a finite
number of states i = 1, . . . , m. The sojourn time of the phase process in state
i is exponentially distributed with mean 1/ω i . If the phase process leaves state
i, it goes to state j with probability p ij , independently of the duration of the
stay in state i. It is assumed that p ii = 0 for all i. The arrival process of cus-
tomers is a compound Poisson process whose parameters depend on the state of
the phase process. If the phase process is in state i, then batches of customers
arrive according to a Poisson process with rate λ i where the batch size has the

∗ This section contains specialized material that is not used in the sequel.