1 P-Cycles

2 What’s a p-Cycle? A preconfigured cycle formed out of the spare capacities in the network –A p-cycle uses one unit of spare capacity on each span Protect both on-cycle spans and straddling spans –On-cycle span: a span on the cycle –Straddling span: a span with two end nodes on cycle but itself not on cycle A p-cycle can protect one unit of working capacity on each on-cycle span and two units of working capacity on each straddling span

3 Redundancy of p-Cycles Redundancy = # spare capacity units used / # working capacity units that can be protected = l /(2*s+l) –s: # straddling spans –l: # on-cycle spans A Hamiltonian p-Cycle can protect every span in the network –Redundancy = N/(N+2(S-N))=1/d-1, where S is the number of spans in the network and d is the average node degree

4 Advantages of p-Cycles Real-time switching simplicity and speed of rings Mesh-like efficiency: very close to span restorable mesh networks P-cycle networks can support dynamic service provisioning under the protected working capacity envelope concept [1] [1] W.D. Grover, "The Protected Working Capacity Envelope Concept: An Alternative Paradigm for Automated Service Provisioning," IEEE Communications Magazine, January 2004, pp

5 Protected Working Capacity Envelope Concept A given distribution of spare channels determines a corresponding maximum number of protected working channels (W i ) on each span As long as a new service path is routed over spans with S i < W i, the service path is inherently protected –S i : total number of channels used by existing service paths on span i Greatly simplify dynamic provisioning of protected services

6 Optimal Capacity Design of p- Cycle Networks Given the working capacities on each span, determine a set of p-cycles that minimizes the total spare capacity required for 100% restorability Solution approach –Generate a set P that contains all elementary cycles of the network graph –Solve an ILP with p-cycles chosen from P

7 An ILP for Optimal Capacity Design Inputs to the ILP: S: set of spans in the network P: set of elementary cycles of the graph  k,j : 1 if cycle j includes span k, 0 otherwise w i : number of working channels on span i c k : cost or length of span k x i,j : number of useful paths cycle j can provide for restoration of span i. x i,j =0 if one or both end nodes of span i are not on cycle j x i,j =1 if span i is on cycle j x i,j =2 if span i straddles cycle j

8 An ILP for Optimal Capacity Design Variables to be solved by ILP: s i : number of spare channels on span i n j : number of unit-capacity copies of cycle j Minimize Subject to

9 Heuristic Approaches to p-Cycle Design An Iterative Design Algorithm Provide a reduced number of p-cycle candidates to ILP –Preselection of candidate p-cycles –Algorithmic approaches for generating candidate p-cycles

10 An Iterative Design Algorithm A priori efficiency of p-cycle j AE(j): –AE is a measure of the potential efficiency of a p-cycle Demand weighted efficiency of p-cycle j DWE(j): –w i : the uncovered amount of working capacity –DWE measures the actual efficiency of a p-cycle

11 An Iterative Design Algorithm 1.Enumerate all cycles up to a circumference limit; initialize w i to the total working capacities on span i 2.For every cycle j, compute DWE(j) 3.Find the p-cycle with the highest DWE score and place the p-cycle into the design 4.Update the uncovered w i values: p = winning cycle For all on-cycle span of p: w i = w i –1 For all straddling span of p: w i = w i –2 5.Repeat 2-4 until w i = 0 for all i

12 Preselection of Candidate p- Cycles 1.Enumerate all elementary cycles of the network graph 2.Choose the K best cycles ranked by AE score and provide them to the ILP Drawback: enumerating all cycles is time consuming –Number of elementary cycles grows exponentially with the network size

13 Algorithmic Approaches for Generating Candidate Cycles Goal: find a set of p-cycles that protect every span in the network without enumerating all cycles Four algorithms –SSA –SP-Add –Expand –Grow

14 Straddling Span Algorithm (SSA) For each span in the network, create a cycle such that the span straddles the cycle Three cases of a span A-B –Case a: both A and B are degree 3 or higher; two node-disjoint paths exist between A and B –Case b: both A and B are degree 3 or higher; there are no two node-disjoint paths between A and B –Case c: at least one of A and B is of degree 2

15 SSA For each span in the network: –Find the shortest path between its end nodes and the next shortest node-disjoint path if the latter exists –Case a: form a p-cycle on the two node-disjoint paths –Case b: form a p-cycle on the span plus the one path between the two end nodes –Case c: no action If a span of case c is unprotected, add a p-cycle of type b # created p-cycles  # spans in the network

16 Advanced p-Cycle Operations Cycles produced by SSA are generally inefficient  Want transform the cycles created by SSA into a set of higher efficiency cycles Key idea: convert an on-cycle span into a straddling span –SP-Add, Expand, Grow

17 SP-Add Start with the set A of cycles created by SSA For each cycle p in A –For each on-cycle span of p, a new cycle is constructed by replacing the span with the shortest path between the span’s end nodes that is node-disjoint from the original cycle, if such a path exists # cycles created  m 2, m is the number of spans in the network

18 Performance of SP-Add J. Doucette, D. He, W. D. Grover, O. Yang, “Algorithmic approaches for efficient enumeration of candidate p-cycles and capacitated p-cycle network design,” Proc. Fourth International Workshop on the Design of Reliable Communication Networks (DRCN 2003), Banff, Alberta, Canada, Oct. 2003, pp

19 Expand Operation Similar to SP-Add except that –After processing a span of the cycle, carry the new cycle along when checking the next span on the original cycle –Each time when expanding the cycle by adding a route to it, ensure that the route is not only disjoint from the original cycle, but also disjoint from every previously added routes –Can keep or discard the intermediate cycles

20 Grow Operation Start with the set A of cycles produced by SSA Use the SP-Add algorithm on A to produce cycle set B For each cycle in B –Visit the first span in the cycle and find the shortest route between its end nodes that is node-disjoint from the cycle –If such a route doesn’t exist, move on to the next span and try again

21 Grow Operation –If such a route exists Replace the span with the route to create a new cycle Visit the first span of the new cycle and start over, continue until no further route additions can be made Each intermediate cycle found along the way is retained, including cycles in A and B –Idea: produce a wide range of cycles Grow generates no more than m 2 n cycles, where m/n is the number of spans/nodes in the network

22 Performance of Expand and Grow J. Doucette, D. He, W. D. Grover, O. Yang, "Algorithmic Approaches for Efficient Enumeration of Candidate p-Cycles and Capacitated p-Cycle Network Design," Proceedings of the Fourth Intl Workshop on the Design of Reliable Communication Networks, Banff, October 2003, pp