Mathematical Models Used To Model Telecommunication Design Problems

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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 Olinick
• Southern Methodist University
• Augustyn Ortynski, Gheorghe Spiride
• Nortel 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,
• 11 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
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 Probability Pt-to-Pt Demand Matrix
1 0.15 D1
2 0.20 D2
3 0.30 D3
4 0.20 D4
5 0.15 D5
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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
2.51E08
2.18E08
2.01E08
1.51E08
Regret
1.23E08
1.01E08
5.40E07
5.10E07
1.40E07
1.00E06
0
2000
4000
6000
8000
10000
12000
14000
16000
Positive underprovisioning
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Regret example
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Basic design model

• Minimize cx (equip. cost)
• Subject to
• Ax b (structural const)
• Bx r (demand const)
• 0 lt x lt u (bounds)
• xj integer for some j (integrality)
• Integer Linear Program

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Decision variables
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Constant definitions
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Routing for scenario s
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Robust model
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Robust model (cont.)
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Mean-Value model
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Stochastic Programming model
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Worst Case model
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Test problems overview

• Regional US network DA problem
• European multinational network KL problem

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DA method comparison
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DA results
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DA under/over-provisioning

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
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KL method comparison
20,264
14,168
996
2,644,620,000
0.2
20.8
1.52








3,032,69
0,000
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








No Feasible

Mean Value



?
100








Solution


Stoch. Prog.
12,154
7456
666
1,537,150,000
2.7
42.7
1.19








1,539,360,000
Wors
t 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
0
1598



over A
57
57
0
0
0
114









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