1.2 Basic Queueing Models
usc.den.ee555-200901-silvester
A. Basic Queueing Concepts
We think of a stream of packets arriving to a multiplexer as the customers in a
queueing system.
x
n
is service time for customer
n
t
n+1
is inter-arrival time for customer
n
(between customer
n
and customer
n+1)
w
n
is waiting time for customer
n
s
n
= w
n
+x
n
is total system time for customer
n
See diagram on next page.
Messages
at a rate
λ
Server at
rate
μ
=
C/L
Users
1.2 Basic Queueing Models
usc.den.ee555-200901-silvester
c
n
c
n+1
c
n+2
c
n+3
c
n
c
n+1
c
n+2
c
n+3
Departures
Arrivals
Queue
In Service
x
n
t
n+1
w
n+2
x
n+2
x
n+1
s
n+2
IDLE
BUSY PERIOD
We note that:
1
1
,
0
max{
+
+
−
+
=
n
n
n
n
t
x
w
w
} which is useful for studying system
behaviour since
x
and
t
are random variables that can be simulated to drive system
evolution.

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1.2 Basic Queueing Models
usc.den.ee555-200901-silvester
)
(
)
(
)
(
t
n
t
a
t
n
−
=
which is the number in system at time
t
Consider the area:
∫
=
t
dt
t
n
t
A
0
)
(
)
(
Then:
)
(
/
)
(
t
N
t
t
A
=
is the average number (packets) in system in (0,
t
).
We can compute this area another way:
∑
=
=
)
(
1
)
(
t
a
n
n
s
t
A
(assume for simplicity
that
t
is in an idle period).
a
(
t
)=
number of arrivals in
(0,
t
)
d
(
t
)=
number of departures in
(0,
t
)
n
(
t
)
s
n
1.2 Basic Queueing Models
usc.den.ee555-200901-silvester
So,
( )
1
( )
1
( )
( )
( )
( )
( )
a t
n
n
n
s
A t
a t
N t
s
t T t
t
t
t
a t
λ
=
=
=
=
=
∑
∑
= Arrival rate in (0,
t
) * Average time in system for packets up to time
t
Dropping the time parameter
t
, we have:
T
N
λ
=
which is known as
Little’s Result
.
This is a basic and extremely valuable result. It can be applied to almost any system
in which the arrivals and departures occurs in single steps (i.e. one at a time).
Paraphrasing: the average time spent in a system is equal to the average number in
the system divided by the rate of customers passing through the system.
The system can be quite general, the queue, the system as a whole, just the server, a
whole network of queues, etc.