Page 121 - A First Course In Stochastic Models

COMPUTATION OF THE EQUILIBRIUM PROBABILITIES 113

where N(z) and D(z) are functions that have no common zeros. The functions N(z)
and D(z) are analytic functions that can be analytically continued outside the unit
circle |z| ≤ 1.
(b) Letting R > 1 be the largest number such that both functions N(z) and D(z)
are analytic in the region |z| < R in the complex plane, the equation
D(x) = 0 (3.4.7)

has a smallest root x 0 on the interval (1, R).
In specific applications the denominator D(z) in (3.4.6) is usually a nice function
that is explicitly given (this is usually not true for the numerator N(z)). It is only
the denominator D(z) that is needed for our purposes. Theorem C.1 in Appendix C
shows that under Condition A plus some secondary technical conditions the state
probabilities π j allow for the asymptotic expansion (3.4.5) with

1
η = . (3.4.8)
x 0
Condition A is a condition that seems not to have a probabilistic interpretation.
Next we give a probabilistic condition for (3.4.5) to hold. This condition is in
terms of the one-step transition probabilities p ij of the Markov chain.
Condition B (a) There is an integer r ≥ 0 such that p ij depends on i and j only
through j − i when i ≥ r and j ≥ 1.
(b) There is an integer s ≥ 1 such that

p ij = 0 for j > i + s and i ≥ 0.

(c) Letting α j−i denote p ij for i ≥ r and 1 ≤ j ≤ i + s, the constants α k satisfy
s

α s > 0 and kα k < 0.
k=−∞
Under Condition B the equilibrium equation for π j has the form

∞

π j = α j−k π k for j ≥ r + s.
k=j−s
This is a homogeneous linear difference equation with constant coefficients. A stan-
dard method to solve such a linear difference equation is the method of particular
j
solutions. Substituting a solution of the form π j = w in the equilibrium equations
for the π j with j ≥ r + s, we find the so-called characteristic equation
∞
s ℓ
w − α s−ℓ w = 0. (3.4.9)
ℓ=0