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Network Design under Demand Uncertainty

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Title: Network Design under Demand Uncertainty


1
Network Design under Demand Uncertainty
  • Koonlachat Meesublak
  • National Electronics and
  • Computer Technology Center
  • Thailand

2
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 -
3
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 ?
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 experiment
  • T. Telkamp, Traffic characteristics and network
    planning, ISMA Oct 2002.

5
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?
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.

How can we handle the uncertainty?
E.g., using m 3s can cover 99.9 of the area.
m
s
xs
peak
7
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).

8
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.

9
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
10
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
11
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.

12
CCP
  • Demand statistics
  • A random variable x has mean (m ) and variance (
    s2 )
  • Probabilistic guarantee ( ai )
  • The amount of bandwidth to be allocated, x

13
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
14
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.
15
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.
16
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.

17
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.

18
General network design problem
19
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
20
Mathematical Problem
Bandwidth constraints
Flow integrity constraints
Fixed charge constraints
21
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
22
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

23
Net 50 (50 nodes, 82 links)
24
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
25
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.

26
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.

27
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.

28
Questions?
Thank You
29
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

30
SP with Recourse
  1. Minimize cost over all of the possible scenarios
    ? choice of x
  2. Given x, choose y (a different decision for each
    random scenario x)

31
SPR Deterministic Equivalent
Recourse cost
subject to
Recourse function
Choose x to be optimal over all the scenarios
32
Example Regret function 29
Given a regret function, determine the minimum
cost based on tradeoffs among all the scenarios.
33
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
34
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35
Corrective process
36
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

37
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?
38
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.
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