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Network Design under Demand Uncertainty

- Koonlachat Meesublak
- National Electronics and
- Computer Technology Center
- Thailand

Assumptions

- Assumption A Traffic demand with the uncertainty.

Traffic demand matrix

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 -

Possible Approaches

- How to handle the uncertainty?

Bandwidth reservation approaches

mean-rate based

peak-rate based

average cases Pro cheap design Con rejection

of demand requests

worst case Pro could handle large

variation Con the most expensive design (large

safety margin)

Statistical approach ?

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 experiment
- T. Telkamp, Traffic characteristics and network

planning, ISMA Oct 2002.

Assumption B

- Thus, the traffic from large aggregation is not

totally uncharacterized. - Traffic distribution could be useful.

Bandwidth reservation approaches

mean-rate based

peak-rate based

average case Pro cheap design Con rejection

of demand requests

worst case Pro could handle large

variation Con the most expensive design

Based on demand N (m, s2) Benefits? How to

deal with such demand?

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.

How can we handle the uncertainty?

E.g., using m 3s can cover 99.9 of the area.

m

s

xs

peak

Applications

- Possible applications
- A generic network design
- A routing/bandwidth allocation scheme at the

IP/MPLS layer that considers those tradeoffs or

benefits. - 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).

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.

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

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.

Level of guarantee

Bandwidth allocation

Demand volume

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.

CCP

- Demand statistics
- A random variable x has mean (m ) and variance (

s2 ) - Probabilistic guarantee ( ai )
- The amount of bandwidth to be allocated, x

Example CCP Formulation

Minimize total bandwidth cost

Chance constraints -guarantee a random demand

with some level

Bandwidth constraints -set limitation on network

resources

Non-negativity constraints

Deterministic Equivalent

- Deterministic equivalent of stochastic constraint

To guarantee that the link can support the random

demand at least a-level, we need to allocate

bandwidth at least ?-1(a)s beyond the mean m of

the demand volume.

Multiple demands

E.g. Let a 0.95, ?-1(0.95) 1.645 Random

variablesx1, x2,,x10 mk 100, s2

100 Sum-part1 10010 1000 Sum-part2

1.645(1010) 164.5

Each flow is guaranteed with a-level.

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.

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.

General network design problem

Network Formulation

Notation

ak 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

Input Data Set

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 djk,p 1 if path p ? Pk for

flow k uses link j 0

otherwise

Decision and Output Variables

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

0 otherwise yj 1 if link j is used 0

otherwise

Mathematical Problem

Bandwidth constraints

Flow integrity constraints

Fixed charge constraints

Example Bandwidth reservation

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

Experimental studies

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

set) - Max hop 12 (Net50)
- a 0.90, 0.95, 0.95
- Wj, cj, and Fj are given.
- Random demand, s2 50-100 and m 2.857s
- Use CPLEX 9.1 solver to solve a linear

programming part

Net 50 (50 nodes, 82 links)

Example Bandwidth reservation

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

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 a could influence route selection,

especially in limited bandwidth environments.

Conclusions

- Theoretical study
- A new interpretation of the Chance-Constrained

Programming optimization in the communications

networks context, considering both the

uncertainty and service guarantees. - 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.

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.

Questions?

Thank You

References Normal Distn

- Gaussian approximation
- 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. - Truncated Gaussian distribution
- Truncation effect is negligible when m is

sufficiently larger than s , (s ? 0.35m) D.

Mitra

SP with Recourse

- Minimize cost over all of the possible scenarios

? choice of x - Given x, choose y (a different decision for each

random scenario x)

SPR Deterministic Equivalent

Recourse cost

subject to

Recourse function

Choose x to be optimal over all the scenarios

Example Regret function 29

Given a regret function, determine the minimum

cost based on tradeoffs among all the scenarios.

Stochastic Programming Long-term design

Paper Problem considered Demand Model

Sen et al. 28 Determine additional capacity to minimize unserved demands subject to capacity budget in a telephone switching network Number of requested connections SP with recourse Long-term design

Kennington et al. 29 Determine an optimal regret function and use this function to minimize equipment cost in a WDM network Amount of capacity (set of future scenarios and probabilities) Robust optimization Long-term design

Leung and Grover. 30 Design a survivable network (working and backup capacity allocation). Consider tradeoff between initial network cost and robustness to uncertainty. Amount of capacity (set of future scenarios and probabilities) SP with recourse Long-term design

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Corrective process

Long-term vs. Short-term Planning

- Long-term
- Capacity expansion determine network topology

and capacities to be allocated on network links - Time frame months or years
- Based on demand forecasting and estimation of

traffic growth - Considers possible future demand scenarios,

economic factors, expected future technologies,

etc. - Short-term
- Traffic management / Traffic Engineering design
- e.g., VPN design
- Network topology and capacities are given.
- Time frame weeks, days, hours
- Involves dynamic routing / bandwidth allocation.
- Optimizes resource usage, QoS, network

performance

Aggregation of Demands

E.g. The sum of independent normal random

variables x1, x2, and x3 is a normal random

variable with mean m1 m2 m3 variance s12

s22 s32

How much demand uncertainty can the network

tolerate? How can guarantees be provided to an

uncertain demand?

Design problem

E.g. Let a 0.95, ?-1(0.95) 1.645 Random

variablesx1, x2,,x10 mk 100, s2

100 Sum-part1 10010 1000 Sum-part2

1.645(1010) 164.5, reserve too much BW? If

the number of flows is large, this bound may

over-approximate the variability of random

demands.

Each flow is guaranteed with a-level.