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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
Heuristic improvesHeuristic improvesas system size growsas system size grows
0
5
10
15
20
25
30
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
0
50
100
150
200
0 4 8 12 16 20 24
time of day
ca
lls
/ho
ur
RA=1
RA=.1
RA=.5
An Optimal SolutionAn Optimal Solution
0
20
40
60
80
100
120
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160
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
0
100
200
300
400
500
600
700
800
0 1000 2000 3000 4000 5000 6000 7000
Calls/Day
Co
st
Cost versusCost versusRelative AmplitudeRelative Amplitude
0
100
200
300
400
500
600
700
800
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
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
0
50
100
150
0 50 100 150
True Optimal Ceiling
Heu
rist
ic C
eili
ng
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)
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
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