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Lecture Set 3, EE 679, Copyright 2000, Sanjay K. Bose
1
Basic Queuing Theory
Kendall's Notation for Queues:
This is a useful way to represent different types of queues in a compact
and easily understood fashion. Kendall's Notation describes the nature
of the arrival process to the queue, the nature of the service process (or
service time), the number of servers, maximum number in the queue
and some basic queuing disciplines. The notation has been considerably
extended to allow it to represent a wide variety of queues. We give here
a very basic description of this notation.
Following this representation, a queue is represented by a sequence
A/B/C/D/E
with the following meaning
attached to the letters
A
to
E
A
This symbolically represents the nature of the arrival process to the queue. Special letters
are used to symbolize the nature of the
interarrival time distribution
as follows 
M
Exponentially distributed interarrival times (Poisson Process)
D
Deterministic (fixed) interarrival times
E
k
Erlang distribution of order k for the interarrival times
H
k
Hyperexponential distribution of order k for the interarrival times
G
General (any!) distribution for the interarrival times
etc.
B
This symbolically represents the nature of the service time distribution for the customers
getting served in the queue. The same letters as the ones above are used to describe the
nature of the
service time distribution
C
Number of servers in the queue
D
Maximum Number of jobs/customers that can be there in the queue  this includes both
the ones currently being served and the ones waiting for service. Note that the default is
infinity
(
∝
) which is assumed when this is omitted
E
Queuing Discipline such as 
FCFS
First Come First Served
LCFS
Last Come First Served
SIRO
Service In Random Order
etc.
This may also be omitted if the queue is FCFS in nature (default)/
Examples:
M/M/1 or M/M/1/
Poisson Arrivals, Exponential Service Time Distribution, Single
Server, Infinite Number of Waiting Positions
M/E
2
/2/K
Poisson
Arrivals, Erlangian of order2 Service Time Distribution, 2
Severs, Maximum Number K in queue
G/M/2
Generalized Arrivals, Exponential Service Time Distribution, 2
Servers, Infinite Number of Waiting Positions
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View Full DocumentLecture Set 3, EE 679, Copyright 2000, Sanjay K. Bose
2
Equilibrium (Steady State) Solutions for BirthDeath Processes
For state k, birth rate is
l
k
and death rate is
m
k
. Note that this would correspond to an exponential
distribution for the interarrival and service times with means 1/
l
k
and 1/
m
k
., respectively.
To obtain the equilibrium solution to this, we set
0
=
dt
dp
k
for k=0, 1, 2, 3, .
............
This leads to the following equations for the steady state solutions {
p
k
}
1
1
1
1
)
(
0
+
+


+
+
+

=
k
k
k
k
k
k
k
p
p
p
m
l
m
l
where we assume further that
0
=
=
=



i
i
i
p
m
l
for
i=1, 2, .
.......
and
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