1. p-Cycle Network Design: from Fewest in Number to Smallest in Size Diane P. OnguetouWayne D. Grover Diane P. Onguetou and Wayne D. Grover TRLabs and ECE, University of Alberta donguetou@trlabs.ca, grover@trlabs.ca October 8 th, 2007

2. What is a p-Cycle? MIL VIE ZUR PAR LUX BRU LON AMS COP BER PRA on-cycle spans straddling spans

3. p-Cycle Operating Principle MIL VIE ZUR PAR LUX BRU LON AMS COP BER PRA MIL VIE ZUR PAR LUX BRU LON AMS COP BER PRA Loopback in the event of on-cycle span failure Break-in handling a straddling span failure

4. Hamiltonian: One Type of p-Cycle MIL VIE ZUR PAR LUX BRU LON AMS COP BER PRA  Visits all nodes once,  Not necessarily crosses all spans,  Single structure can be enough for full single failure restorability.

5. Hamiltonian p-Cycle Network Design Having only a single structure may be attractive from the network management view: HOWEVER  Some network graphs are not Hamiltonians.  Even if the graph is Hamiltonian, this is only one option for p-cycle network design.  The most capacity-efficient p-cycle network design is not obtained by using a Hamiltonian  The most capacity-efficient p-cycle network design is not obtained by using a Hamiltonian.  Hamiltonians may be very long structures.

6. Recall the “3 Little Bears” "This porridge is too hot!" "This porridge is too cold," "Ahhh, this porridge is just right," Our aim: show how with p-cycle networks you can have just what you want….Fewest cycles, or least capacity, or anything in between,,,,Whatever is “just right” for Goldilocks Networks or your network. This clarifies a misunderstanding of late in part of the field. However, fewest and smallest structures at minimum capacity are some interesting new design goals suggested by the focus on number of structures and their size-circumference as well. number of structures spare capacity size of structures

7. Outlines 1.Motivations 2.Conventional p-Cycle Network Design 3.Design with an Emphasis on Fewest Number of Structures 4.Controlling the Size of p-Cycles 5.Concluding Discussion

8. The COST239 Network  11 nodes and 26 spans, average nodal degree of 4.72.  3531 distinct eligible p- cycles of which 394 are Hamiltonians.  55 demand-pairs uniformly distributed on [1…20].  Shortest distance based routing. MIL VIE ZUR PAR LUX BRU LON AMS COP BER PRA

9. Working Capacities to Be Protected MIL VIE ZUR PAR LUX BRU LON AMS COP BER PRA 3 2 5 Objective: Objective: Minimize spare capacity cost while ensuring full restorability against single span failures.

10. Basic Minimum Capacity Design  55% of redundancy.  9 distinct structures of which 4 are Hamiltonians.  16 unit- channel copies. 2 copies 3 copies 1 copy2 copies1 copy 3 copies

11. Comparison with Hamiltonian Solutions Eligible Cycles All cycles (3531) Hamiltonian cycles (394) One single Cycle The shortest Hamiltonian Redundancy55%58%66%90% Distinct Structures 9611

12. Outline 1.Motivations 2.Conventional p-Cycle Network Design 3.Design with an Emphasis on Fewest Number of Structures 4.Controlling the Size of p-Cycles 5.Concluding Discussion

13. Fewest Structures… A Different Goal in Design Another property of the conventional p-cycle ILP is the fact that it might have multiple solutions for the same capacity cost.  Therefore, using a bi-criterion objective in the ILP design model could help to bias the model towards always using the fewest number of cycle structures without capacity penalty.  Doing so in the COST239 network, we found that there is a solution involving 8 structures (instead of 9) for zero capacity penalty.

14. Set a Fixed Number of Structures It is also possible to force the ILP to design under a given maximum number of structures. Of course this involves capacity penalty, but apparently this is not so significant. So it might be more useful to design with fewest structures and no significant increase in spare capacity. e.g. +1%  5 structures, +5%  3 structures, (versus 8 or 9).

15. R 2 Restorability vs. Fewest Structures  However, be certain that playing with the number of structures matches all your goals.  For instance, selecting fewer structures somewhat harms the robustness under dual failure conditions.

16. Outlines 1.Motivations 2.Conventional p-Cycle Network Design 3.Design with an Emphasis on Fewest Number of Structures 4.Controlling the Size of p-Cycles 5.Concluding Discussion

17. Impacts on Capacity Requirements Well known that limiting the circumference-size of eligible cycles very  Not feasible for very small limits,  Requires some additional capacities (especially for small limits),  Decreasing function in general,  and Steady state for large limits. Already discussed by D. Schupke, C. G. Gruber and A. Autenrieth in ICC’02.

18. Fewest Structures vs. Smallest Sizes  More structures tend to be required when p-cycles are constrained to the smallest sizes.  However, the plot fluctuates between successive values of fewest structures.  For very large maximums, the ILP model keeps the optimal solution and thus, the same number of fewest p-cycle structures.

19. R 2 Restorability vs. Smallest Cycles As the design is forced to use smaller cycles, the R 2 benefits significantly  In fact, as a side-effect of their being more protection structures over which dual failures are in effect dispersed as parts of single failures which are less likely to affect the same cycle.

20. Outlines 1.Motivations 2.Conventional p-Cycle Network Design 3.Design with an Emphasis on Fewest Number of Structures 4.Controlling the Size of p-Cycles 5.Concluding Discussion

21. Conclusion  Hamiltonian Solutions vs. Conventional p-Cycle Network Design.  -Clarifies the misunderstanding in some papers.  Since using a single shortest Hamiltonian cycle is attractive from a management view, study of designs with an Emphasis on Number of Structures.  Small-circumference cycles might be desired to eliminate the need of signal regeneration en-route: controlling the size of p-cycles in the design  Tradeoff between capacity requirements, number of structures and circumference-size of p-cycles.

22. Thank You!!!