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

2. Multi-rate systems 2 Integrated services system

3. Multi-rate systems - parameters 3 arrival rate for class-i call stream, arrival rate for carried call stream of class-i, arrival rate for lost call stream of class-i, service rate for class-i call stream, number of demanded units of service resources for class-i call, number of offered call streams in the system, offered traffic of class-i:

4. Full-Availability Group FAG 4

5. FAG with multi-rate traffic 5 A mixture of different multi-rate traffic streams Microstate: Microstate probability: 1 2 V 1 2 M i PJP

6. Multidimensional Markov process 6 Microstate: Microstate probability: 1 i M i 1 M

7. Reversibility of multi-dimensional Markov process Necessary and sufficient condition for reversibility (Kolmogorov criteria): o The circulation flow (product of streams parameters) among any four neighboring states in a square equals zero. o Flow clockwise = flow counter clockwise State equations: o Reversibility property leads to local balance equations between any two neighboring microstates of the process. 7

8. Reversibility of multi-dimensional Markov process 8

9. Product form solution of multi- dimensional distribution (multi-rate) All offered streams are considered to be mutually independent and the service process in the group is reversible, so we can write each microstate in product form 9

10. Product form solution of multi- dimensional distribution (single-rte) 10

11. Macro-states Macro-state: {n}, where n is the integer number of BBUs in the group. Macro-state probability: where: is the set of such subsets, that the following equation is fulfilled: 11

12. Macro-states and micro-states 12 Example: V=10, t 1 =1, t 2 =2, t 3 =4 {5,0,0} {3,1,0} {1,2,0} {1,0,1} Micro-states associated with macro-states {5}

13. Markov process in FAG – micro-state level 13 1 i M i 1 M

14. Markov process in FAG – macro-state level Solution: 14

15. Kaufman-Roberts recursion One-dimensional Markov chain - graphic interpretation (t 1 =1, t 2 =2): 15

16. Blocking probability in FAG The Kaufman-Roberts model for multi-rate systems is a generalization of the Erlang model for one-rate systems. 16

17. Blocking probability – graphic interpretation 17 Example: V=10, t 1 =1, t 2 =2, t 3 =4 B 1 =P(10) B 3 =P(10)+P(9)+P(8)+P(7) B 2 =P(10)+P(9) {10} {9} {8} {7}

18. Blocking probability – results 18 OFFERED TRAFIC FULL AVAILABILITY GROUP V=30 Stream 1 Stream 2 Stream 3 Simulations Calculations OFFERED TRAFIC BLOCKING PROBABILITY

19. Service streams 19 State equations for state {n}:

20. Service streams 20 Balance equation for state n: This equation is fulfilled when the local balance equations are fulfilled for each stream i :

21. Calculation algorithm for Kaufman- Roberts distribution 21

22. Calculation algorithm for Kaufman- Roberts distribution Let us assume that a full-availability group with capacity V services two traffic classes: t 1 =1, t 2 =2. 22 2)q(2) value calculations: 1)q(2) value calculations:

23. Calculation algorithm for Kaufman- Roberts distribution 23 4) Using normalization procedure we calculate the value q(0): Note that the results of calculation are expressed as coefficients x i multiplied by constant q(0)=1. GVGV 5) Calculation of the real values of probabilities : 3)q(i) values calculations :

24. Calculation algorithms for multi-service distributions 24 Analytical models Recurrence algorithmsConvolution algorithms Poisson traffic modelAny kind traffic model State – independent systems State – dependent systems State – independent systems State – dependent systems ?

25. Convolution operation for two distributions 25 Convolution of two distributions: Normalization of the state space 2V  V

26. Convolution algorithm 3 steps of algorithm: o Calculation of the occupancy distribution for each traffic class o Calculation of the aggregated occupancy distribution [P] V o Calculation of the blocking probability E i for the class i traffic stream 26

27. Convolution algorithm – step 1 27 [p 0 ] 1 4 [p 1 ] 1 4 [p 2 ] 1 4 [p 3 ] 1 4 [p 4 ] 1 4 [p 0 ] 2 4 [p 2 ] 2 4 [p 4 ] 2 4 state

28. Convolution algorithm – step 2 28 [p 3 ] 1 4 [p 1 ] 1 4 [p 0 ] 1 4 [p 2 ] 1 4 [p 4 ] 1 4 * [p 0 ] 2 4 [p 2 ] 2 4 [p 4 ] 2 4 [p 0 ] 12 4 [p 3 ] 12 4 [p 4 ] 12 4 = [p 2 ] 12 8 = [p 0 ] 1 4 [p 2 ] 2 4 + [p 2 ] 1 4 [p 0 ] 2 4 (0+2=2) (2+0=2) [p 1 ] 12 4 [p 2 ] 12 4

29. Convolution algorithm – step 3 29 [p 0 ] 12 4 [p 1 ] 12 4 [p 2 ] 12 4 [p 3 ] 12 4 [p 4 ] 12 4 state E2E2 E1E1

30. Convolution algorithm for M class of traffic 30 Convolution algorithm – step 1 Convolution algorithm – step 2 Convolution algorithm – step 3

31. Convolution algorithm for different distributions 31 Convolution algorithm – step 1 Convolution algorithm – step 2

32. Convolution algorithm for different distributions 32 Blocking / loss probability Convolution algorithm – step 3

33. Example of link dimensioning Offered traffic parameters: To find the number of channels for blocking probabilities B(i) <0.005 33 class 1class 2class 3 t126 a i [Erl.]2110.53.5 a i t i [Erl.]21

34. Example of link dimensioning 34 Variant class 1class 2class 3 2 x 300.0340.10.44 3 x 300.0010.0060.064 4 x 30B<0.0001 0.001 Variant 2: 3 x 30, a=63/90=0.7 Variant 3: 4 x 30, a=63/120=0.525 Variant 1: 2 x 30, a=63/60=1.05

35. FAG – multi-service Erlang-Engset model PROBLEM o Calculation of blocking probabilities E i and loss probabilities B i for M 1 traffic streams of PCT1type and M 2 traffic streams of PCT2 type : 35 PCT1 PCT2

36. FAG – multi-service Erlang-Engset model Assumptions 36 PCT1 stream intensity of class i: 1,i, PCT2 stream intensity of class j: PCT1 traffic of class i offered to the group : PCT2 traffic offered to the group by one free source of class j: PCT2 stream intensity offered by one free source of class j:  2,i

37. Multi-service Erlang-Engset model – recurrence algorithm Idea of the algorithm o It was assumed in the algorithm that the number of occupied BBU’s y 2,j (n) by PCT2 stream of class j in each macro-state {n} is the same as the number of occupied BBU’s by equivalent PCT1 stream with traffic intensity A 2,j =N 2,j  2,j. Approximation rule: the number of serviced calls in the given state of the group is the same for both Erlang and Engset models. 37

38. Recurrence algorithm – step 1 Determination of occupancy distribution under the assumption that all offered streams are PCT1 type (Erlang streams): 38 Erlang

39. Recurrence algorithm – step 2 Determination of busy BBU’s y 2,i (n), occupied by PCT2 calls in each macro-state {n} 39

40. Recurrence algorithm – step 3 Determination of occupancy distribution, under the assumption that offered streams are PCT1 and PCT2 type : 40 Engset

41. Recurrence algorithm – step 4 Calculation of the blocking probability E, and loss probability B, for PCT1 and PCT2 streams PCT1 stream: PCT2 stream: 41

42. Full availability group with Engset traffic 42 S=400 S - infinity