Internet Access: A Two-Provider Cost Model

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Internet AccessA Two-Provider Cost Model

• Andrew M. Ross
• Eastern Michigan University
• Math Dept.
• 2006-12-13

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Dial-Up Internet Access

• Time-of-day patterns
• Build modem banks to handle peaks
• Hourly Company H charges 1 per modem-hour
  used.
• Peak Company P charges 4 per modem in use at
  peak time of day

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Route Traffic to Save

• Minimize 4 Epeak 1 Ehours

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Feasibility StudyClairvoyant System

• Upper bound on possible savings
• Known data
• When each call arrives
• How long each call stays on-line
• Route using an Integer Program
• Also try a heuristic

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Network Version of IP
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Heuristic improvesas system size grows
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Any questions?

• people.emich.edu/aross15/
• andrew.ross_at_emich.edu

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IP Formulation

• Data
• Oij 1 if call i is still online when call j
  arrives, 0 otherwise
• Si duration of call i
• Variables
• Xi 1 if call i given to Company P,
• 0 if given to Company H
• Z height of peak on Company P
• E total elapsed time on Company H

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IP Formulation

• Minimize
• Subject to
• Elapsed hours
• Minimax for all j,

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Arrival Rate Functions

RA1
RA.5
RA.1
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An Optimal Solution

Company P
Company H
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Cost versus Calls per Day
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Cost versus Relative Amplitude
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Example
Call Arrives _at_ Duration
1 1 1.8
2 2 2.5
3 3 0.5
4 4 2.0
Sum6.8
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Minimax constraints

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Strict Ceiling Policy

• Admit a call to P when number on P currently is
  lt Z
• For clairvoyant case, try each Z and choose the
  best
• Not always exactly optimal

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Heuristic Ceiling vs.True Optimal Ceiling
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Course Overview

• What is a Math Model
• Modeling Procedures
• Dynamical Systems (Ch 1)
• Model Fitting Interpolation (Ch 2,3,4)
• Simulation Queueing (Ch 5)
• Linear Programming (Ch 7)
• Non-Linear Programming (Ch 12)
• Differential Equations (Ch 10,11)

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Topic Histogram
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Follow-on courses (419!)

• Math(419!)
• 223 Calc III (some NLP)
• 325 Diff.Eqn 426 Diff. Eqn II
• 418 Modeling with Lin.Alg.
• 416 Adv. Lin.Alg.
• 425 Math for Scientists
• 436 Numerical Analysis

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Stats courses

• Math 419
• Math 360 or 370, then
• 460 Survey Sampling
• 461 Linear Regression
• 462 Design of Experiments
• 471 Prob/Stat II
• 474 Applied Stats

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Computer Science

• 245 Numerical Methods
• 311 Algorithms Data Structures
• 314 Computational Discrete Structures
• 461 Heuristic Programming
• Math 419

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Business/Econ

• POM 374 Production Operations Management
• DS 265 Statistics
• DS 317 Simulation
• Econ 310 Econ. Stats
• Math 419