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Transportation Research Part B 42 (2008) 38–56

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Integrated daily commuting patterns and optimal road tollsand parking fees in a linear city

Xiaoning Zhang a,*, Hai-Jun Huang b, H.M. Zhang a,c

a Key Laboratory of Road and Traffic Engineering of the Ministry of Education, Tongji University, Shanghai 200092, Chinab School of Economics and Management, Beijing University of Aeronautics and Astronautics, Beijing 100083, China

c Department of Civil and Environment Engineering, University of California Davis, CA 95616, USA

Abstract

For decades, the dynamic traffic patterns of morning and evening commutes have been investigated separately, and it isoften assumed that they are simple mirror symmetries. In this paper, we use a two-stage differential method to establish adaily traffic pattern that links the morning and evening commutes as an integrated one. Based on a bi-direction bottlenecknetwork with a spatial pattern of parking, we use analytical models to describe travelers’ behavior in choosing departuretimes in their morning and evening trips, where a commuter’s morning and evening decisions are joined by a parking loca-tion. Given fixed parking locations of commuters, we firstly derive the evening commute pattern, which is a Nash equilib-rium in the sense that no one can reduce her/his travel cost given other commuters’ decisions. Then the individual eveningcommute costs are allocated to different parking locations in modeling the morning commuting behavior, and the morningtravel pattern is a user equilibrium in the sense that everyone has equal daily travel cost and no one can reduce privatetravel cost by unilaterally changing travel decisions. Then we propose a time-varying road toll regime to eliminate queuingdelay and reduce schedule delay penalty. Furthermore, a time-varying road toll and location-dependent parking fee regimeis developed to achieve a system optimum where the morning schedule delay cost is further reduced to the minimum byreversing the spatial order of parking. In view of the fact that road pricing is hard to implement, we propose a location-dependent parking fee regime with no road tolls to optimize the morning commute pattern, without improving the eveningcommute pattern.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Daily commuting pattern; Nash equilibrium; User equilibrium; Road toll; Parking fee

1. Introduction

In most urban areas parking space in downtowns is of high demand and limited supply, and thus the man-agement of parking could have significant impact on the performance of the transportation system in which

0191-2615/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.trb.2007.06.001

* Corresponding author. Tel.: +86 21 65982443; fax: +86 21 65984583.E-mail addresses: xn.zhang@alumni.ust.hk (X. Zhang), hjhuang@mail.nsfc.gov.cn (H.-J. Huang), hmzhang@ucdavis.edu

(H.M. Zhang).

X. Zhang et al. / Transportation Research Part B 42 (2008) 38–56 39

parking is an integral part. Although various aspects of parking, such as parking policy (Verhoef et al., 1995;Bifulco, 1993), parking behavior (Hester et al., 2002), parking demand (Lam et al., 1998), and parking price(Arnott et al., 1991; Glazer and Niskanen, 1992; Miller and Everett, 1982; Shoup, 1982; Arnott and Rowse,1999) have been considered in the literature, an integral analysis of downtown parking considering commutingbehavior, bottleneck dynamics and parking/congestion fees is still lacking. In this paper, we examine the inter-plays among commuters’ travel behavior, congestion pricing policies, parking demand and the resultingdynamic traffic patterns in a simplified linear city, and analyze how parking fees and road tolls are fairlycharged to improve system efficiency.

The basic modeling approach used in our study is dynamic traffic assignment. Dynamic user equilibriumhas been studied for more than two decades, which has accumulated a vast literature. A typical and analyt-ically solvable dynamic user equilibrium (DUE) problem is the morning commute problem with a bottleneckmodel originated by Vickrey (1969): commuters to the central business district (CBD) travel through a bot-tleneck of limited capacity, and experience queuing delay at the bottleneck if traffic arrival rate exceeds bot-tleneck capacity. Travelers therefore choose different departure times so as to minimize their combined cost ofcongestion delay and late/early arrival penalty, and in equilibrium no one can (strictly) reduce his/her trip costby changing his/her departure time unilaterally.

Past research paid more attention to the modeling of morning home-to-work commute than that of eveningwork-to-home commute. It was generally believed that the evening trip-timing pattern is a mirror image of themorning one, since every morning commute trip is matched by a reverse trip in the evening. Some researchersexamined the differences between morning and evening travel patterns. The distinction between arrival-time-determined scheduling preferences in the morning commute and the departure-time-determined schedulingpreferences in the evening commute is frequently noted in the literature. In line with this, Hurdle (1981) iden-tified some differences between the equilibrium patterns of morning and evening commutes in terms of thedeparture time pattern and magnitude of congestion. Moreover, when travelers are not identical, the symme-try between morning and evening commutes breaks down. Vickrey (1969) and Fargier (1983) demonstratedthis in their insightful articles, and recently de Palma and Lindsey (2002) gave a more thorough analysis tothis.

Arnott et al. (1991) constructed a model that incorporates both the spatial distribution of parking and thetemporal peaking of travel pattern on a highway with a single-bottleneck. In their model, such cost compo-nents as queuing time, walking time, schedule delay, road toll and parking fee are considered, but only morn-ing home-to-work commute is modeled. They investigated and compared the efficiency of various road tolland parking fee regimes, and showed that when parking is incorporated, the rush-hour traffic pattern is dif-ferent from that generated by the traditional bottleneck model. Specifically, queuing dead-weight cost can beeliminated and schedule delay can be minimized with the implementation of time-varying road tolls and loca-tion-dependent parking fees. As the authors pointed out, neglected in their paper are several important real-world features of parking. First, the return commute in the evening is ignored, which can be an importantlimitation if the traffic pattern in the evening commute is not a mirror image of the morning commute whenparking is considered in the analysis. Second, in practice the commuting traffic demand is not fixed, as men-tioned by the authors. Rather, it is elastic and is generally believed to be a decreasing function of trip cost.Moreover, the travel demand should be a function of the daily commuting cost rather than that of the singlemorning or evening trip cost.

Motivated by Arnott et al. (1991), in this paper we attempt to give a more thorough and complete inves-tigation of the daily commuting problem with consideration of parking. First, the evening commuting patternis firstly established given fixed parking locations of commuters. Second, the parking location-dependent com-muting cost in the evening is considered as part of the daily commuting cost in deriving the morning commutepattern. Third, several road toll and parking fee regimes are proposed, and their mechanisms and efficienciesare discussed.

As is commonly adopted in similar studies, the simple network depicted in Fig. 1 is used in our study. Ithas a single origin–destination pair connected by a highway corridor of two routes. The origin represents aresidential area and the destination a city business center. Among the residents living in the origin, somewould go to the city center for work in daytime and come back home after work. For simplicity, we supposeall these travelers are behaviorally homogenous. Moreover, each of the two parallel paths connecting the

Home Work

Work to home

1E

2E

Home to work

Fig. 1. A bi-direction bottleneck network.

40 X. Zhang et al. / Transportation Research Part B 42 (2008) 38–56

origin and the destination has a bottleneck with limited capacities, and it is further assumed that the con-stant travel times on the line haul parts of these two paths are zero. The same network is used in Zhanget al. (2005) to study the dynamic scheduling of work trips using stochastic models with no considerationto spatial parking.

In Fig. 1, the service capacity of the bottleneck in the home-to-work direction is s!,1 and that in the work-to-home direction is s . Assuming N commuters travel from the origin to the destination for work everyday, aqueue will develop if the arrival rate exceeds the capacity.2 There is a parking lot around the CBD, where

the parking spots that are treated as continuous variables are indexed by n in order of increasing distance from the CBD. Walking time to the workplace from location n is taken to be W þ wn, where W is the walking timefrom the parking lot to the workplace and w is the time taken for passing one parking spot on foot. Since thecost related to W is the same to everyone, and thus assumed to be zero for simplicity. The in-vehicle travel timewithin the parking area is ignored, so the interference between vehicles being parked, and between parkers andthrough traffic is assumed away.

Individuals are assumed to have a common preferred arrival time at work (e.g. their official work startingtime) t!

�, and a common preferred leaving time from work (their official off work time) t �. The cost of unit

early arrival time in the morning is taken to be b!

, and late arrival for work is not allowed. In the evening,early departure from work is not allowed, and the cost of unit late departure time is c

. The unit cost of in-

vehicle travel time is a (for both morning and evening commutes), and the unit cost of walking time is k.To ensure the existence of a deterministic equilibrium it is assumed that a > b

!and a > c

. Since in general

people prefer to drive rather than to walk, k > a.In modeling the departure time choice behavior, we use two equilibrium assumptions, namely Nash equi-

librium and user equilibrium. In a Nash equilibrium, no one can reduce private travel cost by changing depar-ture time when all other commuters’ departure times are given. In a user equilibrium, every one has the sametravel cost and no one can reduce her/his travel cost by unilaterally changing departure time. Clearly, Nashequilibrium is a more general assumption, of which user equilibrium is a special case.

The paper is organized as follows. In Section 2, we establish a Nash-equilibrium traffic pattern in the even-ing commute without pricing for a given set of assigned parking spaces, and then the user-equilibrium morningtraffic pattern is derived considering daily travel cost by adding the parking location-dependent evening travelcost into the morning trip. In Section 3, we optimize the evening and morning travel patterns with time-vary-ing road tolls. In Section 4, we derive the system optimal commuting pattern under time-varying road tolls andlocation-dependent parking fees. In Section 5, we propose a location-dependent parking fee and no road tollregime to optimize the morning commuting pattern. Section 6 provides some numerical examples. Finally,Section 7 concludes the paper.

1 Note that, to differentiate the morning commute from the evening commute, an arrow ‘‘!’’ is used in this paper to underline somenotations associated with the morning commute, an arrow ‘‘ ’’ for the evening reverse commute.

2 In the bottleneck model, vehicles are assumed as points so that the queue does not take up space. Such a queue is called a vertical orpoint queue.

X. Zhang et al. / Transportation Research Part B 42 (2008) 38–56 41

2. Morning and evening commuting patterns considering daily travel cost without road toll and parking fee

(regime f)

2.1. Nash-equilibrium travel pattern in the evening commute with fixed parking locations

The parking locations of commuters in the evening were decided in their morning commute and are con-sidered given when we model their evening commute. Since all commuters leave work after t

�, the travel costof an individual who leaves the bottleneck at time t with a parking spot n is2 3

Cfð t ; nÞ ¼ aD ð t Þ þ kwnþ c t �

D ð t Þ � wn� t �4 5; ð1Þ

s s

where D ð t Þ is the length of the queue upon the arrival of a commuter who leaves the bottleneck at time t . Onthe right hand side of the above equation, the first term is the cost of waiting in the queue, the second term isthe cost of walking from the office to the parking spot, and the third term is the penalty cost of leaving work

late. Taking partial derivative with respect to t on both sides, we obtain

oC fð t ; nÞ ¼ c þ

a� c d D ð t Þ

: ð2Þ

o t s d t

Since the queue is zero at the beginning of evening rush hour, and a negative queue is impossible, we haved D ð t Þ

d t P 0 at the beginning of the evening rush hour. Hence condition

oC f ð t ;nÞ

o t > 0 holds for a while, which

means some commuters depart as early as possible to avoid a higher late departure penalty and an increasingqueuing delay, since the travel cost of a commuter increases as the increase of her/his leaving time t from the

bottleneck. Due to the first-in-first-out assumption, commuters want to enter the queue as early as possible toreduce travel cost. Therefore, some commuters depart from the office immediately after t

�, the official office

closure time, and those who parked closer to the city center can enter the queue earlier simply because theyhave shorter walking distances. In this paper, it is assumed that w < 1= s , which guarantees a queue growing at the entrance of the bottleneck. In this situation, arrival rate to the bottleneck is r 1

f ¼ 1=w. Since vehicles are

parked in arrays in a parking lot, the equivalent speed of passing a single vehicle is the number of vehicles in an

array times the physical walking speed. Therefore, it is highly possible that the average walking time of passinga vehicle is less than the headway of vehicles in the bottleneck.

However, the queue may decrease if commuters who parked far from the city center are willing to wait inthe office to avoid the long queue. In other words, it reaches a steady state where the travel cost remains

unchanged when postponing the departure time. That isoC

f ð t ;nÞ

o t ¼ 0, or

d D ð t Þ

d t ¼

c

s

c �a. In this situation, arrival

rate to the bottleneck is r 2

f ¼a� c

a s .

Given the conditions that the first and last arrival encounter no queue, the above traffic pattern is shown in

Fig. 2. In the figure, ABC is the arrival curve to the bottleneck, and AC is the departure curve from the bottle-neck. At the turning point, �k ¼ c

N=ðaþ c

w s �aw s Þ, here �k is the last commuter who leaves the office imme-

diately after work. And the queuing delay encountered by this commuter is �kð1s � wÞ, which is also the maximal

queuing delay among all commuters. In the figure, the vertical axis is the order of commuters arriving at (and/or

departing from) the bottleneck, instead of the order of their parking spots, since the condition that commuters with closer parking spots depart earlier applies only for k 6 �k. For commuters k > �k, they do not necessarilydepart strictly in the order they parked their cars. Instead, their departure times only need to produce an arrivalpattern to the bottleneck depicted by curve BC in Fig. 2. Let n(k) denote the parking spot of the kth departure,the necessary and sufficient condition for n(k) to produce an arrival curve ABC in Fig. 2 is

nðkÞ¼ k 8 k 2 ½0; �k�;6

Nw s � aðN�kÞða� c

Þw s

8 k 2 ð�k;N �:

(ð3Þ

0

f1r

t

N

A

B

C

s

k

*t

k

f2r

*t N*t kw+ + s

Fig. 2. Traffic pattern in the evening commute, regime f.

42 X. Zhang et al. / Transportation Research Part B 42 (2008) 38–56

Obviously there may be multiple solutions to n(k) that satisfies the above constraints, with n(k) = k,"k 2 [0,N] as a typical one.

Set �n ¼ �k. For n 6 �n; D ð t Þ ¼ s ð1� w s Þð t � t �Þ, and thus C

fn ¼ an

s � awnþ kwn. For n > �n, we split

C fð t ; nÞ into two parts, C

0fð t ; nÞ, and C 00fð t ; nÞ where2 3

0fD ð t Þ D ð t Þ �4 5

C ð t ; nÞ ¼ a

s þ c

t � s � t ð4Þ

and

C 00fð t ; nÞ ¼ ðk� c

Þwn: ð5Þ

Clearly C 0fð t ; nÞ is independent of parking location, whereas C

00fð t ; nÞ depends on parking location. From

the above analysis, we can see that C 0fð t ; nÞ ¼

N c

s for all n > �n based on the Nash-equilibrium traffic pattern

shown in Fig. 2.Therefore, if condition 3 is satisfied in regime f, the evening travel cost for a commuter with parking loca-

tion n is given by

C fn ¼

ans � awnþ kwn if n 6 �n;

k� c

� �wnþ

N c

s if n > �n:

8><>: ð6Þ

It is easy to verify that this traffic pattern is a Nash equilibrium in the sense that no one can reduce her/hisown travel cost by unilaterally changing departure time given others’ departure times. For a commuter n 6 �n,s/he cannot depart further earlier, and departing later will increase her/his travel cost. For a commuter n > �n,arriving at the bottleneck before the queuing peak is impossible, and arriving at the bottleneck after the end ofthe queue will increase her/his travel cost.

X. Zhang et al. / Transportation Research Part B 42 (2008) 38–56 43

It is also worth noting that the traffic pattern shown in Fig. 2 is the unique Nash-equilibrium solution, sincethe aggregate arrival rates to the bottleneck before and after the peak queue are uniquely determined.However, the decisions of individual commuters are not unique, because for a commuter with parking loca-tion n > �n the departure time is more or less flexible. One can think of the following process as a generatingmechanism of the above Nash equilibrium: the commuters who parked close knew they have a chance to beatthe traffic thus depart as early as they can, which establishes the first branch of the arrival curve ðn 6 �nÞ. Forthose who parked further out ðn > �nÞ, who through experience realized that there is no advantage in departingsoon because they could not beat the commuters who parked closer in reaching the bottleneck, would stay intheir office longer. This produces the second branch of the arrival curve ðn > �nÞ. For the latter commuters, itdoes not matter who leaves ahead of whom as long as their departure maintains a stable demand for the bot-tleneck to sustain the queue until the last commuter. This is similar to the Nash equilibrium of the classicalmorning commute problem except that the commuters in the combined commute problem consider their walk-ing times to their vehicles when they choose when to leave office for home.

2.2. User-equilibrium traffic pattern in the morning considering daily travel cost

In the previous section, it was found that individual travel cost differs and depends on parking location. There-fore travelers do not have identical evening commutes, since the parking locations they chose in the morningaffected their positions in the evening commuting competition. However, in the morning commute, commutersare a priori identical since they have free choices of departure times and parking locations. Would this samenessof travelers gives rise to a user-equilibrium traffic pattern in the morning, if each commuter considers her/hisdaily travel cost when she/he chooses her/his home-to-work departure time? The answer is given as follows.

The daily commuting cost for a commuter who leaves the bottleneck at time t! and chooses a parking spot n

in the morning is

D! ð t!Þ

Cfð t!; nÞ ¼ a

s!þ kwnþ b

!ð t!� � t!�wnÞ þ C n

f : ð7Þ

For n 6 �n, we have

Cfð t!; nÞ ¼ aD!ð t!Þ

s!þ kwnþ b

!ð t!� � t!�wnÞ þ an

s � awnþ kwn ð8Þ

and

oCfð t!; nÞ

on¼ ð2k� a� b

!Þwþ a= s > 0; ð9Þ

and for n > �n, we have

Cfð t!; nÞ ¼ aD!ð t!Þ

s!þ ð2k� c

Þwnþ b

!ð t!� � t!�wnÞ þ

N c

s ð10Þ

and

oCfð t!; nÞ

on¼ ð2k� b

!� c Þw > 0: ð11Þ

Obviously once a commuter leaves the bottleneck, she/he wants to park as close as possible to the city center.Firstly we look at the case of n 6 �n. Since commuters park outwards around the city center, we have

nð t!Þ ¼ pð t!� t!f0Þ, where p is the departure rate from the bottleneck and t!

f0 is the earliest departure time. If

the bottleneck is operating in full capacity or there is a queue, then p ¼ s!, otherwise p < s . Whether thereis a queue or not depends on the relative benefits of finding a convenient parking location and arriving close

to the official work start time. If the benefit of a convenient parking spot is greater than that of having a close arriving time, some commuters depart from home very early to compete for good parking spots and thus pushforward the start of the rush hour. In such a case, there is no queue in the bottleneck and the departure rate is

44 X. Zhang et al. / Transportation Research Part B 42 (2008) 38–56

less than its service capacity. On the contrary, if arriving close to work start time is still more attractive thanoccupying a close parking spot, then the rush hour is still concentrated tightly before the official work starttime subject to the capacity of the bottleneck and a queue is unavoidable.

Let us now take the latter assumption, i.e. p ¼ s!, and check later the validity of this assumption. In a userequilibrium, no one can unilaterally change departure time (from home) to reduce her/his daily travel cost. Sofor an individual who leaves the bottleneck at time t! with parking spot nð t!Þ ¼ pð t!� t!0

fÞ , the travel cost is

Cfð t ; nÞ ¼ aD!ð t!Þ þ ð2k� a� bÞw s ð t � t fÞ þ bð t � � t Þ þ a

s ð t � t fÞ: ð12Þ

! s! ! ! ! !0 ! ! ! s ! ! !0

And user-equilibrium condition requires

dCfð t!; nÞ

d t!¼ a

s!

d D!ð t!Þ

d t!þ ð2k� a� b

!Þw s!� b

!þ a

s s! ¼ 0: ð13Þ

So the varying rate of the queue is

d D!ð t!Þ

d t!¼

s!a

b!�

a s!s þ aþ b

!�2k

� �w s!

0@

1A: ð14Þ

In this paper, we do not consider the unusual case that the bottleneck capacity in work-to-home direction ismuch greater than that in home-to-work direction. In other words, we always maintains!s >

b!

a þ 1þb!

a � 2ka

� �w s!. With this mild and reasonable assumption, we always have

d D!ð t!Þ

d t!< 0. Since the ini-

tial queue is zero, the user-equilibrium condition leads to a negative queue that is impossible in reality. There-fore the assumption of p ¼ s! is incorrect, and the real situation is that the bottleneck is operating undercapacity, i.e. p < s!. Putting D!ð t!Þ ¼ 0 into Eq. (8), and taking derivative with respect to t! on both sides ofEq. (8), we have

dCfð t!; nÞ

d t!¼ ð2k� a� b

!Þwþ a

s

24

35 dn

d t!� b!; for n 6 �n: ð15Þ

f

In user equilibrium,dC ð t!;nÞ

d t!¼ 0, thus the arrival rate and the departure rate of the bottleneck are equal and

given by

p ¼ r!1

f ¼ dnd t!¼

b!

s

ð2k� a� b!Þw s þa

: ð16Þ

Now we look at the situation of n > �n. Let the departure rate from the bottleneck be denoted as q sonð t!Þ ¼ �nþ qð t!� t!

f�nÞ, here t!

f�n is the �nth commuter’s departure time. If there is a queue in this period, then

q ¼ s! and the individual daily travel cost is� �

Cf t!; n� �

¼ aD! t!

s!þ w 2k� c

� ��nþ s! t!� s! t!

f�n

� �þ b!

t!� � t!�w �nþ s! t!� s! t!

f�n

� �� �þ

N c

s :

ð17Þ

In user equilibrium, no one can unilaterally change leaving time (from work) to reduce the travel cost.

Hence

dCf t!; n� �d t!

¼ as!

D! t!

� �d t!

þ 2k� b!� c

� �w s!� b

!¼ 0; ð18Þ

X. Zhang et al. / Transportation Research Part B 42 (2008) 38–56 45

and then

d D! t!

� �d t!

¼s!a

b!� 2k� b

!� c

� �w s!

� �: ð19Þ

Therefore, the prerequisite of a positive queue is b!� 2k� b

!� c!

� �w s! > 0, and we denote this case as

regime f(a). In this case, the departure rate is

r!fa2 ¼

a s!

a� b!þ 2k� b

!� c

� �w s!

: ð20Þ

With the last commuter arriving on time, the traffic pattern in regime f(a) is depicted in Fig. 3. In the figure,ABC is the arrival curve to the bottleneck, ABF is the departure curve from the bottleneck, and ADE is thearrival curve to the workplace. Individual travel cost in this regime is

TCfa ¼ Nb!

s!þ w b

!þ

c

a� aw s þ c

w s

as �

b!

s!þ 2kw� aw� b

!w

0@

1A

24

35: ð21Þ

And the social travel cost in this regime is

SCfa ¼ N 2

b!

s!þ w b

!þ

c

a� aw s þ c

w s

as �

b!

s!þ 2kw� aw� b

!w

0@

1A

24

35: ð22Þ

When b!�ð2k� c

� b!Þw s! 6 0, and we denote the situation as regime f(b). In this case, there is no queue in

the bottleneck, i.e. q < s!. With user-equilibrium conditiondCf ð t!;nÞ

d t!¼ 0, the arrival rate or departure rate of the

bottleneck is

q ¼ r!fb2 ¼

dnd t!¼

b!

2k� b!� c

� �w: ð23Þ

*t0

f1r

t

N

A

B

C

s

n

*t wN−

E

D

F

n

f1

f11

r

wr+

1

s

ws+

f0t

f f0 1t r n+

fa2r

Fig. 3. User-equilibrium traffic pattern in the morning commute, regime f(a).

0 t

N

A

B

C

n

E

D

f1r

f1

f11

r

wr+

*t*t wN−

n

f0t

fnt

fb2r

fb2r

fb2rw+1

Fig. 4. User-equilibrium traffic pattern in the morning commute, regime f(b).

46 X. Zhang et al. / Transportation Research Part B 42 (2008) 38–56

The corresponding traffic pattern in regime f(b) is shown in Fig. 4. In the figure, ABC is the arrival (and alsodeparture) curve of the bottleneck, and ADE is the arrival curve to the workplace. Individual travel cost inregime f(b) is

TCfb ¼ N w 2k� c

� �þ

c

s

24

35: ð24Þ

And the social travel cost in this regime is

SCfb ¼ N 2 w 2k� c

� �þ

c

s

24

35: ð25Þ

Regime f is inefficient in three aspects. First, in the evening every commuter encounters a queuing delay,and the queuing delay can be eliminated if commuters are willing to stay in the office for some time after work.Second, in the whole duration of morning rush hour in regime f(b), the bottleneck capacity has not been fullyused; and the same problem also exists in the former part of morning rush hour in regime f(a). If commuters’arrival times are concentrated more tightly before the work start time, the total unpunctuality cost in themorning can be reduced. Third, the morning queuing delay in regime f(a) is purely dead-weight loss, whichcan be removed by introducing an appropriate departure pattern.

3. Morning and evening commuting patterns with time-varying road tolls (regime r)

In this section, we investigate the situation that time-varying tolls are charged in both morning and eveningcommutes, in order to eliminate the efficiency loss in regime f.

3.1. Travel pattern in the evening commute with a time-varying road toll

With a time-varying road toll, the travel cost of an individual who leaves the bottleneck at time t with aparking spot n is

C r t ; n� �

¼ kwnþ c

t �wn� t �

� �þ s t

� �: ð26Þ

The first term at the right hand side is the walking time cost. The second term is the penalty cost for leavinglate from the workplace. The last term s ð t Þ is the toll to be charged when departing the bottleneck at time t .

X. Zhang et al. / Transportation Research Part B 42 (2008) 38–56 47

In equilibrium, no one can shift her/her departure time to reduce travel cost, the optimal departure time hasto satisfy the following equation:

oC r t ; n� �o t

¼ 0: ð27Þ

That is

d s t

� �d t

¼ � c : ð28Þ

Therefore, to establish a user-equilibrium traffic pattern, a time-varying linear toll has to be charged at theexit of the bottleneck. Namely

s t

� �¼ s

0 � c

t � t �

� �: ð29Þ

To keep a lowest no-negative toll level, the last commuter pays no toll. In other words, the first departure

has to pay s 0 ¼

N c

s . The Nash-equilibrium traffic pattern is depicted in Fig. 5, where AB is both arrival and

departure curve of the bottleneck. With the same definition of n(k), the necessary and sufficient condition onn(k) that can support the arrival curve AB in Fig. 5 is nðkÞ 6 k

w s ; 8k 2 ½0;N �. Obviously there are also multiple

solutions here, with n(k) = k, "k 2 [0,N] as a typical one. In equilibrium a commuter with parking spot n has

an individual travel cost ðk� c Þwnþ

N c

s .

With the evening road toll, the queuing delay is fully eliminated by postponing the departure times of com-muters. But it also increases the waiting time in the office after work, and the total increased waiting time is

equal to the original total queuing delay. Since a > c

, social travel cost decreases compared with regime f.

0

rr s=

* /t N s−t

N

A

B

k

*t

Fig. 5. User-equilibrium traffic pattern in the evening commute, regime r.

48 X. Zhang et al. / Transportation Research Part B 42 (2008) 38–56

3.2. Travel pattern in the morning commute with an optimal time-varying road toll

If a commuter leaves the bottleneck at time t! and parks her/his car at spot n, the individual daily travel costis

Cr t!; n� �

¼ s! t!

� �þ 2k� c

� �wnþ b

!ð t!� � t!�wnÞ þ

N c

s : ð30Þ

And we have

oCr t!; n� �on

¼ 2k� b!� c

� �w > 0; ð31Þ

which means one wants to find a closest park spot to the city center once she/he leaves the bottleneck. Becausethe bottleneck is operating in full capacity and commuters park outwards, the parking spot for a commuterwho arrives at the parking lot at time t! is nð t!Þ ¼ s!ð t!� t!

r0Þ, here t!

r0 is the earliest departure time in this

regime.So for an individual with parking spot n, we have

Cr t!; n� �

¼ s!ð t!Þ þ ð2k� c Þw s!ð t!� t!

r0Þ þ b

!t!� � t!�w s! t!� t!

r0

� �� �þ

N c

s : ð32Þ

In equilibrium, no one can unilaterally change leaving time from home (or t!) to reduce her travel cost.Therefore

dCrð t!; nÞ

d t!¼

d s!ð t!Þ

d t!þ ð2k� b

!� c Þw s!� b

!¼ 0: ð33Þ

So the temporally varying rate of the bottleneck tolls is

d s!ð t!Þ

d t!¼ b!�ð2k� b

!� c Þw s! : ð34Þ

The time dependent tolls are given by

s!ð t!Þ ¼ s!0 þ b

!t!�ð2k� b

!� c Þw s! t!; ð35Þ

here s!0 is the toll to be paid by the first commuter. To deduce a minimal level of non-negative tolls, s!

0 is givenby

s!0 ¼

0 if b!�ð2k� b

!� c Þw s! P 0;

ð2k� c � b!ÞwN � b

!Ns!

if b!�ð2k� b

!� c Þw s! < 0:

8<: ð36Þ

The morning traffic pattern in the bottleneck is shown in Fig. 6. In the figure, AB is the departure and arri-val curve of the bottleneck, and AC is the arrival curve to the workplace. In this regime, daily individual travelcost is

TCr ¼b!

wþ 1s!

� �N þ

c

N

s if b!�ð2k� b

!� c Þw s! P 0;

ð2k� c ÞwN þ

c

N

s if b!�ð2k� b

!� c Þw s! < 0:

8>>><>>>:

ð37Þ

*t0 *t wN N s− −

t

N

A

B C

1s

ws+

n

s

*t wN−

Fig. 6. User-equilibrium traffic pattern in the morning commute, regime r.

X. Zhang et al. / Transportation Research Part B 42 (2008) 38–56 49

Clearly when b!�ð2k� b

!� c Þw s! < 0, individual travel cost in this regime is the same as in regime f(b).

The daily social travel cost includes walking time cost and unpunctuality penalty cost, and is given by

SCr ¼ kwþc

2

1

s � w

0@

1Aþ b

!

2wþ 1

s!

0@

1A

24

35N 2: ð38Þ

The morning road toll results in an optimal arrival traffic pattern to the bottleneck, the economic benefitsare twofolds. On one hand, it makes parking spots less competitive in the morning commute, and thus reducesthe schedule delay cost by shortening the morning rush hour to the minimal value N= s!. On the other hand, italso eliminates the queuing delay in regime f(a).

4. Time-varying road tolls and location-dependent parking fees for a system optimum (regime o)

In the morning commute of regime f, commuters park their cars outwards due to the competition for a con-venient parking spot. Actually parking inwards is more efficient since it can decrease the total unpunctualitypenalty cost. If commuters park inwards, arrival time to the workplace of the first commuter can be postponedby 2wN, and hence the average early arrival time is reduced by wN. The system optimal morning traffic patternwith a reversed parking order is shown in Fig. 7, where AB is the departure and arrival curve of the bottleneck,and CB is the arrival curve to the workplace. In this section, we seek optimal location-dependent parking feesto achieve the system optimum morning commuting pattern, based on the evening traffic pattern in regime rthat is already optimized.

With a parking fee, the individual travel cost for a commuter who leaves the bottleneck at time t! and parksat spot n is

Co t!; n� �

¼ /ðnÞ þ 2k� c

� �wnþ b

!ð t!� � t!�wnÞ þ

N c

s : ð39Þ

Parking inwards requires

oCo t!; n� �on

¼ 2k� c � b!

� �wþ d/ðnÞ

dn< 0: ð40Þ

*t0

*t N s wN− +t

N

A

B

C

1s

ws−

n

s

*t N s−

Fig. 7. User-equilibrium traffic pattern in the morning commute, regime o.

50 X. Zhang et al. / Transportation Research Part B 42 (2008) 38–56

So the spatially varying rate of parking fees has to satisfy d/ðnÞdn < ðb

!þ c �2kÞw. With parking inwards, the

parking location of a commuter who arrives at the parking lot at time t! is nð t!Þ ¼ N � s!ð t!� t!o0Þ, where t!

o0 is

the earliest departure time in regime o. This gives rise todnð t!Þ ¼ � s . In user equilibrium, we have

d t! !� �

dCo t!; n

d t!¼ � d/ðnÞ

dns!� 2k� c

� �w s!þ b

!w s!� b

!¼ 0: ð41Þ

So the varying rate of the parking fees is

d/ðnÞdn

¼ b!þ c �2k

� �w�

b!

s!; ð42Þ

which satisfies the requirement of parking inwards.So the target parking fee scheme is

/ðnÞ ¼ /0 þ ðb!þ c �2kÞwn�

b!

s!n: ð43Þ

To achieve a minimal level of non-negative parking tolls, the parking fee for the closest parking spot is

/0 ¼ ð2k� b!� c ÞwN þ

b!

S!N .

In this regime, individual travel cost is

TCo ¼ N 2k� b� c

� �wþ

b! þ

c

24

35: ð44Þ

! s! s

In system optimum, the social travel cost is

SCo ¼ N 2 kwþc

2

1

s � w

0@

1Aþ b

!

2

1

s!� w

0@

1A

24

35: ð45Þ

X. Zhang et al. / Transportation Research Part B 42 (2008) 38–56 51

By replacing the morning time-varying road tolls with the above location-dependent parking fees, not onlythe arrival pattern to the bottleneck but also the parking order is optimized, and thus a daily system optimumhas been achieved.

5. Optimal location-dependent parking fees without road tolls (regime p)

As discussed in Arnott et al. (1991), road pricing has disadvantages in implementation, such as high oper-ating cost, interfering traffic and political opposition etc. Parking fees are relatively more favorable than roadtolls, since the transaction entailed in making payment generally does not impede traffic flow. That’s why roadpricing has been implemented only in few cities, and almost all cities are collecting parking fees in urban area.In this section, we discuss the location-dependent parking fee regime without both home-to-work and work-to-home road tolls. Since parking fees only affect morning commuting pattern, only morning travel patterncan be optimized based on the evening travel pattern in regime f.

With a location-dependent parking fee, the daily commuting cost for a commuter who leaves the bottleneckat time t! and chooses a parking spot n in the morning is

Cp t!; n� �

¼ /ðnÞ þ kwnþ b!ð t!� � t!�wnÞ þ C

fn: ð46Þ

For n 6 �n, we have

Cp t!; n� �

¼ /ðnÞ þ kwnþ b!ð t!� � t!�wnÞ þ an

s � awnþ kwn ð47Þ

and

oCp t!; n� �on

¼ d/ðnÞdnþ 2k� a� b

!

� �wþ a= s ; ð48Þ

and for n > �n, we have

Cp t!; n� �

¼ /ðnÞ þ 2k� c

� �wnþ b

!ð t!� � t!�wnÞ þ

N c

s ð49Þ

and

oCp t!; n� �on

¼ d/ðnÞdnþ 2k� b

!� c

� �w: ð50Þ

To support an inward parking pattern, oroCpð t!;nÞ

on < 0, the varying rate of parking fees has to satisfy

d/ðnÞdn

<

aþ b!�2k

� �w� a= s if n 6 �n;

b!þ c �2k

� �w if n > �n:

8>>><>>>:

ð51Þ

Since commuters arrive at the parking lot in a rate of s! and park inwards, the parking location of a com-

muter who arrives at the parking lot at time t! is nð t!Þ ¼ N � s! t!� t!p0

� �, where t!

p0 is the earliest departure

time in regime p. Sodnð t!Þ

d t!¼ � s!.

52 X. Zhang et al. / Transportation Research Part B 42 (2008) 38–56

For n 6 �n, user-equilibrium condition requires

dCp t!; n� �d t!

¼ � d/ðnÞdn

s!� 2k� a� b!

� �w s!� b

!� a

s s! ¼ 0; ð52Þ

or

d/ðnÞdn

¼ aþ b!�2k

� �w� a

s �

b!

s!: ð53Þ

For n > �n, user-equilibrium condition requires

dCp t!; n� �d t!

¼ � d/ðnÞdn

s!� 2k� b!� c

� �w s!� b

!¼ 0; ð54Þ

or

d/ðnÞdn

¼ b!þ c �2k

� �w�

b!

s!: ð55Þ

Obviously the varying rate of parking fees in user-equilibrium satisfies the above condition (51) of parkinginwards. We assume that the first departure bears no parking fee, thus the minimal non-negative parking feesfor all locations are given by

/ðnÞ ¼2k� b

!� c

� �w N � �nð Þ þ

b!ðN�nÞ

s!þ 2k� a� b

!

� �wþ a

s

� ��n� nð Þ if n 6 �n;

2k� b!� c

� �wðN � nÞ þ

b!ðN�nÞ

s!if n > �n:

8>>><>>>:

ð56Þ

In this regime, the individual travel cost is

TCp ¼ N w 2k� b!� c

� �þ

b!

s!þ

c

s

24

35: ð57Þ

Comparing the above travel cost with that in Eq. (44), we can see that the individual travel cost in thisregime is the same as in regime o.

And the social travel cost in this regime is

SCp ¼ N 2

22k� b

!� c

� �wþ

b!

s!þ

c

s 2�

c

aþ c

w s �aw s

0@

1A

24

35: ð58Þ

6. Numerical examples

In this section, we present some numerical examples to clearly demonstrate the analytical results obtainedin the above sections. In the network shown in Fig. 1, the service rate of each bottleneck is assumed to bes! ¼ s ¼ 1:0� 102 veh=min, and totally N = 1.0 · 104 people are assumed to live in the residential area. Eachperson drives his/her own car to work in the morning and returns home in the evening. The official work starttime is t!

� ¼ 9 :00 am and the end time is t � ¼ 17 :00 pm. The unit cost of in-vehicle waiting time is a = 0.6 $/

min. The time for passing one parking spot by walk is w = 0.001 min and its unit cost is k = 2.0 $/min. In the

X. Zhang et al. / Transportation Research Part B 42 (2008) 38–56 53

morning, the penalty cost per unit time for arriving early is b!¼ 0:3 $=min and late arriving is not allowed. In

the evening, the penalty cost per unit time for leaving late is c!¼ 0:3 $=min, and early arriving is not allowed.

Given these parameters, the morning and evening traffic patterns are derived for various pricing regimesbased on the models derived earlier.

In regime f, all components of daily travel cost for different parking locations are depicted in Fig. 8. Theevening travel cost is represented by the part below the bold lines (curve AEF), and the morning travel cost isthe upper part. Curve ACD represents the walking time cost of each commuter in the evening. The combina-tion of queuing time cost and schedule delay cost in the evening is represented by area ACDFEA, in whichACE is purely queuing time cost. If commuters depart in the strict order of their parking spots, the verticaldistance between CD and ED is the queuing time cost of the corresponding commuter, and that betweenED and EF is the schedule delay cost. Namely for a commuter nD; �n < nD < N , her/his queuing time is yD,and the schedule delay cost is xD. However, this is not true if commuters do not depart in the order of parkingspots. In the latter case, xD and yD may not be the queuing time cost and the schedule delay cost respectively,but the summation of the two costs of commuter nD must be xD + yD. In the morning, the walking time cost isrepresented by area AEFHGA and the schedule delay cost is the area AGHJIA. Clearly, no one encounters aqueue in the morning commute. This map shows us although commuters have different travel costs in eveningand morning commutes, the daily travel cost is equal for everyone and thus a user-equilibrium travel patterncan be established. In this regime, the competition for close parking spots is very keen, and people are willingto depart very early in the morning to occupy a convenient parking spot.

In regime r, the cost distribution is shown in Fig. 9. In the evening, the area below curve AC representswalking cost, and area ACEDA denotes the summation of schedule delay cost and road toll. Similar as inregime f, if commuters depart in the strict order of parking spots, the vertical distance between AC and AE

67

20

27

20

17.526

10.526

28.422

10.526A B

GF

E

D

C

HJI

Parking Location (n)

Cos

t ($)

0 5263 10000n

x

y

Δ

Δ

Fig. 8. Travel cost combination, regime f.

33

20

27

20

A B

G

F

E

D

C

H

Parking Location (n)

Cos

t ($)

0 10000

30

4

Fig. 9. Travel cost combination, regime r.

54 X. Zhang et al. / Transportation Research Part B 42 (2008) 38–56

is the schedule delay cost, and that between AE and DE is the road toll to pay. Otherwise, if commuters do notdepart in the order of parking spots, we cannot split the two costs with line AE. In the morning, walking cost,schedule delay cost, and road toll are represented by areas DEGD, DGFD, and FGHF respectively. In thisregime, the evening queue is eliminated by a time-varying road toll. And the schedule delay cost in the morn-ing is reduced, since the rush hour is shortened by the morning road toll. From the two figures, we can see thatin both regime f and regime r individual travel cost is $67. However regime r is superior over regime f in termof social travel cost, since the revenue collected in regime r is still useful to the society.

In regime o, the cost distribution is shown in Fig. 10. The evening costs are the same as in regime r. In themorning, the walking cost, schedule delay cost, and road toll are represented by areas DEFD, DFGD, andDGHD respectively. Clearly, by reversing the parking order, the maximal schedule delay cost reduces to$27 from the original $33 in regime r. Achieving this schedule delay cost reduction is at the cost of increasingthe individual travel cost by $27.

In regime p, the cost distribution is shown in Fig. 11. The evening costs are the same as in regime f. In themorning, the walking cost, schedule delay cost, and road toll are represented by areas AEFHG, AGHJIA, and

64

20

27

20

A B

G

F

E

D

C

H

Parking Location (n)

Cos

t ($)

0 10000

30

27

Fig. 10. Travel cost combination, regime o.

9420

27

20

14.211

10.526

28.422

10.526A B

GF

E

D

C

H

J

I

Parking Location (n)

Cos

t ($)

0 5263 10000

K L

2730.315

Fig. 11. Travel cost combination, regime p.

X. Zhang et al. / Transportation Research Part B 42 (2008) 38–56 55

AIJLKA respectively. The superiority of this regime over regime f is that schedule delay cost is minimized inthe morning. We can see that in the morning commute the maximal schedule delay cost reduces sharply from$67 in regime f to $27 in this regime. As derived in the theoretical analysis, this regime has the same individualtravel cost as in regime o, but the social cost in this regime is much higher than that in regime o.

7. Conclusions

In this paper, we investigated the joint morning and evening rush-hour commutes with a two-stage differ-ential method, based on a linear city with one end of residential area and another end of workplace. In the firststage, the evening work-to-home commuting pattern was developed, and travel costs have been derived forindividual commuters with different parking locations. Then in the second stage, by attaching the evening tra-vel costs to different parking locations, the morning home-to-work commuting pattern was established takinginto account total travel cost in a whole day. Furthermore, we optimized various road toll and/or parking feeregimes to improve the system efficiency.

In the case of free pricing, we derived a Nash-equilibrium evening travel pattern, given the parking loca-tions that have been chosen by commuters in the morning commute. In the equilibrium, no one can reduceher/his private travel cost by changing departure time given other commuters’ decisions, but commuters withdifferent parking spots have different travel costs. In the evening, some commuters who parked close to citycenter leave workplace immediately after work, but other commuters are willing to stay in the office for a whileto enjoy a diminishing queue in the bottleneck. We then established a user-equilibrium traffic pattern in themorning commute, considering both home-to-work and work-to-home travel costs. In this equilibrium, com-muters have the same travel cost that cannot be further reduced by unilaterally changing travel decisions. Inmorning rush hour, there are two different situations dependent on the setting of parameters: one is that thebottleneck is always operating under capacity, the other is that the arrival rate is below bottleneck capacityfirstly and then beyond the capacity later. If commuters have a strong intention to occupy a close parking loca-tion, the first situation will occur, otherwise the second situation will occur.

We proposed a pricing regime with time-varying road tolls in both morning and evening commutes. Theevening road tolls eliminate the queuing delay in the work-to-home trip, at the cost of increasing the waitingtime cost in the office after work. The saving of queuing delay cost is greater than the increase of scheduledelay cost. The morning road tolls eliminate the queuing delay (if any) in the home-to-work trip, and mean-while reduces the morning schedule delay cost by shortening the morning rush hour. We then propose a newregime that replaces the morning time-varying road toll scheme with a location-dependent parking fee scheme,and maintains the evening time-varying road toll scheme. The new regime further reduces the morning sche-dule delay cost to the minimum by reversing the spatial order of parking, and thus results in a system opti-mum. Moreover, in a situation that road pricing is hard to implement, we proposed a location-dependentparking fee regime without road tolls that only optimizes the morning commuting pattern based on the ori-ginal evening commuting pattern in the free-pricing regime. This parking fee regime can eliminate queuingdelay in the morning trip and minimize the schedule delay cost with reverse order of parking.

The new findings in this paper have several implications to traffic management. First, morning road tollscan only affect the traffic pattern in the morning rush hour, however evening road tolls can affect both even-ing and morning traffic patterns. This implies pricing the evening work-to-home trip may be more efficientthan pricing the home-to-work trip. But in most cities that implemented road pricing, road tolls are col-lected in the inbound trips. Clearly a comparative study of the efficiency between inbound and outboundroad pricing is needed. Second, parking outwards results in more schedule delay cost than parking inwards,since the rush hour is pushed forward for a certain time. Location-dependent parking fees can reverse theparking order and thus reduce the schedule delay cost, but at the cost of raising individual travel cost. Androad tolls can never change parking order. Third, a location-dependent parking fee can be more efficientthan a morning road toll, since parking fees can obtain not only an optimal departure pattern but alsoan appropriate parking pattern. Fourth, location-dependent parking fees can only affect morning commut-ing pattern, and has no influence on evening commuting pattern. Finally, only parking pricing or road pric-ing cannot achieve system optimum, they have to be implemented jointly in order to obtain an optimal dailytravel pattern.

56 X. Zhang et al. / Transportation Research Part B 42 (2008) 38–56

It should be pointed out that we made an assumption that later arrival and early departure for work are notallowed, in order to derive analytically a Nash-equilibrium traffic pattern in the evening commute. We shallinvestigate in our future work the feasibility of deriving an equilibrium traffic pattern when this assumptionis relaxed, or showing that an Nash equilibrium does not exist under the relaxed assumptions.

Acknowledgements

The authors wish to thank Prof. Kenneth A. Small of University of California at Irvine and an anonymous ref-eree for their helpful comments in improving the paper. This study has been substantially supported by the NationalNatural Science Foundation Council of China through three projects (#70401016, #70571058 and #70521001).

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