Source: J. Appl. Probab. Volume 47, Number 2
(2010), 459-473.
We take a new look at transient, or time-dependent Little laws for queueing
systems. Through the use of Palm measures, we show that previous laws (see
Bertsimas and Mourtzinou (1997)) can be generalized. Furthermore, within this
framework, a new law can be derived as well, which gives higher-moment
expressions for very general types of queueing system; in particular, the laws
hold for systems that allow customers to overtake one another. What is
especially novel about our approach is the use of Palm measures that are
induced by nonstationary point processes, as these measures are not commonly
found in the queueing literature. This new higher-moment law is then used to
provide expressions for all moments of the number of customers in the system in
an M/G/1 preemptive last-come-first-served queue at a time t > 0,
for any initial condition and any of the more famous preemptive disciplines
(i.e. preemptive-resume, and preemptive-repeat with and without resampling)
that are analogous to the special cases found in Abate and Whitt (1987c),
(1988). These expressions are then used to derive a nice structural form for
all of the time-dependent moments of a regulated Brownian motion (see Abate and
Whitt (1987a), (1987b)).
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