Page 375 - A First Course In Stochastic Models

370 ALGORITHMIC ANALYSIS OF QUEUEING MODELS

The following result holds:
(∞)
(1 − ρ)P {W q > τ}
P loss = , (9.4.7)
(∞)
1 − ρP {W q > τ}
(∞)
where the random variable W q is distributed as the steady-state delay in queue of
a customer in the standard M/G/1 queue with service in order of arrival. That is,
(∞)
P {W q ≤ x} = W q (x). The computation of W q (x) is discussed in Section 9.2.2.
The proof of (9.4.7) is very similar to that of (9.4.1). To obtain the formula for P loss ,
it is no restriction on the mathematical analysis to assume that customers finding
an amount of work in system larger than τ upon arrival do not enter the system but
(τ)
are immediately lost. Using this convention, denote by the random variable V t
the amount of work in system at time t and let V (τ) (x) = lim t→∞ P {V t (τ) ≤ x}
for x ≥ 0. Then, using the PASTA property,
P loss = 1 − V (τ) (τ). (9.4.8)

By the same arguments as used to obtain (9.4.4), there is a constant γ so that
V (τ) (x) = γ V ∞ (x), 0 ≤ x ≤ τ. (9.4.9)

To find the constant γ , we use Little’s formula for the average number of busy
servers. Since 1 − V (τ) (0) gives the fraction of time the server is busy,
λ(1 − P loss )µ = 1 − V (τ) (0). (9.4.10)

Since V (τ) (0) = γ V ∞ (0) and V ∞ (0) = 1 − ρ, we obtain from (9.4.10) that
(1 − ρ)(γ − 1)
P loss = . (9.4.11)
λµ
Also, by (9.4.8), P loss = 1 − γ V ∞ (τ) and so
1
γ = . (9.4.12)
1 − ρ [1 − V ∞ (τ)]
Finally, the desired result (9.4.7) follows by substituting (9.4.12) in (9.4.11) and
noting that V ∞ (x) equals the waiting-time distribution function W q (x). Assuming
that the service-time distribution function satisfies Assumption 9.2.1, it follows
from (9.4.7) and the asymptotic expansion (9.2.16) that
(1 − ρ) γ e −δτ −δτ
P loss ∼ −δτ ∼ (1 − ρ) γ e as τ → ∞,
1 − ργ e
where γ and δ are given by (9.2.17). In other words, P loss decreases exponen-
tially fast as τ gets larger. The structural form of (9.4.7) is remarkable. The loss
(∞)
probability is expressed in terms of the waiting-time probability P {W q > τ}.
The latter probability represents for the M/G/1 queue without impatience the