Page 374 - A First Course In Stochastic Models

M/G/1 QUEUES WITH BOUNDED WAITING TIMES 369

1 K
I (K) = K − V ∞ (x) dx. (9.4.6)
V ∞ (K) 0

The proof of (9.4.5) is based on Little’s formula for the average number of busy
servers. The long-run fraction of time the server is busy equals 1 − V K (0) =
1 − V ∞ (0)/V ∞ (K). Hence, by Little’s formula,

V ∞ (0)
λ(1 − f (K))µ = 1 − .
V ∞ (K)
Since V ∞ (0) = 1−ρ, the formula (9.4.5) next follows. Using partial integration, the
result (9.4.6) directly follows by using (9.4.2). The performance measures π(K),
f (K) and I (K) can be calculated by using numerical Laplace inversion for the

x
computation of V ∞ (x), V (x) and V ∞ (y) dy from the corresponding Laplace
′
∞ 0
transforms. The formula (9.4.5) for the overflow probability f (K) has an interesting
form. It is our conjecture that this structural form provides a useful approximation
to the overflow probability in more complex finite-buffer models such as the finite-
buffer fluid model with a Markov modulated Poisson input process determined by
a number of independent on-off sources. The solution of the infinite-buffer version
of this model is given in the classic paper of Anick et al. (1982); see also Schwartz
(1996). In this paper the linear differential equations for the work in system are
solved through eigenvalues and eigenvectors.

9.4.2 An M/G/1 Queue with Impatient Customers

A queueing system often encountered in practice is one in which customers wait for
service for a limited time and leave the system if service has not begun within that
time. Practical examples of queueing systems with customer impatience include
real-time telecommunication systems in which data received after a hard deadline
are useless, telecommunication systems in which subscribers give up due to impa-
tience before the requested connection is established and inventory systems with
perishable goods.
In this subsection we consider an M/G/1 queue in which customers arrive
according to a Poisson process with rate λ. The service or work requirements
of the customers are independent random variables having a general probability
distribution function B(x) with finite mean µ. It is assumed that ρ = λµ is less
than 1. The service discipline is first-come first-served. Each arriving customer
enters the system, but is only willing to wait in queue for a fixed time τ > 0. A
customer who waits for a time τ without his service having begun leaves the system
after that time τ and becomes a lost customer. A basic measure for the quality of
service in such a system is the fraction of customers who are lost. Define the
performance measure P loss by

P loss = the long-run fraction of customers who are lost.