Page 34 - A First Course In Stochastic Models

MARKOV MODULATED BATCH POISSON PROCESSES 25

(i)
discrete probability distribution {a , k = 1, 2, . . . }. It is no restriction to assume
k
(i) (i) (i) (i) (i)
that a = 0; otherwise replace λ i by λ i (1 − a ) and a by a /(1 − a )
0 0 k k 0
for k ≥ 1.
For any t ≥ 0 and i, j = 1, . . . , m, define
P ij (k, t) = P {the total number of customers arriving in (0, t) equals k and
the phase process is in state j at time t | the phase process is in
state i at the present time 0}, k = 0, 1, . . . .

Also, for any t > 0 and i, j = 1, . . . , m, let us define the generating function P ∗
ij
(z, t) by
∞

∗ k
P (z, t) = P ij (k, t)z , |z| ≤ 1.
ij
k=0
∗
To derive an expression for P (z, t), it is convenient to use matrix notation. Let
ij
Q = (q ij ) be the m × m matrix whose (i, j)th element is given by
q ii = −ω i and q ij = ω i p ij for j
= i.
Define the m × m diagonal matrices and A k by

(1) (m)
= diag(λ 1 , . . . , λ m ) and A k = diag(a , . . . , a ), k = 1, 2, . . . .
k k
(1.4.1)
Let the m × m matrix D k for k = 0, 1, . . . be defined by
D 0 = Q − and D k = A k , k = 1, 2, . . . . (1.4.2)

Using (D k ) ij to denote the (i, j)th element of the matrix D k , define the generating
function D ij (z) by

∞
k
D ij (z) = (D k ) ij z , |z| ≤ 1.
k=0

Theorem 1.4.1 Let P (z, t) and D(z) denote the m × m matrices whose (i, j)th
∗
elements are given by the generating functions P (z, t) and D ij (z). Then, for any
∗
ij
t > 0,
∗
P (z, t) = e D(z)t , |z| ≤ 1, (1.4.3)
n n
∞
where e At is defined by e At = n=0 A t /n!.
Proof The proof is based on deriving a system of differential equations for the
P ij (k, t). Fix i, j, k and t. Consider P ij (k, t + t) for t small. By conditioning