Mean sojourn times in Markov queueing networks: Little's formula revisited

It is commonly supposed that,
L=\lambdaW
applies to "almost any" queueing system with
\lambda
some average customer entrance rate,
L
the asymptotic expectation or time average of the number of customers in the system and
W
some average of the sojourn time. This formula is studied for irreducible positive recurrent Markov queueing systems whose state vector
Z
consists of entries representing queue lengths at the respective service stations; blocking, finite capacities, batch arrivals, and variable rates of arrival and service are consistent with
Z
. Sojourn times are defined inan augmented Markov model
Y=(Z,U)
, where the customer marking process
U
describes the service discipline in sufficiently general terms to include most possibilities of interest. It is shown that
L=\lambda W
is universally applicable, if properly interpreted to take account of state-varying entrance rates, batch arrivals, and multiple customer classes.
L,\lambda
, and
W
may each be equivalently viewed as time averages, means over a regeneration cycle, or expectations with respect to the asymptotic probability structure of
Z
. Indeterminate forms of
L=\lambda W
are possible within the scope of Markov queueing networks (MQN); as is shown by some examples, these may take the form
\infty \times 0
for
\lambda W
, or of
\infty= \infty
for
L=\lambda W
.