17 th International Teletraffic Congress Topological design of telecommunication networks Michał Pióro a,b, Alpar Jüttner c, Janos Harmatos c, Áron Szentesi c, Piotr Gajowniczek b, Andrzej Mysłek b a Lund University, Sweden b Warsaw University of Technology, Poland c Ericsson Traffic Laboratory, Budapest, Hungary

1/15 © M.Pioro, A.Jüttner, J.Harmatos, Á.Szentesi, P.Gajowniczek, A.Mysłek Topological Design of Telecommunication Networks Outline Background Network model and problem formulation Solution methods –Exact (Branch and Bound) and the lower bound problem –Minoux heuristic and its extensions –Other methods (SAN and SAL) –Comparison of results Conclusions

2/15 © M.Pioro, A.Jüttner, J.Harmatos, Á.Szentesi, P.Gajowniczek, A.Mysłek Topological Design of Telecommunication Networks Background of Topological Design problem: localize links (nodes) with simultaneous routing of given demands, minimizing the cost of links selected literature: Boyce et al branch-and-bound (B&B) algorithms Dionne/Florian 1979 – B&B with lower bounds for link localization with direct demands Minoux problems’ classification and a descent method with flow reallocation to indirect paths for link localization

3/15 © M.Pioro, A.Jüttner, J.Harmatos, Á.Szentesi, P.Gajowniczek, A.Mysłek Topological Design of Telecommunication Networks Transit Nodes’ and Links’ Localization – problem formulation Given –a set of access nodes with geographical locations –traffic demand between each access node pair –potential locations of transit nodes find –the number and locations of the transit nodes –links connecting access nodes to transit nodes –links connecting transit nodes to each other –routing (flows) minimizing the total network cost

4/15 © M.Pioro, A.Jüttner, J.Harmatos, Á.Szentesi, P.Gajowniczek, A.Mysłek Topological Design of Telecommunication Networks Symbols used constants h d volume of demand d a edj =1 if link e belongs to path j of demand d, 0 otherwise c e cost of one capacity unit installed on link e k e fixed cost of installing link e Bbudget constraint M e upper bound for the capacity of link e variables x dj flow realizing demand d allocated to path j (continuous) y e capacity of link e (continuous)  e =1 if link e is provided, 0 otherwise (binary)

5/15 © M.Pioro, A.Jüttner, J.Harmatos, Á.Szentesi, P.Gajowniczek, A.Mysłek Topological Design of Telecommunication Networks Network model adequate for IP/MPLS LER  access node LSR  transit node LSP  demand flow LER LSR LER LSR LSP L1 L2 L3 L4

6/15 © M.Pioro, A.Jüttner, J.Harmatos, Á.Szentesi, P.Gajowniczek, A.Mysłek Topological Design of Telecommunication Networks Optimal Network Design Problem and Budget Constrained Problem ONDP minimize C =  e c e y e +  e k e  e constraints  j x dj = h d  d  j a edj x dj = y e y e  M e  e BCP minimize C =  e c e y e constraints  e k e  e  B  j x dj = h d  d  j a edj x dj = y e y e  M e  e

7/15 © M.Pioro, A.Jüttner, J.Harmatos, Á.Szentesi, P.Gajowniczek, A.Mysłek Topological Design of Telecommunication Networks Solution methods Specialized heuristics Simulated Allocation (SAL) Simulated Annealing (SAN) Exact algorithms: branch and bound (cutting planes)

8/15 © M.Pioro, A.Jüttner, J.Harmatos, Á.Szentesi, P.Gajowniczek, A.Mysłek Topological Design of Telecommunication Networks Branch and Bound method advantages –exact solution –heuristics’ results verification disadvantages –exponential increase of computational complexity –solving many “unnecessary” sub-problems 101

9/15 © M.Pioro, A.Jüttner, J.Harmatos, Á.Szentesi, P.Gajowniczek, A.Mysłek Topological Design of Telecommunication Networks Branch and Bound - lower bound LB proposed by Dionne/Florian 1979 is not suitable for our network model – with non-direct demands it gives no gain We propose another LB – modified problem with fixed cost transformed into variable cost: minimize C =  e  e y e +  e k e where  e = c e + k e /M e

10/15 © M.Pioro, A.Jüttner, J.Harmatos, Á.Szentesi, P.Gajowniczek, A.Mysłek Topological Design of Telecommunication Networks Minoux heuristics The original Minoux algorithm: step0 (greedy) allocate demands in the random order to the shortest paths: if a link was already used for allocation of another demand use only variable cost, otherwise use variable and installation cost of the link 1calculate the cost gain of reallocating the demands from each link to other allocated links (the shortest alternative path is chosen) 2select the link, whose elimination results in the greatest gain 3reallocate flows going through the link being eliminated 4if improvement possible go to step 2 elimination

11/15 © M.Pioro, A.Jüttner, J.Harmatos, Á.Szentesi, P.Gajowniczek, A.Mysłek Topological Design of Telecommunication Networks Minoux heuristics’ extensions individual flow shifting (H1) individual flow shifting with cost smoothing (H2) C e (y)=c e y + k e ·{1 - (1-  )/[(y-1)  +1]} if y > 0 = 0 otherwise. bulk flow shifting (H3) –for the first positive gain (H3F) –for the best gain (H3B) bulk flow shifting with cost smoothing (H4) –two versions (H4F and H4B)

12/15 © M.Pioro, A.Jüttner, J.Harmatos, Á.Szentesi, P.Gajowniczek, A.Mysłek Topological Design of Telecommunication Networks Other methods Simulated Allocation (SAL) in each step chooses, with probability q(x), between: –allocate(x) – adding one demand flow to the current state x –disconnect(x) – removing one or more demand flows from current x Simulated Annealing (SAN) starts from an initial solution and selects neighboring state: –changing the node or link status –switching on/off a node –switching on/off a transit or access link

13/15 © M.Pioro, A.Jüttner, J.Harmatos, Á.Szentesi, P.Gajowniczek, A.Mysłek Topological Design of Telecommunication Networks Comparison - objective

14/15 © M.Pioro, A.Jüttner, J.Harmatos, Á.Szentesi, P.Gajowniczek, A.Mysłek Topological Design of Telecommunication Networks Comparison - running time

15/15 © M.Pioro, A.Jüttner, J.Harmatos, Á.Szentesi, P.Gajowniczek, A.Mysłek Topological Design of Telecommunication Networks Conclusions proposed modification of Minoux algorithm can efficiently solve TNLLP, especially H4B Simulated Allocation seems to be the best heuristics proposed lower bound can be used to construct branch-and-bound implementations need for diverse methods - hybrids of the best shown here, e.g. Greedy Randomized Adaptive Search Procedure using SAL seems to be a good solution