EE 489
Telecommunication Systems Engineering
University of Alberta
Dept. of Electrical and Computer Engineering
Introduction to Traffic Theory
Wayne Grover
TRLabs and University of Alberta
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
A note on sources of this material
• The following material on traffic theory / traffic engineering was
initially developed as printed handwritten notes from 1998 to
2001 by W. Grover for EE589.
• In 2002 John Doucette set these materials into the present
powerpoint format for use in EE589.
• The ppt versions of the original notes, with updating and some
revisions by W. Grover, 2007, are made available courtesy J.
Doucette for use in EE489.
• Related Reading in Bellamy 3rd Edition:
– Chapter 12, pp. 519-567.
Material prepared by W. Grover (1998-2002)
2
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Traffic Engineering
• One billion+ terminals in voice network alone
– Plus data, video, fax, finance, etc.
• Imagine all users want service simultaneously
• In practice, low overall utilization
– Random duration at random times
• Balance cost and practicality with acceptably low chance of
network blocking.
• We use traffic engineering to “dimension” the network, i.e.
mainly to decide on how many transmission paths (trunks) are
needed between node and the sizes of switches or routers
needed.
Material prepared by W. Grover (1998-2002)
3
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Characterization of Circuit-Switched Traffic
• Calling Rate () – also called Arrival Rate
– Average number of calls or “connections” initiated per unit time
(units. “attempts per hour”)
– Each arrival independent of other calls
– Random in time
If receive  calls from a terminal in time T:
α
γ
T
If receive  calls from m terminals in time T:
Group calling rate
α
γg 
T
Per terminal
calling rate
α
γ
m T
Material prepared by W. Grover (1998-2002)
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EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Characterization of Telephone Traffic (2)
• Calling rate assumption:
– Number of calls in time T is Poisson distributed:
e    x
p( x) 
x!
•
x  0, 1, 2...
 Time between calls is negative exponentially distributed:
f (t )    e t
mean 
0t 
1

Material prepared by W. Grover (1998-2002)
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EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Characterization of Telephone Traffic (3)
• Holding Time (h)
– Mean length of time a call lasts
– Probability of lasting time t or more is exponential in nature:
P(T  t )  e t / h
P(T  t )  0
t 0
t 0
– Real sampled voice data fits very closely to the negative
exponential form above
– As non-voice “calls” begin to dominate, more and more calls have a
constant holding time characteristic
• Departure Rate ():
1
h

Material prepared by W. Grover (1998-2002)
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EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Real Holding Time Sample Data
Material prepared by W. Grover (1998-2002)
7
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Exponential Form of Holding Time
• Memory-less property
– Call “forgets” that it has already survived to time T1
PT  T1  t T  T1   PT  t 
• Proof:
PT  T1  t T  T1 
Recall:
P(T  t )  e t / h
P T  T1  t  T  T1 

PT  T1 
P T  T1  t 
e  (T1 t ) / h

 T1 / h
PT  T1 
e
e T1 / h  e t / h  e  t / h  PT  t 

e T1 / h
Material prepared by W. Grover (1998-2002)
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EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Exponential Form of Holding Time
• Understanding Memorylessness in Call holding times
It means that:
– whether a connection has already existed for 1 minute or one
hour…
– the probabiity that it will last another minute (or any other unit
time)…
– is the same.
• Counterintuitive (?) but very accurate actually.
• Can understand it (or any memoryless process) as being
analogous to repeated coin tossing
Material prepared by W. Grover (1998-2002)
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EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Traffic Volume (V)
 = # calls in time period T
V  h
h = mean holding time
V = volume of calls in time period T
• Units can be “ccs”:
– Hundred call seconds
“c”
“c”
“s”
– 1 ccs is volume of traffic equal to:
– one circuit busy for 100 seconds, or
– two circuits busy for 50 seconds, or
– 100 circuits busy for one second, etc.
Material prepared by W. Grover (1998-2002)
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EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Traffic Intensity (A)
• This is the rate of “traffic flow”.
 = # calls in time period T
 h
V

A

  h 
T
T

Recall:


T
Recall:
1

h
h = mean holding time
T = time period of observations
Recall:
 = calling rate
V  h
 = departure rate
V = call volume
• Units:
– “ccs/hour”, or
– dimensionless (if h and T are in the same units)
“Erlang” unit
Material prepared by W. Grover (1998-2002)
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EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
The “Erlang”
• Dimensionless unit of traffic intensity. Characterizes the intensity
of any stream of demands for circuit-switched (or connectionoriented data) connections
• Named after Danish mathematician A. K. Erlang (1878-1929)
• Usually denoted by symbol E.
• 1 Erlang is equivalent to the traffic intensity that keeps:
– one circuit busy 100% of the time, or
– two circuits busy 50% of the time, or
– four circuits busy 25% of the time, etc.
• e.g., 26 Erlangs is equivalent to traffic intensity that keeps :
– 26 circuits busy 100% of the time, or
– 52 circuits busy 50% of the time, or
– 104 circuits busy 25% of the time, etc.
Material prepared by W. Grover (1998-2002)
12
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Erlang (2)
• How does the Erlang unit correspond to ccs?
1 ccs hour 
100 call  seconds
 0.027E
1 hour × 60
60 min hr × 60 sec min
3600 call  seconds
 1E
36 ccs hour 
1 hour × 60 min hr × 60 sec min
• Percentage of time a terminal is busy is equivalent to the traffic
generated by that terminal in Erlangs, or
• Average number of circuits in a group busy at any time
• Typical usages:
– residence phone -> 0.02 E
– business phone -> 0.15 E
– interoffice trunk -> 0.70 E
Material prepared by W. Grover (1998-2002)
13
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Traffic Offered, Carried, and Lost
• Offered Traffic (TO ) equivalent to Traffic Intensity (A)
– Takes into account all attempted calls, whether blocked or not,
and uses their expected holding times
• Also Carried Traffic (TC ) and Lost Traffic (TL )
• Consider a group of 150 terminals, each with 10% utilization (or
in other words, 0.1 E per source) and dedicated service:
1
150
each terminal has an
outgoing trunk
(i.e. terminal:trunk ratio = 1:1)
1
150
TO = A = 150 x 0.10 E = 15.0 E
TC = 150 x 0.10 E = 15.0 E
TL = 0 E
Material prepared by W. Grover (1998-2002)
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EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Traffic Offered, Carried, and Lost (2)
• A = TO = TC + TL
Traffic
Intensity
Offered
Traffic
Lost
Traffic
Carried
Traffic
• TL = TO x Prob. Blocking (or congestion)
= P(B) x TO = P(B) x A
• Circuit Utilization () - also called Circuit Efficiency
– proportion of time a circuit is busy, or
– average proportion of time each circuit in a group is busy

TC

# of Trunks
Material prepared by W. Grover (1998-2002)
15
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Example #1
• Traffic Engineered solution for the 150 terminals at 0.1 E ...
Material prepared by W. Grover (1998-2002)
16
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Grade of Service (gos)
• In general, the term used for some traffic design objective
• Indicative of customer satisfaction
• In systems where blocked calls are cleared, usually use:
gos 
TL
TL

 P( B)
TO TL + TC
• Typical gos objectives:
– in busy hour, range from 0.2% to 5% for local calls, however
– generally no more that 1%
– long distance calls often slightly higher
• In systems with queuing, gos often defined as the probability of
delay exceeding a specific length of time
Material prepared by W. Grover (1998-2002)
17
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Grade of Service Related Terms
• Busy Hour
– One hour period during which traffic volume or call attempts is the
highest overall during any given time period
• Peak (or Daily) Busy Hour
– Busy hour for each day, usually varies from day to day
• Busy Season
– 3 months (not consecutive) with highest average daily busy hour
• High Day Busy Hour (HDBH)
– One hour period during busy season with the highest load
Material prepared by W. Grover (1998-2002)
18
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Hourly Traffic Variations
Material prepared by W. Grover (1998-2002)
19
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Daily Traffic Variations
Material prepared by W. Grover (1998-2002)
20
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Seasonal Traffic Variations
Material prepared by W. Grover (1998-2002)
21
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Seasonal Traffic Variations (2)
Material prepared by W. Grover (1998-2002)
22
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Grade of Service Related Terms (2)
• Average Busy Season Busy Hour (ABSBH)
– One hour period with highest average daily busy hour during the
busy season
– For example, assume days shown below make up the busy season:
00:00 to 01:00
01:00 to 02:00
02:00 to 03:00
03:00 to 04:00
04:00 to 05:00
05:00 to 06:00
06:00 to 07:00
07:00 to 08:00
08:00 to 09:00
09:00 to 10:00
10:00 to 11:00
11:00 to 12:00
12:00 to 13:00
13:00 to 14:00
14:00 to 15:00
15:00 to 16:00
16:00 to 17:00
17:00 to 18:00
18:00 to 19:00
19:00 to 20:00
20:00 to 21:00
21:00 to 22:00
22:00 to 23:00
23:00 to 00:00
1-Apr 2-Apr 3-Apr 4-Apr 5-Apr 6-Apr 7-Apr 8-Apr 9-Apr 10-Apr 11-Apr 12-Apr 13-Apr 14-Apr 15-Apr 16-Apr 17-Apr 18-Apr 19-Apr 20-Apr 21-Apr Mean
1.4
1.4
1.2
1.5
1.1
1.5
1.7
1.5
1.0
1.0
1.8
1.5
1.8
1.6
1.2
1.9
1.8
1.6
1.4
1.5
1.2
1.5
1.2
1.8
1.6
1.3
1.0
1.6
1.1
1.1
1.0
1.2
1.7
2.0
2.0
1.8
1.3
1.7
1.4
1.9
1.1
1.4
1.5
1.5
1.4
1.8
1.5
1.9
1.2
1.0
1.2
1.1
1.1
1.7
1.5
1.5
1.9
1.9
1.3
1.5
1.8
1.1
1.1
1.2
1.5
1.4
1.2
1.8
1.7
1.4
1.7
1.1
1.5
1.6
1.1
1.9
1.0
1.0
1.4
1.5
1.6
1.1
1.4
1.9
1.4
1.2
1.1
1.4
1.8
1.8
2.3
2.2
2.0
1.7
2.3
1.6
2.2
1.5
2.1
1.6
2.3
2.1
1.7
2.5
1.6
2.0
1.7
1.5
2.3
1.9
2.2
2.3
1.9
2.4
2.5
2.0
2.0
1.7
1.8
1.6
2.0
2.0
2.2
2.2
2.1
1.8
1.6
1.7
2.0
2.3
2.1
2.0
1.7
2.2
1.7
2.5
2.2
2.1
2.2
2.0
2.3
1.6
2.4
2.2
1.5
2.1
2.2
1.8
1.8
1.7
2.1
2.0
2.1
2.0
2.0
2.8
2.2
2.4
2.3
2.4
2.9
2.0
2.4
2.4
2.1
2.9
2.3
2.1
2.9
2.7
2.8
2.3
2.1
2.1
2.7
2.4
3.4
3.1
2.8
2.9
2.5
2.7
2.9
3.0
3.4
3.4
3.1
2.9
2.9
2.9
3.3
3.2
3.5
3.1
3.1
3.1
2.5
3.0
3.4
3.4
4.0
3.2
3.5
3.4
3.1
3.7
3.3
3.3
3.5
3.9
3.4
4.0
3.7
3.7
3.1
3.4
3.9
3.9
3.4
3.5
5.0
4.4
4.8
4.9
4.1
3.0
4.0
4.9
4.2
4.9
4.7
4.2
3.8
3.0
4.6
4.9
4.4
5.0
4.7
3.6
3.8
4.3
4.8
5.0
4.7
4.3
4.5
3.8
3.4
4.2
5.0
4.6
5.0
4.7
3.2
3.4
5.0
4.8
4.1
4.3
4.4
3.6
3.7
4.3
4.5
4.2
4.1
4.8
4.6
3.8
3.3
4.0
4.2
4.6
4.7
4.0
3.3
3.1
5.0
4.9
4.6
4.1
4.2
3.2
3.6
4.1
4.3
4.2
4.7
4.5
4.8
3.2
3.1
4.1
4.5
4.6
4.9
4.7
3.6
3.6
4.8
4.2
4.8
4.9
4.4
3.3
3.0
4.2
4.8
4.7
4.5
4.1
4.4
3.6
3.7
4.5
4.3
4.3
4.9
4.5
3.5
3.5
4.3
4.3
4.3
4.5
4.3
3.3
3.2
4.2
4.4
4.9
4.4
4.8
4.5
3.8
3.2
4.1
4.8
4.4
4.5
4.2
3.3
3.9
4.3
4.9
4.4
4.3
4.5
3.7
3.3
4.2
3.2
3.2
3.8
3.5
3.7
3.1
3.5
3.5
3.2
3.2
3.8
3.4
3.2
4.0
3.3
4.0
3.9
3.0
3.3
3.5
3.3
3.5
indicates
2.7
2.6Note:
2.7Red2.9
3.3
3.1
3.4
2.9
3.2
2.8
2.7
3.0
3.3
3.2
2.5
2.9
2.8
3.4
3.5
2.9
3.2
3.0
3.0
2.9
3.0
2.7
2.9
3.4
3.3
3.4
2.7
3.3
3.5
3.5
2.7
3.1
3.1
3.3
3.4
3.1
3.0
3.3
3.3
3.1
daily busy hour
3.3
3.3
2.6
3.4
3.2
2.7
2.7
3.4
3.4
3.0
3.0
3.4
3.1
2.8
3.2
3.4
3.0
3.4
3.4
3.1
2.9
3.1
2.9
2.3
2.1
2.9
2.9
3.0
3.0
2.4
2.3
2.9
3.0
2.1
2.2
2.9
3.0
2.6
2.4
2.5
2.7
2.7
2.6
2.6
2.1
1.6
2.3
1.6
2.2
2.1
2.4
1.9
1.6
2.1
2.4
1.7
1.8
2.4
1.8
1.9
2.2
1.9
2.2
2.2
1.6
2.0
1.5
2.1
1.9
1.6
1.7
1.6
2.3
2.5
2.4
1.7
2.1
1.8
2.0
2.4
1.7
1.9
2.2
2.3
1.7
2.4
1.8
2.0
1.5
1.0
1.1
1.1
1.5
1.8
1.5
1.4
1.8
1.1
1.9
1.2
1.6
1.9
1.8
1.1
1.5
2.0
1.8
1.6
1.4
1.5
ABSBH
Highest
Material prepared by W. Grover (1998-2002)
23
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Grade of Service Related Terms (3)
• Ten High Day Busy Hour (10HDBH)
– One hour period with highest average load for the 10 highest day
loads for that hour
– For example:
00:00 to 01:00
01:00 to 02:00
02:00 to 03:00
03:00 to 04:00
04:00 to 05:00
05:00 to 06:00
06:00 to 07:00
07:00 to 08:00
08:00 to 09:00
09:00 to 10:00
10:00 to 11:00
11:00 to 12:00
12:00 to 13:00
13:00 to 14:00
14:00 to 15:00
15:00 to 16:00
16:00 to 17:00
17:00 to 18:00
18:00 to 19:00
19:00 to 20:00
20:00 to 21:00
21:00 to 22:00
22:00 to 23:00
23:00 to 00:00
1-Apr 2-Apr 3-Apr 4-Apr 5-Apr 6-Apr 7-Apr 8-Apr 9-Apr 10-Apr 11-Apr 12-Apr 13-Apr 14-Apr 15-Apr 16-Apr 17-Apr 18-Apr 19-Apr 20-Apr 21-Apr Mean
1.4
1.4
1.2
1.5
1.1
1.5
1.7
1.5
1.0
1.0
1.8
1.5
1.8
1.6
1.2
1.9
1.8
1.6
1.4
1.5
1.2
1.66
1.2
1.8
1.6
1.3
1.0
1.6
1.1
1.1
1.0
1.2
1.7
2.0
2.0
1.8
1.3
1.7
1.4
1.9
1.1
1.4
1.5
1.76
1.4
1.8
1.5
1.9
1.2
1.0
1.2
1.1
1.1
1.7
1.5
1.5
1.9
1.9
1.3
1.5
1.8
1.1
1.1
1.2
1.5
1.71
1.2
1.8
1.7
1.4
1.7
1.1
1.5
1.6
1.1
1.9
1.0
1.0
1.4
1.5
1.6
1.1
1.4
1.9
1.4
1.2
1.1
1.67
1.8
1.8
2.3
2.2
2.0
1.7
2.3
1.6
2.2
1.5
2.1
1.6
2.3
2.1
1.7
2.5
1.6
2.0
1.7
1.5
2.3
2.22
2.2
2.3
1.9
2.4
2.5
2.0
2.0
1.7
1.8
1.6
2.0
2.0
2.2
2.2
2.1
1.8
1.6
1.7
2.0
2.3
2.1
2.21
1.7
2.2
1.7
2.5
2.2
2.1
2.2
2.0
2.3
1.6
2.4
2.2
1.5
2.1
2.2
1.8
1.8
1.7
2.1
2.0
2.1
2.23
2.0
2.8
2.2
2.4
2.3
2.4
2.9
2.0
2.4
2.4
2.1
2.9
2.3
2.1
2.9
2.7
2.8
2.3
2.1
2.1
2.7
2.71
3.4
3.1
2.8
2.9
2.5
2.7
2.9
3.0
3.4
3.4
3.1
2.9
2.9
2.9
3.3
3.2
3.5
3.1
3.1
3.1
2.5
3.25
3.4
3.4
4.0
3.2
3.5
3.4
3.1
3.7
3.3
3.3
3.5
3.9
3.4
4.0
3.7
3.7
3.1
3.4
3.9
3.9
3.4
3.78
5.0
4.4
4.8
4.9
4.1
3.0
4.0
4.9
4.2
4.9
4.7
4.2
3.8
3.0
4.6
4.9
4.4
5.0
4.7
3.6
3.8
4.82
4.8
5.0
4.7
4.3
4.5
3.8
3.4
4.2
5.0
4.6
5.0
4.7
3.2
3.4
5.0
4.8
4.1
4.3
4.4
3.6
3.7
4.80
4.5
4.2
4.1
4.8
4.6
3.8
3.3
4.0
4.2
4.6
4.7
4.0
3.3
3.1
5.0
4.9
4.6
4.1
4.2
3.2
3.6
4.61
4.3
4.2
4.7
4.5
4.8
3.2
3.1
4.1
4.5
4.6
4.9
4.7
3.6
3.6
4.8
4.2
4.8
4.9
4.4
3.3
3.0
4.74
4.8
4.7
4.5
4.1
4.4
3.6
3.7
4.5
4.3
4.3
4.9
4.5
3.5
3.5
4.3
4.3
4.3
4.5
4.3
3.3
3.2
4.54
4.4
4.9
4.4
4.8
4.5
3.8
3.2
4.1
4.8
4.4
4.5
4.2
3.3
3.9
4.3
4.9
4.4
4.3
4.5
3.7
3.3
4.62
3.2
3.2
3.8
3.5
3.7
3.1
3.5
3.5
3.2
3.2
3.8
3.4
3.2
4.0
3.3
4.0
3.9
3.0
3.3
3.5
3.3
3.73
2.7
2.6
2.7
2.9
3.3
3.1
3.4
2.9
3.2
2.8
2.7
3.0
3.3
3.2
2.5
2.9
2.8
3.4
3.5
2.9
3.2
3.26
3.0
2.9
3.0
2.7
2.9
3.4
3.3
3.4
2.7
3.3
3.5
3.5
2.7
3.1
3.1
3.3
3.4
3.1
3.0
3.3
3.3
3.37
3.3
3.3
2.6
3.4
3.2
2.7
2.7
3.4
3.4
3.0
3.0
3.4
3.1
2.8
3.2
3.4
3.0
3.4
3.4
3.1
2.9
3.36
2.9
2.3
2.1
2.9
2.9
3.0
3.0
2.4
2.3
2.9
3.0
2.1
2.2
2.9
3.0
2.6
2.4
2.5
2.7
2.7
2.6
2.91
2.1
1.6
2.3
1.6
2.2
2.1
2.4
1.9
1.6
2.1
2.4Note:
1.7 Red
1.8indicates
2.4
1.8
1.9
2.2
1.9
2.2
2.2
1.6
2.26
1.5
2.1
1.9
1.6
1.7
1.6
2.3
2.5
2.4
1.7
2.1
1.8
2.0
2.4
1.7
1.9
2.2
2.3
1.7
2.4
1.8
2.27
101.2highest
hourly
1.5
1.0
1.1
1.1
1.5
1.8
1.5
1.4
1.8
1.1
1.9
1.6
1.9
1.8
1.1
1.5
2.0
1.8
1.6
1.4
1.78
10HDBH
Highest
loads for each hour
Material prepared by W. Grover (1998-2002)
24
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Grade of Service Related Terms (4)
• Examples of grade-of-service type specification statements:
–
–
–
–
–
–
1.5% of calls in ABSBH have dial tone delay more than 3 seconds
blocking on trunk groups < 3%
blocking through switch matrix < 0.1%
probability of packet delay > x msec less than 5%
probability of dropped connection in progress < 1% per minute
etc.
• Note implications of designing to “busy hour” g.o.s. objectives:
– simplifies design and forecasting problems
– busy hour may change (unpredictably!)
– the resulting network is “peak engineered” - same as the power
network …may be greatly underutilized at off busy-hour times
• Q. What could you do with this?
Material prepared by W. Grover (1998-2002)
25
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Typical Call Attempts Breakdown
• Calls Completed - 70.7%
• Called Party No Answer - 12.7%
• Called Party Busy - 10.1%
• Call Abandoned - 2.6%
• Dialing Error - 1.6%
• Number Changed or Disconnected - 0.4%
• Network Blockage or Failure - 1.9%
Material prepared by W. Grover (1998-2002)
26
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Traffic Theoretic Models for Blocked Calls
• Blocked Calls Cleared (BCC)
– Blocked calls leave system and do not return
– Good approximation for calls in 1st choice trunk group with overflow
available.
• Blocked Calls Held (BCH)
– Blocked calls remain in the system for the amount of time it would
have normally stayed for
– If a server frees up, the call picks up in the middle and continues
– Not a good model of real world behaviour (mathematical
approximation only)
– Tries to approximate call reattempt efforts
• Blocked Calls Wait (BCW)
– Blocked calls enter a queue until a server is available
– When a server becomes available, the call’s holding time begins
Material prepared by W. Grover (1998-2002)
27
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Blocked Calls Cleared (BCC)
2 sources
10 minutes
Source #1
Offered Traffic
1
3
Source #2
Offered Traffic
2
4
1st call arrives and is served
Only one server
2nd call arrives but
server already busy
Traffic
Carried
Total Traffic Offered:
TO = 0.4 E + 0.3 E
TO = 0.7 E
1
2
3
4
Total Traffic Carried:
TC = 0.5 E
2nd call is cleared
3rd call arrives and is served
4th call arrives and is served
Material prepared by W. Grover (1998-2002)
28
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Blocked Calls Held (BCH)
2 sources
10 minutes
Source #1
Offered Traffic
1
3
Source #2
Offered Traffic
2
4
Total Traffic Offered:
TO = 0.4 E + 0.3 E
TO = 0.7 E
1st call arrives and is served
Only one server
2nd call arrives but server busy
Traffic
Carried
1
2 2
3
4
2nd call is held until server free
2nd call is served
Total Traffic Carried:
TC = 0.6 E
3rd call arrives and is served
4th call arrives and is served
Material prepared by W. Grover (1998-2002)
29
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Blocked Calls Wait (BCW)
2 sources
10 minutes
Source #1
Offered Traffic
1
3
Source #2
Offered Traffic
2
4
Total Traffic Offered:
TO = 0.4 E + 0.3 E
TO = 0.7 E
1st call arrives and is served
2nd call arrives but server busy
Only one server
2nd call waits until server free
Traffic
Carried
1
2
2
3
4
Total Traffic Carried:
TC = 0.7 E
2nd call served
3rd call arrives, waits, and
is served
4th call arrives, waits, and
is served
Material prepared by W. Grover (1998-2002)
30
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Blocking Probabilities
• System must be in a Steady State
– Also called state of statistical equilibrium
– Arrival Rate of new calls equals Departure Rate of
disconnecting calls
– Why?
• If calls arrive faster that they depart?
• If calls depart faster than they arrive?
Material prepared by W. Grover (1998-2002)
31
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Binomial Distribution Model
• Assumptions:
– m sources
– A Erlangs of offered traffic
• per source: TO = A/m
• probability that a specific source is busy: P(B) = A/m
• Can use Binomial Distribution to give the probability that a
certain number (k) of those m sources is busy:
 m  A   A 
P(k )     1  
 k  m   m 
k
mk

 A   A 
m!
  1  
 
 k!(m  k )!  m   m 
k
mk
Material prepared by W. Grover (1998-2002)
32
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Binomial Distribution Model (2)
• What does it mean if we only have N servers (N<m)?
– We can have at most N busy sources at a time
– What about the probability of blocking?
• All N servers must be busy before we have blocking
P( B)  P(k  N )  P(k  N )  P(k  N  1)  ...  P(k  m)
k
mk
m
  A   A 
     1  
m
k  N  k  m  
m
 m  A   A 
 1      1  
m
k 0  k  m  
N 1
k
Remember:
 m  A 
P(k )    
 k  m 
k
 A
1  
 m
mk
Material prepared by W. Grover (1998-2002)
33
mk
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Binomial Distribution Model (3)
• What does it mean if k>N?
– Impossible to have more sources busy than servers to serve them
– Doesn’t accurately represent reality
• In reality, P(k>N) = 0
– In this model, we still assign P(k>N) = A/m
– Acts as good model of real behaviour
• Some people call back, some don’t
• Which type of blocking model is the Binomial Distribution?
– Blocked Calls Held (BCH)
Material prepared by W. Grover (1998-2002)
34
EE489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering
Time Congestions vs. Call Congestion
• Time Congestion
– Proportion of time a system is congested (all servers busy)
– Probability of blocking from point of view of servers
• Call Congestion
– Probability that an arriving call is blocked
– Probability of blocking from point of view of calls
• Why/How are they different?
Time Congestion:
Call Congestion:
P( B)  P(k  N )
P( B)  P(k  N )
Probability that all
servers are busy.
Probability that there are
more sources wanting service
than there are servers.
Material prepared by W. Grover (1998-2002)
35