Mathematics and Computers in Simulation 28 (1986) 305-309

North-Holland

305

A MODEL OF VEHICLES MOVEMENTS IN PARKING FACILITIES *

William YOUNG

Department of Cwrl Engineering, Monush Unwer.uty, Clayton, Victoriu 3168, Austruba

This paper presents an outline of a discrete event simulaticn model of vehicle movements in parking facilities. It describes the rmcdel development and components.

1 . INTRODUCTION

Recommended practices for the design of parking lots are plentiful il ,2,3,41. These provide

usefu I i “format ion on survey and design

procedures for the various components of the

parking system. Procedures for gather i ng these

components into an overal I systems design are

not so clearly described. Research into the

development of design tools to estimate the

overa I I performance of particular parking

I ayouts is limited [5]. Creation of tools has

been hindered by the difficulty in developing

mathematical models of camp lex networks. The

considerable developments in co~lputers, both

micro and macro, and associated computer

graphics may provide a solution to this problem

and enable useful design models of park i ng

facilities to be developed.

This paper describes a model developed for the

study of the movement of vehicles in parking

facilities. It describes the components of the

process resulting in the decision to park and

the model development. The components of the

model are then outlined. The paper closes witt?

some concluding remarks and some directions for

future research.

Parking decision making process. To understand

the decision making process the considerable literature published in the behavioural model I ing area is used. A general view of the decision making process is presented in Figure 1 . The overall objective of the process is for the decision maker to make a choice after consideration of the alternatives available to him. Before the final choice can be made, however, it is postulated that there are a number of discrete steps through which the choice process moves. Each decision-maker is represented in the process by an importance hierarchy. Initially each decision maker be I ieves that certain attributes are of importance in the decision at hand. Not all

decision makers wi I I have the same importance hierarchy. For instance, some drivers entering a

parking facility will take the first reasonably accessible parking place (Pessimistic decision maker 1. Another driver may go to the parking space closest to his final location (Optimistic

decision maker) then start to look for a parking place. For each individual those attributes he

considers to be relevant form the attribute set.

2. STUDY APPROACH

The understanding of the transport decision

process and the modelliny of the transport

system has made considerable advances irs the

last decade. These advances have not, as yet, been applied to the study of parking facilities. This section briefly describes the drivers

decision process in the context of parking

procedures and the model development process.

* This paper was presented at the 11th IMACS World Con-

gress, Oslo, Norway, August 1985. Fiyure 1 Transport decision making process

0378-4754/86/$3.50 0 1986, IMACS/Elsevier Science Publishers B.V. (North-Holland)

306 W. Young / Vehicles movements in parkrng fucrltties

Before proceeding too far in the choice

modelling process, it is essential to determine

whether there are any constraints which

significantly influence its outcome. The choice

set for the drivers is essentially ail free

parking spaces. The individuals knowledge of the

availability will however result in the driver

perceiving a smaller number of places available.

Given the final choice set and a finite

attribute set, the next step in the process is

the perception of attribute levels for each

attribute for each individual. It must be

emphasised that an individual may have a

different perception concerning a given physical

level of an attribute, and it is the perceived

level that will influence the decision. Further, given the perceived level of an attribute, each

individual must make an evaluation of the degree

of satisfaction associated with the attribute. For instance, the evaluation of walking distance

is likely to differ for a mother with children

and a young male.

The importance and satisfaction rankings can

then be conlbined into a composite evaluation for

each alternative. The composite evaluations are

then entered into the decision rule anti a

measure of behavioural intention obtained. The

behavioural intention only results in a choice

if various forms of choice inertia (e.g. habit

and time lags) can be overcome. Having made a

choice there are a series of feedback loops, 3s

shown by the dashed lines in Figure 1, which may

effect future choice situations.

The previous discussion has illustrated that a

decision is not a single action but rather one

part of a process. Modelling this process by a

single mathematical relationship is therefore

difficult. Similarly, heuristic, physical,

ana logue and macroscopic computer simulation

models do not give the flexibility required to

mode I the interactions present. The discrete

event simulation model Iing approach was

therefore chosen for this study.

Model development process. The development of a

simulation model takes a number of interrelated

steps to its conclusion. These steps have been

described [6! in the appropriate order as the

problem definition or objective statement,

systems analysis, systems synthes is, program development, program verification, program refinement, program validation and application. Each step can interact with another and are

constrained by the resources available.

The first and one of the most important steps is the statement of a clear definition of the

objectives to be achieved. This statement of objectives should lead to a concise statement of the requirements of the model. The objectives of the project, of which this paper forms part, is to determine the feasibility of developing a

design tool for the study of parking facilities.

The second step in the model development process is the study of the system; its components, interactions and interrelationships. In this study observation of the system isolated the following elements;

Boundary conditions; the parking lot includes all roads inside the street network.

Components; Vehicle speed, parking duration, parking times, unparking times, gap acceptance when unparking and gap acceptance at intersections.

interactions; Car-following, parking, unparking,

leaving parking street, entering parking streets.

After the systems analysis, it may be possible to begin the systems synthesis. Flow charts and

the development of program algorithms form the essence of this stage. The systems synthesis results in the computer program.

The fifth stage is the verification of the mode I. This step consisted of tracing vehicles through the parking system and observing their behaviour. Irregularities in their behaviour are corrected. This step in the model deve I opment process represents the present stage of development of the project. Further steps I ike the validation, refinement and application of the model will be carried out at a later date.

It should, however, be reiterated that each of the steps in the model development process are interrelated and it is often necessary after a fault in found to move back in the process.

3. MODEL DESCRIPTION

3.1 Introduction

The development of the model started with a very

simple system and progressively introduced new dimensions. This process of developing simulation mode I s from the “bottom-up” enables the model builder to develop his thoughts with the development of the nlodel. It also enables detailed investigation of each of the components as they are developed. One possible problem with this approach is that a decision to develop a particular algorithm may not be appropriate at a I ater stage in model development. This could require a rewrite of particular components of the model. The other.approach of developing a simulation rlode I (“top-down”) requires the analyst to have an overall idea of the working of the system and to develop a model to suit this specification. This approach was not chosen here since this study was’s research exercise and it was not possible to fully specify the problem at the start of model development.

The first step in the model development process was the construction of a model to simulate the

movement of vehicles along a link. Parking was

then introduced into the model. The extension of

the model into a network model was the next

step. In order to facilitate a network it was necessary to introduce a intersection simulation

sub-model. The final stages of development was

the introduction of a probabilistic parking

place choice model.

3.2 Movement along a link

The general philosophy behind this model is to

order the vehicles with respect to their

position and to consider the vehicle furthest

along the link first. This vehicle is moved

forward a certain distance. The distance moved

is a function of the time interval chosen to

update the model. After the first vehicle is

moved the second is moved. If the second

vehicles position, after movement, is too close

to the first it is necessary to introduce a car-

following process. The process of considering each vehicle in turn is continued until the last

vehicle on the I ink is considered. The

information required for the simulation of

vehicles in this model are the initial spacing

of vehicles entering the facility, their desired

speed and the car-following procedure.

Arrival of vehicles. The arrival of vehicles can

be expected to exhibit all the characteristics

of a time series [6j. The trend variation

reflects, long term changes in traffic flow.

Seasona I variations occur throughout the year.

Cyclic variation occur throughout the day and

random variations result from short terril

fluctuations in traffic flow. The first three

characteristics are long term variations and are essential information for the application of the

mode I to real wor Id analyses. They can be

incorporated exogenously. The random arrival c’f

vehicles at the parking facility depend on the

conditions present in the roads surrounding it.

lf, for instance, there is a set of traffic

signals in close proximity to the facility the

vehicles may arrive in bunches. Further, if

traffic flows are light the arrival rate nlay be

random. The choice of arrival distribution is

therefore determined by the surrounding road

conditions. In the development of the model the

arrival distribution used was a displaces

exponential. This distribution has been found to

replicate gaps in traffic at medium traffic

flows and to take into account the desire of

people not to travel to close together [Gl.

Speed of vehicles. The distribution of the speed

of vehicles in parking facilities has not been

presented in the available literature. There has

however been considerable research directed a

determining free speeds of vehicles on arterial

and residential streets. Gipps [7: found that

the free speed adopted by vehicles on arterial roads are normally distributed with coefficient

of variation of between 0.16 lnd 0.17 for cars. Studies for residential streets ‘81 have shown

simi lar results. The distribution adopted in

this study was therefore the same as that used

to describe desired speeds on arterial roads:

the normal distribution. It should however be

noted that validation of the appropriateness of

this distribution for vehicle movements in

parking lots is required before application of

the model

Car-following process. Considerable effort has

been directed at developing car-following

procedures for simulation models [71. The

average speeds in car parks is very low and

these interactions are of little importance. In this model the procedure adopted is to have the

following vehicle adopt the speed of the vehicle in front and remain at a safe spacing.

Updating the simulation time. Three methods are available for updating the time interval associated with the temporal variations in the model [61. These are the “vehicle update”, the “time update” and the “event update” procedures. The “vehicle update” procedure traces the movement of each vehicle through the system in turn. This approach is useful where an individual vehicle can only i nf I uence the movement of vehicles behind it. Since this

approach breaks down when overtaking occurs it is not used here. The “time update” approach updates the simulation time in regular discrete intervals of time. This approach is most suited to situations where a large number of events have to be considered or the events are not discrete (e.g. in car-f01 lowing situations). Since there are a large number of interaction between the vehicles in this model this approach was initially used to update the simulation time. The “event update” approach updates the simulation time to when the next event occurs.

This approach used when there are a small number of discrete events to be considered. This discussion will return to this approach latter.

3.3 Parking on a road link

The major element of link parking simulation model is the movement along a I ink and the accepting of a parking place. The driver of the vehicle enters the street and moved along it search i ng for a parking place. When an appropriate parking place is found the vehicle is manoeuvred into it. The duration of stay is calculated. When the stay ends the vehicle exits the parking space, blocking other vehicle movements for a prescribed period. The components of this process and how they are model led will be discussed next.

308 W. Young / Vehicles rnouemen~~ rn parkrng facdities

Variable update interval. The model of movement

along a I ink had an “time update” procedure and

the update time interval was constant. If this

approach is used when model I ing parking

behaviour it could result in inaccuracies, if

the size of the time interval is less than the

time to travel between decision points. Even if

sma I I update intervals are used errors will br

present. To provide a more accurate answer it

was decided to introduce an option where the

update interval is a function of the next event

to take place. This could be either the time

when a car reached a parking place or the end of

the link. This “event update” approach decreased run time considerably and is the normal node for

running the program

Parking procedure. The parking procedure adopted in the model assumed that the vehicle travel led

at its desires speed, or the desired speed of

its platoon leader, until it reached the parking

place. It then stopped and waited for a defined

period before parking. This period of time is called the parking time. The average parking

time and its distribution are important factors

in determining the performance of the parking

system. Hobbs [9: provides estimates of the

parking and unparking times for particular

parking space angles. These were obtained from

controlled experiments and represent the

manoeuvre times from a mark 6.1 meters fron the

parking bay. Farrow [5j presents data on the

distribution of the parking times and the spread of the distribution (standard deviation). He found that the distribution could be adequately

represented by a normal distribution.

Parking duration. In studies of the capacity

requirements of parking lots the average

duration and the shape of the parking duration

distribution are all important in ascertaining

how many spaces must be supplied. Published

reports on the duration distribution are scanty.

The Highway Research Board [lOI summarises the results of 111 parking studies carried out in

the USA in the 1950”s and summarises the

duration distributions for various trip

purposes. More recently, Richardson Cl11 studied

data from 8 sites in Sydney, Australia. He

concluded that where parking was of a common

purpose the appropriate distribution varied

with the coefficient of variation. A Gamma

distribution was appropriate for a coefficient

of variation I ess than one and a

Hyperexponential distribution is appropriate

when the coefficient of variation is greater

than one. Richardsons study however collected data at the entry and exit points of parking facilities. Hence the travel time in the

facility was included. Further, he found that

for most parking lots the coefficients of

variation were close to one. In which case both

the Gamma and the Hyperexponential simplify into an Negative Exponential distribution. It is

therefore necessary to incorporate all three

distributions into the model.

Unparking procedure. The unparking manoeuvre consists of two parts. The first, is a gap acceptance problem where the vehicle leaving a

parking space looks for a gap in the through traffic. The second, is the action of unparking.

The distribution of unparking times was also

found to be adequately described by a normal

distribution [5]. Unfortunately, no research into this gap acceptance process could be found.

It was therefore assumed that the gap accepted by the unparking driver equals the time required

to exit the parking space. The unparking process in the model is as follows. If there is an

acceptable gap between the parked vehicle and the first moving vehicle closer to the start of

the link, then the parked vehicle will start to

unpark. If there is not an appropriate distance

the vehicle will wait for the moving vehicle to

pass. It can occur, however, that a passing

vehicle wou I d like to park in the space made

vacant by the unparking vehicle. In such a case

the moving vehicle will stop and let the

unparking vehicle leave.

3.4 Description of network

Initially a parking lot is an open space. Movement in this space can be in any direction.

To develop the simulation model of vehicle

movements it is necessary to develop a network

on which the vehicles move. This r;ay not

represent actua I movements in partly filled

parking stations but will become more realistic

as the parking facility approaches capacity. Since the parking facility which is near capacity is likely to be more critical with respect to vehicle movements this restriction

kas not considered a limitation.

Link ordering. The basic philosophy behind the

consideration of each link is related to the

link type and the position of the link in the

network. First al I the major I inks are considered in order. The links closest to the

exits first. Once all the major links have been considered the minor links are considered. This approach IS consistent with that chosen for the consideration of vehicles on each link.

Intersection simulation. The development of the model from a link to a simple network simulation

required an intersection sub-model. This

involves two parts: the queuing model and a gap

acceptance model.

The queuing process occurs in two manoeuvres. The first type of manoeuvre that may result in queuing occurs when a vehicle moves from a major

link to another link in search of a parking

place. In this case if there is a v,e,hicle in the

required link which is too close to the

intersection the searching vehicle cannot enter

the link. The searching vehicle and following vehicles must queue. The second type of vehicle that may block the intersection is a vehicle leaving the parking lot. This vehicle always

moves into a major link. However, if the major link to be entered has a vehicle stopped too close to the intersection the entering vehicle

will be delayed.

The gap acceptance process can also influence queuing. This can occur as described in the

previous paragraph but can also occur when a vehicle exits a link. Major road vehicles are given priority in this simulation. If a appropriate gap is not present, the vehicle exiting the minor road must queue until there is

an appropriate gap.

3.5 Decision to park

A major component of the modelling is the choice

mode I . The model is used at every decision point

in the network. The functional form of the model of behavioural intention can take many forms.

The decision of the most appropriate will depend on the comparison of reality and the model. The model used here can be classed as a logit model. It determines the probabi I ity of choosing alternative a (p(a)) considering the utility

gained from alternative a (Ua) and takes the

form

p(a)=exp(Ua)/(exp(Ua)+exp(Ub)+.....) (1)

For the choice of which intersection to take the decision maker considers each link as providing

a given utility, or in this model an opportunity

to park. The utility gained from each leg is

therefore a function of the number of parking

spaces better than those available on each leg.

This approach can also be used for the

consideration of a particular parking place on a

I ink. The model for a particular link can

however be simplified by recognising that the

denominator of the logit model is a constant for a particular choice situation.

The simulation model developed in this study is

however a discrete choice simulation model and

requires a definite decision not just a

probability. The discrete choice is obtained by

using the probabilities determined in choice

model as a basis for sampling.

4. CONCLUS I ON

This paper has described a model that simulates the movements of vehicles through a parking

faci I ity. The models main application is to compare parking lot layouts to determine the one most appropriate for a site. The model is still in the early state of development and requires a number of further developments before the model

can reach its full potential. The first is the development of a computer graphics capability. This will aid in the verification of the model

as well as enabling the designer to gain an idea of the workings of the facility. The second is the validation of the model using data from existing facilities.

REFERENCES

[l] N.A.A.S.R.A. (1982). “Guide to traffic engineering practice”. Nat. Ass. of Aust. State Road Authorities, Sydney, Australia.

CZ] Brierley J. (1982). “Parking of motor vehicles”. (Applied science: New York).

[3] Institute of Traffic Engineers (1982). “Transport and traffic engineering handbook”.

[4] Ogden K.W. and Bennett D.W. (1984). “Traffic engineering practice”. Department of Civi I Engineering, Monash University Australia.

[5] Farrow D. (1984). “A simulation model of a simple parking system”. Master of Eng. SC.,

Dept. of Civil Eng., Monash Un i . , Aust.. [6] Young W. (1984). “Traffic simulation”.

Dept. of Civi I Eng., Monash Uni ., Australia.

[7] Gipps P.G. (1981). “A behavioural car-

following model for computer simulation Transportation Research, Vol 158, pp 105- 111.

[8] Armour M. (1982). Vehicle speeds on residential streets”. Proc. Austra I i an Road Research Board Conf., Vol. 11, pp. 190-205.

[9; Hobbs F.D. (1974). “Traffic planning and engineering”. (Birmingham Uni. : England).

[lo] HRB (1971). “Parking principles”. Hi ghway Research Board Special report 125.

[ll] Richardson, A.J. (1974). “An improved parking duration study”. Proc. Australian

Road Research Board Conference, Vol. 7.