Page 169 - A First Course In Stochastic Models

162 CONTINUOUS-TIME MARKOV CHAINS

that Theorem 4.4.1 remains valid with the same matrix A(z). For the particular case
of the unloader problem, we find that (4.4.9) reduces to the polynomial equation

2
(λ + β − λz) (λz + µ − (λ + µ + δ)z) + δβz = 0.

Special case of linear birth-death rates
Suppose that the transition rates λ s , µ s , β s and δ s have the special form

λ s = b 1 × (m − s) + c 1 s, µ s = b −1 × (m − s) + c −1 s
β s = a 0 × (m − s) and δ s = d 0 s (4.4.10)

for constants a 0 , b 1 , b −1 , c 1 , c −1 and d 0 . Then the numerical problem of computing
the roots of det A(z) = 0 can be circumvented. The decay factor η in (4.4.2) is
then the unique solution of the equation

2
B(x) + C(x) − x[A(1) + B(1) + C(1) + D(1)] + F(x) + 4A(x)D(x) = 0
on the interval (0,1), where

2
2
A(x) = a 0 x, B(x) = b 1 + b −1 x , C(x) = c 1 + c −1 x , D(x) = d 0 x,
F(x) = [A(1) + B(1) − C(1) − D(1)]x + C(x) − B(x).
In a more general context this result has been proved in Adan and Resing (1999). It
also follows from this reference that Assumption 4.2.1 holds when d 0 (b −1 − b 1 ) +
a 0 (c −1 −c 1 ) > 0. The condition (4.4.10) is satisfied for several interesting queueing
models. For example, take a queueing model with m traffic sources which act inde-
pendently of each other. Each traffic source is alternately on and off, where the on-
times and off-times have exponential distributions with respective means 1/δ and
1/β. The successive on- and off-times are assumed to be independent of each other.
During on-periods a source generates service requests according to a Poisson pro-
cess with rate λ. There is a single server to handle the service requests and the server
can handle only one request at a time. The service times of the requests are inde-
pendent random variables that have a common exponential distribution with mean
1/µ. This queueing problem can be modelled as a continuous-time Markov chain
whose state space is given by (4.4.1) with i denoting the number of service requests
in the system and s denoting the number of sources that are on. This Markov chain
has the property (4.4.10) with λ s = λs, µ s = µ, β s = β × (m − s) and δ s = δs.

4.5 TRANSIENT STATE PROBABILITIES

In many practical situations one is not interested in the long-run behaviour of a
stochastic system but in its transient behaviour. A typical example concerns airport
runway operations. The demand profile for runway operations shows considerable