Page 35 - A First Course In Stochastic Models

26 THE POISSON PROCESS AND RELATED PROCESSES

on what may happen in (t, t + t), it follows that

P ij (k, t + t) = P ij (k, t)(1 − λ j t)(1 − ω j t) + P is (k, t)[(ω s t) × p sj ]
s
=j
k−1

(j)
+ P ij (ℓ, t) (λ j t) × a + o( t).
k−ℓ
ℓ=0
Using the definition of the q ij , we rewrite this relation as
m

P ij (k, t + t) = P ij (k, t)(1 − λ j t) + P is (k, t)q sj t
s=1
k−1
(j)

+ P ij (ℓ, t)λ j a t + o( t),
k−ℓ
ℓ=0
which implies that
m k−1
d (j)
P ij (k, t) = −λ j P ij (k, t) + P is (k, t)q sj + λ j P ij (ℓ, t)a k−ℓ .
dt
s=1 ℓ=0
Letting P (k, t) be the m × m matrix whose (i, j)th element is P ij (k, t), we have
in matrix notation that

k−1
d
P (k, t) = P (k, t)(Q − ) + P (ℓ, t)A k−ℓ .
dt
ℓ=0
Using the definition of the matrices D k , we find next that

k−1
d
P (k, t) = P (k, t)D 0 + P (ℓ, t)D k−ℓ
dt
ℓ=0
k

= P (ℓ, t)D k−ℓ .
ℓ=0
k
Multiply componentwise both sides of this matrix equation by z and sum over k.
Since the generating function of the convolution of two sequences is the product
of the generating functions of the two sequences, it follows that
d
∗
P (z, t) = P (z, t)D(z).
∗
dt
For each fixed i this equation gives a system of linear differential equations in
∗
P (z, t) for j = 1, . . . , m. Thus, by a standard result from the theory of linear
ij