1. Modeling and Dimensioning of Mobile Networks: from GSM to LTE
Models of Links Carrying Single-Service Traffic
Chapter 7

2. 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

3. Two-dimensional Erlang distribution
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

5. 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,

6. Statistical equilibrium equations

7. Two-dimensional Erlang distribution
Blocking probability:
where

8. 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

9. 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}.

10. 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

11. 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 :

12. Example of the two-dimensional Erlang distribution1
0,0
1,0
2,0
0,1
0,2
1,1

13. Macrostate probability
where:

14. Blocking probability – macrostate level
Example:

15. 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

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

17. 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:

18. State transition diagram for multi-dimensional Markov chainx 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.

19. Statistical equilibrium equations

20. Reversibility of multi-dimensional Markov process

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:
where:

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

23. 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:

24. Recurrent form of multidimensional Erlang distribution
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/ ):

26. 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

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