1. 1 Presented by Sarbagya Buddhacharya

2. 2 Increasing bandwidth demand in telecommunication networks is satisfied by WDM networks. Dimensioning of WDM networks with conventional steady state blocking probability results in overprovisioning problem (Nayak et al., 2002). To minimize the overdimensioning problem in WDM network various approaches are being introduced. Introduction  Absorption probability model (Nayak et al., 2002)  Time dependent blocking probability model (Tian, 2006)

3. 3 Problem Statement All the approaches (Nayak et al., 2002; Tian, 2006) to reduce the overdimensioning problem have considered fixed routing. Several literatures (Lin et al., 1978; Birman 1996; Zhu et al., 2000) have discussed different methods for the computation of blocking probability with alternate routing. We propose a method to compute the absorption probability with fixed alternate routing, and demonstrate its use in network capacity dimensioning. Introduction

4. 4 Objectives To develop a method for the computation of absorption probability with fixed alternate routing. To compare the value of absorption probability obtained from fixed alternate routing with previously used fixed routing. To study the impact of alternate routing on the first passage time ( network upgrade time) of the network. To analyze capacity allocation of a network using the absorption probability with fixed alternate routing. Introduction

5. 5 Scope and Limitations We consider only full wavelength conversion, although there are other wavelength conversion techniques such as sparse wavelength conversion. We do not analyze dynamic alternate routing techniques and focus on fixed alternate routing. We do not include traffic growth model. Introduction

6. 6 Computation and simulation of absorption probability for A single link Fixed routing Fixed alternate outing Methodology

7. 7 Absorption probability for a single link Assumptions: Poisson arrival rate and exponential service time. Absorption probability for a single link with capacity K at a given time t (Nayak et al., 2002). where are the negative eigenvalues of ( K +2) ×(K+2) matrix A, arranged in the descending order. where and Methodology

8. 8 Absorption Probability for Fixed Routing For the given network topology, fixed route for each S-D pair is obtained using shortest path algorithm. Shortest path is selected based on the number of hop counts. Arrival rate for each link is thinned using Erlang fixed point approximation. Methodology where, R = set of all possible routes. A jr {0, 1}: equal to one if and only if a lightpath on route uses a circuit from link j. = Poisson arrival rate of lightpath requests on route.

9. Using link absorption probability, absorption probability of route can be obtained. 9 Methodology For the given link capacity and arrival rate, absorption probability of all the links at time t, can be obtained using single link formula. The average absorption probability of the whole network can be obtained using equation (1). (1)

10. 10 Absorption Probability for Fixed Alternate Routing Augmented route tree for each S-D pair is obtained and path-loss sequence is represented in terms of link set U i. A E B D C E L D D C P 1 (ABD) P 2 (ACED) P 3 (AECD) L(AL) U 1 U 2 U 3 U 4 Pr {U i. used} is obtained based on the call completion rule. Pr {U i. used}= Pr {U i. is available and U 1, U 2.,…, U i-1. are unavailable } = Pr {U i. available}× Pr{U 1 U 2., …, U i.-1 are unavailable| U i. available} (2) To solve equation 2 reliability formulae are used (Lin et al.,1978). Methodology

11. We define following terms Using these terms following reliability formulae are defined where, 11 Methodology

12. Link absorption probability calculated using an iterative method.  Initialize P asb,j,n (t ) = 0  Calculate Pr { P s,k used } for each alternate path of each S-D pair using equation 2.  Find carried traffic of each alternate routes  Find carried traffic for each link  Find offered traffic for each link  Calculate P asb,j,n+1 (t ) using single link formula.  Taking P asb,j,n+1 (t ) as the starting point start next iteration. 12 Methodology

13. Compute carried traffic of each link Compute offered traffic of each link Absorption Probability (Initially assumed) Compute Pr { P s,k used } Compute carried traffic of each alternate route 13 Methodology

14.  We continue iteration until the difference between P asb,j,n (t ) and P asb,j,n+1 (t ) for each link is below certain threshold value. Absorption probability of S-D pair s can be computed as, 14 Methodology The average absorption probability of the whole network can be obtained using equation (1).

15. Simulation Model MATLAB software is used. Event based simulation. Simulation outputs are used to validate the computational results. Poisson arrival rate and exponential service time are assumed. Simulation is done for  Single link.  Fixed routing.  Fixed alternate routing. Methodology: Simulation 15

16. For each event (arrival or termination): Update link status. When link status is greater than the link capacity : Increment the number of blocked call. At the end of time t : if the number of blocked call is greater than zero, then increment the number of absorption count. Absorption probability=absorption count/number of iteration For a Single Link Methodology : Simulation 16

17. For each event (arrival or termination):Find the path on which event take place and update the link status. For each path: if the link status of any one of its link is greater than the link capacity, then increment the number of blocked call on that path. For each path: if the number of blocked call is greater than zero at time t, then increment the number of absorption count on that path. Absorption probability of a path= absorption count of the path/ number of iterations. For Fixed Routing Methodology : Simulation 17

18. For each event (arrival or terminations): Find the S-D pair on which event takes place and update the link status. If Arrival on S-D: search for the free path If the path is not found, then increment the number of blocked call on the S-D pair. For each S-D pair: if blocked call is greater than zero, then increment the absorption count on the S-D pair. Absorption probability for S-D pair = Absorption count on S-D pair/ number of iterations. For Fixed Alternate Routing Methodology : Simulation 18

19. Simulation Results For a single link Capacity K =32, arrival rate =24 arrival/year and mean service time 1/  =1 year (Nayak et al., 2002). Outputs obtained from our method are close to output of (Nayak et al., 2002). Simulation results 19 Plot of absorption probability for a single link (a) Using our simulation program (b) Plot from (Nayak et al., 2002).

20. For Fixed Routing Network Topology : Pan-European COST 239 (O’Mahony et al., 1996) The numerical value above each link indicates the link capacity (in wavelength channels). 9 11 10 6 8 7 3 2 4 5 1 40 41 43 44 Simulation results 20

21. Average absorption probability of the whole network is obtained using equation(1). Plot of absorption probability for fixed routing Simulation results 21 ParametersValue S-D pairs20 pairs randomly selected Arrival rate ( ) 10 arrival/year mean service time (1/  ) 1 year RoutingShortest path routing based on hop count Computational values are higher than the simulation output due to link independence assumptions in EFPA

22. For Fixed Alternate Routing Simulation results 22 ParametersValue Network topologyPan-European COST 239 (O’Mahony et al., 1996) S-D pairs20 pairs randomly selected Number of sets of S-D pairs5 Arrival rate ( ) 10 arrival/year mean service time (1/  ) 1 year RoutingFixed alternate routing Alternate path calculationk-shortest path algorithm (Yen, 1971). Number of alternate routes (P)2

23. Plot of absorption probability for P=2 Simulation results 23

24. Comparison of absorption probability with Fixed Routing and Fixed Alternate Routing Results Obtained 24 Results obtained ParametersValue Network topologyPan-European COST 239 (O’Mahony et al., 1996) S-D pairs20 pairs randomly selected Number of sets of S-D pairs5 Arrival rate ( ) 8 arrival/year,10 arrival/year and 12 arrival/year mean service time (1/  ) 1 year Number of alternate routes (P)1, 2, 3 Absorption probability calculated using purposed computational method. Absorption probability is reduced as the number of alternate paths are increased.

25. Plot of absorption probability with 20 SD pairs for (a) =8 arrival/year (b) =10 arrival/year (c) =12 arrival/year 25 Results Obtained

26. 26 Results Obtained ParametersValue Network topologyPan-European COST 239 (O’Mahony et al., 1996) S-D pairs10, 15, 20 (randomly selected) Number of sets of S-D pairs5 Arrival rate ( ) 10 arrival/year mean service time (1/  ) 1 year Number of alternate routes (P)1, 2, 3 Absorption probability calculated using purposed computational method. Absorption probability is reduced as the number of alternate paths are increased. Plot is obtained for different number of S-D pairs

27. Plot of absorption probability with arrival rate =10 arrival/year for (a) 10 S-D pairs (b)15 S-D pairs (c) 20 S-D pairs 27 Results Obtained

28. Refers to the time period during which there is high probability that at least one lightpath request will not be served (Nayak et al., 2002). At the end of this time, operators need to upgrade the capacity of the network. Alternate routing increase first passage time as compared to fixed routing. First passage time is plotted for 3 different arrival rates. First Passage Time Results Obtained 28 ParametersValue Network topologyPan-European COST 239 (O’Mahony et al., 1996) S-D pairs20 pairs randomly selected Number of sets of S-D pairs5 Arrival rate ( ) 8 arrival/year,10 arrival/year and 12 arrival/year mean service time (1/  ) 1 year Number of alternate routes (P)1, 2, 3 Absorption probability0.01, 0.001

29. Plot of first passage time with 20 SD pairs for (a) =8 arrival/year (b) =10 arrival/year (c) =12 arrival/year 29 Results Obtained

30. 30 Results Obtained First passage time is also computed for 3 different numbers of S-D pairs ParametersValue Network topologyPan-European COST 239 (O’Mahony et al., 1996) S-D pairs10, 15, 20 (randomly selected) Number of sets of S-D pairs5 Arrival rate ( ) 10 arrival/year mean service time (1/  ) 1 year Number of alternate routes (P)1, 2, 3 Absorption probability0.01, 0.001

31. Plot of first passage time with arrival rate =10 arrival/year for (a) 10 S-D pairs (b) 15 S-D pairs (c) 20 S-D pairs 31 Results Obtained

32. Capacity Allocation Capacity allocation is done based on a heuristic algorithm from (Gunawardena et al., 2009) This algorithm has mainly two phase : capacity increment phase and capacity decrement phase. In capacity increment phase, links are selected based on three link criticalities given below and link capacity of the selected links is incremented by 1. N j : number of S-D pairs which use link j that have the absorption probability greater than the threshold value. In capacity decrement phase, all the links are selected in some random order and capacity is decremented by 1. This is repeated until any route absorption probability exceeds the threshold. Results Obtained 32

33. Capacity allocation for Pan-European COST 239 Network parameters for capacity allocation of Pan-European COST 239 network are given below. ParametersNumerical values Arrival rate ( ) 6 arrival/year Service time (1/  ) 1 year S-D pairs55 Absorption probability threshold ( P th )0.01, 0.001 Time period (years)0.5, 1, 1.5, 2 Number of routes (P )1, 2, 3 Results Obtained 33

34. t0.51 P th 0.010.0010.010.001 LinkP=1P=2P=3P=1P=2P=3P=1P=2P=3P=1P=2P=3 1242021302622382930413034 2212018252419323129353430 326242128272636 33414043 Total 425394367485455404613588567694635590 Results of capacity allocation are shown below P th : Absorption probability threshold t: Observation period Results Obtained 34 t1.52 P th 0.010.0010.010.001 LinkP=1P=2P=3P=1P=2P=3P=1P=2P=3P=1P=2P=3 1443735493941494440544145 23836374340364240 464539 3444339484741484743545246 Total734712683807745697814777756888833779

35. Total capacity is reduced as the number of alternate paths are increased. Capacity allocation is done based on the absorption probability obtained from the simulation program. Results obtained are shown below. t1 P th 0.01 (Comp)0.01 (Sim)0.001(Comp)0.001(Sim) LinkP=1P=2P=3P=1P=2P=3P=1P=2P=3P=1P=2P=3 1382930352732413034383433 2323129 3026353430312831 336 33363335414043393637 Total 613588567578527505694635590641560548 P th : Absorption probability threshold t : Observation period Comp: Computation Sim: Simulation Results Obtained 35

36. Plots are obtained for the capacity allocation with arrival rate of   arrival/year,  arrival/year, and  arrival/year using computational  method.  = 4 arrival/year (a) P th =0.01 (b) P th = 0.001 36 Results Obtained

37.  = 6 arrival/year (a) P th =0.01 (b) P th = 0.001  = 8 arrival/year (a) P th =0.01 (b) P th = 0.001 37 Results Obtained

38. Discussions Absorption probability of the network is reduced with fixed alternate routing as compared to the fixed routing. First passage time of the network is increased with the implementation of fixed alternate routing instead of fixed routing. The capacity required to maintain the absorption probability of the network below certain threshold is reduced with the increased number of alternate paths. 38

39. Summary of Contributions A numerical computational method for calculating the absorption probability with fixed alternate routing. Simulation program for the computation of the absorption probability with fixed routing and fixed alternate routing. Fixed alternate routing is better than the fixed routing for network dimensioning based on absorption probability.  Decrease absorption probability of the Network  Increase first passage time of the Network  Reduce the capacity allocation 39

40. Recommendations We have considered full wavelength conversion. This work can be further explored with several other wavelength conversion techniques such as sparse wavelength conversion technique. We have considered fixed alternate routing so this work can be further extended for adaptive routing. We have considered constant traffic, so this thesis can be further explored with traffic growth model. 40

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