Performance Modeling of Intelligent Car Parking Systems

Intelligent Car Parking System Mean field approximation Performance of Intelligent Car Parking System

Performance Modeling of Intelligent Car Parking Systems

Károly Farkas Gábor Horváth András Mészáros Miklós Telek Technical University of Budapest, Hungary

EPEW 2014, Florence, Italy Károly Farkas, Gábor Horváth , András Mészáros, Miklós Telek Performance [6mm] Technical Modeling University of Intelligent of Budapest, Car Parking Hungary Systems


Outline

Intelligent Car Parking System Mean Field Model Performance of Intelligent Car Parking System

Károly Farkas, Gábor Horváth , András Mészáros, Miklós Telek Performance [6mm] Technical Modeling University of Intelligent of Budapest, Car Parking Hungary Systems


Parking assistance A mobile application based parking assistance is available in the Alle Mall in Budapest. Entrances Examples of transit only fields (Np=0)

Field with Np=8

Exits

Escalators (targets)



Parking assistance The assistance system is aware of the floor plan (i. e., routs, entrances, targets, exits), the available parking fields, the position and the destination of the guided car, and guides the car to a recommended parking field. The optimal guidance is a complex and dynamic optimization problem: available parking fields can be taken by others, the are un guided drivers, guided drivers can make own decisions. Károly Farkas, Gábor Horváth , András Mészáros, Miklós Telek Performance [6mm] Technical Modeling University of Intelligent of Budapest, Car Parking Hungary Systems


Performance problem

Our goal is the approximate quantitative assessment of the guidance based on the floor plan (i. e., routs, entrances, targets, exits), car behavior motion model (uninformed, distance aware, guided /w, wd/), destination, parking time, patience, ...



Car motion models

uninformed: randomly chooses free parking field distance aware: searches free parking field close to destination, guided w: driver assistance optimized for walking distance, guided wd: driver assistance optimized for walking and driving time.



Performance model

A Markovian model of individual car behavior destination based (memoryless) navigation and parking, Ph distributed parking time,



Mean field models

Performance analysis of real systems often results discrete state systems with dependent identical entities → very large number of states. The mean field methodology gives an exact solution when the size of the system is infinite and it gets less accurate as the system size decreasing.



Properties of mean field models

Basic model: a given number (finite or infinite) of identical entities, the entities behaves according to a Markov chain, the state transition of an entity depends on the state of all other entities but only through the number of entities which stays in different states.



Notations S – state space of an entity, s – number of states of S, N – number of entities, Ni (t) – number of entities which stays at state i at time t. N(t) = (N1 (t), N2 (t), . . . , Ns (t)) state vector of the system at time t. ⇒ number of possible states N+s−1 s−1 Transition rate of an entity staying in state i K ij (N(t)) = Pr (X (t +1) = j |X (t) = i, N(t)) if Ni (t) > 0, 0 if Ni (t) = 0, Károly Farkas, Gábor Horváth , András Mészáros, Miklós Telek Performance [6mm] Technical Modeling University of Intelligent of Budapest, Car Parking Hungary Systems


Normalized measures To avoid handling large numbers we introduce occupancy measures: ni (t) = Ni (t)/N – ratio of entities which stays at state i at time t. n(t) = (n1 (t), n2 (t), . . . , ns (t)) state vector of the system at time t. . . . and the associated transition probabilities kij (n(t)) =  1  Pr (X (t +∆) = j |X (t) = i, n(t)) if ni (t) > 0, lim ∆→0 ∆ 0 if ni (t) = 0, The occupancy measures make possible the handle finite and infinite number of objects in the same framework. Károly Farkas, Gábor Horváth , András Mészáros, Miklós Telek Performance [6mm] Technical Modeling University of Intelligent of Budapest, Car Parking Hungary Systems


Mean field method Theorem As N tends to infinity the normalized state vector, n(t), tends to be deterministic and satisfies the following differential equation X nj (t) kji (n(t)) ⇔ n(t + 1) = n(t) k(n(t)). ni (t + 1) = j∈S

Corollary When N is sufficiently large, the normalized state vector n(t), is a random vector whose mean can be approximated by the following differential equation P E (ni (t + 1)) ≈ j∈S E (nj (t)) kji (E (n(t))) m E (n(t + 1)) ≈ E (n(t)) k(E (n(t))). Károly Farkas, Gábor Horváth , András Mészáros, Miklós Telek Performance [6mm] Technical Modeling University of Intelligent of Budapest, Car Parking Hungary Systems


Mean field model

identical Markovian objects: cars, state of a car at time k Sk ∈ {(search, p, j , o, n, f )} ∪ {(parked , p, m)} ∪ {(leaving , p, x)}, interactions: based only on the number of objects at different positions, a parking field is available if no other car is parking there. independent of the particular car.



State of a car

For searching cars the current position p, the desired target destination j , the current orientation of the car o, and, in case of the distance-aware strategy, the phase of the DPH distribution representing the patience n. a flag f indicating that the car lost patience and gave up optimizing on distance



State of a car For parked cars we have to follow the the position of the car p, the phase of the DPH corresponding to the parking time m. Finally, for leaving cars we have to include into the state space the current position of the car p, and the selected exit where the car is heading to x. Thus, the state of a car at time k can be represented by Sk ∈ {(search, p, j , o, n, f )} ∪ {(parked , p, m)} ∪ {(leaving , p, x)}. Károly Farkas, Gábor Horváth , András Mészáros, Miklós Telek Performance [6mm] Technical Modeling University of Intelligent of Budapest, Car Parking Hungary Systems


Performance measures mean driving time to parking, LS , the mean walking distance from the selected parking field to the target destination, LW , the mean of the total latency including the driving and walking time, LT , the ratio of cars moving in the garage at the same time (either in search or leaving phase), C . For example C (k) =

X

Ni (k)

i ∈{(search,p,j,o,n,f )}∪{(leaving ,p,x)}



0.11 uninformed 0.1 distance aware 0.09 assisted, case 1. 0.08 assisted, case 2. 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 100 150 200 250 300 350 400 450 500 550

120 Mean searching time

Ratio of cars moving

Results 100 80 60 40 20 0 100 150 200 250 300 350 400 450 500 550

Number of cars in the garage (N)


9 8

uninformed distance aware assisted, case 1. assisted, case 2.

Mean time to the target

Mean walking distance

11 10

7 6 5 4 3 2 100

150

200

250

300

350

400

450



500

550

140 120 100


80 60 40 20 0 100 150 200 250 300 350 400 450 500 550 Number of cars in the garage (N)



Results



Conclusions

A quick model for quantitative performance assessment of the intelligent car parking system. Basic model properties are evaluated with approximate (imprecise) Markovian motion models. The system behavior is visualized based on the mean field approximation.


Performance Modeling of Intelligent Car Parking Systems

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