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Lecture 21

Mathematical Models Used To Model Telecommunication Design Problems

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Robust Designs for WDM Routing and Provisioning

Jeff Kennington, Karen Lewis, Eli OlinickSouthern Methodist University

Augustyn Ortynski, Gheorghe SpirideNortel Networks

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Objective of the work

Develop a robust design procedure for WDM routing and provisioning problems.

These problems come in three varieties based upon the protection requirements no protection, 1+1 protection shared protection

So far we have studied the “no protection” case

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The problem

Given The network topology An estimate of the traffic demands, and routing

assumptions Equipment capacity, modularity, and unit cost

assumptions Determine

Working and protection channel routing Required number of network elements at nodes and

on links. Several versions of this problem

Depending on protection requirements

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The goal

Design for given point forecast However,

Traffic growth is difficult to predict Uncertain point forecasts to start with

Therefore, An optimal design for an erroneous forecast

may prove to be inferior. The goal is to develop a network design

that will be robust over a variety of demand forecasts.

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The proposed approach

Consider a set of scenarios, each with a given probability of occurrence

A fixed budget to cover cap expenses Create a network design that

minimizes the regret over the range of scenarios, while the total equipment cost is below the budget

The regret associated with a design penalizes non-robust designs

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Equipment modeling – sample network link

LTE LTE

LTE LTE

LTE LTE

… …

LTE LTE

LTE LTE

LTE LTE

… …

LTE LTE

LTE LTE

LTE LTE

… …

R R

R R

R R

… …

R R

R R

R R

… …

R R

R R

R R

… …

A A

A A

… …A A

A A

… …A A

A A

… …A A

A A

… …

R R

R R

R R

… …

R R

R R

R R

… …

R R

R R

R R

… …

A A

A A

… …A A

A A

… …

LTE LTE

LTE LTE

LTE LTE

… …

LTE LTE

LTE LTE

LTE LTE

… …

LTE LTE

LTE LTE

LTE LTE

… …

A A

A A

… …A A

A A

… …

A A

A A

… …A A

A A

… …

Note: the cost of WDM couplers is included in the LTE/R cost

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Equipment modeling

Nodal equipment LTEs have a given modularity

Line equipment Regenerators have given modularity Optical amplifiers have a larger

modularity

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Other assumptions

Demand is expressed in DS3 Line capacity is OC192 Routing candidate paths are

computed and fed into the model In this analysis we consider the first k-

shortest paths as candidates for each demand

A given maximum number of candidate routings is considered for each demand

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Modeling uncertainty

Scenario #

ProbabilityPt-to-Pt Demand Matrix

1 0.15 D[1]

2 0.20 D[2]

3 0.30 D[3]

4 0.20 D[4]

5 0.15 D[5]

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Solution approaches

Robust optimization Design a network that minimizes regret

Other approaches from the literature Stochastic Programming

Minimize overall cost (equip. + penalty) Worst-Case

Minimize the maximum cost Mean-Value

Compute expected value of demand and use the basic design approach

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What is regret?

Time 0 – Build Network

Time t later – Demand is known

Case 1: Under Provision

(can not meet demand for some (o,d) pairs)

Case 2: Over Provision

(there is excess capacity)

Regret is a piece-wise linear approximation to a quadratic

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Regret example

1.40E+07

5.40E+07

1.23E+08

2.18E+08

1.00E+06

5.10E+07

1.01E+08

1.51E+08

2.01E+08

2.51E+08

0 2000 4000 6000 8000 10000 12000 14000 16000

Positive underprovisioning

Reg

ret

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Regret example

3.00E+06

1.30E+07

3.10E+07

5.40E+07

1.00E+06

5.10E+07

1.01E+08

1.51E+08

2.01E+08

2.51E+08

0 2000 4000 6000 8000 10000 12000 14000 16000

Positive overprovisioning

Reg

ret

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Basic design model

Minimize cx (equip. cost)

Subject to

Ax = b (structural const)

Bx = r (demand const)

0 < x < u (bounds)

xj integer for some j (integrality)

Integer Linear Program

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Decision variables

Scenarios Model Variables

Robust Model Variables

Variable Type

Description

xsp xp continuous number of DS3s assigned to path p

sn n continuous number of LTEs assigned to node n

tse te continuous number of LTEs assigned to link e

ase ae continuous number of optical amplifiers assigned to link e

rse re continuous number of regens assigned to link e

fse fe integer number of fibers assigned to link e

cse ce integer number of channels assigned to link e

zse ze continuous number of DS3s assigned to link e

- z+ods continuous

positive infeasibility for demand (o,d) and scenario s

- z-ods continuous

negative infeasibility for demand (o,d) and scenario s

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Constant definitions

Constant Value or Range Description

Rsod 300-1500 traffic demand for pair (o,d) in scenario s in units of DS3s

MLTE 192 number of DS3s that each LTE can accommodate

MR 192 number of DS3s that each regen can accommodate

MA 15,360 number of DS3s that each optical amplifier can accommodate

CLTE 50,000 unit cost for an LTE

CR 80,000 unit cost for a regen

CA 500,000 unit cost for an optical amplifier Fe 24 max number of fibers available on link e

R 80km max distance that a signal can traverse without amplification, also called the reach

Q 5 max number of amplified spans above which signal regeneration is required

Be 2-1106 the length of link e

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Routing for scenario s

(9)

(8)

rs)regenerato and amplifiers into channels andfiber (convert

(7)

(6)

channels) and fibers intocapacity link (convert

(5)

links)on LTEs e(accumulat

(4)

LTEs) ocapacity tlink (convert

(3)

capacity)link ocapacity tpath (convert

(2) ),( R

on)satisfacti (demand

(1) )( Minimize

EercG

EeafG

EecMz

EefMz

Nnlt

EetMz

Eezx

Ddox

aCrClC

se

se

Re

se

se

Ae

se

Rse

se

Ase

sn

Ae

se

se

LTEse

Lp

se

sp

sod

Jp

sp

Nn Ee

se

Ase

Rsn

LTE

n

e

od

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Robust model

(16)

LTEs) tolinkson DS3s ofn (conversio

(15)

flows)link toflowspath ofn (conversio

(14) -

s)constraint (demand

(13)

)constraint(budget

(12) ,),(

(11) ,),(

pieces)function regret ofion (accumulat

(10) P Minimize

e

4

1

4

1

Ss

4

1s

EetMz

Eezx

EezzRx

BudgetaCrClCE

SsDdozz

SsDdozz

zczc

LTEe

eLp

p

odsodssod

Jp

sp

Eee

Ae

R

Nnn

LTE

kodskods

kodskods

Dod kodsk

okodsk

uk

e

od

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Robust model (cont.)

(24)

fibers)on (bounds

(23) 4,...1,,),( 4

R0

(22) 4,...1,,),( 4

R0

pieces) individualon (bounds

(21)

(20)

regens) and amplifiers tochannels andfiber ofn (conversio

(19)

(18)

channels) and fibers tolinkson DS3s ofn (conversio

max

max

Eecf

kSsDdoz

kSsDdoz

EercG

EeafG

EecMz

EefMz

ee

odsk

odsk

eeRe

eeAe

eR

e

eA

e

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Mean-Value model

(24)-(15) (13), sconstraint and

),(

Subject to

Minimize

),(

bygiven be scenarios demand theofmean Let the

DdoRx

E

DdoRPR

odJp

p

Ss

sodsod

od

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Stochastic Programming model

(24)-(13) sconstraint

Subject to

min

ityinfeasibilfor cost penalty thebe Let

),(

Ss Ddoodsodss zzdPE

d

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Worst Case model

(24)-(13) sconstraint

Subject to

max Minimize),(

Ss Ddo

odsodsSs

zzdE

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Regional US network – DA problem

European multinational network – KL problem

Test problems overview

Source Total Nodes 67 Total Links 107 Total Demand Pairs 200 Number of Paths/Demand 4 Total Demand Scenarios 5

Source Total Nodes 18 Total Links 35 Total Demand Pairs 100 Number of Paths/Demand 4 Total Demand Scenarios 5

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DA – method comparison

Scenario Prob. LTEs Rs As CPU Seconds Equipment Cost 1 0.15 24,996 3962 563 0.5 1,848,000,000 2 0.20 39,456 6502 864 0.5 2,925,000,000 3 0.30 51,882 8074 1101 0.5 3,791,000,000 4 0.20 65,086 10,122 1355 0.6 4,742,000,000 5 0.15 76,848 12,447 1584 0.5 5,630,000,000

Expected Value

— 51,749 8,208 1096 — 3,792,000,000

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DA – results

Budget Method LTEs Rs As

Equipment Cost

CPU Seconds

Unrouted Demand

Scaled Regret

Mean Value 51,800 8117 1081 3,780,000,000 0.7 15.5% 1.40 Stoch. Prog. 44,373 7446 918 3,273,000,000 1.8 20.4% 1.82 5,630,000,000 Worst Case 39,098 5495 757 2,773,000,000 4.6 27.2% 3.75 Robust Opt. 63,122 10,813 1425 4,734,000,000 2.7 5.2% 1.00 Mean Value 51,800 8117 1081 3,780,000,000 0.2 15.5% 1.11 Stoch. Prog. 44,373 7446 918 3,273,000,000 0.6 20.4% 1.44 3,787,000,000 Worst Case 39,098 5495 757 2,773,000,000 2.1 27.2% 2.95 Robust Opt. 52,159 8108 1061 3,787,000,000 4.5 12.6% 1.00

Mean Value — — — No Feasible

Solution 0.3 100% —

Stoch. Prog. 25,583 3696 515 1,832,000,000 3.9 42.3% 1.15 1,848,000,000 Worst Case 27,180 2960 505 1,848,000,000 6.6 42.3% 1.51 Robust Opt. 25,856 3575 539 1,848,000,000 5.6 43.3% 1.00

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DA – under/over-provisioningBudget Provisioning Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 Totals

under LTE 11 381 2908 7838 16,536 27,674 under R 0 0 91 504 2076 2671 under A 1 4 49 126 254 434

5,630,000,000 over LTE 38,137 24,048 14,149 5875 2811 85,020 over R 6851 4311 2830 1195 442 15,629 over A 863 565 373 196 95 2092 under LTE 253 566 3969 13,832 25,322 43,942 under R 119 101 724 2080 4419 7443 under A 12 17 124 307 535 995

3,787,000,000 over LTE 27,482 13,334 4312 971 699 46,798 over R 4205 1647 698 6 20 6576 over A 513 217 87 16 15 848 under LTE 2731 14,180 26,145 39,157 50,922 133,135 under R 720 3082 4552 6555 8880 23,789 under A 82 341 572 822 1051 2868

1,848,000,000 over LTE 3663 653 191 0 3 4510 over R 352 147 45 0 0 517 over A 52 10 4 0 0 66

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KL – individual scenarios

Scenario Prob. LTEs Rs As CPU Seconds Equipment Cost 1 0.15 12,767 7275 638 0.3 1,539,356,770 2 0.20 17,493 11,691 958 0.3 2,288,919,583 3 0.30 24,020 15,783 1178 0.3 3,052,619,167 4 0.20 29,295 19,196 1455 0.2 3,727,940,417 5 0.15 35,732 23,606 1760 0.3 4,554,614,375

Expected Value

— 23,837 15,545 1196 — 3,033,250,000

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KL – method comparison

Budget Method LTEs Rs As Equip. Cost CPU

Seconds Unrouted

Demand Scaled Regret

Mean Value 25,124 15,350 1221 3,094,700,000 1.1 15.4% 1.41 Stoch. Prog. 20,264 14,168 996 2,644,620,000 0.6 20.8% 1.94

4,554,610,000 Worst Case 17,977 11,812 872 2,279,830,000 1.1 27.8% 4.05 Robust Opt. 27,520 21,348 1614 3,890,840,000 1.0 5.6% 1.00 Mean Value 23,978 15,382 1198 3,028,460,000 0.5 15.4% 1.11 Stoch. Prog. 20,264 14,168 996 2,644,620,000 0.2 20.8% 1.52

3,032,690,000 Worst Case 17,977 11,812 872 2,279,830,000 0.4 27.9% 3.20 Robust Opt. 23,967 15,548 1181 3,032,690,000 200.0 13.1% 1.00

Mean Value — — — No Feasible

Solution ? 100% —

Stoch. Prog. 12,154 7456 666 1,537,150,000 2.7 42.7% 1.19 1,539,360,000 Worst Case 12,782 7222 645 1,539,360,000 1.9 44.4% 1.71

Robust Opt. 13,562 7172 575 1,539,360,000 5.6 43.3% 1.00

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KL – under/over-provisioning

Budget Provisioning Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 Totals under LTE 965 1484 3993 7311 12,263 26,016 under R 294 722 1793 3500 6676 12,985 under A 22 48 112 259 442 883

4,554,610,000 over LTE 15,718 11,511 7494 5536 4060 44,319 over R 14,367 10,379 7358 5652 4418 42,174 over A 998 704 548 418 296 2964 under LTE 102 295 2302 6207 12,602 21,508 under R 52 263 1932 3963 8592 14,802 under A 14 34 139 296 615 1098

3,032,690,000 over LTE 11,151 6619 2099 729 696 21,294 over R 8369 4164 1741 359 578 15,211 over A 565 265 150 30 44 1054 under LTE 1372 5473 10,518 15,733 22,161 55,257 under R 918 5272 8641 12,024 16,434 43,289 under A 120 440 603 880 1185 3228

1,539,360,00 over LTE 2166 1542 60 0 0 3768 over R 815 753 30 0 0 1598 over A 57 57 0 0 0 114

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