17 th International Teletraffic Congress Topological design of telecommunication networks Michał Pióro a,b, Alpar Jüttner c, Janos Harmatos c, Áron Szentesi

17th International Teletraffic Congress

Topological design of telecommunication networks

Michał Pióroa,b, Alpar Jüttnerc, Janos Harmatosc,Áron Szentesic, Piotr Gajowniczekb, Andrzej Mysłekb

a Lund University, Swedenb Warsaw University of Technology, Polandc Ericsson Traffic Laboratory, Budapest, Hungary

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© 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

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Background of Topological Design

problem:localize links (nodes) with simultaneous routing of

given demands, minimizing the cost of links

selected literature:Boyce et al1973 - branch-and-bound (B&B) algorithmsDionne/Florian1979 – B&B with lower bounds for link

localization with direct demandsMinoux1989 - problems’ classification and a descent

method with flow reallocation to indirect paths for link localization

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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

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Symbols used

constantshd volume of demand d

aedj =1 if link e belongs to path j of demand d, 0 otherwise

ce cost of one capacity unit installedon link e

ke fixed cost of installing link eB budget constraintMe upper bound for the capacity

of link e

variablesxdj flow realizing demand d allocated to path j (continuous)

ye capacity of link e (continuous)

e =1 if link e is provided, 0 otherwise (binary)

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Network model adequate for IP/MPLS

• LER access node• LSR transit node• LSP demand flow

LER

LSR

LSR

LERLSR

LSR

LSP

L1

L2L3

L4L4

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Optimal Network Design Problemand Budget Constrained Problem

ONDP

minimize

C = e ce ye + e kee

constraints

j xdj = hd

dj aedj xdj = ye

ye Ł Mee

BCP

minimize C = e ce ye

constraints

e kee Ł B

j xdj = hd

dj aedj xdj = ye

ye Ł Mee

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Solution methods

• Specialized heuristics

• Simulated Allocation (SAL)

• Simulated Annealing (SAN)

• Exact algorithms: branch and bound (cutting planes)

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Branch and Bound method

• advantages– exact solution– heuristics’ results verification

• disadvantages– exponential increase of computational complexity– solving many “unnecessary” sub-problems

1 0 1

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Branch and Bound - lower bound

• LB proposed by Dionne/Florian1979 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 eye + eke

wheree = ce + ke /Me

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Minoux heuristics

The original Minoux algorithm:step 0 (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 link1 calculate the cost gain of reallocating the demands fromeach link to other allocated links (the shortest alternative path is chosen) 2 select the link, whose elimination results in the greatest gain3 reallocate flows going throughthe link being eliminated4 if improvement possiblego to step 2

elimination

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Minoux heuristics’ extensions

• individual flow shifting (H1)• individual flow shifting with cost smoothing (H2)

Ce(y) =cey + ke ·{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)

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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

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Comparison - objectiveRelative cost difference for ONDP with respect to the optimal solution [% ]network n H1 H2 H3F H3B H4F H4B SAN SALN7 0 0 0 0 0 0 0 0 0N7 1 0 0 0 0 0 0 0 0N7 2 0 0.90 0 0 0 0 0 0N7 3 4.90 7.78 4.90 4.90 3.39 3.39 0 1.55N7 5 114.23 110.84 20.29 20.29 13.02 0 11.66 0N7 6 125.61 125.82 19.64 19.64 5.99 0 12.71 0N14 0 0 0 0 0 0 0 0 0N14 1 0.02 0.05 0.02 0.02 0.03 0.03 0 0N14 2 0.91 1.15 0.63 0.63 0.26 0.35 0 0.44N14 3 10.27 5.65 8.11 8.11 3.03 2.95 1.31 2.26N14 5 128.35 17.92 43.86 37.73 10.70 10.70 25.4 4.39

Relative cost difference for TNLLP with respect to the optimal solution [% ]network n k H1 H2 H3F H3B H4F H4B SAN SALN14 4 4 24.13 11.59 24.13 24.13 3.43 2.28 41.24 0N14 4 5 23.72 11.39 23.72 23.72 3.37 2.24 39.63 0N14 4 6 22.73 11.97 22.73 22.73 4.97 3.98 25.21 3.55

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Comparison - running time

ONDP

0,01

0,1

1

10

100

1000

10000

0N7

2N7

5N7

0N14

2N14

5N14

0N28

2N28

5N28

run

nin

g t

ime

[s

]

TNLLP

0,1

1

10

100

1000

10000

(4,4)N14

(4,5)N14

(4,6)N14

(5,4)N14

(5,5)N14

(5,6)N14

(4,4)N28

(4,5)N28

(4,6)N28

(5,4)N28

(5,5)N28

(5,6)N28

run

nin

g t

ime

[s

]

H1

H2

H3F

H3B

H4F

H4B

SAN

SAL

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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

Documents

17 th International Teletraffic Congress Topological design of telecommunication networks Michał Pióro a,b, Alpar Jüttner c, Janos Harmatos c, Áron Szentesi