TELETRAFFIC ENGINEERING
HANDBOOK
ITUD SG 2/16 & ITCDraft 2001-06-20
Contact:Villy B. IversenCOM Center
Technical University of DenmarkBuilding 343, DK-2800 Lyngby
Tlf.: 4525 3648 Fax.: 4593 0355Email: [email protected]
www.tele.dtu.dk/teletraffic
June 20, 2001
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NOTATIONS
Page/Formulaa Carried traffic pr. source or per channelA Offered traffic = Ao ??Ac Carried traffic = Y ??A` Lost traffic ??B Call congestion ??B Burstiness (??)c ConstantC Traffic congestion = load congestion ??Cn Catalans number ??d Slot size in multi-rate trafficD Probability of delay or ??
Deterministic arrival or service process ??E Time congestion ??E1,n(A) = E1 Erlangs Bformula = Erlangs 1. formula 99E2,n(A) = E2 Erlangs Cformula = Erlangs 2. formula 194F Improvement function ??, 199g Number of groups ??h Constant time interval or service timeH(k) PalmJacobus formula ??I Inverse time congestion I = 1/EJ(z) Modified Besselfunction of order ??k Accessibility = hunting capacity ??
Maximum number of customers in a queueing system ??K Number of links in a telecommuncation network or
number of nodes in a queueing networkL Mean queue length 197Lk Mean queue length when the queue is greater than zero 197L Stochastic variable for queue length 197m Mean value (average) = m1 ??mi ith (non-central) moment ??mi ith centrale moment ??mr Mean residual life time (3.13)M Poisson arrival process ??n Number of servers (channels)N Number of traffic streams or traffic typesp(i) State probabilities, time averagesp{i, t | j, t0} Probability for state i at time t given state j at time t0 ??P (i) Cumulated state probabilities P (i) =
ix= p(x)
q(i) Relative (non normalised) state probabilitiesQ(i) Cumulated values of q(i): Q(i) =
ix= q(x)
Q Normalisation constant
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Page/Formular Reservation parameter (trunk reservation)R Mean response time ??s Mean service timeS Number of traffic sources ??, ??t Time instantT Stochastisc variable for time instantU Load function ??v VarianceV Virtuel waiting time ??w Mean waiting time for delayed customers ??W Mean waiting time for all customers ??W Stochastisk variable for waiting time 193y Arrival rate. Poissonprocess: y = ??Y Carried traffic ??Z Peakedness ??
Offered traffic per source (8.9) Offered traffic per idle source (8.3) Palms form factor (3.10) Lagrange-multiplicator 186(s) Laplace/Stieltjes transform (??)i ith cumulant ?? Arrival rate of a Poisson process ?? Total arrival rate to a system Death rate, inverse mean service time 94(i) State probabilities, customer mean values ??% Service ratio ??2 Variance, = standard deviation ?? Time-out constant or constant time-interval ??
Contents
1 Introduction to Teletraffic Engineering 1
1.1 Modelling of telecommunication systems . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 System structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 The Operational Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Statistical properties of traffic . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.4 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Conventional Telephone Systems . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 System structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 User behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.3 Operation Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Communication Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 The telephone network . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.2 Data networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.3 Local Area Networks LAN . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.4 Internet and IP networks . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Mobile Communication Systems . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.1 Cellular systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.2 Third generation cellular systems . . . . . . . . . . . . . . . . . . . . . 16
1.5 The International Organisation of Telephony . . . . . . . . . . . . . . . . . . 16
1.6 ITU-T recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Traffic concepts and variations 17
2.1 The concept of traffic and the unit erlang . . . . . . . . . . . . . . . . . . . 17
2.2 Traffic variations and the concept busy hour . . . . . . . . . . . . . . . . . . . 20
2.3 The blocking concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Traffic generation and subscribers reaction . . . . . . . . . . . . . . . . . . . . 27
3 Probability Theory and Statistics 35
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3.1 Distribution functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.1 Characterisation of distributions . . . . . . . . . . . . . . . . . . . . . 36
3.1.2 Residual lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.3 Load from holding times of duration less than x . . . . . . . . . . . . . 40
3.1.4 Forward recurrence time . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Combination of stochastic variables . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.1 Stochastic variables in series . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.2 Stochastic variables in parallel . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Stochastic sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4 Time Interval Distributions 49
4.1 Exponential distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.1 Minimum of k exponentially distributed stochastic variables . . . . . . 51
4.1.2 Combination of exponential distributions . . . . . . . . . . . . . . . . 51
4.2 Steep distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 Flat distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3.1 Hyper-exponential distribution . . . . . . . . . . . . . . . . . . . . . . 55
4.4 Cox distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4.1 Polynomial trial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4.2 Decomposition principles . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4.3 Importance of Cox distribution . . . . . . . . . . . . . . . . . . . . . . 61
4.5 Other time distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.6 Observations of lifetime distribution . . . . . . . . . . . . . . . . . . . . . . . 63
5 Arrival Processes 65
5.1 Description of point processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.1.1 Basic properties of number representation . . . . . . . . . . . . . . . . 67
5.1.2 Basic properties of interval representation . . . . . . . . . . . . . . . . 68
5.2 Characteristics of point process . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2.1 Stationarity (Time homogeneity) . . . . . . . . . . . . . . . . . . . . . 70
5.2.2 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2.3 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.3 Littles theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6 The Poisson process 75
6.1 Characteristics of the Poisson process . . . . . . . . . . . . . . . . . . . . . . 75
6.2 The distributions of the Poisson process . . . . . . . . . . . . . . . . . . . . . 76
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6.2.1 Exponential distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2.2 The Erlangk distribution . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2.3 The Poisson distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2.4 Static derivation of the distributions of the Poisson process . . . . . . 83
6.3 Properties of the Poisson process . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.3.1 Palms theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.3.2 Raikovs theorem (Splitting theorem) . . . . . . . . . . . . . . . . . . 87
6.3.3 Uniform distribution - a conditional property . . . . . . . . . . . . . . 87
6.4 Generalisation of the stationary Poisson process . . . . . . . . . . . . . . . . . 88
6.4.1 Interrupted Poisson process (IPP) . . . . . . . . . . . . . . . . . . . . 88
7 Erlangs loss system, the Bformula 93
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.2 Poisson distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.2.1 State transition diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.2.2 Derivation of state probabilities . . . . . . . . . . . . . . . . . . . . . . 96
7.2.3 Traffic characteristics of the Poisson distribution . . . . . . . . . . . . 97
7.3 Truncated Poisson distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.3.1 State probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.3.2 Traffic characteristics of Erlangs B-formula . . . . . . . . . . . . . . . 99
7.4 Standard procedures for state transition diagrams . . . . . . . . . . . . . . . . 105
7.4.1 Evaluation of Erlangs B-formula . . . . . . . . . . . . . . .
