2. Maciej Stasiak, Mariusz Głąbowski Arkadiusz Wiśniewski, Piotr Zwierzykowski Groups models Modeling and Dimensioning of Mobile Networks: from GSM to LTE

3. Erlang Model Modeling and Dimensioning of Mobile Networks: from GSM to LTE2

4. Full-availability group (FAG) Assumptions: o V channels in the full-availability trunk group. Each of them is available if it is not busy; o Arrival process is the Poisson process; o Service time has exponential distribution with parameter 1/μ; o Rejected call is lost Modeling and Dimensioning of Mobile Networks: from GSM to LTE3

5. State transition diagram state „0” - all channels are free, state „1” - one channel is busy, others are free,..., state „i” - i channels are busy and (V-i) are free,..., state „V” - all channels are busy. Modeling and Dimensioning of Mobile Networks: from GSM to LTE4

6. Statistical equilibrium equations Modeling and Dimensioning of Mobile Networks: from GSM to LTE5

7. Interpretation λ/μ Modeling and Dimensioning of Mobile Networks: from GSM to LTE6 /  determines the average number of arrivals within average service time

8. Erlang’s distribution Modeling and Dimensioning of Mobile Networks: from GSM to LTE7 Distribution of busy channels in the FAG, capacity V=5, offered traffic: A=1 Erl. (a); A=3 Erl. (b); A=8 Erl. (c). a)b)c)

9. Erlang formula Modeling and Dimensioning of Mobile Networks: from GSM to LTE8 Blocking probability = f ( offered traffic, capacity)

10. Recurrence property of Erlang formula Modeling and Dimensioning of Mobile Networks: from GSM to LTE9

11. Characteristics of carried traffic Mean value of carried traffic (average number of simultaneously busy channels) Variance of carried traffic Modeling and Dimensioning of Mobile Networks: from GSM to LTE10

12. Characteristics of carried traffic Variance of carried traffic Modeling and Dimensioning of Mobile Networks: from GSM to LTE11

13. Characteristics of carried traffic Variance of carried traffic o Taking into account: o we obtain: Modeling and Dimensioning of Mobile Networks: from GSM to LTE12

14. Characteristics of carried traffic NOTE ! o Variance of offered traffic o is equal to o mean value of offered traffic Modeling and Dimensioning of Mobile Networks: from GSM to LTE13

15. Erlang tables Two kinds of Erlang tables in engineering practice: o o N A1 A2 A3 N N B1 B2 B3 N o 1 B11 B21 B31 1 1 A11 A21 A31 1 o 2 B12 B22 B32 2 2 A12 A22 A32 2 o 3 B13 B23 B33 3 3 A13 A23 A33 3 Modeling and Dimensioning of Mobile Networks: from GSM to LTE14

16. Erlang table Modeling and Dimensioning of Mobile Networks: from GSM to LTE15 Capacity VBlocking probability B E=0.02E=0.01E=0.005E=0.001 Offered traffic intensity A 10.020.010.0050.001 51.701.401.100.80 105.104.504.003.10 159.008.107.386.08 2013.2012.0011.109.41 2517.5016.1015.0013.00 3021.9020.3019.0016.70 3526.4024.6023.2020.50 4031.0029.0027.4024.40 4535.6033.4031.7028.40 5040.3037.9036.0032.50 6049.6046.9044.8040.80 7059.156.153.7049.20 8068.7065.4062.7057.80 9078.3074.7071.8066.50 10088.0084.1080.9075.20

17. Group principle Modeling and Dimensioning of Mobile Networks: from GSM to LTE16 Two groups joint group

18. Group principle - example Modeling and Dimensioning of Mobile Networks: from GSM to LTE17 group 1 group 2 joint group 0,001)( 0,01)(0,02)( 20,Erl.12,010,Erl.5,1 21)( 21 1211 21 21     AAE AEAE VAVA VV VV

19. Poisson distribution Border case of Erlang distribution The number of channels is infinite, so there is no blocking in the system Modeling and Dimensioning of Mobile Networks: from GSM to LTE18

20. Poisson distribution Approximation of blocking probability o If the number of servers is equal to V, the blocking probability can be approximated by the Poisson model: Modeling and Dimensioning of Mobile Networks: from GSM to LTE19

21. Channel load – random hunting Traffic carried by V channels: Traffic carried by any channel: For V=10, A=10 Erl.: Modeling and Dimensioning of Mobile Networks: from GSM to LTE20

22. Channel load – random hunting Modeling and Dimensioning of Mobile Networks: from GSM to LTE21 group : A=10 Erl., V=10 Load Channel number

23. Channel load – successive hunting Traffic carried by i channels: Traffic carried by i-1 channels: Traffic carried by channel i: Modeling and Dimensioning of Mobile Networks: from GSM to LTE22

24. group : A=10 Erl., V=10 Load Channel number Channel load – successive hunting Modeling and Dimensioning of Mobile Networks: from GSM to LTE23

25. Palm – Jacobaeus formula Formula defines occupancy probability of x exactly determined servers Modeling and Dimensioning of Mobile Networks: from GSM to LTE24 occupancy probability of any i channels Conditional occupancy probability of x exactly determined servers under condition that i servers are busy:

26. Engset Model Modeling and Dimensioning of Mobile Networks: from GSM to LTE25

27. Full availability group – Engset model Assumptions: o V channels in the full availability trunk group. Each of them is available if it is not busy; o Arrivals create a stream generated by a finite number of N (N>V) traffic sources. Each free source generates arrivals with intensity γ; o Service time has exponential distribution with parameter 1/μ; o Rejected call is lost Modeling and Dimensioning of Mobile Networks: from GSM to LTE26

28. State transition diagram state „0” - all channels are free, N sources are free state „1” - one channel is busy, (N-1) sources are free,..., state „i” - i channels are busy and (N-i) sources are free,..., state „V” - all channels are busy (V-N) sources are free Modeling and Dimensioning of Mobile Networks: from GSM to LTE27

29. Statistical equilibrium equations Modeling and Dimensioning of Mobile Networks: from GSM to LTE28 traffic offered by one free source

30. Blocking / loss probability Blocking probability Loss probability: o The loss probability in the group with traffic generated by N sources is equal to the blocking probability in the group with traffic generated by N-1 sources Modeling and Dimensioning of Mobile Networks: from GSM to LTE29

31. Recurrence property of Engset formula Modeling and Dimensioning of Mobile Networks: from GSM to LTE30

32. Engset formula – another form of notation Blocking probability: Parameter a expresses the ratio of the average time of source activity (occupancy) to the sum of the average time of source activity and the average time between the moment of terminating the activity and the moment of activity related to the generation of the next call. Therefore, the parameter a can be interpreted as the mean traffic offered by one source. Note that parameter  is the mean traffic offered by one free source. Modeling and Dimensioning of Mobile Networks: from GSM to LTE31

33. Engset model – carried traffic Mean value of carried traffic is equal to the average number of simultaneously busy channels: o where y is traffic carried by one source It can be proved: Modeling and Dimensioning of Mobile Networks: from GSM to LTE32

34. Engset model – offered traffic Mean value of offered traffic is equal to the average number of busy channels in the group with capacity of N channels (system without losses): Modeling and Dimensioning of Mobile Networks: from GSM to LTE33

35. Engset model – variance Variance of Engset distribution: Peakedness factor: Modeling and Dimensioning of Mobile Networks: from GSM to LTE34

36. Engset model – lost traffic Lost traffic intensity: Traffic loss probability (traffic congestion ) – relation of lost traffic to offered traffic: Modeling and Dimensioning of Mobile Networks: from GSM to LTE35

37. Engset model – paradox of call stream Stream parameter averaging all over the states (expresses mean number of calls per mean service time, i.e. mean call intensity) Mean call intensity resulting from evaluation of offered traffic: Modeling and Dimensioning of Mobile Networks: from GSM to LTE36

38. Engset model – paradox of call stream Product AC determines the lost traffic intensity, i.e. the average number of sources which should be free as a result of blocking. The g parameter is the traffic intensity per one free source. If we assume that each blocked source within mean service time 1/m is not active, then Δ=0 and L A =L. The parameter L determines the mean call intensity under the assumption that each lost call (as a results of blocking) immediately causes the source to be free within hypothetical service time. The parameter L A determines the mean call intensity under the assumption that each lost call (as a results of blocking) immediately causes the source to be blocked within hypothetical service time Modeling and Dimensioning of Mobile Networks: from GSM to LTE37

39. Palm – Jacobaeus formula for Engset Formula defines occupancy probability of x exactly determined servers Modeling and Dimensioning of Mobile Networks: from GSM to LTE38 occupancy probability of any i channels Conditional occupancy probability of x exactly determined servers under condition that i servers are busy:

40. Erlang and Engset Models Modeling and Dimensioning of Mobile Networks: from GSM to LTE39

41. Erlang and Engset model Engset formula is a generalization of the Erlang formula when the number of traffic sources N tends to infinity, and parameter γ is decreased in such a way that the product N γ remains constant. Modeling and Dimensioning of Mobile Networks: from GSM to LTE40 Engset distribution Erlang distribution