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**Modeling and Dimensioning of Mobile Networks: from GSM to LTE**

Models of Links Carrying Single-Service Traffic Chapter 7

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**Two-dimensional Erlang distribution Assumptions**

V channels in the full-availability group 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

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**Two-dimensional Erlang distribution**

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

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**State transition diagram for two-dimensional Markov chain**

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**

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**Two-dimensional Erlang distribution**

Blocking probability: where

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**Reversibility of the two-dimensional Markov process**

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**

Reversibility property leads to local balance equations between any two neighboring microstates of the process. If there exists a possibility to achieve the microstate {x2 y2} outgoing from the microstate {x1 y1}, then there exists the possibility to achieve the microstate {x1 y1}, outgoing from the microstate {x2 y2}. Reversibility property leads to local balance equations between any two neighboring microstates of the process. If there exists a possibility to achieve the microstate {x2 y2} outgoing from the microstate {x1 y1}, then there exists the possibility to achieve the microstate {x1 y1}, outgoing from the microstate {x2 y2}.

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**Reversibility of two-dimensional Markov process**

Note 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

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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 GV is the normalization constant :

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**Example of the two-dimensional Erlang distribution**

1 0,0 1,0 2,0 0,1 0,2 1,1

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**Macrostate probability**

where:

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**Blocking probability – macrostate level**

Example:

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**Multi-dimensional Erlang distribution**

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

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**Multi-dimensional Erlang distribution**

offered traffic: Full-availability group with M call streams

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**Microstates in multi-dimensional Erlang distribution**

state {x1,..., xi ,..., xM } x1 channels are servicing calls of class 1, . . . xi channels are servicing calls of class i, xM channels are servicing calls of class M. Total number of busy channels:

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**State transition diagram for multi-dimensional Markov chain**

x M 1 , . + - m l ( ) State interpretation: state (x1,..., xi,..., xM) - group services x1 calls of class 1, ..., xi calls of class i, ..., xM calls of class M.

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**Statistical equilibrium equations**

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**Reversibility of multi-dimensional Markov process**

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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: where:

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**Macro-state probability**

where: is the set of such subsets in which the following equation is fulfilled:

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**Interpretation of macrostates distribution**

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

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**Recurrent form of multidimensional Erlang distribution**

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/ ):

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**Interpretation of recurrent notation form of multidimensional Erlang distribution**

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

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**Service streams Balance equation for state n:**

This equation is fulfilled when the local balance equations are fulfilled for each stream i :

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