1. Mariusz Głąbowski, Adam Kaliszan, Maciej Stasiak A New Convolution Algorithm of Blocking Probability Calculation in Full-Availability Group with Bandwidth Reservation Poznań University of Technology

2. M. Głąbowski et all: A New Convolution Algorithm of Blocking Probability Calculation... 2 Plan of the presentation  Full-availability group  Analytical models  One-dimensional Markov chain  Convolution algorithm  Full-availability group with reservation  Numerical results  Conclusions

3. M. Głąbowski et all: A New Convolution Algorithm of Blocking Probability Calculation... 3 Full availability group 1 2 3 4 5 6 7 … V Shared Bandwidth Class 1: λ 1, μ 1, t 1 Class 2: λ 2, μ 2, t 2 Class 3: λ 3, μ 3, t 3 Class M: λ M, μ M, t M

4. M. Głąbowski et all: A New Convolution Algorithm of Blocking Probability Calculation... 4 Multi-dimensional Markov process

5. M. Głąbowski et all: A New Convolution Algorithm of Blocking Probability Calculation... 5 Multi-rate systems Analytical models One-dimensional Markov chain Convolution algorithm Poisson traffic modelBPP traffic model State – independent systems State – dependent systems State – independent systems State – dependent systems ?

6. M. Głąbowski et all: A New Convolution Algorithm of Blocking Probability Calculation... 6 One-dimensional Markov chain

7. M. Głąbowski et all: A New Convolution Algorithm of Blocking Probability Calculation... 7 Full-availability group 1 2 3 4 5 6 7 … V Shared Bandwidth

8. M. Głąbowski et all: A New Convolution Algorithm of Blocking Probability Calculation... 8 Convolution algorithm  3 steps of algorithm:  Calculation of the occupancy distribution  Calculation of the aggregated occupancy distribution [P] V  Calculation of the blocking probability B i for the class i traffic stream

9. M. Głąbowski et all: A New Convolution Algorithm of Blocking Probability Calculation... 9 Convolution algorithm – step 1 [p 0 ] 1 V [p 1 ] 1 1 [p 2 ] 1 V [p 3 ] 1 V …[p V ] 1 V [p 0 ] 2 V [p 2 ] 2 V …[p V ] 2 V state

10. M. Głąbowski et all: A New Convolution Algorithm of Blocking Probability Calculation... 10 Convolution algorithm – step 2 [p 0 ] 1 V [p 1 ] 1 V [p 2 ] 1 V [p 3 ] 1 V …[p V ] 1 V * [p 0 ] 2 V [p 2 ] 2 V …[p V ] 2 V [p 0 ] 12 V [p 2 ] 12 V [p 3 ] 12 V …[p V ] 12 V = [p 2 ] 12 V = [p 0 ] 1 V [p 2 ] 2 V + [p 2 ] 1 V [p 0 ] 2 V (0+2=2) (2+0=2)

11. M. Głąbowski et all: A New Convolution Algorithm of Blocking Probability Calculation... 11 Convolution algorithm Step 2 & 3 Convolution algorithm – step 3 Convolution algorithm – step 2

12. M. Głąbowski et all: A New Convolution Algorithm of Blocking Probability Calculation... 12 Full-availability group with reservation Reservation spaceReservation treshold

13. M. Głąbowski et all: A New Convolution Algorithm of Blocking Probability Calculation... 13 Full-availability group with reservation  One-dimensional Markov chain

14. M. Głąbowski et all: A New Convolution Algorithm of Blocking Probability Calculation... 14 FAGR Convolution algorithm 0 1 2 3 4 0 2 4 0 1 2 3 4 0 2 4

15. M. Głąbowski et all: A New Convolution Algorithm of Blocking Probability Calculation... 15 FAGR Convolution algorithm p0p1p2p3p4 p0 0 p2 0 p4 p0 0 p2 0 p4 p0p1p2p3p4 4-th word of conditional distributions [Q] {1,2},{2} and [Q] {1,2},{1}

16. M. Głąbowski et all: A New Convolution Algorithm of Blocking Probability Calculation... 16 FAGR Convolution algorithm p0p1p2p3p4 p0 0 p2 0 p4 p0 0 p2 0 p4 p0p1p2p3p4 + PP ** [Q 1 ] 2+1 [Q 0 ] 2+1 [Q 4 ] 2+1 [Q 2 ] 1+2 [Q 0 ] 1+2 [Q 1 ] 1+2 [Q 3 ] 1+2 [Q 4 ] 1+2

17. M. Głąbowski et all: A New Convolution Algorithm of Blocking Probability Calculation... 17 FAGR Convolution algorithm [p] 1 [p] 2 [p]1 P(2) * * + P(1) [P] {1,2} [p] 3 [P] {1,2} P(3) * * + P(1,2) [P] {1,2,3} [p] 4 [P] {1,2,3} P(4) * * + P(1,2,3) …

18. M. Głąbowski et all: A New Convolution Algorithm of Blocking Probability Calculation... 18 Results

19. M. Głąbowski et all: A New Convolution Algorithm of Blocking Probability Calculation... 19 Results

20. M. Głąbowski et all: A New Convolution Algorithm of Blocking Probability Calculation... 20 Conclusions  This paper presents an approximate method of blocking probability calculations in the full availability group with reservation carrying a mixture of different traffic streams  The proposed method is based on convolution algorithm  The obtained calculation results have been compared with the results of simulation experiments. This research has confirmed a high accuracy of the proposed model