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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

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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

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

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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

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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,

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Statistical equilibrium equations 6

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

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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

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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 }.

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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

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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

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

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Macrostate probability 13 where:

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Blocking probability – macrostate level 14 Example:

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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

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

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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

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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

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Statistical equilibrium equations 19

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Reversibility of multi-dimensional Markov process 20

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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:

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

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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:

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Recurrent form of multidimensional Erlang distribution 24 Calculation algorithm:

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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

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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

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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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