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Maciej Stasiak, Mariusz Głąbowski Arkadiusz Wiśniewski, Piotr Zwierzykowski Models of Links Carrying Single-Service Traffic Chapter 7 Modeling and Dimensioning.

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Presentation on theme: "Maciej Stasiak, Mariusz Głąbowski Arkadiusz Wiśniewski, Piotr Zwierzykowski Models of Links Carrying Single-Service Traffic Chapter 7 Modeling and Dimensioning."— Presentation transcript:

1 Maciej Stasiak, Mariusz Głąbowski Arkadiusz Wiśniewski, Piotr Zwierzykowski Models of Links Carrying Single-Service Traffic Chapter 7 Modeling and Dimensioning of Mobile Networks: from GSM to LTE

2 Two-dimensional Erlang distribution Assumptions V channels in the full-availability group o each of them is available if it is not busy Arrivals create two Poisson streams with intensities 1 and 2 Service times has exponential distribution with parameters  1 and  2 Rejected call is lost 2

3 Two-dimensional Erlang distribution 3 Offered traffic: Full-availability group with two call streams

4 Microstate definition Operation of the system is determined by the so-called two-dimensional Markov chain with continuous time: {x(t), y(t)},where x(t) and y(t) are the numbers of channels occupied at the moment t by calls of class 1 and 2, respectively Steady-microstate probabilities 4

5 State transition diagram for two- dimensional Markov chain 5 state {0,0} – all channels are free, state {x,y} – x channels are servicing calls of class 1, y channels are servicing calls of class 2,

6 Statistical equilibrium equations 6

7 Two-dimensional Erlang distribution Distribution: Blocking probability: 7 where

8 Reversibility of the two-dimensional Markov process 8 Necessary and sufficient condition for reversibility (Kolmogorov criteria): The circulation flow (product of streams parameters) among any four neighboring states in a square equals zero. Flow clockwise = Flow counterclockwise

9 Reversibility of the two-dimensional Markov process 9 Reversibility property leads to local balance equations between any two neighboring microstates of the process. If there exists a possibility to achieve the microstate {x 2 y 2 } outgoing from the microstate {x 1 y 1 }, then there exists the possibility to achieve the microstate {x 1 y 1 }, outgoing from the microstate {x 2 y 2 }. Reversibility property leads to local balance equations between any two neighboring microstates of the process. If there exists a possibility to achieve the microstate {x 2 y 2 } outgoing from the microstate {x 1 y 1 }, then there exists the possibility to achieve the microstate {x 1 y 1 }, outgoing from the microstate {x 2 y 2 }.

10 Reversibility of two-dimensional Markov process Note o Between any two neighboring microstates of the process we can (as in the case of one-dimensional birth and death process) write local balance equations 10

11 Product form solution of two- dimensional distribution Independently on the chosen path between microstates {x, y} and {0, 0}, we always obtain: Where G V is the normalization constant : 11

12 Example of the two-dimensional Erlang distribution ,0 1,0 2,0 0,1 0,2 1,1

13 Macrostate probability 13 where:

14 Blocking probability – macrostate level 14 Example:

15 Multi-dimensional Erlang distribution Assumptions: o V channels in the full availability trunk group; each of them is available if it is not busy; o Arrivals create M Poisson streams with intensities 1, 2,..., M o Service times have exponential distribution with parameters  1,  2,...,  M o Rejected call is lost 15

16 Multi-dimensional Erlang distribution 16 Full-availability group with M call streams offered traffic:

17 Microstates in multi-dimensional Erlang distribution state {x1,..., xi,..., xM } o x 1 channels are servicing calls of class 1, o... o x i channels are servicing calls of class i, o... o x M channels are servicing calls of class M. Total number of busy channels: 17

18 State transition diagram for multi- dimensional Markov chain 18 State interpretation: state (x 1,..., x i,..., x M ) - group services x 1 calls of class 1,..., x i calls of class i,..., x M calls of class M. xx M1,..., xx M1 1 ,..., xx M1 1,...,  xx M1 1 ,..., xx M1 1,...,   11 x 1  MM x M   11 1x  1   MM x  1 M

19 Statistical equilibrium equations 19

20 Reversibility of multi-dimensional Markov process 20

21 Multi-dimensional Erlang distribution All offered streams are considered to be mutually independent and the service process in the group is reversible, so we can rewrite each microstate in product form: 21 where:

22 Macro-state probability 22 where: is the set of such subsets in which the following equation is fulfilled:

23 Interpretation of macrostates distribution 23 call stream with intensity: Service time – hyper-exponential distribution (weighed sum of exponential distributions) with average value: Blocking probability: Multi-dimensional distribution is the Erlang distribution for traffic: This distribution can be treated as a model of the full-availability group with parameters:

24 Recurrent form of multidimensional Erlang distribution 24 Calculation algorithm:

25 Birth and death process calibration (calibration constant) A section of a state transition diagram for the birth and death process in the full availability group: A section of the calibrated state transition diagram for the birth and death process in the full availability group (calibration constant 1/  ): 25

26 Interpretation of recurrent notation form of multidimensional Erlang distribution 26 A fragment of a state transition diagram which interprets the recurrent form of multidimensional Erlang distribution Calibration constant: Each component process is calibrated by „own” calibration constant 1/  i

27 Service streams 27 Balance equation for state n: This equation is fulfilled when the local balance equations are fulfilled for each stream i :


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