# Parking Management Efficiency Analysis through Various Charge Schemes for Day-Long Commuting considering Elastic Travel Demand

Authors: Jing Wang; Chaowu Tan; Zhigang Liu
Journal: Journal of Advanced Transportation
 (2018)

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## Abstract

In this paper, we investigate the travel pattern of the day-long commuting in a bisection bottleneck network and the efficiency of pricing schemes with elastic travel demand. We extend the Vickrey model to morning and evening commutes and allow commuters to arrive at workplace late and depart from workplace early. The parking searching time is considered in the morning commute. Next, we derive the independent morning and evening commuting travel patterns without road toll and parking fee. Then, we propose three pricing regimes: duration dependent parking fees; optimal time-varying road tolls; optimal time-varying road tolls with duration dependent parking fees. We compare the efficiency of the four schemes with elastic demand. Theoretical analysis and the numerical examples show that optimal time-varying road toll is the most efficient pricing scheme. Charging a duration dependent fee neither improves nor deteriorates the scheme of time-varying road toll, if the toll rates are appropriately set. The regime of duration dependent parking fee only is less efficient than the regime of independent morning and evening commuting travel patterns without road toll and parking fee. In the regime of duration dependent parking fee, the social surplus decreases with the increase of duration dependent parking fee rate.


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## Body

## 1. Introduction

With the development of metropolis and process of urbanization, traffic demand is increasing steadily. Expanding road network is not efficient, due to the limitation of budget and urban land. Sometimes network expansion is very complicated due to the involvement of various sectors [1]. Parking management provides a new method for urban traffic management. In morning commuting, parking supply cannot meet the traffic demand especially in downtown area and people have to spend much cost for acquiring parking spot. Many scholars have investigated the parking management in morning commuting. Arnott et al. [2] proposed a model that considers spatial distribution of parking in morning commuting and compared the efficiency of road toll and parking fees regimes. Bifulco [3] presented a stochastic user equilibrium assignment model to evaluate parking policies Verhoef et al. [4] studied the parking policies as a substitute to road pricing. Zhang et al. [5] used parking permits to improve the traffic efficiency by eliminating the competition for inadequate parking spots. Qian et al. [6] studied the economics of parking provision for the morning commuting. However, there are few researches in integral analysis of parking searching time for parking space and bottleneck dynamic. In this paper, we analyze how the parking searching time and parking fees affect the commuter’s behavior and examine the efficiency of road toll and parking fee to improve the social surplus with elastic demand.The basic model used in our paper is originated by Vickrey [7]; in the morning, commuters need go to workplace in central business district through a bottleneck with limited service capacity. In the equilibrium, no one can reduce individual travel cost by changing the departure time. Later, many scholars made a lot of research based the bottleneck model. Daganzo [8] assumed commuters have different working start time and proved uniqueness of the time-dependent equilibrium distribution of arrivals at a single bottleneck. Arnott et al. [9] extended the bottleneck model from fixed demand to elastic demand. Zhang et al. [10] investigated how travelers behave and react on bottlenecks with time-varying capacities, examined the user equilibrium and system optimal traffic patterns, and derived pricing regimes that lead to the system optimum pattern. Recently, many scholars started to measure traffic flows of downtown area. Geroliminis and Daganzo [11] demonstrated the existence of the macroscopic fundamental diagram (MFD), correlating the rate of ending trips with the number of vehicles in an urban area. Arnott [12] assumed that the outflow and the travel time from the downtown area depend on the vehicle accumulation at bottleneck and developed a bathtub model of downtown traffic congestion on the basis of MFD. Fosgerau [13] proposed a similar bathtub model considering the heterogeneity in trip length of population. Geroliminis [14] integrated the traffic dynamic of cruising for parking with a spatially aggregated model of urban hypercongestion, the macroscopic fundamental diagram. Following the study of Geroliminis [14], Liu and Geroliminis [15] explored how the interactions between cruising for parking and congestion reshape the morning commuting considering travelers’ scheduling cost and time of departure choice. Ji et al. [16] considered the parking behavior of mixed autonomous and traditional vehicles in a bottleneck and investigated the effect of fare rate of autonomous vehicles. Jiang [17] found that automatic driving can substantially reduce the queuing delay at the intersection bottlenecks. In this paper, we use the traditional queuing model and Vickrey model and treat capacity of bottleneck as a constant value.The past researches mainly focused on the morning commuting; the evening commuting was not often considered. It was generally believed that the traffic pattern in evening commuting is a mirror image of that in morning commuting. de Palma and Lindsey [18] compared the morning and evening commutes and differed them in just one respect that the schedule preference in morning commuting is in terms of arrival time in destination and the preference in evening is in terms of departure time from destination. Daganzo [19] established the system optimum of day-long commuting and minimized the general social cost of whole day with two travel modes: car and transit. He assumed that the commuters have different wished arrival time and departure time. Gonzales and Daganzo [20] showed that the user equilibrium for isolated morning and evening commutes are asymmetric for different schedule penalty in morning and evening commutes. Zhang et al. [5], Yang et al. [21], and Wang et al. [22] developed methods of parking permit distribution and trading to improve travel efficiency and reduce traffic emission.Motivated by Arnott et al. [2], Zhang et al. [23] derived a model integrating morning and evening commutes and developed a time-varying road toll and location-dependent parking fee regime to achieve system optimum. Zhang and van Wee [24] compared efficiency of several parking schemes and solved the optimal parking fee rate by minimizing the social cost or maximizing social surplus when considering parking searching time.In this paper, Vickrey’s model is developed from a single morning peak to morning and evening peaks. The morning and evening commutes are treated as two independent user equilibrium traffic patterns for the day-long commuting. The analysis of this paper is based on a single bottleneck; it is expected that we can consider day-long parking behavior with network effect with resorting to some approximation methods [25]. Besides, we relax the assumption in the study of Zhang and van Wee [24] by allowing the commuters to arrive at workplace late in morning commuting and leave workplace early in evening commuting. Moreover, traffic demands are often stochastic [26] and dynamic [25]; therefore it is very difficult to precisely predict the travel demands. Travel demands are often influenced by some social and economic factors and affected by travel costs. In the paper, we extend the analysis to considering elastic demands.Figure1 shows the network of day-long commuting. In Figure 1, home is represented by “H” and workplace is represented by “W.” Let N denote the number of commuters who travel from home to workplace in the morning and return from workplace to home in the evening. The service capacity of bottleneck in home-to-work direction is s→ and that in work-to-home direction is s←. We assume the parking lot is near to workplace and ignore the commuters’ walking time between parking spots and workplace. However, searching time of parking spots is considered in morning commuting. We assume that there is no information provided for auto drivers in the parking lot and the parking searching time increases as the number of occupied parking spots increases with a fixed rate π. Here, π is the increasing searching time rate of unit parking spot. Then the searching time for parking spots in morning commuting with n occupied parking spots is πn. So the first commuter has no parking searching time and the last commuter’s parking searching time is πN. When commuters leave from workplace in evening commuting, their searching time for parking spots is zero because they have known where their cars are parked in the morning.Figure 1
A bidirection bottleneck network in morning and evening commutes.In morning and evening commutes, the commuters have an identical desired arrival time to workplacet∗⟶ and a departing time from workplace t∗⟵. In the morning, if commuters cannot arrive at workplace on time, there are schedule delay costs for them. Let β1 be the cost of unit early arrival time and γ1 be the cost of unit late arrival time. In the evening, the commuters also have schedule delay costs for early departing or late departing from workplace. The cost of unit early departing time is β2 and the cost of unit late departing time is γ2. The cost of unit travel time is α for both morning and evening commutes. Following previous related studies [18, 27], it is assumed that γ1>α>β1 and β2>α>γ2. For simplifying the model in this paper, we assume the morning commuting and evening commuting are symmetric. It means that the cost of unit early arrival time in morning commuting β1 is equivalent to the cost of unit late departing time in evening commuting γ2; the cost of unit late arrival time in morning commuting γ1 is equivalent to the unit early departing time in evening commuting β2 and the capacity of bottleneck in home-to-work direction is equivalent to that in work-to-home direction, s→=s←.When there is no toll and parking fee, the individual travel time includes queuing time and searching time for parking spots in morning commuting. However, in evening commuting, the travel time only comprises queuing time. As commuter has known the parking spot, searching time is zero in evening commuting. Besides, we ignore the walking time from parking lot to working place. The free flow travel time spent in home to bottleneck is assumed to be zero in morning, and that in the reverse direction is also zero. It means that commuter arrives to the bottleneck as soon as he leaves home in morning commuting and commuter arrives to bottleneck as soon as he leaves from workplace in evening commuting.In this paper, we investigate the traffic patterns in morning and evening commutes under various regimes and compare their respective efficiency with elastic demand. The four regimes proposed in this paper are listed as follows:Regimef: user equilibrium without road toll and parking fee;Regimeu: duration dependent parking fees;Regimer: optimal time-varying road tolls;Regimeo: optimal time-varying road tolls with duration dependent parking fees.We present the notations used in this paper to describe traffic pattern of four regimes in Notations.


## 2. User Equilibrium without Road Toll and Parking Fee (Regimef)

### 2.1. User Equilibrium in Evening Commuting

In evening commuting, commuters do not need to search parking spots; the individual travel costCef is a function of the departing time from workplace tf⟵. In user equilibrium, the individual travel cost cannot further decrease by changing departing time from workplace. So, we first give the individual travel cost of a commuter departing workplace early:(1)Ceftf⟵=αDtf⟵s←+β2t∗⟵-tf⟵t0f⟵≤tf⟵≤t∗⟵,where t0f⟵ is the time at which the first commuter leaves from workplace and D(tf⟵) is the queue length. The first term of right hand of (1) is the cost of waiting time at bottleneck and the second term is the schedule delay cost.For a commuter departing workplace late, the individual travel cost is(2)Ceftf⟵=αDtf⟵s←+γ2tf⟵-t∗⟵t∗⟵≤tf⟵≤t1f⟵.Here t1f⟵ is the departing time from workplace of the last commuter. According to user equilibrium condition in evening commuting, dCef(tf⟵)/dtf⟵=0, we can get(3)dDtf⟵dtf⟵=s←αβ2t0f⟵≤tf⟵≤t∗⟵-s←αγ2t∗⟵≤tf⟵≤t1f⟵.According to (3), we can easily obtain the departure rate from workplace r1f⟵,  r2f⟵,  r1f⟵=(β2/α+1)s←,r2f⟵=(1-γ2/α)s←. Obviously, in the evening commuting in regime f, the departure rate from workplace of early departing r1f⟵ is larger than capacity of bottleneck in work-to-home direction, r1f⟵>s←, and the departure rate from workplace of late departing r2f⟵ is less than s←, r1f⟵<s←. Then the capacity of bottleneck is fully used in the evening commuting and the beginning time and ending time of evening peak satisfy the equation as follows:(4)t1f⟵-t0f⟵=Ns←.The user equilibrium depicted in Figure 2 requires that the individual travel cost of the first commuter is equivalent to that of the last commuter, Cef(t0f⟵)=Cef(t1f⟵), so the departure times of the first and last commuter are(5)t0f⟵=t∗⟵-γ2β2+γ2Ns←,t1f⟵=t∗⟵+β2β2+γ2Ns←.The individual travel cost in evening commuting of regime f is(6)Cef=β2γ2β2+γ2Ns←.The system cost in evening commuting of regime f is(7)SCef=β2γ2β2+γ2N2s←.In Figure 2, the departing rate from workplace of early departing commuters is r1f⟵ and the departing rate of late departing commuter is r2f⟵. The capacity of bottleneck is s←. The number of early departing commuters is N⟵~f. For the commuter who departs from workplace on time, the queue length at bottleneck is D(t∗⟵) and the waiting time is D(t∗⟵)/s←.Figure 2
User equilibrium in evening commuting of regimef.


### 2.2. User Equilibrium in Morning Commuting

In morning commuting, the travel time includes queuing time and the searching time for parking spot. The individual travel cost is a function with respect to the leaving time from bottlenecktf⟶. The individual travel cost for the commuter arriving to workplace early is(8)Cmftf⟶=αDtf⟶s→+πs→tf⟶-t0f⟶+β1t∗⟶-tf⟶-πs→tf⟶-t0f⟶.Here, the arrival time of the first commuter to parking lot is t0f⟶. The queuing time at bottleneck is D(tf⟶)/s→ and the searching time for parking spot is πs→(tf⟶-t0f⟶).The individual travel cost for the commuter who arrives to the workplace late is(9)Cmf=αDtf⟶s→+πs→tf⟶-t0f⟶+γ1tf⟶+πs→tf⟶-t0f⟶-t∗⟶.According to the equilibrium condition of morning commuting dCmf(tf⟶)/dtf⟶=0, we can get(10)dDtf⟶dtf⟶=s→αβ1+β1-απs→t0f⟶≤tf⟶≤t¯f⟶-s→αγ1+γ1+απs→t¯f⟶<tf⟶≤t1f⟶.Here, we hold the same assumption β1(1+πs→)>απs→ to the study of Arnott et al. [2] to ensure a growing queue at the bottleneck. t¯f⟶ is the leaving time from bottleneck of the commuter who arrives at workplace on time. The arrival rate to bottleneck r1f⟶ and r2f⟶ can be obtained easily, r1f⟶=αs→/α-[β1+(β1-α)πs→],  r2f⟶=αs→/α+[γ1+(γ1+α)πs→].Figure3 shows user equilibrium in the morning commuting of regime f. The length of rush hour interval is N/s→, t1f⟶-t0f⟶=N/s→. The equilibrium condition requires the travel cost of the first commuter to be equivalent to that of the last commuter:(11)β1t∗⟶-t0f⟶=γ1t1f⟶+πN-t∗⟶+απN.So we can easily get the leaving times from bottleneck of the first commuter and last commuter:(12)t0f⟶=t∗-γ1+α+γ1πs→β1+γ1Ns→,t1f⟶=t∗+β1-α+γ1πs→β1+γ1Ns→.The individual travel cost in morning commuting of regime f is(13)Cmf=β1γ1+β1α+γ1πs→β1+γ1Ns→.The system cost in morning commuting of regime f is(14)SCmf=β1γ1+β1πs→α+γ1β1+γ1N2s→.In Figure 3, the arrival rate to bottleneck of early arrival commuters is r1f⟶ and the arrival rate to bottleneck of late arrival commuters is r2f⟶. The number of early arrival commuters is N⟶~f. The capacity of bottleneck is s→ and the arrival rate to the workplace is s→/1+πs→. For the commuter who arrives at workplace on time, the queue length is D(t¯f⟶) and the searching time for parking is πs→(t¯f⟶-t0f⟶). In the user equilibrium, the commuter cannot reduce individual travel cost by changing departing time.Figure 3
User equilibrium in morning commuting of regimef.


## 2.1. User Equilibrium in Evening Commuting

In evening commuting, commuters do not need to search parking spots; the individual travel costCef is a function of the departing time from workplace tf⟵. In user equilibrium, the individual travel cost cannot further decrease by changing departing time from workplace. So, we first give the individual travel cost of a commuter departing workplace early:(1)Ceftf⟵=αDtf⟵s←+β2t∗⟵-tf⟵t0f⟵≤tf⟵≤t∗⟵,where t0f⟵ is the time at which the first commuter leaves from workplace and D(tf⟵) is the queue length. The first term of right hand of (1) is the cost of waiting time at bottleneck and the second term is the schedule delay cost.For a commuter departing workplace late, the individual travel cost is(2)Ceftf⟵=αDtf⟵s←+γ2tf⟵-t∗⟵t∗⟵≤tf⟵≤t1f⟵.Here t1f⟵ is the departing time from workplace of the last commuter. According to user equilibrium condition in evening commuting, dCef(tf⟵)/dtf⟵=0, we can get(3)dDtf⟵dtf⟵=s←αβ2t0f⟵≤tf⟵≤t∗⟵-s←αγ2t∗⟵≤tf⟵≤t1f⟵.According to (3), we can easily obtain the departure rate from workplace r1f⟵,  r2f⟵,  r1f⟵=(β2/α+1)s←,r2f⟵=(1-γ2/α)s←. Obviously, in the evening commuting in regime f, the departure rate from workplace of early departing r1f⟵ is larger than capacity of bottleneck in work-to-home direction, r1f⟵>s←, and the departure rate from workplace of late departing r2f⟵ is less than s←, r1f⟵<s←. Then the capacity of bottleneck is fully used in the evening commuting and the beginning time and ending time of evening peak satisfy the equation as follows:(4)t1f⟵-t0f⟵=Ns←.The user equilibrium depicted in Figure 2 requires that the individual travel cost of the first commuter is equivalent to that of the last commuter, Cef(t0f⟵)=Cef(t1f⟵), so the departure times of the first and last commuter are(5)t0f⟵=t∗⟵-γ2β2+γ2Ns←,t1f⟵=t∗⟵+β2β2+γ2Ns←.The individual travel cost in evening commuting of regime f is(6)Cef=β2γ2β2+γ2Ns←.The system cost in evening commuting of regime f is(7)SCef=β2γ2β2+γ2N2s←.In Figure 2, the departing rate from workplace of early departing commuters is r1f⟵ and the departing rate of late departing commuter is r2f⟵. The capacity of bottleneck is s←. The number of early departing commuters is N⟵~f. For the commuter who departs from workplace on time, the queue length at bottleneck is D(t∗⟵) and the waiting time is D(t∗⟵)/s←.Figure 2
User equilibrium in evening commuting of regimef.


## 2.2. User Equilibrium in Morning Commuting

In morning commuting, the travel time includes queuing time and the searching time for parking spot. The individual travel cost is a function with respect to the leaving time from bottlenecktf⟶. The individual travel cost for the commuter arriving to workplace early is(8)Cmftf⟶=αDtf⟶s→+πs→tf⟶-t0f⟶+β1t∗⟶-tf⟶-πs→tf⟶-t0f⟶.Here, the arrival time of the first commuter to parking lot is t0f⟶. The queuing time at bottleneck is D(tf⟶)/s→ and the searching time for parking spot is πs→(tf⟶-t0f⟶).The individual travel cost for the commuter who arrives to the workplace late is(9)Cmf=αDtf⟶s→+πs→tf⟶-t0f⟶+γ1tf⟶+πs→tf⟶-t0f⟶-t∗⟶.According to the equilibrium condition of morning commuting dCmf(tf⟶)/dtf⟶=0, we can get(10)dDtf⟶dtf⟶=s→αβ1+β1-απs→t0f⟶≤tf⟶≤t¯f⟶-s→αγ1+γ1+απs→t¯f⟶<tf⟶≤t1f⟶.Here, we hold the same assumption β1(1+πs→)>απs→ to the study of Arnott et al. [2] to ensure a growing queue at the bottleneck. t¯f⟶ is the leaving time from bottleneck of the commuter who arrives at workplace on time. The arrival rate to bottleneck r1f⟶ and r2f⟶ can be obtained easily, r1f⟶=αs→/α-[β1+(β1-α)πs→],  r2f⟶=αs→/α+[γ1+(γ1+α)πs→].Figure3 shows user equilibrium in the morning commuting of regime f. The length of rush hour interval is N/s→, t1f⟶-t0f⟶=N/s→. The equilibrium condition requires the travel cost of the first commuter to be equivalent to that of the last commuter:(11)β1t∗⟶-t0f⟶=γ1t1f⟶+πN-t∗⟶+απN.So we can easily get the leaving times from bottleneck of the first commuter and last commuter:(12)t0f⟶=t∗-γ1+α+γ1πs→β1+γ1Ns→,t1f⟶=t∗+β1-α+γ1πs→β1+γ1Ns→.The individual travel cost in morning commuting of regime f is(13)Cmf=β1γ1+β1α+γ1πs→β1+γ1Ns→.The system cost in morning commuting of regime f is(14)SCmf=β1γ1+β1πs→α+γ1β1+γ1N2s→.In Figure 3, the arrival rate to bottleneck of early arrival commuters is r1f⟶ and the arrival rate to bottleneck of late arrival commuters is r2f⟶. The number of early arrival commuters is N⟶~f. The capacity of bottleneck is s→ and the arrival rate to the workplace is s→/1+πs→. For the commuter who arrives at workplace on time, the queue length is D(t¯f⟶) and the searching time for parking is πs→(t¯f⟶-t0f⟶). In the user equilibrium, the commuter cannot reduce individual travel cost by changing departing time.Figure 3
User equilibrium in morning commuting of regimef.


## 3. Duration Dependent Parking Fees (Regimeu)

A continuous duration parking fee usually equals the parking duration time multiplied by charge rateμ [28]. We assume that parking fees are charged based on the parking duration given a uniform parking fee rate μ. Parking costs are assigned to the morning and evening trips, respectively, using an intermediate time point tΔ,  tΔ=t∗⟵-t∗⟶/2,  [5];  tΔ is denoted as the midday time in this section.Using the analytic method shown in AppendixA, we can obtain the departure rate from workplace in evening commuting r1u⟵(μ),  r2u⟵(μ),  r1u⟵(μ)=s←(1+β2-μ/α),  r2u⟵(μ)=s←(1-γ2+μ/α). Here, we require the parking fee rate μ not to exceed β2, μ<β2, to ensure the growing queue at bottleneck. Then, in regime u, we have r1u⟵(μ)>s←,  r2u⟵(μ)<s←.The first commuter’s departure time and the last commuter’s departure time are(15)t0u⟵μ=t∗⟵-γ2+μβ2+γ2Ns←,t1u⟵μ=t∗⟵+β2-μβ2+γ2Ns←.In evening commuting, because commuters in regime u may depart earlier than regime f to reduce their parking fees, the leaving time from workplace is advanced as the commuter’s parking fee rate μ increases. From (15), we can find that, in the user equilibrium of regime u, the starting time and ending time of leaving from workplace will be advanced μ/β2+γ2N/s← on the basis of that in regime f, t0u⟵(μ)-t0f⟵=μ/β2+γ2N/s←.The individual travel cost in evening commuting of regimeu is(16)Ceuμ=β2γ2+β2-γ2μ-μ2β2+γ2Ns←+μt∗⟵-tΔ.The system travel cost in evening commuting of regime u is(17)SCeuμ=122αβ2γ2+β2γ2μ+β2-γ2μ2+αμ-μ3αβ2+γ2N2s←.Similarly, the arrival rate of early and late arrival to bottleneck in morning commuting, r1u⟶μ and r2u⟶(μ) are r1u⟶(μ)=αs→/(α-β1-μ)(1+πs→),  r2u⟶(μ)=αs→/(α+γ1-μ)(1+πs→). Obviously, the departure rates from home r1u⟶μ,r2u⟶(μ) increase as the parking fee rate μ increases, and the departure rate in regime u is not less than that in regime f, r1u⟶(μ)≥r1f⟶,  r2u⟶(μ)≥r2f⟶.In regimeu, if μ>α-β1, the departure rate r1u⟶(μ) is negative and if μ=α-β1, the departure rate r1u⟶(μ) is infinity. So we can set 0≤μ<a-β1 to ensure the existence of equilibrium of morning commuting. Then we have r1u⟶μ>s→,  r2u⟶μ<s→.The arrival times to parking lot of the first and last commuter are(18)t0u⟶μ=t∗⟶+μ-γ1β1+γ1Ns→+μ-γ1β1+γ1πN-απNβ1+γ1,t1u⟶μ=t∗⟶+β1+μβ1+γ1Ns→+μ-γ1β1+γ1πN-απNβ1+γ1.We can find that, in morning commuting of regime u, commuters can delay their arrival time to workplace to reduce the parking fee. So, commuters will put the arrival time backward till no commuter can reduce individual travel cost. The interval of morning rush hour in regime u will be delayed by μ(1+πs→)/β1+γ1N/s→ comparing with that in regime f, t0u⟶(μ)-t0f⟶=μ(1+πs→)/β1+γ1N/s→.The individual travel cost in morning commuting of regimeu is(19)Cmuμ=β1+μγ1-μβ1+γ1Ns→1+πs→+απNβ1+γ1β1+μ+μtΔ-t∗→.The system cost in morning commuting of regime u is(20)SCmuμ=β1γ1-μβ1+γ1N2s→1+πs→+αβ1πN2β1+γ1+12μN2s→1+πs→.According to (16) and (19), we can find that the midday time tΔ cannot affect the total individual travel cost of day-long commuting and the total system cost. However, the midday time can determine the ratio of morning commuting individual travel cost Cmu(μ) (or evening commuting) to the total individual travel cost Cmu(μ)+Ceu(μ).


## 4. Optimal Time-Varying Road Tolls (Regimer)

In regimer, the queue delay at bottleneck can be eliminated by time-varying road toll. Then, the system cost is equivalent to the total schedule delay cost and the optimal time rush hour can be obtained by adjusting road toll to minimize the total schedule delay cost.Using a similar analytic method shown in AppendixB, we can get the toll function τ(tr⟵) with respect to the departing time from workplace tr⟵ in the evening commuting:(21)τtr⟵=β2γ2β2+γ2Ns←+β2tr⟵-t∗⟵t0r⟵≤tr⟵≤t∗⟵,β2γ2β2+γ2Ns←-γ2tr⟵-t∗⟵t∗⟵<tr⟵≤t1r⟵.In the equilibrium shown in Figure 4, the departing times of first and last commuters are(22)t0r⟵=t∗⟵-γ2β2+γ2Ns←,t1r⟵=t∗⟵+β2β2+γ2Ns←.In Figure 4, with the existence of time-varying road toll, the queue is eliminated and the departure rates of early and late departing commuter are equivalent to the capacity s←.Figure 4
User equilibrium in evening commuting of regimer.The individual travel cost in evening commuting of regimer is(23)Cer=β2γ2β2+γ2Ns←.The system cost in evening commuting of regime r is(24)SCer=12β2γ2β2+γ2N2s←.Using a similar method, we can also obtain the optimal toll function in morning commuting which is given as follows:(25)τtr⟶=απN+β11+πs→-απs→tr⟶-t0r⟶t0r⟶≤tr⟶≤t¯r⟶απs→+γ1+γ1πs→β1-γ1πs→β1+γ1Ns→-απs→+γ1+γ1πs→tr⟶-t∗⟶t¯r⟶<tr⟶≤t1r⟶.In the user equilibrium shown in Figure 5, the arrival times to parking lot of the first and last commuter are(26)t0r⟶=t∗⟶-γ11+πs→β1+γ1Ns→,t1r⟶=t∗⟶+β1-γ1πs→β1+γ1Ns→.In Figure 5, the queue at bottleneck in morning commuting is also eliminated by time-varying road toll. With the existence of parking searching time, the arrival rate to workplace is s→/1+πs→.Figure 5
User equilibrium in morning commuting of regimer.The individual travel cost in morning commuting of regimer is(27)Cmr=β1γ11+πs→β1+γ1Ns→+απN.The system cost in morning commuting of regime r is(28)SCmr=12απN2+12β1γ1β1+γ1N2s→1+πs→.


## 5. Optimal Time-Varying Road Tolls with Duration Dependent Parking Fees (Regimeo)

In regimeo, commuters need to pay for the road tolls and duration dependent parking fees in day-long commuting. One aim of time-varying road tolls is to eliminate the queue delay at bottleneck and another aim of time-varying road tolls is to minimize the total schedule delay cost and get the optimal starting and ending time of morning and evening commutes.In Section3, we have found that commuters are willing to delay the arrival time at workplace in morning commuting and putting forward departing time from workplace in evening commuting reduces duration dependent parking fees in regime u. So, in regime u, the total schedule delay cost is not optimal. Then, in regime o, we can determine the optimal rush hour interval by time-varying road tolls to achieve system optimum. However, the system cost in day-long commuting in regime o cannot be reduced further on the basis of the system cost in regime r, because the queue delay is eliminated by road toll and the two regimes have the same and optimal total schedule delay costs.Using an analytic method shown in AppendixC, we can obtain the optimal time-varying road toll function in evening commuting which can be given as(29)τto⟵=μNs←+β2-μγ2β2+γ2Ns←+β2-μto⟵-t∗⟵t0o⟵≤to⟵≤t∗⟵γ2+μβ2β2+γ2Ns←-γ2+μto⟵-t∗⟵t∗⟵<to⟵≤t1o⟵;the optimal starting time and ending time of rush hour in evening commuting are(30)t0o⟵=t∗⟵-γ2β2+γ2Ns←,t1o⟵=t∗⟵+β2β2+γ2Ns←.The individual travel cost in evening commuting of regime o is(31)Ceo=β2γ2β2+γ2Ns←+β2μβ2+γ2Ns←+μt∗⟵-tΔ.The system cost in evening commuting of regime o is(32)SCeo=12β2γ2β2+γ2N2s←.Similarly, the optimal time-varying road toll in morning commuting can be obtained as follows:(33)τto⟶=απN-μNs→+πN+β1+μ1+πs→-απs→to⟶-t0o⟶t0o⟶≤to⟶≤t¯o⟶γ1-μ1+πs→+απs→t∗⟶+β1-γ1πs→β1+γ1Ns→-to⟶t¯o⟶≤to⟶≤t1o⟶.The optimal beginning time and ending time of rush hour in morning commuting are(34)t0o⟶=t∗⟶-γ11+πs→β1+γ1Ns→,t1o⟶=t∗⟶+β1-γ1πs→β1+γ1Ns→.The individual travel cost in morning commuting of regime o is(35)Cmo=β1γ11+πs→β1+γ1Ns→+απN+μtΔ-t∗⟶-μβ11+πs→β1+γ1Ns→. The system cost in morning commuting of regime o is(36)SCmo=12απN2+12β1γ1β1+γ1N2s→1+πs→.


## 6. Elastic Travel Demand considering Daily Travel Cost

In this section, we investigate the efficiency of each pricing charge scheme proposed from Sections2 to 5 in the elastic travel demand. The demand function for travel is given as follows:(37)N=DP,dDdP<0,where N is the number of commuters and P is the private daily travel cost. We assume the demand function D(P) is a strictly decreasing linear function with respect to P. Let D-1(N) be the inverse demand function.A commuter’s surplus with travel demandN is(38)CSN=∫0ND-1xdx-ATCNN,where the first term is the total gross benefit and the second term is the total travel cost. The total travel cost is(39)ATCNN=SCN+RN,where the first term is the total social cost and the second term is the total revenue. Then, we can obtain the social surplus SS(N) which is the sum of the commuter’s surplus, the total revenue, and total externality(40)SSN=∫0ND-1xdx-SCN,where SS(N) is the function of travel demand N.Maximizing the social surplus leads to the following optimality condition:(41)D-1N=MSCN,where MSC(N) represents the marginal social cost.Now we can apply our elastic demand case to evaluate four different parking pricing regimes proposed from Sections2 to 5. We first formulate the linear daily individual travel cost, ATCk(N)=Ik+KkN for regime k, k=f,u,r,o. If=Ir=0, Iu=Io=μ(t∗⟵-t∗⟶). The formulas of parameter Kk in four regimes are given in Appendix D.In market equilibrium, the implemented demandNek can be obtained by solving the two equations representing demand function N=D(P) and cost function P=Ik+KkN, respectively. And the total social cost is proportional to the square of the demand, SCk(N)=LkN2. The average social cost ASCk(N) is LkN, ASCk(N)=LkN.The marginal social cost becomesMSCk(N)=2LkN and the optimal demand Nsk can be obtained by solving the demand function N=D(P) and cost function P=2LkN. If the equilibrium demand Nek is not less than the optimal demand Nsk, Nek≥Nsk, to derive the travel demand Nek to Nsk, the commuter must be charged to internalize the externality Ek which is the difference between marginal social cost and individual travel cost in pricing regime k. The externality Ek is(42)Ek=MSCkNsk-ATCNsk=2Lk-KkNsk-Ik,where Nsk is the optimal demand level in regime k.If the equilibrium demandNek is less than the optimal demand Nsk, Nek<Nsk, to achieve system optimum, the commuter should be subsidized the negative externality Ek.Next, we could compare the number of optimal travel demandNsk in four regimes by determining the sequence of parameter Lk at first. The process of comparing parameter Lk is given in Appendix D.In Figure6, D-1(N) is the reverse demand function. MSCk represents marginal social cost and ATCk is the daily individual travel cost function. ASCk is the average social cost. When there is no constant toll, the equilibrium appears at point e. Maximizing the social surplus can determine the optimal demand Nsk shown at the point s. SH represents the externality for each commuter; it can be charged by a constant road toll or parking fee to induce system optimum. At the optimal demand level, we present the constitution of commuter’s surplus, total externality, total revenue, and total social cost. Here, total revenue is comprised of time-varying road toll and duration dependent parking fees. In regime f, because of no road toll and parking fee, curve ASC and curve ATC are overlapped and the total revenue is zero.Figure 6
Demand and supply function of day-long commuting.In the case of elastic demand, we denoteNso, Nsr,Nsf, Nsu as the optimal travel demand in regimes o,r,f,u. Let SSo(Nso), SSr(Nsr), SSf(Nsf), SSu(Nsu) be the social surplus in regimes o,r,f,u. For the same demand function, we can easily find that Nso=Nsr>Nsf≥Nsu. In Figure 7, SSo(Nso) and SSr(Nsr) are area ABE, SSf(Nsf) is area ACE, and SSu(Nsu) is area ADE, so we can easily obtain that SSo(Nso)=SSr(Nsr)>SSf(Nsf)≥SSu(Nsu). When the parking fee rate μ is zero, μ=0, the social surplus of regime u  SSu(Nsu) is equivalent to that of regime f  SSf(Nsf). If we set a positive parking fee rate μ, μ>0, the social efficiency of regime u is less than that of regime f, SSu(Nsu)<SSf(Nsf). We find that the best pricing regimes are regimes o and r, followed by regime f; and the worst regime is regime u.Figure 7
Comparisons of optimal travel demand and total social surplus in four regimes.


## 7. Numerical Examples

In the network shown in Figure1, the service rate of each bottleneck is assumed to be s→=s←=500veh/h and the total number of commuters N=1000; they go to workplace from home in the morning and go back to home in the evening. We assume each commuter drives his/her car to workplace in the morning and returns home in the evening. The desired arrival time to workplace in the morning t∗⟶ = 9:00 and the desired departing time from workplace t∗⟵=17:00. The unit cost of travel time is α=10$/h, the cost of unit early arrival time in the morning is β1=5$/h, and the cost of unit late arrival time in the morning is γ1=20$/h.In evening commuting, the cost of unit early departing time isβ2=20 $/h and the cost of unit late departing time is γ2=5 $/h. And the searching time of unit parking spot π is 0.72 s/spot. In regime o, the parking fee rate μ is 0.5 $/h and, in regime u, we set parking fee rate from 1 $/h to 4 $/h. The demand function is specified as N(P)=2000-20P, where P is the daily private travel cost.Table1 gives the main computing results. We can find the regimes r and o have the highest social surplus. In regime r, the externality is zero and it means that the time-vary road toll can automatically lead to the system optimum. In regime o, the equilibrium demand is less than the optimal demand and the externality is negative; the decision maker needs to subsidize commuters to achieve system optimum. In regime u, when parking fee rate μ is less than 2.822 $/h, the externality is charged to balance the optimal demand and supply. Otherwise, when parking fee rate is greater than 2.822 $/h, the commuters should be subsidized to achieve system optimum due to the negative externality. Besides, we also find that the social surplus decreases as the parking fee rate increases in regime u and the parking fee has a negative social effect. It means regime u has the highest social surplus when the parking fee rate is zero.Table 1
Main computing results of various regimes in day-long commuting with elastic demand.
Regime
r
o
f
u
1
1.5
2
2.5
3
3.5
4
Equilibrium demand
14
53
1384
1
488
1
321
1
244
1
171
1
101
10
34
970
908
Optimal demand
1453
1453
118
5
113
9
111
6
1093
107
2
1050
1029
1009
Externality
0
-
4.84
20.38
12.72
9.10
5.57
2.15
-
1.17
-
4.42
-
7.60
Individual travel cost
27.3
3
32.17
20.3
8
30.35
35.12
39.7
6
44.27
48.6
7
52.9
5
57.14
Social surplus (103)
72.674
72.674
59.24
2
56.92
7
55.789
54.671
53.576
52.507
51.468
50.460
Marginal social cost
27.3
3
27.3
3
40.7
6
43.07
44.21
45.3
3
46.4
2
47.49
48.53
49.5
4
Social cost (103)
19.85
9
19.85
9
24.14
6
24.520
24.66
5
24.78
2
24.872
24.937
24.978
24.99
8
Revenue (103)
1
9.859
26.897
0
10.036
14.52
3
18.690
22.564
26.1
70
29.53
1
32.669In Table2, r1k⟶ is the departing rate of early arrival commuter to workplace and r2k⟶ is the departing rate of late arrival commuter to workplace in regime k, k={r,o,f,u}. The number of early arrival commuters in regime k in morning commuting is N⟶~k. t0k⟶ is the departure time of the first commuters and t1k⟶ is the departure time of the last commuter in regime k.Table 2
Main computing results in regimek (k=f,r,o,u) in morning commuting under optimal demand.
Regimek (k=f,r,o,u)
r
o
f
u with the following parking fee rate
1
1.5
2
2.5
3
3.5
4
r
1
k
⟶
500
500
909
1136
1299
1515
1818
2273
3030
4545
r
2
k
⟶
500
500
152
157
159
162
165
168
172
175
N
⟶
~
k
1163
1163
991
907
866
827
789
752
717
683
δ
→
k
0.80
0.80
0.84
0.8
0.78
0.76
0.74
0.72
0.70
0.68
t
0
k
⟶
6:26
6:26
6:49
7:00
7:05
7:10
7:15
7:21
7:25
7:30
t
1
k
⟶
9:21
9:21
9:11
9:17
9:20
9:22
9:24
9:27
9:29
9:31In morning commuting, the departure rates from home of early arrival and late arrival of regimesr and o are equivalent to the capacity of bottleneck due to the time-varying road toll. In regimes r and o, the optimal starting time is 6:26 and the optimal ending time is 9:21. In regime u, as the duration dependent parking fee rate μ increases, the departure rates from home r1u⟶ and r2u⟶ increase and the commuters push back their starting and ending time of the morning commuting, t0u⟶ and t1u⟶ to reduce the duration dependent parking fee. Figure 8 shows the ratio of early departing commuters of four schemes. When the parking fee rate μ increases from 0 $/h to 4 $/h, the ratio of the number of early departing commuters to total demand δ→k decreases from 0.84 to 0.68. It means that more commuters depart late from home as the parking fee rate increases in the morning commuting of regime u.Figure 8
The ratio of early and late departing commuters to total demand of four pricing regimes in the morning commuting.Table3 lists the main numerical results in regime k, k={r,o,f,u} in the evening commuting. Here, r1k⟵ is the departing rate of early departing commuters from workplace and r2k⟵ is the departing rate of late departing commuters from workplace. The number of early departing commuters in evening commuting is N⟵~k. t0k⟵ is the departure time of the first commuter and t1k⟵ is the departure time of the last commuter.Table 3
Main computing results in regimek (k=f,r,o,u) in evening commuting under optimal demand.
Regimek (k=f,r,o,u)
r
o
f
u with the following parking fee rate μ
1
1.5
2
2.5
3
3.5
4
r
1
k
⟵
500
500
1500
1450
1425
1400
1375
1350
1325
1300
r
2
k
⟵
500
500
250
200
175
150
125
100
75
50
N
⟵
~
k
291
291
711
792
827
857
884
907
927
945
δ
←
k
0.20
0.20
0.60
0.70
0.74
0.78
0.82
0.86
0.90
0.94
t
0
k
⟵
16:25
16:25
16:32
16:27
16:25
16:23
16:21
16:20
16:18
16:16
t
1
k
⟵
19:20
19:20
18:53
18:44
18:39
18:34
18:30
18:26
18:22
18:18In the evening commuting of regimesr and o, the queues are eliminated by the road toll, so the departure rates from workplace of early departing commuters and late departing commuters r1k⟵, r2k⟵, k=r,o, are same to the capacity of bottleneck s←. In regimes r and o, the optimal starting time of rush time t0r⟵, t0o⟵ is 16:25 and the optimal ending time t1r⟵, t1o⟵ is 19:20. In regime u, as the duration dependent parking fee rate μ increases, the departure rates from workplace r1u⟵, r2u⟵ are reduced, and the commuters put forward their starting and ending time of the evening commuting t0u⟵, t1u⟵ to reduce the duration dependent parking fee. Figure 9 shows the ratio of early departing commuters of four schemes in the evening commuting. When the parking fee rate μ increases from 0 $/h to 4 $/h, the ratio of the number of early departing commuters to total demand δ←k also increases from 0.6 to 0.94. It means that more commuters depart from workplace early with the growth of parking fee rate in the evening commuting of regime u.Figure 9
The ratio of early and late departing commuters to total demand of four pricing regimes in the evening commuting.Figures10 and 11 show the road toll curves of evening and morning commutes in regimes r and o. The two pricing schemes have the same and optimal starting time and ending time in evening and morning commutes. In the evening commuting, the optimal starting and ending times of rush time of regimes r and o are 16:25 and 19:20. The road tolls of first commuter and last commuter are zero in regime r. The road toll of first commuter in regime o is 1.45$ and the road toll of last commuter is zero. The commuter who departs from workplace at 17:00 should pay for the maximum road toll 12.79$ in regime o and the maximum road toll 11.62$ in regime r.Figure 10
Time-varying road toll of evening commuting in regimesr and o.Figure 11
Time-varying road toll of morning commuting in regimesr and o.In the morning commuting, the optimal starting and ending times of rush time are 6:26 and 9:21. In regimer, the road toll of first commuter is 2.91$ and the road toll of last commuter is zero. In regime o, the road toll of first commuter is 1.31$ and the road toll of last commuter is zero. The commuter who arrives at workplace on time should pay for the maximum road toll 13.05$ in regime o and the maximum road toll 13.37$ in regime r.


## 8. Conclusions

In this paper, we extend the bottleneck model from a single morning commuting to a day-long commuting. The morning commuting and evening commuting are treated as two independent user equilibriums. On the basis of previous study of Zhang et al. [23], we relax the assumption by allowing commuters to arrive later than the desired arrival time in morning commuting and depart earlier than the desired time in evening commuting. Besides, we assume that the morning and evening commutes are symmetric.We investigated the day-long commuting of four regimes (regimesf,u,r,o) and derived the traffic pattern of each regime. In user equilibrium of regime f and u, no one can reduce the individual travel cost by changing the departure time. In regimes r and o, the queues at the bottleneck of morning and evening commutes are eliminated by road toll. We first determine the optimal starting and ending time of rush hour by minimizing the total schedule delay cost. Then, we utilize road toll to achieve system optimum.Next, we analyze efficiency of four schemes with elastic demand and find that the regimesr and o have the highest social surplus, followed by regime f and the regime u has the lowest social surplus. In regime u, the duration parking fee may drive commuters to narrow their work duration for reducing the parking fee and the duration parking fee cannot increase social surplus. Furthermore, as the parking fee rate increases, the social surplus decreases. In regimes r and o, the optimal travel demands are equivalent and the two pricing schemes have the highest social surplus. However, in regime o, the duration dependent parking fee should be subsidized to commuters to balance the optimal travel demand and supply as a result of negative externality.These findings also have several implications for traffic management; first, the duration dependent parking fee may reduce the average parking time per car to relieve the pressure of parking demand, but it will reduce the social surplus under optimal travel demand. Second, the time-varying road toll can eliminate the queue delay and minimize schedule delay cost by optimizing the arrival and departure time interval in morning and evening commutes in regimer and o. Third, if the time-varying road toll is carried out, it is inefficient to charge duration dependent parking fee; the duration dependent parking fee cannot increase the social surplus in basis of time-varying road toll except for refunding a part of revenue as a subsidy to commuters.The current paper considers the morning commuting and evening commuting are symmetric. However, the symmetry between morning and evening commutes is easily broken when commuters are heterogeneous [20]. If the commutes have different desired working starting and ending times, then the traffic pattern could be FIFO (first in first out) in day-long commuting and the morning and evening commutes may not be independent. So if we relax this assumption, the single morning or evening commuting is not an independent user equilibrium anymore, and the problem may be more complicated and worth investigating in the future.



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*Source: 1580287-2018-01-23.xml*