Network Design under Demand Uncertainty

Koonlachat MeesublakNational Electronics and

Computer Technology Center

Thailand

2

Assumptions

Assumption A: Traffic demand with the uncertainty.

Known pattern

Unknown

Partiallyknown

1 2 3 4

1 - 12

+ ∆1

44 10

2 15

+ ∆2

- 34 45

3 40 22

+ ∆3

- 55

4 18

+ ∆4

45 50

+ ∆5

-

Traffic demand matrix

3

Possible Approaches

How to handle the uncertainty?

mean-rate based peak-rate based

Bandwidth reservation approaches

Statistical approach ?

“average cases”Pro: cheap design

Con: rejection of demandrequests

“worst case”Pro: could handle large variationCon: the most expensive design

(large safety margin)

4

Distribution of Random Demands

Assumption B: Traffic between a node pair comes from

many independent sources By CLT, the distribution of large

aggregate traffic Normal distribution.J. Kilpi and I.Norros, “Testing the Gaussian approximation of aggregate traffic,” in Proc. Internet Measurement Workshop, 2002, pp. 49-61.R. G. Addie, M. Zukerman, and T.D. Neame, “Broadband traffic modeling: simple solutions to hard problems,” IEEE Commun. Mag., vol. 36,  Issue 8, pp. 88-95,  Aug. 1998.

Measurement experimentT. Telkamp, “Traffic characteristics and network planning,” ISMA Oct 2002.

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Assumption B

Thus, the traffic from large aggregation is not totally uncharacterized.

Traffic distribution could be useful.

mean-rate based peak-rate based“average case”

Pro: cheap designCon: rejection of demand

requests

“worst case”Pro: could handle large variationCon: the most expensive design

Bandwidth reservation approaches

Based on demand ~ Benefits?

How to deal with such demand?

6

Benefits

Why use statistical allocation? Bandwidth is not unlimited, and is not free consider

the tradeoffs between cost and ability to handle the variation. These are the benefits between mean and peak schemes.

x

How can we handle the uncertainty?

peak

E.g., using + 3 can cover 99.9% of the area.

7

Applications

Possible applications1. A generic network design

2. A routing/bandwidth allocation scheme at the IP/MPLS layer that considers those tradeoffs or benefits.

3. A routing design that guarantees the traffic base on its demand statistics, and also based on the resource limitation along the path (as will be explained later).

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Related optimization models Deterministic model

Demand is known or easily estimated Uses mean value or worst-case value Extension: time dimension, e.g., multi-hour design.

Stochastic model Demand can be treated as a random variable Typical Long-term / multi-period planning design:

Stochastic Programming with Recourse [28], [30] Robust Optimization [29] involves forecasting of future events.

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Example: Scenario-based demand

Scenario Probability of

occurrence

Demand

1 0.25 1→2: 2 Gb/s

2→3: 4.5 Gb/s

2 0.25 1→2: 10 Gb/s

2→3: 3.4 Gb/s

3 0.20 1→2: 5 Gb/s

2→3: 4 Gb/s

3→1: 7 Gb/s

4 0.30 1→3: 3 Gb/s

2→3: 4 Gb/s

2 3

1

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Alternative approach

Chance-constrained programming (CCP) is a SP variation. It uses different probabilistic assumption, and does not assume the future events

Input: statistical information on a random demand To handle the random demand, levels of probabilistic

guarantee can be specified.“Probability that the allocated bandwidth exceeding the volume of random demand is greater than or equal to 0.95.”

0.95P x Level of guaranteeBandwidth

allocationDemand volume

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CCP

Medova [31] studies routing and link bandwidth allocation problem in an ATM network

Level of guarantee = 1 - Probability of blocking ATM connection request

Assumes that this Prob. is very small the approximation eliminates statistical information of random demands.

Our work Levels of guarantee are used. Each demand has its own

guarantee value. Aggregate traffic carried on each link is composed of two

parts: certain and uncertain parts.

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CCP

Demand statistics A random variable has mean () and variance ( 2 )

Probabilistic guarantee ( i )

The amount of bandwidth to be allocated, x

11

22

( )

( ) 0.9

( ) 0.8

i i iP x

P x

P x

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Example: CCP Formulation

Chance constraints-guarantee a random demand with some level

i ix W

, 1i N

1i

N

i ix

i

cMinimize x

( )ii iP x

0ix

Subject to:

(3.2)

(3.3)

(3.4)

, 1i N

, 1i N Bandwidth constraints-set limitation on network resources

Non-negativityconstraints

Minimize total bandwidth cost

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Deterministic Equivalent

Deterministic equivalent of stochastic constraint

P x

1x

is N , 2cumulative distribution function

-1 = inverse transform of (.)

“uncertain factor”“certain factor”

To guarantee that the link can support the random demand at least -level, we need to allocate bandwidth at least -1()beyond the mean of the demand volume.

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Multiple demands

1

1

K

j k k kk

x

1x

E.g.Let = 0.95, -1(0.95) = 1.645Random variables:x1, x2,…,x10

k = 100, 2 = 100Sum-part1 = Sum-part2 =1.645*(10*10)= 164.5

... Link j

N(1,2

1)

N(2,2

2)

N(K,2K)

Each flow is guaranteed with -level.

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Proposed research Goal

To develop a methodology for network design under demand uncertainty.

Need to solve a routing and bandwidth allocation problem based on CCP so achieve the benefits from statistical guarantee.

Research Approach To develop mathematical models for a routing and

bandwidth allocation problem with uncertainty constraints.

This is intended for usage in the IP layer, and will not solve the traffic glooming problem in the physical layer.

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Basic design problem

“Given network information and a demand volume matrix, determine routes and the amount of bandwidth to be allocated on such routes so that the total network cost subject to network constraints is minimized.”

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General network design problem

RoutingBandwidthAllocation

Chance-constrainedapproximation

Uncertainty bounds

Demand Statistics {mean, variance}

+Uncertainty-guarantee

levels

Network Information- Topology- Costs {bandwidth cost,uncertain-routing cost}- Resource constraints- Uncertainty-guaranteelevels

Route selection andBandwidth assignment

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Network Formulation

Notation

D Set of random demands

A Set of links (arcs)

Pk Set of predefined candidate paths

Wj Bandwidth bound for total traffic demand on link j

jk,p= 1 if path p Pk for flow k uses

link j = 0 otherwise

k Level of guarantee of demand k

Fj Fixed cost for routing on link j A

cj Variable cost of adding one unit of bandwidth to link j A

f k,p = 1 if flow k selects path p Pk

= 0 otherwise

yj = 1 if link j is used = 0 otherwise

Input Data Set

Decision and Output Variables

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Mathematical Problem

1 , ,

k

k p k pj j

j A k D p Pk k kc f

, ,

mink p

jf yj j

j A

F y

Subject to:

1 , ,

k

k p k pj j

k D p Pk k k f W

, j A

, 1k

p k

p P

f

, k D

, ,k p k pj jf y , , , kj A k D p P

, , 0,1k pjf y

(4.3)

(4.4)

(4.5)

(4.6)

Bandwidth constraints

Flow integrity constraints

Fixed charge constraints

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Case Reservation Type

Guarantee Level

Bandwidth for 10 demands

(Mbps)

Bandwidth for 200 demands

(Mbps)

1 Mean rate mean level 2250.00 45,000.00

2 Statistical guarantee

95% 2661.25 53,225.00

3 Statistical guarantee

99% 2831.50 56,630.00

4 Statistical guarantee

99.9% 3022.50 60,450.00

5 Peak rate Peak level 3420.00 68,400.00

Example: Bandwidth reservation

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Experimental studies

Three network topologies: Net50 Pre-calculated candidate path set (8 paths per

set): Max hop: 12 (Net50)

= {0.90, 0.95, 0.95} Wj, cj, and Fj are given. Random demand, 2: 50-100 and ≥ 2.857 Use CPLEX 9.1 solver to solve a linear

programming part

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Net 50 (50 nodes, 82 links)

35

21 4

19

6

10

24

8

15

7

11 20

29

23

22

30

32

21 33

31

27

28

12

149

13 17

16

39

18

26

25

40

38

46

4745

48

4449

43

50

4241

36

3435

37

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Case Guarantee Level

Mean

Cost

Uncertainty

Cost

Fixed-charge

Cost

Total cost

1 90% 17428.92 7379.86 2960.00 27768.78

2 95% 17428.92 9469.48 2960.00 29858.40

3 99% 17342.73 13324.95 3120.00 33787.68

Example: Bandwidth reservation

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Note

The number of demands and network size are crucial factors for an optimization problem.

For this network size and demand input set, computational times are in the order of hundreds of milliseconds, which are still acceptable for these studies.

Parameter could influence route selection, especially in limited bandwidth environments.

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Conclusions Theoretical study

1. A new interpretation of the Chance-Constrained Programming optimization in the communications networks context, considering both the uncertainty and service guarantees.

2. A mathematical formulation for network design under traffic uncertainty is developed. This framework is expected to be applied to the virtual network design at the IP layer.

The uncertainty model is based on short-term routing and bandwidth provisioning.

Uses Chance-constraints to capture both the demand variability and levels of uncertainty guarantee.

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Conclusions

Future work: improvement of accuracy of the model

Simulation studies on the relation between different traffic patterns and the benefit of the Chance-constraint approximation are needed.

Traffic measurement: An investigation on other traffic distributions and their effects on the uncertainty bound.

A study on the benefits of the scheme with real traffic input from measurement.

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Questions?

Thank You