Math 171 Homework #5
3.10
The continuous-time system obeys the eigenvector equation πA = 0. Ex-
panded out, this gives
X
k6=x
π(k)α(k, x)−π(x)α(x) = 0
Solving for π(x) yields
π(x) = 1
α(x)X
k6=x
π(k)α(k, x)
Let the invariant probability vector for the discrete-time system be φ(x) and
the transition matrix be Bx,y =α(x, y)/α(x). Then φ(x) satisfies φB =φ:
φ(x) = X
k6=x
φ(k)α(k, x)
α(k)
Divide the above equation by α(x) to get
φ(x)
α(x)=1
α(x)X
k6=x
φ(k)
α(k)α(k, x)
Comparing this last equation with the equation for π(x) we see that π(x) =
φ(x)/α(x).
This has a nice physical interperetation. It says that the amount of time
spent in state x(π(x)) is the number of visits to x(φ(x)) times the length
of time to leave state x(1/α(x)).
3.12
We require that
∞
X
n=1
µ1· · · µn
λ1· · · λn
<∞
Because λn=nλ and µn=nµ, we have
∞
X
n=1
n!λn
n!µn<∞
∞
X
n=1
λ
µn
<∞
which is true when λ < µ.