Notation and Terminology
N
(
t
) = # of customers in the system at time
t
³ 0
P
k
(
t
) = probability exactly
k
customers in system at
time
t
, given # in system at time 0
s
= # of parallel servers
λ
k
= mean arrival rate (expected # of arrivals per unit
time)
μ
k
= mean service rate (expected # of departures per
unit time)
(Both λ
k
and μ
k
assume
k
customers are in system)
If λ
n
does not depend on # of customers in system, λ
n
=
λ
.
If there are
s
servers, each with the same service rate,
then
μ
n
=
s
μfor
n
>=
s
and μ
n
=
n
μ for 0 <=
n
<
s
.
s
μ= customer service capacity per unit time
ρ= λ/
s
μ= utilization factor (traffic intensity)
The systems we study will have ρ < 1 because
otherwise the # of customers in the system will grow
without bound.
We will be interested in the steadystate behavior of
queuing systems (the behavior for
t
large).
Obtaining analytical results for
N
(
t
),
P
n
(
t
), . . .
for arbitrary values of
t
(the transient
behavior) is much more difficult.
Notation for steadystate analysis
π
n
= probability of having exactly
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Staff
 Probability theory, Queueing theory, arrival rate

Click to edit the document details