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Robust Designs for WDM Routing and Provisioning
Jeff Kennington & Eli Olinick
Southern Methodist University
Augustyn Ortynski & Gheorghe Spiride
Nortel Networks
1
LTE LTE
LTE LTE
LTE LTE
… …
LTE LTE
LTE LTE
LTE LTE
… …
TE TE
TE TE
TE TE
… …
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
… …A A
A A
… …A A
A A
… …
LTE LTE
LTE LTE
LTE LTE
… …
LTE LTE
LTE LTE
LTE LTE
… …
TE TE
TE TE
TE TE
… …
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
… …
Basic Building Block for the WDM Network
Optical Amplifier
Regenerator
2
Unmet demandSatisfied demandExcess capacity
UnderprovisionedCase
Dallas Atlanta
Perfect MatchCase
LA Phoenix
OverprovisionedCase
Boston NYC
3
0
50
100
150
200
250
300
350
400
450
-12.5 -10 -7.5 -5 -2.5 0 2.5 5 7.5 10 12.5
OverprovisionedCase
UnderprovisionedCase
Perfect MatchCase
Underprovisioning Overprovisioning
Regret
4
7
4
3
1
2
5
6
Atlanta
San Francisco
Los Angeles
Dallas
Chicago Boston
New York
Example
5
ATL BOS CHI DAL LA NY SF
ATL - - 1126 1287 3540 1368 -
BOS - 1609 - - 322 -
CHI - 1448 - 1287 3540
DAL - 2253 - 2816
LA - - 644
NY - -
SF -
Distance Matrix (KM)
6
ATL BOS CHI DAL LA NY SF
ATL - 5000 3700 8800 3000 9500 -
BOS - - 8600 7100 - 7400
CHI - - - - -
DAL - - - -
LA - - -
NY - 5400
SF -
Scenario #1 Demand Matrix (DS3s)
7
ATL BOS CHI DAL LA NY SF
ATL - 15000
13700
18800
13000
19500
-
BOS - - 18600
17100
- 17400
CHI - - - - -
DAL - - - -
LA - - -
NY - 15400
SF -
Scenario #2 Demand Matrix (DS3s)
8
ATL BOS CHI DAL LA NY SF
ATL - 25000 23700 28800 23000 29500 -
BOS - - 28600 27100 - 27400
CHI - - - - -
DAL - - - -
LA - - -
NY - 25400
SF -
Scenario #3 Demand Matrix (DS3s)
9
Scenario 2Scenario 1
Scenario 3 Robust Solution
Figure 5. Solutions 10
Basic design model
Minimizecx (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
Decision VariablesScenarios Model
Variables Robust Model
Variables Variable
Type Description
xsp xp continuous number of DS3s assigned to path p
sn n continuous number of TEs assigned to node n
tse te continuous number of TEs 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
ConstantsConstant Value or Range Description
Rsod 300-1500 traffic demand for pair (o,d) in scenario s in units of DS3s
MTE 192 number of DS3s that each TE can accommodate
MR 192 number of DS3s that each regen can accommodate
MA 15,360 number of DS3s that each optical amplifier can accommodate
CTE 50,000 unit cost for an TE
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
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 TEs e(accumulat
(4)
TEs) 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
TEse
Lp
se
sp
sod
Jp
sp
Nn Ee
se
Ase
Rsn
TE
n
e
od
Robust model
(16)
TEs) 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
TEe
eLp
p
odsodssod
Jp
sp
Eee
Ae
R
Nnn
TE
kodskods
kodskods
Dod kodsk
okodsk
uk
e
od
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
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
Stochastic Programming Model
(24)-(13) sconstraint
Subject to
min
ityinfeasibilfor cost penalty thebe Let
),(
Ss Ddoodsodss zzdPE
d
Worst Case Model
(24)-(13) sconstraint
Subject to
max Minimize),(
Ss Ddo
odsodsSs
zzdE
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
2960
2437
21
3119
2327 34
4 50
18 17 49
404865
3
14
69
66
9 22 6821
11 1643
10
635626
5125 53
5 2861
33
45 47 42
4135
59
5412
58
44
55 8
64
713
62
38
39 57
52
67
3632
3046
6
20
DA Test Problem 16
11 12
3
15
4
6
16
1
10
9
2
8
13
14 18
7
17
5
Legend
1 Brussels
2 Copenhagen
3 Paris
4 Berlin
5 Athens
6 Dublin
7 Rome
8 Luxembourg
9 Amsterdam
10 Oslo
11 Lisbon
12 Madrid
13 Stockholm
14 Zurich
15 London
16 Zagreb
17 Prague
18 Vienna
European Problem17
DA – method comparison
Scenario Prob. TEs 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
DA – results
Budget Method TEs 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
KL – individual scenarios
Scenario Prob. TEs 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
KL – method comparison
Budget Method TEs 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
Network Protection
Dedicated Protection – 1 + 1 Protection
P-Cycle Protection – Grover Stamatelakis (restoration speed of bi-directional rings at the cost of shared protection)
Shared Protection – Path Restoration
25
2
6
15
4
3No Protection
TE = 2 A = 11 R = 5 Cost = 6.00
2
2
6
15
4
3P-Cycle
TE = 8 A = 32 R = 20 Cost = 18.00
2
6
15
4
3Dedicated
TE = 6 A = 32 R = 15 Cost = 17.5
2
6
15
4
3Shared
TE = 6 A = 32 R = 15 Cost = 17.50
Example 1 Demand: (1,4) of 192 DS3s (1 )
26
2
6
15
4
3No Protection
TE = 6 A = 20 R = 13 Cost = 11.34
2
2
6
15
4
3P-Cycle
TE = 18 A = 32 R = 43 Cost = 20.34
2
6
15
4
3Dedicated
TE = 18 A = 32 R = 45 Cost = 20.50
2
6
15
4
3Shared
TE = 16 A = 32 R = 39 Cost = 19.92
Example 2 Demands: (1,4) 192 DS3s (1 ), (1,3) 384 DS3s ( 2 )
2
2
2
2
2
22
2
2
2
2
27
2
6
15
4
3No Protection
TE = 34 A = 45 R = 91 Cost = 31.48
2
2
6
15
4
3P-Cycle
TE = 70 A = 88 R = 185 Cost = 62.30
2
6
15
4
3Dedicated
TE = 70 A = 47 R = 181 Cost = 66.48
2
6
15
4
3Shared
TE = 72 A = 88 R = 194 Cost = 63.12
Example 3 Demands: (1,4) of 1 , (1,3) of 2 , (2,5) of 4
2
2
2
44
4
4
22
2
4
4
4
4
6
6
2
2
2
2
22
2 2
2
4
4
4
44
28
Protection Type
(1,4) = 1 (1,4) = 1(1,3) = 2
(1,4) = 1(1,3) = 2(2,5) = 4
None 6.00(working)
11.34(working)
31.48(working)
1+1 17.50 20.50 66.48
P-Cycle 18.00(same
working)
20.34(same
working)
62.30(new
working)
Shared 17.50 19.92 63.12
29