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Introduction Analytical Model Simulation Results Conclusion

Estimating Parking Spot Occupancy

David M.W. Landry and Matthew R. Morin

Department of Electrical and Computer EngineeringBrigham Young University

Stochastic Processes Project5 December 2013

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

Outline

1 Introduction

2 Analytical ModelSpatial and Temporal DistributionExpected Time to Destination

3 Simulation Results

4 Conclusion

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

Outline

1 Introduction

2 Analytical ModelSpatial and Temporal DistributionExpected Time to Destination

3 Simulation Results

4 Conclusion

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

Everybody Parks

Figure: Google Earth. BYU Parking Lot 1A. 40◦15′4.49′′N and111◦38′58.37′′W. Image Taken: Jun 17, 2010. Accessed: Nov 11,2013.

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

Outline

1 Introduction

2 Analytical ModelSpatial and Temporal DistributionExpected Time to Destination

3 Simulation Results

4 Conclusion

Landry and Morin Brigham Young University

Parking Occupancy

byu.png

Introduction Analytical Model Simulation Results Conclusion

Spatial and Temporal Distribution

Outline

1 Introduction

2 Analytical ModelSpatial and Temporal DistributionExpected Time to Destination

3 Simulation Results

4 Conclusion

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

Spatial and Temporal Distribution

Bernoulli Process

Each parking space is a Bernoulli random variableThe probability of occupancy, p, changes with

distance from the point of interesttime of day

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

Spatial and Temporal Distribution

Distance

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

Spatial and Temporal Distribution

Time of Day

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

Spatial and Temporal Distribution

Joint Distribution

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

Spatial and Temporal Distribution

Fermi-Dirac - An Alernate Distribution

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

Expected Time to Destination

Outline

1 Introduction

2 Analytical ModelSpatial and Temporal DistributionExpected Time to Destination

3 Simulation Results

4 Conclusion

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

Expected Time to Destination

Total Time

Ttotal = pTo + (1 − p)Tu (1)

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

Expected Time to Destination

To

To is the wait time for a space to open up plus the walking timeto reach the point of interest

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

Expected Time to Destination

To

Y = min(X1,X2...,XN) (2)

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

Expected Time to Destination

To - Derived pdf for Wait Time

fY (x) = NfX (x)(1 − FX (x))N−1 (3)

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

Expected Time to Destination

To - Expected Wait Time

E [Y ] =

∫ ∞−∞

xNfX (x)(1 − FX (x))N−1dx (4)

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

Expected Time to Destination

To - Expected Distance

do =1N

N∑i

di (5)

Landry and Morin Brigham Young University

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Introduction Analytical Model Simulation Results Conclusion

Expected Time to Destination

To - Total

To = E [Y ] +do

v(6)

Landry and Morin Brigham Young University

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Introduction Analytical Model Simulation Results Conclusion

Expected Time to Destination

Tu - Which Spot?

pu = (1 − pn)n−1∏i=1

pi ,n > 1 (7)

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

Expected Time to Destination

Tu - Expected Distance

E [du] =N∑

j=1

djpu,j (8)

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

Expected Time to Destination

Tu

Tu =E [d ]

v(9)

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

Outline

1 Introduction

2 Analytical ModelSpatial and Temporal DistributionExpected Time to Destination

3 Simulation Results

4 Conclusion

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

Retail Parking Example

µ = 20kT = 5

Landry and Morin Brigham Young University

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Introduction Analytical Model Simulation Results Conclusion

Street Parking - Time to Point of Interest

kT = 30µ1 = 30µ2 = 100

Landry and Morin Brigham Young University

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Introduction Analytical Model Simulation Results Conclusion

So which street should I park on?

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

Outline

1 Introduction

2 Analytical ModelSpatial and Temporal DistributionExpected Time to Destination

3 Simulation Results

4 Conclusion

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

Conclusion

Who knew you could have so much fun with parking lots?

Landry and Morin Brigham Young University

Parking Occupancy

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Introduction Analytical Model Simulation Results Conclusion

And Have Fun Parking

Figure: Munroe, Randall. “Parking”. xkcd. Licensed underCC BY-NC 2.5

Landry and Morin Brigham Young University

Parking Occupancy