Internet Access:Internet Access:A Two-Provider Cost ModelA Two-Provider Cost Model

Andrew M. RossAndrew M. Ross

Eastern Michigan UniversityEastern Michigan University

Math Dept.Math Dept.

2006-12-132006-12-13

Dial-Up Internet AccessDial-Up Internet Access

• Time-of-day patternsTime-of-day patterns• Build modem banks to handle Build modem banks to handle

peakspeaks• Hourly: “Company H” charges $1 Hourly: “Company H” charges $1

per modem-hour used.per modem-hour used.• Peak: “Company P” charges $4 per Peak: “Company P” charges $4 per

modem in use at peak time of daymodem in use at peak time of day

Route Traffic to Save $Route Traffic to Save $

• Minimize $4 E[peak] + $1 E[hours]Minimize $4 E[peak] + $1 E[hours]

Feasibility StudyFeasibility Study“Clairvoyant System”“Clairvoyant System”

• Upper bound on possible savingsUpper bound on possible savings

• Known data:Known data:– When each call arrivesWhen each call arrives– How long each call stays on-lineHow long each call stays on-line

• Route using an Integer ProgramRoute using an Integer Program

• Also try a heuristicAlso try a heuristic

Network Version of IPNetwork Version of IP

Heuristic improvesHeuristic improvesas system size growsas system size grows

0

5

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0 200 400 600 800

Ceiling

Per

cen

t C

ost

Incr

ease

Any questions?Any questions?

• people.emich.edu/aross15/people.emich.edu/aross15/

• andrew.ross@emich.eduandrew.ross@emich.edu

IP FormulationIP Formulation

• Data:Data:• OOijij = 1 if call = 1 if call ii is still online when call is still online when call jj arrives, arrives,

0 otherwise0 otherwise• SSii = duration of call = duration of call ii

• Variables:Variables:• XXii = 1 if call = 1 if call ii given to Company P, given to Company P,

0 if given to Company H0 if given to Company H• Z = height of peak on Company PZ = height of peak on Company P• E = total elapsed time on Company HE = total elapsed time on Company H

IP FormulationIP Formulation

• MinimizeMinimize

• Subject toSubject to– Elapsed hours:Elapsed hours:

– Minimax: for all Minimax: for all jj,,

N

iii xSE

1

)1(

j

i

iij xOZ1

EZ 1$4$

Arrival Rate FunctionsArrival Rate Functions

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time of day

ca

lls

/ho

ur

RA=1

RA=.1

RA=.5

An Optimal SolutionAn Optimal Solution

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0 4 8 12 16 20 24time of day

nu

mb

er

on

-lin

e

Company P

Company H

Cost versusCost versusCalls per DayCalls per Day

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Calls/Day

Co

st

Cost versusCost versusRelative AmplitudeRelative Amplitude

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0 0.2 0.4 0.6 0.8 1

Relative Amplitude

Co

st

ExampleExample

Call#Call# Arrives @Arrives @ DurationDuration

11 11 1.81.8

22 22 2.52.5

33 33 0.50.5

44 44 2.02.0

Sum=6.8Sum=6.8

Minimax constraintsMinimax constraints

Zx

1010

0110

0011

0001

Strict Ceiling PolicyStrict Ceiling Policy

• Admit a call to P when number on P Admit a call to P when number on P currently is < Zcurrently is < Z

• For clairvoyant case, try each Z and For clairvoyant case, try each Z and choose the bestchoose the best

• Not always exactly optimal Not always exactly optimal

Heuristic Ceiling vs.Heuristic Ceiling vs.True Optimal CeilingTrue Optimal Ceiling

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True Optimal Ceiling

Heu

rist

ic C

eili

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Course OverviewCourse Overview

• What is a Math ModelWhat is a Math Model

• Modeling ProceduresModeling Procedures

• Dynamical Systems (Ch 1)Dynamical Systems (Ch 1)

• Model Fitting & Interpolation (Ch 2,3,4)Model Fitting & Interpolation (Ch 2,3,4)

• Simulation & Queueing (Ch 5)Simulation & Queueing (Ch 5)

• Linear Programming (Ch 7)Linear Programming (Ch 7)

• Non-Linear Programming (Ch 12)Non-Linear Programming (Ch 12)

• Differential Equations (Ch 10,11)Differential Equations (Ch 10,11)

Topic HistogramTopic Histogram

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Follow-on courses Follow-on courses (419!)(419!)

• MathMath(419!)(419!)::– 223 Calc III (some NLP)223 Calc III (some NLP)

– 325 Diff.Eqn & 426 Diff. Eqn II325 Diff.Eqn & 426 Diff. Eqn II

– 418 Modeling with Lin.Alg.418 Modeling with Lin.Alg.

– 416 Adv. Lin.Alg.416 Adv. Lin.Alg.

– 425 Math for Scientists425 Math for Scientists

– 436 Numerical Analysis436 Numerical Analysis

Stats coursesStats courses• Math 419Math 419

• Math 360 or 370, then:Math 360 or 370, then:– 460 Survey Sampling460 Survey Sampling

– 461 Linear Regression461 Linear Regression

– 462 Design of Experiments462 Design of Experiments

– 471 Prob/Stat II471 Prob/Stat II

– 474 Applied Stats474 Applied Stats

Computer ScienceComputer Science

• 245 Numerical Methods245 Numerical Methods

• 311 Algorithms & Data Structures311 Algorithms & Data Structures

• 314 Computational Discrete 314 Computational Discrete StructuresStructures

• 461 Heuristic Programming461 Heuristic Programming• Math 419Math 419

Business/Econ:Business/Econ:

• POM 374 Production & Operations POM 374 Production & Operations ManagementManagement

• DS 265 StatisticsDS 265 Statistics

• DS 317 SimulationDS 317 Simulation

• Econ 310 Econ. StatsEcon 310 Econ. Stats• Math 419Math 419