1. Traffic and routing

2. Network Queueing Model Packets are buffered in egress queues waiting for serialization on line Link capacity is C bps Average packet length is P bits Service is modelled as independent with exponential distribution Service rate is ¹ = P/C Ingress and egress flows are homogeneous Poisson

3. Queue and Delay Calculations Each link is modelled as and M/M/1 queue Average queue length Q= ½ /(1- ½ ) = ¸ /( ¹ - ¸ ) ½ = ¸ / ¹ <1 Average waiting time (delay): T=1/( ¹ - ¸ ) Little: ¤ T = Q (also for the entire network) Q=  i ¸ i /( ¹ i - ¸ i ) (for entire network) T=1/ ¤  i ¸ i /( ¹ i - ¸ i ) (average delay for entire network)

4. Delay optimal routing Given a set of edge inflows ¤ sd s,d: source, destination ¤ =  s,d ¤ sd A delay optimal route mimizes T under flow constraints and given inflows. Nonlinear cost Difficult non-convex problem.

5. Afine (linear cost) For a given link ”i” the cost of a flow is: b i + a i ¸ i Overall cost: C=  i b i + a i  sd ¸ sdi =  B +  i a i  sd ¸ sdi =  B +  sdi a i ¸ sdi ¸ sdi : is the flow from s to d routed over i Minimization of C w.r.t. { ¸ sdi } min{C}=B + min  i a i  sd ¸ sdi  B + min  sd  i a i ¸ sdi  B +  sd min  i a i ¸ sdi

6. Afine cost min{C}  B +  sd min  i a i ¸ sdi min  i a i ¸ sdi for fixed s,d s.t. { ¸ sdi } routes ¸ sd from s to d ¸ sd ½¸ sd ¸ sd ½¸ sd (1- ½ ) ¸ sd a1a1 a2a2 a3a3 a4a4 a5a5 a6a6 a 10 a 11 a 12 a9a9 a8a8 a7a7

7. Afine cost C  B + ¸ sd (a 1 + a 2 + a 3 + a 10 + a 11 + a 12 ) + ½¸ sd (a 4 + a 5 + a 6 ) + (1- ½ ) ¸ sd (a 7 + a 8 + a 9 ) Assume (a 4 + a 5 + a 6 ) < (a 7 + a 8 + a 9 ) Then C’ = B + ¸ sd (a 1 + a 2 + a 3 + a 10 + a 11 + a 12 ) + ¸ sd (a 4 + a 5 + a 6 ) < C Routing everything through a 4,a 5,a 6 is better Optimal routes are single path Optimal routes are optimal paths where link costs are {a i } ¸ sd ½¸ sd ¸ sd ½¸ sd (1- ½ ) ¸ sd a1a1 a2a2 a3a3 a4a4 a5a5 a6a6 a 10 a 11 a 12 a9a9 a8a8 a7a7

8. Afine approximation of delay T=1/ ¤  i ¸ i /( ¹ i - ¸ i ) ¸ i =  sd ¸ sdi T({ ¸ sdi }’) = T({ ¸ sdi })+  sdi ( ¸ sdi ’ - ¸ sdi )  T/  ¸ sdi  T/  ¸ sdi =  T/  ¸ i = a i =1/ ¤ ¹ i /( ¹ i - ¸ i ) 2 T({ ¸ sdi }’) = T({ ¸ sdi })+  sdi ( ¸ sdi ’ - ¸ sdi )  T/  ¸ sdi = T({ ¸ sdi })+  sdi ( ¸ sdi ’ - ¸ sdi ) a_i = T({ ¸ sdi })-  sdi ¸ sdi a_i +  sdi ¸ sdi ’ a_i = B +  sdi ¸ sdi ’ a_i

9. Small Flow Devitation Let ¸ sdi and ¸ sdi ’ both be a flows routing { ¤ sd } Then for 0<a<1, ¸ sdi ’’ = a ¸ sdi +(1-a) ¸ sdi ’ is also a flow routing { ¤ sd } (The set of flows is convex) ¸ sdi ’’ may not comply to capacity contraints, but if ¸ sdi does we can find a ¼ 1 so that ¸ sdi ’’ also does

10. The Flow Deviation Method Fratta, Gerla and Kleinrock (1973) Given a flow { ¸ sdi } For each s,d take a portion of the flow and re- route Re-routing will be along shortest paths w.r.t. link costs:  T/  ¸ sdi = a i =1/ ¤ ¹ i /( ¹ i - ¸ i ) 2 Since cost is approximately affine for small deviations, the new flow will have lower cost

11. The Flow Deviation Method Fratta, Gerla and Kleinrock (1973) Let old flow be: { ¸ sdi } and new flow: { ¸ sdi ’} Find a: 0<a<1, mimimizing T({ ¸ sdi ’’}) = T({a ¸ sdi +(1-a) ¸ sdi ’}) Fratta, et. al. suggest: The portion rerouted is the portion previously routed along highest cost route w.r.t {a i }

12. Miniproject Suggest a network flow problem for which the Flow Deviation Method is meaningfully illustrated. Write a program implementing the Flow Deviation Method for the suggested network. Initial routes could be shortest paths (hop- count)

13. Wavelength Division Multiplexing WDM

14. Routing and wavelength assignment Since wavelengths are not mixed, traffic sources do not interfere. Queueing is limited to edge nodes. Routing delay is purely propagation – proportional to hop count. Routes may be assigned initially and wavelengths assigned afterwoods. Routes may be assigned as shortest path and in traffic load order

15. Wavelength assignment Given routes R={R i } previously assigned. Create the auxiliary graph G V(G): routes in R (R i,R j ) 2 E(G) iff R i share fibre with R j Assign colours to vertices (routes), such that no neighbouring vertices have identical colours. Classical graph colouring problem. The problem: is it possible i n colours ? is generally NP-complete The 4-colour problem for planar graphs is solved. (colouring of countries on map)

16. Wavelength assignment Sequential policy (colours are used in order): given a vertice order v 1,..,v n 1) select a colour for v 1 2) j=2 3) If (all assigned colours are in the neighbours of v j ) assign new colour to v j else Assign v j an already assigned colour not within its neighbours. 4) j=j+1 5) if (j<=n) goto 3

17. Wavelength assignment Smallest number of colours  (G): chromatic number of G Th: Some sequential assignment uses  (G) colours Proof:Given a  (G) assignment Define a colour ordering Order vertices in colour order v 1,..,v n defines a sequential assignment There are n! (!!!!) different orderings.

18. Operational sequential policy  (G) · deg(G)+1 = max {deg(R i )} +1 If deg(G)+1 colours assigned, there is allways 1 non-neighbouring color assigned and no new colour is needed Heuristic: assign colours to highest degree nodes first.

19. Operational sequential policy Ordering: deg(v 1 ) ¸ deg(v 2 ).. ¸ deg(v n ) Â (G) · max 0 <= i <= n min{i,1+ deg(v i )} Proof: i deg(v i )+1 i At step i we consider the subgraph v 1,..,v i As long as i · deg(v i )+1 assign i colours – one to each vertice Let i* be the smallest i so that i ¸ deg(v i )+1 Then i* ¸ deg(v i )+1 for all i ¸ i* For i ¸ i* we need no new colours

20. Alternate order Given a vertice order v 1,..,v n Number of colours needed: · max 1<= i <= n (1+deg i (v i )) deg j (v i ): the degree of v i in the subgraph (v 1,..,v j ), where i · j Proof: We assign colours in in order v 1,..,v n. When assigning colour to v i, you will never need more colours than 1+deg i (v i ) Note: deg i (v i ) · deg(v i ) 8 i

21. Alternate order 1) select minimum degree vertice v n 2) j=n-1 3) Select v j so that deg j (v j ) is minimum (deg j (v j ): degree in G-(v n,..,v j+1 )) 4) j=j-1 5) if (j ¸ 1) goto 3) The order obtained mimimizes deg j (v j ) for all j

22. Miniproject Suggest a network flow problem for which the colouring schemes may be meaningfully illustrated Write a program that: - Finds shortest path routes for s,d pairs - Constructs the auxiliary graph - Estimates the number of colours needed for colouring in degree order - Performs the colouring in degree order - Finds the alternate order - Estimate number of colours needed in alternate order - Performs colouring in alternate order

23. Linear programming (ILP) Number of colours All lines have at most Fmax colours Flow equations pr colour -- Wavelength continuity Number of paths required from s to d Number of LP over link i,j between s and d, at colour w At most 1 LP over link i,j at colour w Free variables:

24. ILP example ¤ 1 = ¤ 2 = ¤ 3 = 1 F i,w j 2 {0,1} (s=j 2 {1..3}, d fixed)  w,j F i,w j · F  w ¸ j,w = ¤ j  j F i,w j · 1 j=1: s=1=j, d=3: F 3,w 1 - F 1,w 1 = - ¸ 1,w s=2, d=1=j: F 3,w 2 - F 1,w 2 = ¸ 2,w s=3, d=2: F 3,w 3 - F 1,w 3 = 0 F1F2 F3 ¤1¤1 ¤2¤2 ¤3¤3 1 2 3

25. YALMIP %YALMIP program for exercise in Network Performance %A number of integer F matrices - one for each colour F=3; %number of colours F1=intvar(3,3); F2=intvar(3,3); F3=intvar(3,3); %lambdas lambda=binvar(3,F); %Lambdas; L=ones(3,1); %problem formulation P1=[lambda*ones(F,1)==L]; P2=[(F1+F2+F3)*ones(3,1) <= F*ones(3,1)] P3=[F1*ones(3,1)<=ones(3,1),F2*ones(3,1)<=ones(3,1),F3*ones(3,1)<=ones(3,1)]; P=[P1 P2 P3]; S=solvesdp(P);

26. Miniproject Encode flow constraints in the YALMIP program for the example network

27. Miniproject Write a summary on the paper: Wavelength-Routed Optical Networks: Linear Formulation, Resource Budgeting Tradeoffs, and a Reconfiguration Study by: Dhritiman Banerjee and Biswanath Mukherjee Objective ? Method ? Results ?