Maciej Stasiak, Mariusz Głąbowski Arkadiusz Wiśniewski, Piotr Zwierzykowski Modeling of Overflow Traffic Systems Modeling and Dimensioning of Mobile Networks: from GSM to LTE

Simplified division of groups in a network High usage groups o Blocking probability of the group is higher than the acceptable level of blocking in the network Low usage groups o Blocking probability of the group is lower or equal than the acceptable level of blocking in the network 2Modeling and Dimensioning of Mobile Networks: from GSM to LTE

Hierarchical network configuration Modeling and Dimensioning of Mobile Networks: from GSM to LTE3 Low usage group High usage group Transit level Local level

Fixed hierarchical alternative routing Public networks generally use hierarchical alternative call routing strategy with fixed sequences of alternative choices Sequences o Direct (first choice links) Calls not accepted are offered to the second choice links o Alternative choice links Calls not accepted are offered to the next choice link o Last choice links Calls not accepted are lost Modeling and Dimensioning of Mobile Networks: from GSM to LTE4

A network example AB – direct path o the first choice path o origin – destination pair ACB – second choice path ADB – third choice path ADCB – last choice path 5

Overflow traffic Because of the paekedness of overflow traffic, a larger number of links is required to carry the same amount of traffic, at an acceptable blocking probability, than it would be required for Poisson traffic Modeling and Dimensioning of Mobile Networks: from GSM to LTE6

Overflow traffic parameters Mean value of overflow traffic R Variance of overflow traffic σ 2 Peakedness (variance/mean) Spreadness (variance-mean) Modeling and Dimensioning of Mobile Networks: from GSM to LTE7 A... R, D (σ 2,Z) V

Overflow traffic parameters 8 Parameteroffered trafficcarried traffic overflow traffic Peakedness factor Z = 1Z < 1Z > 1 Spreadness factor D = 0D < 0D > 0

Markov process in the overflow system Modeling and Dimensioning of Mobile Networks: from GSM to LTE9 State probability: A R, D (σ 2,Z) V  Primary groupSecondary group where: i(t) is the number of occupied links in the primary group, j(t) is the number of occupied links in the secondary group. Primary (direct) group Secondary (overflow) group

Two-dimensional Markov process Modeling and Dimensioning of Mobile Networks: from GSM to LTE10 State equations

Two-dimensional Markov process Modeling and Dimensioning of Mobile Networks: from GSM to LTE11 State equations

Parameters of overflow traffic Results - Riordan formulae NOTE! o For Poisson traffic stream the variance is equal to the mean value, thus: Modeling and Dimensioning of Mobile Networks: from GSM to LTE12

Dimensioning of overflow groups (one- parameter model) Determination of overflow traffic parameters: Required capacity of the alternative group: Modeling and Dimensioning of Mobile Networks: from GSM to LTE13

Equivalent Random Traffic method step 1 ERT Modeling and Dimensioning of Mobile Networks: from GSM to LTE14 V ov

ERT method - step 2 Modeling and Dimensioning of Mobile Networks: from GSM to LTE15 Traffic stream overflowing from equivalent group is the same as the sum of traffic streams overflowing from real groups V ov V1V1 A1A1 R 1, D 1 VkVk AkAk R 1, D k V ov V* A* R, D equivalent group

ERT method - step 2 Modeling and Dimensioning of Mobile Networks: from GSM to LTE16 Iteration method can be used to solve this equations V ov V1V1 A1A1 R 1, D 1 VkVk AkAk R 1, D k V ov V* A* R, D equivalent group

ERT method - step 2 Modeling and Dimensioning of Mobile Networks: from GSM to LTE17 Rapp approximate solution V ov V1V1 A1A1 R 1, D 1 VkVk AkAk R 1, D k V ov V* A* R, D equivalent group

ERT method - step 2 Blocking probability of the total offered traffic in the system Modeling and Dimensioning of Mobile Networks: from GSM to LTE18 V ov V* A* R

ERT method - step 3 Blocking probability in the overflow group Modeling and Dimensioning of Mobile Networks: from GSM to LTE19

ERT methodPoisson traffic + overflow traffic) Determination of overflow traffic parameters: Modeling and Dimensioning of Mobile Networks: from GSM to LTE20 V ov V1V1 A1A1 R 1, D 1 A3A3 V2V2 A2A2 R 2, D 2

Hayward’s equivalence method Hayward’s equivalence method Modeling and Dimensioning of Mobile Networks: from GSM to LTE21 V ov V1V1 A1A1 R1R1 VkVk AkAk RkRk

Interpretation of the Hayward method Modeling and Dimensioning of Mobile Networks: from GSM to LTE22 (V, R, Z) - system (V, R, Z)-system transformation into Z equal subsystems (V/Z, R/Z, 1): Blocking probability in each subsystem is the same and can be evaluated by Erlang formula: R, Z V R/Z,1 V/Z R/Z,1 V/Z R/Z,1 V/Z 1 2 Z (V/Z, R/Z, 1) - subsystems transformation

Dimensioning of the networks links Erlang FormulaHayward Formula Modeling and Dimensioning of Mobile Networks: from GSM to LTE23 Hayward Formula is used for dimensioning network groups with unrecognized traffic pattern. Frequently, it is assumed that Z = (2 - 3).

Dimensioning of groups in the network Two problems are needed to be solved for dimensioning of groups in the network with hierarchical alternative routing o Determination of optimal capacity for high usage group o Dimensioning of overflow groups Attention o Traffic stream offered to alternative groups is not an Erlang traffic stream Modeling and Dimensioning of Mobile Networks: from GSM to LTE24