1 Continuous Time Processes - IPM

customers in queue, q◦(t) is the probability that the number of customers
arrived is equal to the number of customers serviced. Therefore
q◦(t) = e −(λ+µ)t
∞
l=0
(λµ) k t 2k
(k!) 2 .
In principle an equation of the form (1.3.7), with a known function q◦, can
be solved by using Laplace transforms, a subject which we will discuss in
connection with renewal theory later. We will not pursue the solution of this
equation any further here. ♠
Example 1.3.2 The birth process described above can be easily generalized
to a birth-death process by introducing a positive parameter µ > 0 and
replacing equations (1.3.1) and (1.3.2) with the requirement
⎧
⎪⎨ nµh + o(h), if a = −1;
P [Xt+h = n + a | Xt = n] = nλh + o(h),
⎪⎩
o(h),
if a = 1;
if |a| > 1.
(1.3.8)
The probability generating function for Xt can be calculated by an argument
similar to that of pure birth process given above and is delegated to exercise
1.3.5. It is shown there that for λ = µ
−t(λ−µ)
µ(1 − ξ) − (µ − λξ)e
FX(ξ, t) =
λ(1 − ξ) − (µ − λξ)e−t(λ−µ) N . (1.3.9)
where N is given by the initial condition X◦ = N. For λ = µ this expression
simplifies to
µt(1 − ξ) + ξ N
FX(ξ, t) =
.
µt(1 − ξ) + 1
(1.3.10)
From this it follows easily that E[Xt] = Ne (λ−µ)t . Let ζ(t) denote the probability
that the population is extinct at time t, i.e., ζ(t) = P [Xt = 0 | X◦ = N].
Therefore ζ(t) is the constant term, as a function of ξ for fixed t, of the generating
function FX(ξ, t). In other words, ζ(t) = FX(0, t) and we obtain
lim
t→∞ =
1, if µ ≥ λ;
µ N
λ N , if µ < λ;
for the probability of eventual extinction. ♠
27
(1.3.11)