Page 73 - A First Course In Stochastic Models

64 RENEWAL-REWARD PROCESSES

∗ ∞ −sx
where b (s) = e b(x) dx denotes the Laplace transform of the probabil-
0
ity density b(x) of the service time. To prove this, note that D n , S n and τ n are
independent of each other. This implies that, for any x > 0,

−s(D n +S n −τ n ) +
E e | D n + S n = x
x ∞
−s(x−y) −λy −s×0 −λy
= e λe dy + e λe dy
0 x
λ −sx −λx −λx 1 −sx −λx
= (e − e ) + e = (λe − se )
λ − s λ − s
for s
= λ (using L’Hospital’s rule it can be seen that this relation also holds for
s = λ). Hence, using (2.5.10),

(λ − s)E e −sD n+1 = λE e −s(D n +S n ) − sE e −λ(D n +S n ) .

+
Since P {(D n + S n − τ n ) = 0 | D n + S n = x} = e −λx , we also have

P {D n+1 = 0} = E e −λ(D n +S n ) .

−s(D n +S n )
The latter two relations and E e = E e −sD n E e −sS n lead to (2.5.11).
The steady-state waiting-time distribution function W q (x) is defined by
W q (x) = lim P {D n ≤ x}, x ≥ 0.
n→∞
The existence of this limit can be proved from Theorem 2.2.4. Let the random vari-
able D ∞ have W q (x) as probability distribution function. Then, by the bounded con-
vergence theorem in Appendix A, E(e −sD ∞ ) = lim n→∞ E(e −sD n ). Using (2.5.6), it
follows from lim n→∞ P {D n+1 = 0} = π 0 and q 0 = 1 − ρ that lim n→∞ P {D n+1 =
0} = 1 − ρ. Letting n → ∞ in (2.5.11), we find that

(1 − ρ)s

E e −sD ∞ = . (2.5.12)
s − λ + λb (s)
∗
Noting that P {D ∞ ≤ x} = W q (x) and using relation (E.7) in Appendix E, we get
from (2.5.12) the desired result:
∞ ρs − λ + λb (s)

∗
e −sx 1 − W q (x) dx = . (2.5.13)
∗
0 s(s − λ + λb (s))
Taking the derivative of the right-hand side of (2.5.13) and putting s = 0, we obtain
2
∞
λE(S )
1 − W q (x) dx = ,
0 2(1 − ρ)
in agreement with the Pollaczek–Khintchine formula (2.5.1).