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
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This note was uploaded on 12/19/2011 for the course M E 366l taught by Professor Staff during the Fall '08 term at University of Texas.
 Fall '08
 Staff

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