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TECHCOM
Consulting
Fundamentals of Traffic TheoryFundamentals of Traffic Theory
General remarks to traffic theory:
The traffic theory in general uses mathematical models to describe and to optimise traffic systems.In telecommunication traffic theory (also called teletraffic theory) it is a telecommunication system
which is considered with the help of appropriate mathematical models. Since real systems are quite
complex systems, simplifications and assumptions have to be performed to not deal with too
complicated and sophisticated mathematics. Later on these assumptions and simplifications have tobe justified.
Since a bad model can never lead to good results, the problem is to find a good and easy modelto get reliable results. Some mathematical ideas, models and formulas which are used in traffic
theory are presented now on the following pages.
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TECHCOM
Consulting
Definitions and TermsDefinitions and Terms
General:
Each telephone system must be dimensioned in such a way that even during periods of high traffic(offered), the subscribers still have a good chance of success in making calls. Those subscriberswho do not succeed in making a call will either be lost (in a pure lost-call telephone system) orthe calls will be delayed (in a waiting-call telephone system). Usually, real telephone systems arecombined lost-/ waiting-call systems.
Even during the so called busy hour the percentage of non successful subscribers should notexceed a predefined value. This means for the network operator that the dimensioning of his
telephone system must be driven on the one hand by guaranteeing some Quality of service (QOS)and on the other hand by economical aspects.
From economical point of view, the amount of necessary equipment (switches, base stations,
multiplexers, cross-connectors, ...) and also the number of links between this equipment should be
kept to a minimum.
From QOS point of view, the more trunks are offered by the telephone system, the higher theprobability for the subscribers to succeed in making calls.
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TECHCOM
Consulting
Definitions and TermsDefinitions and Terms
Traffic OfferedTraffic Offered Traffic CarriedTraffic Carried
TrafficTraffic LostLost
TelephoneTelephone systemsystem::
n =n = number ofnumber of
devicesdevices ((trunkstrunks))
KK
KK
KK
KK
KK
KK
KK
LL
JJJJ
JJJJ
JJJJ
N =N = number ofnumber of
traffic sourcestraffic sources
KKKK
KKJJ
JJJJ
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TECHCOM
Consulting
Definitions and TermsDefinitions and Terms
Traffic flow units:
In honour of A. K. Erlang (1878-1929), a Danish mathematician who was the founder of traffictheory, the unit of the traffic flow (or traffic intensity) is called Erlang (Erl).
The traffic flow is a measure of the size of the traffic. Although the traffic flow is a dimensionless
quantity, the Erlang was assigned as unit of the traffic flow in traffic theory.
By definition:
1 trunk occupied for a duration t of a considered period T carries t / T Erlang.
From this definition it follows already that the traffic carried in Erlang cannot exceed the number
of trunks.
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TECHCOM
Consulting
Definitions and TermsDefinitions and Terms
Several definitions can be given for the traffic flow:
Especially for traffic measurements it is useful to consider the traffic flow as averaged number of
trunks which are occupied (busy) during a specified time period:
Traffic intensity = Mean number of busy trunks in a time period
If this is a long time period, ongoing calls at the beginning and at the end of this period can be
neglected. The traffic flow then can be considered as call intensity (number of trunk occupations per
time unit) times the mean holding time (which is the average holding time per trunk occupation):
Traffic intensity = Call intensity x Mean holding time
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TECHCOM
Consulting
Erlang-B FormulaErlang-B Formula
Assumptions:
Pure loss system Infinite number of traffic sources
Finite number of devices (trunks) n
Full availability of all trunks
Exponentially distributed holding times
Constant call intensity, independent of the number of occupations
Time and call congestion are equal:
This formula is called Erlang`s formula of the first kind (or also Erlang loss formula or ErlangB formula).
=
===n
i
i
n
n
i
A
n
A
AEBE
0
,1
!
!)(
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TECHCOM
Consulting
Erlang-B Look-up TableErlang-B Look-up Table
The Erlang B formula describes the congestion as function of the Traffic Offered and the numberof available trunks.
In real life the situation is mostly different. People often want to calculate the number of neededtrunks for a certain amount of traffic offered and a maximum defined congestion.
That means the Erlang B formula must be rearranged:
n = function of (B and A)
This rearrangement cannot be done analytically but only numerically and will be performed mosteasily with the help of a computer. Another possibility is the usage of special tables, namely so
called Erlang B look-up tables. On the following page an example of such an Erlang B look-up
table is presented.
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TECHCOM
Consulting
Erlang-B Look-up TableErlang-B Look-up Table
0.080.471.061.752.503.304.145.005.886.787.698.619.5410.4811.43
12.3913.3514.3215.2916.2717.2518.2419.2320.2221.21
0.050.380.901.532.222.963.744.545.376.227.087.958.849.3710.63
11.5412.4613.3914.3115.2516.1917.1318.0819.0319.99
0.030.280.721.261.882.543.253.994.755.536.337.147.978.809.65
10.5111.3712.2413.1114.0014.8915.7816.6817.5818.48
0.010.150.460.871.361.912.503.133.784.465.165.886.617.358.11
8.889.6510.4411.2312.0312.8413.6514.4715.2916.13
123456789101112131415
16171819202122232425
Offered Traffic Afor
B=E=0.07
7 % blocking)
Offered Traffic Afor
B=E=0.05
5 % blocking)
Offered Traffic Afor
B=E=0.03
(3 % blocking)
Offered Traffic Afor
B=E=0.01
(1 % blocking)
Number oftrunks n
Erlang B look-up table for an infinite number of traffic sources and full availability:
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TECHCOM
Consulting
Erlang-B Look-up TableErlang-B Look-up Table
22.2123.2124.2225.2226.2327.2428.2529.2630.2831.2932.3133.3334.3535.3736.4037.4238.4539.4740.5041.5342.5643.5944.6245.6546.69
20.9421.9022.8723.8324.8025.7726.7527.7228.7029.6830.6631.6432.6233.6134.6035.5836.5737.5738.5639.5540.5441.5442.5443.5344.53
19.3920.3121.2222.1423.0623.9924.9125.8426.7827.7128.6529.5930.5331.4732.4133.3634.3035.2536.2037.1738.1139.0640.0240.9841.93
16.9617.8018.6419.4920.3421.1922.0522.9123.7724.6425.5126.3827.2528.1329.0129.8930.7731.6632.5433.4334.3235.2236.1137.0037.90
26272829303132333435363738394041424344454647484950
Offered Traffic Afor
B=E=0.07
7 % blocking)
Offered Traffic Afor
B=E=0.05
5 % blocking)
Offered Traffic Afor
B=E=0.03
(3 % blocking)
Offered Traffic Afor
B=E=0.01
(1 % blocking)
Number oftrunks n
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TECHCOM
Consulting
Erlang-C FormulaErlang-C Formula
Assumptions:
Pure delay system Infinite number of traffic sources N
Finite number of devices (trunks) n
Full availability of all trunks
Exponentially distributed inter-arrival times between calls which corresponds to a
constant call intensity y, i.e. the probability of a new offered call is the same at all time
points, independent of the number of occupations
Exponentially distributed holding times (s)
Time congestion is defined as the probability that all devices are used:
This formula is called Erlang`s formula of the second kind (or Erlang delay formula or ErlangC formula).
= +
==1
0
,2
!!
!)(n
i
ni
n
n
An
n
n
A
i
A
An
n
n
A
AEE
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TECHCOM
Consulting
Erlang-C FormulaErlang-C Formula
Call congestion is defined as the probability that a call has to wait:
The traffic carried and traffic offered are:The mean number of waiting calls is:
The mean waiting time for calls, which have to wait is:
The mean waiting time for all the calls is:
The waiting time distribution depends on the queue discipline, whereas the mean waiting time is in
general independent of the queue discipline.
EB =
An
stwait
=
An
s
An
n
n
A
i
A
An
n
n
A
Tn
i
ni
n
wait
+
=
=
1
0 !!
!
An
A
An
n
n
A
i
A
An
n
n
A
Nn
i
ni
n
wait
+
=
=
1
0 !!
!
syAAAofferedcarried
===
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TECHCOM
Consulting
Trunking GainTrunking Gain
Exercise: Use the Erlang B look-up table to find out the meaning of trunking gain:Which traffic offered can be handled by an Erlang B system assuming 32 trunks
and 1 % blocking?
Which traffic offered can be handled by 2 Erlang B systems for each assuming
16 trunks and 1 % blocking?
Which traffic offered can be handled by 4 Erlang B systems for each of them
assuming 8 trunks and 1 % blocking?
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TECHCOM
Consulting
Traffic DistributionTraffic Distribution
Time Dependency:
The traffic in a telecommunication network as a function of time will not be constant but will showsignificant fluctuations. Variations of the traffic during a single day, from day to day, for differentweekdays, or even for different seasons can be observed. Also on a long time scale the averaged
traffic will not remain constant but will increase in most telecommunication networks.
0 12 24 hours
0 %
100 %
50 %
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TECHCOM
Consulting
Traffic DistributionTraffic Distribution
Location Dependency:
The traffic in a telecommunication network will not be location independent but will showsignificant location dependencies. For example, in rural areas there will be less traffic compared tocity areas.
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TECHCOM
Consulting
Traffic ForecastingTraffic Forecasting
An important aspect in dimensioning a telecommunication network is the expected traffic in the
future. Therefore, an analysis of the expected traffic is of great interest. Even in case that the
penetration (number of traffic sources) saturates, the amount of traffic does not necessarilysaturates too. Traffic forecasts are not easy and may be influenced by many aspects: e.g. price
politics, offered services,
The more the important dependencies are realised and taken into account, the more precise the
forecasts will be.
For a detailed analysis it is useful to:
Split the total PLMN into subareas
Categorise the subscribers: e.g. into business, residential,
Analyse: e.g. the number of subscribers per area, the development of the penetration
depth, the expected penetration depth
Analyse also economic dependencies like e.g. any correlation between the demand oftelephone service and e.g. the economic activities in a special region, the economic
situation in general (measured e.g. by the economic growth), the income of the people,
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TECHCOM
Consulting
Traffic MeasurementsTraffic Measurements
It is of great interest for the network operator to measure the real traffic situation in his network.
To perform such measurements, in former telecommunication systems special traffic
measurement equipment (e.g. the so called electromechanical meter) was needed. Since in themeantime most telecommunication systems are digital, this kind of equipment is not needed any
more: The call and device concerning data are stored in the memory of the system processor. It isonly a question of software to read them out.
The traffic measurements are usually part of the so called Performance Data Measurements.
Performance Data Measurements can be run continuously, periodically or sporadically, for a long
time or a short time, observing smaller or greater parts of the network.
Concerning the traffic measurements, either special events are counted (e.g. the number of
successful calls, the number of lost calls, ...) or special time intervals are recorded (e.g. holdingtimes, waiting times,...).
The corresponding counters could in principle be actualised continuously during the observationperiod, but mostly a scanning method is used. Scanning method means that the system counts
the number of events not continuously but only at particular times. This leads to some uncertaintyfor the measurement results. Nevertheless, the error performed can be estimated using statistical
methods. In general, the smaller the scanning interval the higher the precision of the
measurement. Typical scanning intervals are 100 ms or 500 ms.
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TECHCOM
Consulting
Dimensioning TRXsDimensioning TRXs
The dimensioning of the number of TRXs per cell should be based on traffic estimations for this
area and should be performed for the busy hour.
Using:
the number of subscribers in the corresponding area (for the busy hour)
the expected averaged traffic per subscriber (for the busy hour)
the offered traffic A results from:
A = No of subscribers x traffic load per subscriber
Using the Erlang B look up table the number of TRXs can be derived.
Hint:
This number also depends on the amount of half rate being used in the cell
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TECHCOM
Consulting
Dimensioning Control ChannelsDimensioning Control Channels
Initial signaling sequence for MTC, MOC, LOCUPD, SMS,...
Paging Request (PCH)
Channel Request (RACH)
Immediate Assignment (AGCH)
SDCCH-Signaling
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TECHCOM
Consulting
Dimensioning Control ChannelsDimensioning Control Channels
combined BCCH: includes 4 SDCCH subchannels
uncombined BCCH: requires additional SDCCH timeslot(each one containing 8 SDCCH subchannels)
SACCH multiframe (containing 2 BCCH multiframes)
S
11
S
1
F
10
F
0
BCCH
2 3 4 5
CCCH
6 7 8 9
CCCH
1 2 1 3 1 4 1 5
CCCH
1 6 1 7 1 8 1 9
S
21
S
31
F
30
F
20
SDCCH0
2 2 2 3 2 4 2 5
SDCCH1
2 6 2 7 2 8 2 9
SDCCH2
3 2 3 3 3 4 3 5
SDCCH3
3 6 3 7 3 8 3 9
S
41
I
50
F
40
SACCH0
42 43 4 4 4 5
SACCH1
46 47 48 4 9
1.
1.
S11
S1
F10
F0
BCCH2 3 4 5
CCCH6 7 8 9
CCCH1 2 1 3 1 4 1 5
CCCH1 6 1 7 1 8 1 9
S21
S31
F30
F20
SDCCH02 2 2 3 2 4 2 5
SDCCH12 6 2 7 2 8 2 9
SDCCH23 2 3 3 3 4 3 5
SDCCH33 6 3 7 3 8 3 9
S41
I50
F40
SACCH242 43 4 4 4 5
SACCH346 47 48 4 92.
uplink R = RACH + SDCCH / 4
downlink BCCH + CCCH + 4 SDCCH / 4, F = FCCH, S = SCH
R
5
R
4
SDCCH3
0 1 2 3
SACCH2
6 7 8 9
SACCH3
1011 1213
R
15
R
14
R
17
R
16
R
19
R
18
R
21
R
20
R
23
R
22
R
25
R
24
R
27
R
26
R
29
R
28
R
31
R
30
R
33
R
32
R
36
R
35
R
34
SDCCH0
3 7 3 8 3 9 4 0
SDCCH1
4 1 4 2 4 3 4 4
R
46
R
45
SDCCH2
4 7 4 8 4 9 5 0
2.R5
R4
SDCCH30 1 2 3
SACCH06 7 8 9
SACCH11011 1213
R15
R14
R17
R16
R19
R18
R21
R20
R23
R22
R25
R24
R27
R26
R29
R28
R31
R30
R33
R32
R36
R35
R34
SDCCH03 7 3 8 3 9 4 0
SDCCH14 1 4 2 4 3 4 4
R46
R45
SDCCH24 7 4 8 4 9 5 0
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TECHCOM
Consulting
Dimensioning Control and Traffic ChannelsDimensioning Control and Traffic Channels
Exercise:
Consider 1 BTS with 2 TRXs and full rate channels. Assume 1% blocking.Assume a typical TCH load of 25 mErl per subscriber per hour.
Furthermore, assume a typical SDCCH load of 10 mErl per subscriber per hour.
Compare configurations A and B: Which one offers the higher capacity?
BTS
Combined BCCH15 TCH full
Configuration A
TRX-0
TRX-1
BTS
Uncombined BCCH1 SDCCH/814 TCH full
Configuration B
TRX-0
TRX-1
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TECHCOM
Consulting
Dimensioning Control and Traffic ChannelsDimensioning Control and Traffic Channels
Total blocking probability:
SDCCH
TCH
A*
Y
B1 (A*+A) B2 (1-B1)A
Y=(1-B1)(1-B2)A
A
Y*=A*(1-B1)
B1 (A*+A)+B2(1-B1)A
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TECHCOM
Consulting
Capacity and Cell RadiusCapacity and Cell Radius
In capacity limited areas of the network:
Cell radius is smaller than would be for coverage limited situation to satisfy the traffic demand.
Coverage limited area
Capacity limited area
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TECHCOM
Consulting
Dimensioning terrestrial interfacesDimensioning terrestrial interfaces
Um
Abis Asub/Ater A
LAPD for O&MLAPD for TRX
LAPDfor O&M
CCSS7
Base
Station
ControllerTranscoder MSC
SGSN
Gb
BSSGP
MS
Signalling links: overview
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TECHCOM
Consulting
Dimensioning terrestrial interfacesDimensioning terrestrial interfaces
LAPD signalling links:
O&M signalling for BTSM: LPDLMTRX signalling: LPDLR
Signalling for TRAU: LPDLS
Rules of thumb:
1. LPDLM and LPDLR are counted as ONE LAPD-link
2. In case of 1 or 2 TRX 16 kbit/s are sufficient for LPDLM+LPDLR
3. Otherwise 64 kbit/s are required
4. LPDLS always uses 64 kbit/s
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TECHCOM
Consulting
Dimensioning terrestrial interfacesDimensioning terrestrial interfaces
Processor capacity:
Traditional BSC: One PPLD processor handles up to 8 LAPD linksOne PPCC processor handles up to 4 CCSS7 links
High cap. BSC: Two PPXX processors handle 248 signalling linksload sharing (LAPD and CCSS7)
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TECHCOM
Consulting
Dimensioning terrestrial interfacesDimensioning terrestrial interfaces
BTSE0
TRX-0-0TRX-0-1TRX-0-2TRX-1-0
BTSE1
TRX-2-0TRX-2-1
BTSE2
TRX-3-0TRX-3-1
Um
Signaling channel on Um
Traffic channel
LAPD = LPDLM + LPDLR
empty
31
Abis
5 1 5 1 5 1 5 14 0 4 4 0 4
7 3 7 3 7 3 7 36 2 6 2 6 2 6 2
5 1 5 1 5 1 5 14 4 0 4 0 4
7 3 7 3 7 3 7 36 2 6 2 6 2 6 2
LAPD
LAPD
FAW BSC
7 6 5 4 3 2 1 7 6 5 4 3 2 1 7 6 5 4 3 2 1
TRX-3-0 TRX-2-0 TRX-0-0
7 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0
TRX-3-1 TRX-2-1 TRX-0-1
7 6 5 4 3 2 1 0
TRX-0-1
7 6 5 4 3 2 1
TRX-1-0
1817
16 14
LAPD
10 2 01
TRX-3-1
TRX-3-0
TRX-2-1
TRX-2-0
TRX-1-0
TRX-0-2
TRX-0-1
TRX-0-0
LAPD signalling on Abis
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TECHCOM
Consulting
Dimensioning terrestrial interfacesDimensioning terrestrial interfaces
01630
31
TRAUBSC
01630
31
TRA
U
MSC
Asub
A
empty
CCSS7
LPDLS
OMAL
CCSS7
FAW
FAW
OMAL
LPDLS
LAPD signalling on Asub
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TECHCOM
Consulting
Dimensioning terrestrial interfacesDimensioning terrestrial interfaces
CCSS7 dimensioning:
Traffic Model with following assumptions for busy hour (standard subscriber):
2%SMS
0.025Erlang per subscr.
1 (65% MOC; 33% MTC; 2% MtMC)Busy Hour Call Attempts
0.7Attach/Detach
1Location registration/updates
0.6Handovers
AssumptionParameter
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TECHCOM
Consulting
Dimensioning terrestrial interfacesDimensioning terrestrial interfaces
CCSS7 dimensioning:
Traffic Model with following assumptions for busy hour (highly mobile subscriber):
2%SMS
0.025Erlang per subscr.
1 (65% MOC; 33% MTC; 2% MtMC)Busy Hour Call Attempts
0.9Attach/Detach
3.5Location registration/updates
1.6Handovers
AssumptionParameter
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TECHCOM
Consulting
Dimensioning terrestrial interfacesDimensioning terrestrial interfaces
SS7 Signalling load per subscriber
600 byteStandard profile
1100 byteHigh mobility profile
Signalling LoadProfile
Signalling load per BSC
Number of subscr. = Traffic capacity / Traffic per subscriber
Example: 2000 Erlang / (0.025 mErlang/subscriber) = 80 000 subscribers
Total Signalling load = Number of subscribers * Signalling load per subscriber
Example 1: 80 000 subscribers * 600 byte / 3600 sec = 13.3 kbyte/sec
Example 2: 80 000 subscribers * 1100 byte / 3600 sec = 24.5 kbyte/sec
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TECHCOM
Consulting
Dimensioning terrestrial interfacesDimensioning terrestrial interfaces
CCSS7 links per BSC
Example 1:
Required CCSS7 link capacity: 13.4 kbyte/sec
CCSS7 link single capacity 64 kbit/s = 8 kbyte/sec
Consequence: 2 CCSS7 links required
Example 2:
Required CCSS7 link capacity: 24.5 kbyte/sec
CCSS7 link single capacity 64 kbit/s = 8 kbyte/sec
Consequence: 4 CCSS7 links required
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TECHCOM
Consulting
Dimensioning terrestrial interfacesDimensioning terrestrial interfaces
CCSS7 dimensioning: Message transfer delay
Message transfer
delay
Load on link
(in Erlang)0.4 0.8
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TECHCOM
Consulting
Dimensioning terrestrial interfacesDimensioning terrestrial interfaces
01630
31
TRAUBSC
01630
31
TRA
U
MSC
Asub
A
empty
CCSS7
LPDLS
OMAL
CCSS7
FAW
FAW
OMAL
LPDLS
CCSS7 signalling on A
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TECHCOM
Consulting
Dimensioning terrestrial interfacesDimensioning terrestrial interfaces
Transcoder capacity: depends on cell sizes
13151275.6291201220240
12133633.434540640240
32502.5071001100100
Trans-
coders
Erlang /
BSC
Erlang / cell
(1% blocking)
TCH /
cell
SitesTRX /
cell
Cells /
BSC
TRX /
BSC
Assumption: 120 traffic channels per Transcoder
Blocking of A interface has to be taken into account
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TECHCOM
Consulting
ExercisesExercises
1) Consider a call rate of 1000 calls per hour. The mean holding time is 90 sec. What is the
Traffic Offered in Erlang?
2) Consider a Traffic Offered of 30 Erlang and a mean holding time of 120 sec. How many callsper hour do you expect?
3) Consider a telephone system with N=6 trunks and a time period of 10 time units (0,1,...,10).
Subscriber 1 makes a call from t1 to t3. Subscriber 2 makes a call from t2 to t4.
Subscriber 3 makes a call from t3 to t7. Subscriber 4 makes a call from t4 to t8.
Subscriber 5 makes a call from t4 to t9. Subscriber 6 makes a call from t5 to t9.
Subscriber 7 makes a call from t6 to t8. Subscriber 8 makes a call from t7 to t10.
a) Draw the number of used trunks as function of time.
b) Draw the number p of simultaneous occupations in the trunk group as function of
the total time with exactly p occupations.c) What is the traffic offered in Erlang?
d) What is the traffic carried in Erlang?
e) What is the lost traffic in Erlang?
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Consulting
ExercisesExercises
4) Consider a pure delay system and a group of 10 trunks belonging to a trunk group. Assume
that all these trunks are available (full availability). Assume a traffic offered of 4 Erlangs and
a mean holding time of 100 seconds. The queue discipline shall be first come, first served(ordered queue).
a) What is the probability to be queued?
b) What is the mean waiting time of queued calls?
c) What is the mean waiting time of offered calls?
d) What is the probability that call are queued for longer than 1 minute?
5) Consider a pure loss system and a group of 10 trunks belonging to a trunk group. Assumefull availability. What is the traffic in Erlangs which can be offered to this system if the
probability to be blocked should be maximum 1%, 3%, 5% and 7% ?
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Consulting
ExerciseExercise
Nominal Cell Plan
Consider a suburban area (clutter correction term = 5 dB) of 1000 km
2
with an expected traffic of20 Erlang/km2. The standard deviation 1 sigma in building was measured as 9 dB. The planningtarget was decided as 99% cell area probability.
Consider also an adjacent rural area (dense forest, clutter correction term = 9 dB) of 5000 km 2
with an expected traffic of 1 Erlang/km2. The standard deviation 1 sigma for outdoor coverage was
measured as 6 dB. The planning target was decided as 95% cell edge probability.
The blocking rate for both areas was defined as maximum 1%.
Assume that in total 60 RF channels are available. Assume also a typical antenna height of 30 m,
a C/I>21dB being required for the BCCH and a C/I>15 dB being required for the TCH. No tower
mounted amplifier is used. The antenna gain is 15 dBi / 17 dBi for 900 / 1800 MHz.
Assume 1 SDCCH is required for up to 2 TRX per cell, 2 SDCCH are required for up to 4 TRX per
cell, 3 SDCCH are required for up to 6 TRX per cell and 4 SDCCH in further cases.
How many sites are needed for a 900 / 1800 MHz system in case frequency hopping is used / notused?