Network Design under Demand Uncertainty Koonlachat Meesublak National Electronics and Computer Technology Center Thailand
Jan 06, 2016
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)
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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?
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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.
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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
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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