Teletraffic

1 Introduction.- 1.1 Features of Teletraffic Systems.- 1.2 Modeling of Teletraffic Systems.- 1.2.1 Traffic Load.- 1.2.2 Call Origination Process.- 1.2.3 Service Time Distribution.- 1.2.4 Classification of Traffic Models.- 1.3 Fundamental Relations.- 1.3.1 Markov Property.- 1.3.2 PASTA.- 1.3.3 Little Formula.- Exercises.- 2 Markovian Models.- 2.1 Markovian Loss Systems.- 2.1.1 M/M/s(0).- 2.1.2 Statistical Equilibrium.- 2.1.3 M(n)/M/s(0).- 2.1.4 State Probabilities at Call Arrival.- 2.1.5 Blocking Probability.- 2.1.6 Ordered Trunk Hunting.- 2.2 Markovian Delay Systems.- 2.2.1 M/M/s.- 2.2.2 Mean Waiting Time.- 2.2.3 Waiting Time Distribution.- 2.3 Extended Markovian Models.- 2.3.1 Birth-Death Process.- 2.3.2 M(n)/M/s(m,?).- 2.3.3 M/M/s(m,?).- 2.3.4 M(n)/M/s.- 2.3.5 M/M/s(m).- Exercises.- 3 Non-Markovian Models.- 3.1 Renewal Process.- 3.1.1 Residual Time Distribution.- 3.1.2 Mean Residual Time.- 3.1.3 Rate Conservation Law.- 3.2 Poisson Input General Service Time Models.- 3.2.1 M/G/1.- 3.2.2 Embedded Markov Chain.- 3.2.3 State Probability at Call Departure.- 3.2.4 Waiting Time Distribution.- 3.2.5 M/G/1(m).- 3.3 Poisson Input Constant Service Time Model.- 3.3.1 M/D/s.- 3.3.2 State Probability Generating Function.- 3.3.3 Mean Waiting Time.- 3.3.4 Waiting Time Distribution.- 3.4 Renewal Input Exponential Server Models.- 3.4.1 GI/M/s(0).- 3.4.2 Blocking Probability.- 3.4.3 GI/M/s.- 3.4.4 Waiting Time Distribution.- 3.4.5 Derivation of Boundary Probability.- 3.5 Renewal Input Single Server Models.- 3.5.1 Spectral Solution for GI/G/1.- 3.5.2 H2/G/1.- 3.5.3 Ek/G/1.- Exercises.- 4 Multi-Class Input Models.- 4.1 Batch Arrival Models.- 4.1.1 M[X]/M/s(0).- 4.1.2 M[X]/G/1.- 4.1.3 GI[X]/G/1.- 4.2 Priority Models.- 4.2.1 Non-Preemptive Priority Model.- 4.2.2 Preemptive Priority Model.- 4.2.3 Work Conservation Law.- 4.3 Multi-Dimensional Traffic Models.- 4.3.1 Multi-Dimensional Traffic.- 4.3.2 Trunk Reservation System.- 4.3.3 Individual Performance Measures.- 4.4 Mixed Loss and Delay Systems.- 4.4.1 M1+M2/M/s(0,?).- 4.4.2 GI+M/M/s(?,0).- 4.4.3 GI+M/M/s(0,?).- 4.5 Multi-Queue Models.- 4.5.1 Modeling of Multi-Queue.- 4.5.2 Exhaustive Model.- 4.5.3 Gated Model.- 4.5.4 Limited Model.- Exercises.- 5 Alternative Routing Systems.- 5.1 Overflow System.- 5.1.1 Overflow Traffic Model.- 5.1.2 Moment and LST of Overflow Process.- 5.1.3 Mean and Variance of Overflow Calls.- 5.2 Approximate Methods for Overflow Systems.- 5.2.1 Equivalent Random Theory.- 5.2.2 Interrupted Poisson Process.- 5.2.3 GI Approximation.- 5.3 Optimum Design of Alternative Routing.- 5.3.1 Conventional Method.- 5.3.2 Given Blocking Probability.- 5.3.3 Trunk Reservation Scheme.- 5.4 Numerical Analysis of State Equations.- 5.4.1 State Equation for Alternative Routing.- 5.4.2 Gauss-Seidel Iteration.- 5.4.3 Solution for Trunk Reservation Scheme.- Exercises.- 6 Advanced Teletraffic Models.- 6.1 Renewal Input Multi-Server Model.- 6.1.1 Diffusion Model.- 6.1.2 Modified Diffusion Approximation.- 6.1.3 GI/G/s.- 6.2 PH-MRP Input Models.- 6.2.1 Phase-Type Markov Renewal Process.- 6.2.2 PH-MRP/M/s(m).- 6.2.3 M$$
M{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{ + }}PH - MRP{\text{ }}/{\text{ }}{{M}_{1}},{{M}_{2}}{\text{ }}/{\text{ }}s(\infty ,0)PPP(n)
$$.- 6.2.4 Special Cases.- 6.3 MMPP Input Model.- 6.3.1 Markov Modulated Poisson Process.- 6.3.2 MMPP/G/1.- 6.3.3 Mean Waiting Times.- 6.4 Statistical Packet Multiplexer.- 6.4.1 Modeling of Packetized Process.- 6.4.2 Indices of Dispersion and Skewness.- 6.4.3 MMPP Approximation.- 6.4.4 MMPP + M/G/1.- 6.5 Local Area Networks.- 6.5.1 Token Ring LAN.- 6.5.2 Approximation for Asymmetric System.- 6.5.3 CSMA/CD LAN.- 6.5.4 Mean Access Time.- Exercises.- 7 Traffic Simulation.- 7.1 Introduction.- 7.2 Methods of Simulation.- 7.2.1 Problem Formulation and Modeling.- 7.2.2 Programming for Simulation.- 7.2.3 Simulation Language, GPSS.- 7.2.4 General Purpose Programming Languages.- 7.3 Generation of Random Numbers.- 7.3.1 Random Number Generation Methods.- 7.3.2 Uniform Random Number.- 7.3.3 Arbitrary Random Numbers.- 7.4 Analysis of Output Result.- 7.4.1 Confidence Interval.- 7.4.2 Replication Method.- 7.4.3 Batch Mean Method.- 7.4.4 Regenerative Method.- Exercises.- Appendices.- A Basic Teletraffic Formulas.- A.1 Notation.- A.2 Markovian Models.- A.3 Poisson Input Non-Markovian Models.- A.4 Renewal Input Non-Markovian Models.- A.5 Non-Renewal Input Models.- B Programs for Erlang B Formula.- B.1 Integer Number of Servers.- B.2 Real Number of Servers.- B.3 Derivatives of Erlang B Formula.- B.4 BASIC Program Lists.- C Basis of Probability Theory.- C.1 Events and Probability.- C.1.1 Events.- C.1.2 Probability.- C.1.3 Conditional Probability.- C.2 Distribution Functions.- C.2.1 Random Variables.- C.2.2 Joint Distribution Function.- C.2.3 Convolution.- C.3 Expectation and Transforms.- C.3.1 Expectation.- C.3.2 Laplace-Stieltjes Transform.- C.3.3 Probability Generating Function.- C.4 Examples of Probability Distributions.- C.4.1 Probability Distribution.- C.4.2 Discrete Distributions.- C.4.3 Continuous Distributions.- C.5 Stochastic Processes.- C.5.1 Markov Process.- C.5.2 Discrete-Time Markov Chain.- C.5.3 Continuous-Time Markov Chain.- C.5.4 Birth-Death Process.- D Solutions to Exercises.