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Telecommunications
Traffic
An Introduction
© JEREMY HARVEY 2010 All Rights Reserved
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What is it?
Telecommunications Traffic or ³teletraffic´ is: The combined messages carried between
two or mor e points via transmission links
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TeletrafficWe ar e inter ested in the ³intensity of that traffic´ so that we can provide the
right number of transmission links
³Traffic Intensity´ depends on:
The total number of simultaneousmessages in progr ess on the link It is measur ed in ³Er langs´
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TeletrafficWhat is an ³Er lang´
If one line is occupied for one hour, it issaid to carry one Er lang of traffic
So if a line carries 0.5 Er lang of Traffic,it is only busy for 50% of the hour, or 30minutes
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TeletrafficEr langs ar e used to calculate the percentage of time each hour, that aline will actually be in use.
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TeletrafficThe volume of traffic depends on:
The number of messages (or calls)
The duration of each call
For example 10 calls (each lasting 3minutes) is the same as 1 call lasting 30minutes
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TeletrafficMathematically we say that:
Traffic Volume is the integral of trafficintensity with r espect to time, and we expr ess it in ³Er lang-hours´
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Integrals
Integration is an ar ea of mathematicsthat is used to measur e the ar ea under the line on a graph
Erlangs
Time
(0-3600
seconds
_
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Integrals
Traffic Intensity (Utilisation of line each second)
Erlangs
Time
(0-3600
seconds
_
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Integrals
Traffic Volume (Traffic intensitymeasur ed each second for an hour)
Erlangs
Time
(0-3600
seconds
_
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Integrals
The integral is found by dividing the graph into r ectangles and finding the
ar ea of r ectangle (then adding them all up)
Erlangs
Time
(0-3600
seconds
_
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Integrals
Mathematically this is:
v = I(t) dt
Traffic
VolumeIntegral (the
area under the
graph)
Traffic
Intensity
Over time (eg over
3600 seconds)
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Integrals
We can have a function that produces a copy of the curve of the line on the graph
v = f(x) dx
Traffic
VolumeIntegral (the
area under the
graph)
Function that produces a
copy of the curve on the
graph
Over time (eg a
changeIn the x axis)
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Integrals
We can have a function that produces a copy of the curve of the line on the graph
v = f(x) dx
This is read as ³delta
x´ or change in x(delta means change)
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Integrals
We can also limit our ar ea to between twopoints (a and b, or say 10:00 am and 11:00
amb
v = f(x) dxa
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Traffic Patterns The traffic can vary from instant to instant ±
as calls ar e established (set up) or
terminated (ended)
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Traffic Patterns It can also f luctuate hour ly, often it is:
Less during the night Rises rapidly in the morning (when offices,
shops & factories open for business)
Goes down at lunch time
Falls again in the afternoon between 5 and 7pm as people leave work for the day
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Traffic Patterns And it can have seasonal
f luctuations e.g:
Higher just befor e Easter &
Ch
ristmas
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Traffic PatternsFinally ther e ar e long term
patterns as the traffic grow year by year
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Traffic PatternsHowever of gr eatest inter est to
the traffic planner is the busiesthour of the day (known as the TCBH or Time Constant BusyHour)
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Traffic PatternsBecause if we cater for the
busiest hour, ther e will be sufficient lines available for the quieter times of the day
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Busy Hour Traffic
Call arriving at a switch may arrive:
At random
In mor e or less r egular patterns
In bursts
Separated by periods in which no callsarrive (eg to an ACD (Automatic Call Distributor) or Call Centr e)
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Busy Hour Traffic
Random arrival of calls is typical at a:Subscriber Switching Stage in
Medium to large public exchanges Large PABXs
Calls to an ACD ar e not random asthey ar e usually in r esponse to anevent (eg an advert on TV)
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Busy Hour Traffic
How long each call lasts (knownas a ³holding time´ is asimportant as how many callsarrive each second
The
combination of numbe
r of calls and holding time dictatesthe ³Traffic Intensity´
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Busy Hour Traffic
The ³messages´ sent on atransmission link can include:
Conversations
Data messages Associated control messages
(data)
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Busy Hour Traffic
The ³messages´ holding time´ (itsduration) can vary from a f ewseconds to a whole hour
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Busy Hour Traffic
The distribution (statistically) of these individual holding time ±that is the number of calls of each duration in the hour ± is importantin determining the number of
outside lines (trunks) and the
waiting time in a queue (eg ACD)
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Fundamental
Relationships Our problem is that we can only measur e
the traffic that is actually carried on a link
(known as the Carried Traffic) which hasthe symbol ³´
What we actually want to know is the Offered Traffic (including that which didnot get through) with symbol ³A´
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Fundamental
Relationships The traffic that did not get through
(because ther e wer e insufficient lines) is
known as Blocked Traffic and has the symbol ³´
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Fundamental
Relationships To calculate the percentage of blocked
traffic:
B = (A-Y) / A
Which is (off er ed traffic ± carried traffic) /(off er ed Traffic)
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Fundamental
Relationships And to calculate carried traffic:
Y = A(1-B)
Which is (off er ed traffic) ± (1 ± Blocked
Traffic)
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Fundamental
Relationships And to calculate off er ed traffic (A in Er langs) the
formula is:
A = (nh) /T
Wher e n is number of calls
h is avg holding time of each call (in seconds)
T is the number of seconds in the measur ement period(eg 3600 seconds in an hour)
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Measuring Teletraffic
The Traffic Engineer is primarily inter estedin the:
Average Traffic intensity at various points in the telephone exchange and the
³Dispersion´ of traffic (wher e the callers ar e
dialling)
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Measuring Teletraffic
Measuring the Offered Traffic is difficult (if not impossible)
So we measur e the Carried Traffic anduse it to estimate the Offered Traffic
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Measuring Teletraffic
We do this by first measuring the Traffic
Dispersion (wher e the callers ar e dialling)
This is done by r ecording the first 4,5 or 6digits dialled by the callers and the total calls duration.
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M
easuring TeletrafficWhen the call is completed the telephone
system pr epar es an SMDR r ecord.
This is a text message, wher e details such as extension, called number, duration etcar e in certain field in the text message.
The text message is sent via a serial port
to a PC running a spe
cial
program. The program extracts the information and
places them into a database format.
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Measuring Teletraffic
By running queries on the database it ispossible to compile r eports on the numbers
callers wer e dialling, as well as individual,total, and ther efor e average holding times(call durations) to each of the destinations.
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Measuring Teletraffic
Since it would be uneconomical to provide equipment for every peak every
encounter ed by the telephone system«
The equipment is instead designed to carrythe traffic in the average Time ConstantBusy Hour (TCBH) , which is the twobusiest consecutive 30 minute periods,during the busy season
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Measuring Teletraffic
The busy season is defined as:
T he 4 consecutive busiest weeks in theyear
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System Types
Having identified wher e callers ar e dialling(and for how long) it is then necessary to
make a decision as to what callers whoexperience congestion should do:
Do they wait (queue) or r eceive busy tone
(and be forced to hang up and dial again)
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System Types
Callers that wait (eg when dialling a call centr e) in a delay will be placed in a
F.I.F.O queue (First In First Out)
This ensur es that callers who have beenwaiting the longest, ar e answer ed first
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System Types
The other consideration is the type of switching system being used (eg the type of exchange or PABX
The system can be dir ectly controlled bythe dial impulses (digit dialled) such as in
Ste
p by Ste
p Ex
ch
ange
s Or impulses can be stor ed and analysedlater (as in Crossbar, AXE, Digital PABXs)
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Congestion
In a switching system (eg a PABX) ther e ar e often internal traffic r eports (eg
JUNCTOR r eports) that can be run todetermine whether congestion is occurringbetween shelves or cabinets.
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System Types
The ratio of incoming or outgoing lines, tothe number of devices or people wanting to
access those lines is known as a Grade of Ser vice
To be mor e specific the Grade of Ser vice
the percentage of calls that we expect to
fail in trying to access an incoming or outgoing line.
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System Types
For example a Grade of Service of 0.01 (or 1%) indicates that:
We expect 1 out of every 100 calls toexperience congestion when trying toaccess either an incoming or outgoing line.
This is the ³normal´ public network Grade of Service (GOS)
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System Types
The calls may fail because ther e ar e:
Insufficient incoming or outgoing lines or Insufficient signalling devices (eg r egisters)
or
Insufficie
nt control
e
quipme
nt (or path
) inthe switching system to set up the call or
A combination of all of the above
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Congestion Types
Ther e ar e two types of congestion:
Call Congestion Time Congestion
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Call Congestion
The probability that a call will encounter congestion
Eg 1% of calls ar e expected to r eceive busy tone
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Time Congestion
The proportion of time during which congestion exists:
Eg for 10% of the hour congestion is likelyto exist
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Congestion
When calls arrive at random, and all trunks(lines) in the route ar e fr eely available:
Call Congestion and Time Congestion ar e equal
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Congestion
But in small switching systems (or wher e ther e is limited availability of trunks, eg
when the calls have to get between cabinet/ shelves in the system) then:
Call Congestion & Time Congestion ar e
diff er ent (usually the diff er ence is small butcan be significant)
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Congestion
If the congestion is inside the switchingsystem (eg getting between cabinets /
shelves) adding extra external lines will notcur e the problem (in fact it will make itworse)
So it is important to understand why the
congestion is occurring, and where it isoccurring
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Congestion
In a switching system (eg a PABX) ther e ar e often internal traffic r eports (eg
JUNCTOR r eports) that can be run todetermine whether congestion is occurringbetween shelves or cabinets.
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Congestion
If no detailed r eports ar e available it ispossible to estimate traffic from Peg
Count Reports
Peg Count r eports provide details on the number of times a piece of equipment was
accessed (by then estimating the average holding time, the traffic can be calculated inEr langs)
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Delay Systems
A delay system (eg a Call Centr e) allows callers towait (queue) rather than r eceive busy tone.
The delay depends on The Traffic Intensity
The number of circuits provided
The distribution of holding times (diff er ent queues to the same system may have diff er ent call durations
de
pe
nding on the
natur e
of the
e
nquiry and the
numbe
r of operators (Agents) available to take the calls
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Delay Systems
But if the mean (average) Offered Traffic
does exceed the number of service
devices (eg equipment, trunks etc) then:
The service standard is expr essed in termsof either: Average Delay
Or Percentage of Calls delayed more than
some specified time
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Delay Systems
It is often better to use the ³Percentage of
Calls delayed more than some specified
time´ as this is aff ected mor e by what isknown as the Ser vice Discipline
The Service Discipline r ef ers to the way
calls ar e taken out of the queue andpr esented to Agents (operators) eg FIFO
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Availability
To gain access to trunks (outside lines) or equipment in the exchange it is necessary
to consider ³availability´
Availability is defined as the number of outlets in a switching stage which can be
r eached from its inlets.
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Availability
In an exchange with a single switchingstage (such as Step by Step) the traffic
load has no eff ect on availability, as it isfixed by the mechanical design of the exchange
Likewise if traffic is always sent to a pr e-
assigned route (I.e ther e ar e no alternative path) then traffic load will also not have anyeff ect on availability
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Availability
But modern systems connect to a partial switching stage that is shar ed by others,and ther e can be multiple switching stagesthen:
The actual availability under heavy load may be less than some nominal pr e-set value (eg if it
was designed to carry 99% of traffic it may onlycarry 70% because intermediate switchingstages ar e congested).
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Availability
Traffic ther efor e is carried most efficientlyby group of circuits having full availability
wher e:
Every idle circuit can be r each all the time from all inlets. However it is often
uneconomical to provide full availability(especially in large systems)
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Availability
We ther efor e have to calculate the pr obability that each part of the path will
be available, based on call patterns, trafficloads and traffic intensity to design asystem that will have the r equir ed Grade of Service
And that is wher e Traffic Engineering isused
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Factorial
We first have to determine how many waysour circuit (eg trunk or equipment ports)
can be accessed If we have 5 callers and 5 trunks (outside lines) how many ways ar e ther e of arranging the callers on the available
trunks?
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Factorial
We count each possible way of arrangingthe callers on the 5 trunk as ³n´
The first arrangement would be n1
The 2nd arrangement would be n2
The 3rd arrangement would be n3
The
4th
arrange
me
nt woul
d be
n4 The 5th arrangement would be n5
The total n = n1 n2 n3 n4 n5
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Factorial
So ther e ar e:
5 ways to arrange the 1st caller (on 5
trunks) 4 ways to arrange the 2nd (ther e ar e only 4lines left)
3 ways for the 3rd caller
2 ways for the 4th caller
1 way for the 5th caller
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Factorial
This mean that ther e ar e the followingpossible way of arranging the callers:
n = 5 x 4 x 3 x 2 x 1= 120 ways
We call this a ³factorial´ with the symbol ³!´
known as ³shriek´
So 5! = 120
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Permutation
When we are not worried about the order in which things ar e selected it is a
per mu
tatio
n
A permutation just uses a factorial calculation to calculate the number of ways
ways that things can be arranged but we can choose how many of the ³things we select at a time´
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Permutation
For example if we have 5 caller (Fr ed, Joe, Alice, Frank and Kathy) but only two
trunks,h
ow many ways can the
calle
rs be
arrange on the line, if only to can getthrough at a time?
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Permutation
We say that we take:
n things r at a time
So we take 5 callers (n=5), 2 at a time (r=2)
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Permutation
The formula for a permutation is: n! / (n-r)!
So 5 callers taken 2 at a time is:
5! / (5-2)!
= 120 / (3 x 2 x 1)
= 120/6
= 20 possible ways (permutations) five callers canaccess 2 trunks, two callers at a time
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Permutation
This is written mathematically as:
nPr
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Combination
When we are worried about the order inwhich thing ar e selected (and we do notwant to count thing twice) we use a³combination calculation´
For example if Fr ed accesses Trunk 1 and Joe Trunk 2
It is the same as if Joe accesses Trunk 1 and Fr ed Trunk 2
We still the same two callers accessing 2 lines at the same time, sowe only count it as one way of accessing the trunks rather than 2ways
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Combination
The formula for a combination is: n! / (n-r)! r!
So 5 callers taken 2 at a time is:
5! / (5-2)! 2! = 120 / (3 x 2 x 1) 2x1 = 120/6 x2 = 120 / 12
= 10 possible ways (permutations) five callers canaccess 2 trunks, two callers at a time (each beinga unique arrangement of callers)
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Combination
This is written mathematically as:
nCr or (nr )
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Probability Theory
Probability theory deal with events thatcannot be pr edicted in advance
For example if I have 3 trunks, which one will a telephone call choose if all ar e equally likely to be chosen?
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Probability Theory
We can however list all the possible outcomes
So if ³s´ means the line was chosen and ³f ³means it was not chosen the possible outcomes ar e:
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Probability Theory
Lines s f f
f s f
f f s
The list of all possible outcomes is known as a ³sample space´
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Probability Theory
Lines s f f
f s f
f f s
The list of individual outcomes (eg s f f , are known as samplepoints
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Probability Theory
Lines s f f
f s f
f f s
The sample space is Discrete because the outcomes can be listedone by one, and Finite because there are a finite (limited) number
of sample points
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Events
Events ar e groups of sample points(individual outcomes) that can be
r ecorded on a VENN DIAGRAM
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Events
Consider Event E as ³throw a 1 or 2with an unbiased dice´
E
1 2
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Events
_
Event E (r ead as NOT E) is ³every
other number except 1 or 2´
E
1 2 3 45 6
_
E
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Events
If Event E is ³the call is a toll (STD) call´
And Event B is ³that a call has been
progr ess for 2 minutes´We can ask ³what is the probability thatthe call is an STD call and has been in
progr ess for longer than 2 minutes?´
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Events
If Event E occurs 0.07 (7%) of the time
And Event B occurs 0.09 (9%) of the
time
A and B occur 0.03 (3%) of the time
We draw it as the following:
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Events
E F
0.07 0.090.03
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Events
E F
0.07 0.090.03
This is known as conditional probability, it is the probability that
Event F has occurred given that Event E has also occurred.
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Events
E F
0.07 0.090.03
We represent it as P(E | F)
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EventsE F
0.07 0.090.03
If 3% of calls are
STD and last
longer than 2
minutes
What is the
probability that acall is STD given
it has
been in progress
for longer than 2
minutes
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Events
E F
0.07 0.090.03
We calculate P(E|F) = P(EF) / P(F) where mean ³and´
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Events
E F
0.07 0.090.03
We calculate P(E|F) = 0.03 / 0.03 + 0.09) = 0.25 (or 25%)
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Events
E F
0.07 0.09
If however nocalls were STD
and longer than 2
minutes we
say the Events E
and F aremutually
exclusive (they
have no
sample points incommon)
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Independent Events
To calculate the probability of callsbeing ³blocked´ and not getting through
(to determine the number of lines for aparticular Grade of Service´) usingEr langs Poisson formula ± the events(the call) need to be independent of
each other
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Independent Events
Two events ar e independent if one (sayE) has the same probability of occurring
whether or not another event F hasoccurr ed.
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Independent Events
As an example ± what is the probabilitythat a household with a car, also has a
phone«in other words can we determine how many phones ther e ar e in an ar ea by measuring the number of households that have a car?
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Independent Events
Households in town
Phone No Phone
Car 844 284
No Car 1477 497
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Independent Events
P (Car|Phone) = 844 (Car and phone) /(844 (Car and Phone) + 1477 (No Car
and Phone) = 4/11
P (Car) = (844 (Car and phone) +
284(Car + no phone))/ (844 + 1477 +284 + 497) = 4/11
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Independent Events
Since both probabilities ar e the same (4/11) they ar e said to be independent
of each other
Possessing a car does not make it mor e
or less likely that the household will have a phone
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Probability Distribution
We can graph the individual probabilities of an eventoccurring, we end up with a pr obability distributioncur ve
This can be described mathematically with a functionP(X = x ) wher e X is random variable and x is the value that variable can take
If the value if x can be any random number, known asa continuously random variable (eg time spent ona call), then it is known as a Pr obability DensityFunction
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The story of e
Befor e looking at the Poisson ProbabilityDistribution curve (used by Er lang) we firsthave to look at a curious curve made by
plotting ³e´
³e´ is a number that has the approximate
val
ue
of 2.71828
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The story of e
In 1683 Jacob Bernoulli (a Swissmathematician) got to work in trying tocalculate compound inter est
Imagine that you invest $1 over 1 year, andyou get a massive 100% inter est on your
inve
stme
nt
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The story of e
At the end of the first six month you will have $1.50($1 principal and $0.50 inter est)
But at the next six months you will have $1.50 plusanother $0.75 giving you $2.25
You made an extra 25c just by compounding. Now if
our original investment was 1000 times bigger youwould make an extra 25,000c or $250 than if the inter est was paid just once a year.
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The story of e
³e´ is the amount the $1 would grow to if the compounding took place continuously ± it isknown as exponential growth
It is used to calculate population growth,radioactive decay and much much mor e
e it turns out can be calculated by:
e = 1 + 1/1! + (1 / 2!)) + (1/3!) + (1/4!)«. And equals 2.717828 (approx)
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The story of e
e = 1 + 1/1! + (1 / 2!)) + (1/3!) + (1/4!)«.might look familiar
The 1!..2!«3!«. Ar e all factorials..and couldbe saying how many way ar e ther e to seize agroup of lines 1 at a time, 2 at a time, 3 at atime etc.
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How to calculate
Traffic
An Introduction
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Weighting
Probabilities
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Weighting Probabilities
We first have to decide on how likely anevent is to occur.
Using ³e´ for what is known as³exponential distribution´ we can do the following?
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Weighting Probabilities
For calls of 2 minutes average durationwe can work out how likely calls will be
in 30 second intervals from 0.5 to 3minutes
It is known that call durations ar e
³ex
pone
ntiall
y distribute
d´ (th
ish
asbeen shown by experimentation)
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Weighting Probabilities
We first take the call duration (t) and divide it by the average conversation time ±h (2 minutes)
So it becomes t/h
The formula to find the probability of each call duration t is:
P(T <=t) = 1 ± e-t/h
t (min) (h) min ± t/h P(T<=t)
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avg call duration
0.5 2 0.25 0.221
1.0 2 0.5 0.393
1.5 2 0.75 0.528
2.0 2 1 0.6323.0 2 1.5 0.777
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Weighting Probabilities
We then multiply the weightedprobabilities
So for our pr evious example the weighted probability is:
0.221(0.5) + 0.393(1.0) + 0.525(1.5) +0.632(2.0) + 0.777(3.0) = 4.886 %