Macroscopic modeling of on-street and garage parking: Impact on traffic performance

1 The short-term interactions between on-street and garage parking policies and the associated parking pricing can be 2 highly influential to the searching-for-parking traffic and the overall traffic performance in the network. In this paper, 3 we develop a macroscopic on-street and garage parking decision model and integrate it into a traffic system with an 4 on-street and garage parking search model over time. 5 We formulate an on-street and garage parking-state-based matrix that describes the system dynamics of urban traffic 6 based on different parking-related states and the number of vehicles that transition through each state in a time slice. 7 This macroscopic modelling approach is based on aggregated data at the network level over time. This leads to data 8 collection savings and a reduction in computational costs compared to most of the existing parking/traffic models. 9 This easy to implement methodology can be solved with a simple numerical solver. 10 All parking searchers face the decision to drive to a parking garage or to search for an on-street parking space in the 11 network. This decision is affected by several parameters including the on-street and garage parking fees. Our model 12 provides a preliminary idea for city councils regarding the short-term impacts of on-street and garage parking policies 13 (e.g., converting on-street parking to garage parking spaces, availability of garage usage information to all drivers) 14 and parking pricing policies on: searching-for-parking traffic (cruising), the congestion in the network (traffic 15 performance), the total driven distance (environmental conditions), as well as the revenue created for the city by the 16 hourly on-street and garage parking fee rates. This model can be used to analyze how on-street and garage parking 17 policies can affect the traffic performance; and how the traffic performance can affect the decision to use on-street or 18 garage parking. The proposed methodology is illustrated with a case study of an area within the city of Zurich, 19 Switzerland. 20 Manuel Jakob, Monica Menendez Page 3


Introduction
garage parking. These empirical methods usually focus on collecting data for both on-street and garage parking, e.g.,

30
For modelling both on-street and garage parking and the associated parking fees, [1] and [16] illustrate how the actual 31 full price of parking contains both the interaction between garage operators and the cruising costs for on-street parking.

32
They develop a spatial competition model to eliminate cruising by allocating excessive cruising demand to garage 33 parking and focus on social optimum suggestions concerning the relationship between curbside and garage fares. [20] 34 models variable on-street and garage pricing in real-time for effective parking access and space utilization by using a 35 dynamic Stackelberg leader-follower game theory approach.
[24] develops a real-time pricing approach for a parking 36 lot based on its occupancy rate as a system optimal parking flow minimization problem. They assume a user 37 equilibrium travel behavior and only focus on garage parking without analyzing its interdependency with on-street 38 parking in the network.
[27] studies park-and-ride (P+R) networks with multiple origins and one destination and focus 39 on an optimal parking pricing strategy. They only focus on setting optimal parking fees for P+R terminals and do not consider the interaction with on-street parking.
[29] models multi-modal traffic with limited on-street and garage parking option (e.g., the garage parking pricing) or on both parking options (e.g., the number of parking spaces of 23 each kind, and the desired parking duration). Drivers with desired long parking durations are more likely to choose 24 garage parking. All drivers are assumed to be rational during their parking decision and only compare the relevant 25 parking costs between on-street and garage parking, i.e., all drivers are treated as risk-neutral.

Data inputs for decision model 1
The decision between on-street and garage parking is dependent on the input variables shown in Tables 1 and 2. The 2 model parameters and all variables that are required to define the traffic network are presented in Table 1. These 3 variables can either be directly measured, or estimated based on simulation results and/or the macroscopic fundamental 4 diagram.
5 Table 1. Independent variables for parking decision (inputs to the model): Traffic network and model parameters.
Given the homogeneous network, parking searchers are assumed to be homogenously distributed within the overall 23 driving traffic. This is reasonable, as we also assume that all on-street parking spaces (not only the available ones) are 24 uniformly distributed on the network. Recall that we focus on small compact areas with standard parking policies 25 (e.g., downtown areas or portions thereof), and we are only interested in whether there is on average at least one car 26 that takes each available parking space. As a result, we do not need to record the location of individual cars and parking 27 spots throughout the different time slices in the system, i.e., only average numbers of vehicles during a time slice and 28 total/average searching times and distances are tracked.

29
Parking garages are also assumed to be uniformly distributed within the network, and without loss of generality, all 30 associated garage parking capacities are assumed to be equal. The distribution of desired parking durations is Manuel Jakob, Monica Menendez Page 6 In theory, however, any distribution can be used, e.g., poisson, negative binomial ( [10]). It is assumed that during the 1 period of one working day drivers do not repark their car after the on-street parking time limit has expired.

Notation Definition
Total number of existing on-street parking spaces (for public use) in the area.
Number of parking garages in the network.
Total capacity of all parking garages, i.e., total number of all garage parking spaces.
Parking duration of vehicles (independently of on-street and garage parking).
Price per kilometer driven on the network (i.e., external costs as petrol, wear and tear of vehicles).
Proportion of new arrivals during time slice that corresponds to traffic that is not searching for parking.

/
Distance that must be driven by a vehicle from user group ∈ before it starts to search for parking.

/
Distance that must be driven by a vehicle from user group ∈ before it leaves the area without having parked.

/
Distance that must be driven by a vehicle from user group ∈ before it leaves the area after it has parked on-street or in a garage.   Average cruising time for on-street parking in time slice .
Average driving distance to closest garage location.
Average travel speed in time slice , including stopped time at intersections.
Maximum driven distance per vehicle in time slice .
Number of available on-street parking spaces at the beginning of time slice .
Garage parking availability of all parking garages in time slice .
Total revenue resulting from hourly on-street and garage parking fee rates for the city.

10
We model the parking decision between on-street and garage parking macroscopically in Eq. (1)-(3). This will then 11 be incorporated into the on-street and garage parking-state-based matrix from [10]; Notice that the decision of some drivers is restricted by and . This is taken into account when calculating 6 , in Eq. (2), which described the proportion of vehicles deciding for garage parking, and , in Eq.
(3) which 7 describes the proportion of vehicles deciding to search for on-street parking. (4) Term 1 represents the hourly on-street parking fee rate which, in the remainder of this paper, is assumed to be 15 constant. In theory, however, the on-street parking fee could also be modelled as a responsive parking pricing scheme drivers search for on-street parking, the higher the average cruising time is, and consequently also the , .
Page 8 abstracted network was a ring, we may assume without loss of generality that the real network is a square grid, where 1 the average length of a block in the network is known. The total length of the ring network, , is then equivalent 2 to joining all blocks of length together. As on-street parking spaces are uniformly distributed throughout the 3 network, the walking costs can be determined using the average distance traveled (Eq. (5)) between two random points 4 in the square grid ( [21]).
Term 1 represents the side length of the square grid.  Term 1 represents the hourly garage parking fee rate which, in the remainder of this paper, is assumed to be constant.
Remember that the actual garage locations are assumed to be uniformly distributed on the network and that we assume 15 that traffic on the abstracted ring moves in a single direction.

18
Term 4 in Eq. (6) represents the cost of walking from the garage parking to the destination expressed in price units

19
for ∈ . As the number of garages is limited, they are expected to require, on average, some walking distance. The walking speed is assumed to be a constant input. To estimate the area served by each parking garage we take the in term 2 are both defined in section 3.1 (Table 5).

13
The on-street and garage parking-state-based matrix describes the system dynamics of urban traffic based on multiple 14 parking-related states as in [10]. The matrix is used to incorporate our parking decision model into a macroscopic 15 traffic system framework that emulates the interactions over time between the on-street and garage parking systems.

16
This section shows an overview of all on-street and garage parking-related traffic states (section 3.1), and the analytical formulations for the transition events between those states (section 3.2).

25
The total traffic demand entering the network is divided into two groups; through-traffic, and vehicles searching for 26 parking. The first group of vehicles represents the proportion of traffic that is driving through this area but does not 27 want to park or has a destination outside (i.e., through-traffic). Therefore, it only experiences two transition events as 28 seen in Fig. 3(a). The second group of vehicles needs to decide between searching for on-street parking or driving   Table 4. The initial conditions of all traffic state variables are model input variables that 16 can be measured, assumed or simulated.

On-street parking
Number of vehicles in the state "on-street parking" for user group ∈ at the beginning of time slice .
, Driving to garage parking Number of vehicles in the state "driving to garage parking" for user group ∈ at the beginning of time slice .

Garage parking
Number of vehicles in the state "garage parking" for user group ∈ at the beginning of time slice .
These parking-related states are determined using the information on the transition events. We introduce the transition 20 events in Table 5.
Page 11 Table 5. All transition event variables for the on-street and garage parking-state-based matrix per time slice.
Enter the area Number of vehicles that enter the area and transition to "non-searching" for user group ∈ during time slice (i.e., travel demand per VOT user group).
/ , Go to parking (Decision to park: Driving to garage parking) Number of vehicles that transition from "non-searching" to "driving to garage parking" (depending on their parking decision) for user group ∈ during time slice .
/ , Go to parking (Decision to park: Searching for on-street parking) Number of vehicles that transition from "non-searching" to "searching for on-street parking" (depending on their parking decision) for user group ∈ during time slice .
/ , Switch to garage parking Number of vehicles that transition from "searching for on-street parking" to "driving to garage parking" for user group ∈ during time slice .
/ , Find and access on-street parking Number of vehicles that transition from "searching for on-street parking" to "on-street parking" for user group ∈ during time slice .
/ , Access garage parking Number of vehicles that transition from "driving to garage parking" to "garage parking" for user group ∈ during time slice .
/ , Not access garage parking Number of vehicles that transition from "driving to garage parking" to "searching for onstreet parking" for user group ∈ during time slice .
/ , Depart on-street parking Number of vehicles that transition from "on-street parking" to "non-searching" for user group ∈ during time slice .
/ , Depart garage parking Number of vehicles that transition from "garage parking" to "non-searching" for user group ∈ during time slice .
/ , Leave the area Number of vehicles that leave the area and transition from "non-searching" for user group ∈ during time slice .

2
Eq. (10) to (14) update the number of "non-searching", "searching for on-street parking", "on-street parking", "driving 3 to garage parking", and "garage parking" vehicles, respectively. Notice that all equations need to be determined for every user group ∈ , where is the total number of user groups for the demand input of the network.
Eq. (10) updates the number of "non-searching" vehicles for each ∈ before aggregating them to +1 . Vehicles entering the area (i.e., / , ), and vehicles that depart from on-street or garage parking (i.e.,

20
We assume that the vehicles from user group ∈ make their parking decision (searching for on-street parking or 21 driving to garage parking) after driving a distance / since they enter the area.   The vehicles from user group searching for on-street parking that find and access a parking space is determined in Notice that all drivers decide to access the first available on-street parking space in the network, as all parking spaces 25 have the same price. As previously stated, details on / can be found in [10].
Term 1 in Eq. (20) represents the portion of vehicles trying to access garage parking that belong to user group .  , surpass , the 17 remaining vehicles need to return to searching-for-parking state; otherwise all vehicles can successfully enter a garage. the drivers' decision to stay in the "drive to garage parking" state due to a low in comparison to + . This term 20 is not time-dependent since there is no real-time usage information available. This constraint is relaxed later (section 21 4.5) when real-time information is available. Notice that for more realistic applications, the capacity of garage parking will not be an active constraint. It is included here, however, for the sake of completeness.
The on-street parking availability is updated in Eq. (24) after vehicles access or depart from on-street parking.  The vehicles leave the area after having driven for a given distance / or / depending on whether they have parked or not. Notice that the distances / and / analogously to / can be fixed or taken out of any given probability

21
The parking demand (Fig. 4(b)), parking durations, and initial conditions are extracted from an agent-based model in

22
In this section, we validate the garage parking occupancy rate using empirical data collected by the city of Zurich. The

3
The curve reflecting the estimated garage parking occupancy rate shows a rather similar pattern to that of the real data.

4
The approximation is more accurate compared to the validation in [11], where no differentiation between on-street

7
In this section, we present some valuable insights with respect to on-street and garage parking. Table 6 illustrates the 8 average/total time and driven distance for the vehicles in the states "Searching for on-street parking", "Drive to garage 9 parking" and "Non-searching" during a typical working day.

12
On average each vehicle spends 9.7 minutes in the network (excluding the time spent parked). Not surprisingly, 13 vehicles spend on average longer in the "Searching for on-street parking"-state (3.7 minutes) than in the "Drive to 14 garage parking"-state (3.3 minutes). A similar behavior can be detected when looking at the average driven distance 15 in the network (Table 6). What is interesting, however, is that the absolute difference in average travel time between 16 the two parking options is less than a minute. This happens because of two reasons. First, the area itself is rather small.
Second, based on our decision framework in Eq. (2) on average only 48.8% of the parking vehicles are able to make that the on-street parking duration limit is set to = 180 min.

20
Following the parking demand ( Fig. 4(b)), the number of vehicles searching for on-street parking increases drastically 21 between the 9th and the 13th hour, and the number of available on-street parking spaces goes down ( Fig. 7(a)). After the 9.5th hour the number of available garage parking spaces gets close to zero (Fig. 7(b)). The vehicles that cannot 23 access garage parking then return back to the searching-for-on-street-parking state. This leads to more searching Page 19 5th and the 20th hour ( Fig. 7(b)). Given the distribution of garage parking durations and the resulting turnover, the 1 number of available garage parking spaces decreases drastically between the 9.5th and the 14th hour (see also

5
Once there are no available on-street parking spaces anymore ( Fig. 7(a)), the average cruising time increases (Fig. 8).

6
This leads to an increase in the costs associated with cruising-for-on-street-parking.   behaves analogously to the parking demand ( Fig. 4(b)). It increases between the 5th 2 and the 20th hour. / , is negligibly small. /s , increases from approximately the 9.5th hour since the garage 3 parking occupancy rate is close to 100% (Fig. 6). Thus, not enough available garage parking spaces are left ( Fig.   4 7(b)) and vehicles are not able to access the parking garages.

15
Remember that both the hourly on-street and garage parking fee rates, and , are part of the decision related 16 cost variables for on-street and garage parking. Based on these cost variables the drivers decide for on-street or garage parking, affecting the average travel time in each parking-related state as illustrated in Fig. 10. Increasing the ratio 18 leads to a higher cost variable , (section 2.2.1) and the drivers are more likely to drive to garage parking.

19
Thus, the average time for vehicles driving to garage parking increases, while the average searching time decreases  In reality, the actual garage parking availability also influences the drivers' decision to park on-street or to drive 6 towards a parking garage. This garage usage information can be made available to the drivers by providing real-time 7 smartphone applications or garage information signs in the traffic network.

8
In this section, we include the availability of garage parking information into our on-street and garage parking model 9 and assume that the garage parking availability is known to all drivers at all times. Since this garage parking availability has an influence on the driver's parking decision, we replace all by in Eq. (1) and Eq. (22). Table   11 7 illustrates the average time and driven distance for the scenario with available garage usage information for all 12 drivers in the network during a typical working day.

18
The average driven distance in the network reduces similarly (Table 7). Allowing drivers to make their on-street or 19 garage parking decision based on real-time occupancy data leads to a better traffic performance on the network, and 20 on average, a faster journey for drivers searching for parking.

21
The parking choice for garage over on-street parking decreases drastically for drivers with available garage usage information between the 9.5th and the 14th hour compared to drivers who have no garage information available ( Fig.   23   11). Since the increase in the average cruising time (Fig. 8) has an impact on the drivers' decision, more drivers without 24 any available garage usage information drive to garage parking between the 9.5th and the 13th hour. Due to the lack 25 of garage information this parking choice is made even if the garage occupancy rate is low. Note that this parking 26 choice only affects the portion of the parking demand that can make a decision between on-street and garage parking 27 due to the on-street parking duration limit. By including the garage usage information into the decision framework the 28 drivers react towards the garage occupancy rate. The garage occupancy rate (Fig. 6) is then reflected in Fig. 11 and the parking choice for garage parking increases from the 14th hour analogously to the decrease of the garage

10
The outputs in Fig. 12

17
The more on-street parking spaces are converted to garage parking spaces, the less drivers drive to on-street parking.

18
This reflects the drivers' parking decision that is dependent on the total number of existing on-street and garage 19 parking spaces in the network. It leads to a decreasing average searching time and an increasing average time driving 20 to garage parking in the short-term assuming that drivers are used to their old on-street parking habits ( Fig. 12(a)).
Page 23 becomes the same for all levels (based on the combination of Fig. 5(a) and Fig. 5(b)) as in [11]. Fig. 12(b) shows the 1 impact of the on-street parking conversion on the total revenue created by on-street/garage parking. While a decreasing 2 number of on-street parking spaces leads to a decreasing total on-street parking revenue, it leads to an increasing total 3 revenue. A conversion of on-street parking to garage parking spaces might lead to a higher average time driving to 4 garage parking and a lower average searching time in the short-term with an increase in the total parking revenue for 5 the city. In this study, we develop a dynamic macroscopic on-street and garage parking model such that the short-term 8 influences of different on-street and garage parking policies on the traffic system can be studied and illustrated. The 9 macroscopic model is built on a traffic system with a parking search model over time. It is incorporated into the on-10 street parking framework from [10]. We validate this model based on real data for a case study of an area within the 11 city of Zurich, Switzerland.

12
The main contributions of this paper are three-fold.

13
First, we model garage parking macroscopically, including the parking searchers' decision between driving to a 14 parking garage or searching for an on-street parking space in the network. This includes the influences on the 15 searching-for-parking traffic (cruising), the congestion in the network (traffic performance), the total driven distance 16 (environmental conditions), and the revenue created by on-street and garage parking fees for the city.
Second, we analyze the relationship between on-street and garage parking, but also their interdependency on cruising-18 for-parking traffic and traffic performance with respect to different parking fees. Different hourly on-street and garage 19 parking fee rates can lead not only to more vehicle time/distance in the network, but also to various financial revenue 20 outputs. Thus, this analysis can be used for city councils or private agencies to find reasonable hourly on-street and

21
garage parking fees such that the average vehicle time/distance is not negatively affected and additionally, acceptable 22 financial revenues are obtained. Our methodology provides the tools to do a cost benefit analysis and to study the 23 trade-off between the revenue and the average travel time. In the long-term, drivers might avoid paying high on-street 24 or garage parking fees and quit their journeys. This could affect the demand, but long-term effects are out-of-scope of 25 this paper.

26
Third, our model allows us to analyze parking policies in city center areas, e.g., the short-term effects of converting 27 on-street to garage parking spaces on the traffic system can be simulated and recommendations for city councils can 28 be made. In the city of Zurich, a conversion of on-street parking to garage parking spaces might lead to a higher 29 average time driving to garage parking and a lower average searching time in the short-term with an increase in the 30 total parking revenue. Additionally, the influences of the availability of garage usage information to all drivers can be 31 analyzed. This might lead to a better traffic performance on the network and an on average faster and shorter journey 32 for each driver searching for parking.

33
The general framework provides an easy to implement methodology to macroscopically model on-street and garage 34 parking. All methods are based on very limited data inputs, including travel demand, VOT, number of garages with 35 their capacity, the traffic network, and initial parking specifications. Only aggregated data at the network level over 36 time is required such that there is no need for individual on-street and garage parking data. This macroscopic approach 37 saves on data collection efforts and reduces the computational costs significantly compared to existing literature.

38
Additionally, there is no requirement of complex simulation software and the model can be easily solved with a simple 39 numerical solver.

Manuel Jakob, Monica Menendez
Page 24 Overall, the usage of the model is far beyond the illustration in the case study of an area within the city of Zurich. In 1 reality, vehicles often prefer parking possibilities in a central street or area of the network, while discarding other 2 parking opportunities elsewhere. A further consideration is tiered parking pricing that can be included into the model.

3
Certain cities have tiered pricing for both on-street and garage parking such that the driver may pay a low rate for the 4 first hours, and then the rate jumps up significantly to increase turnover and promote higher parking availability. We 5 can also study the usage of a responsive parking pricing scheme. In addition, we can include a traffic demand split 6 with a fixed (low subsidized) parking fee for all on-street and/or garage parking spaces. All remaining portions of 7 demand could be treated responsively, reflecting the external costs for parking. This approach can be motivated by, 8 e.g., the subsidy by a company or a city for their residents.

9
In summary, the model can be used to efficiently analyze the influence of different on-street and garage parking 10 policies on the traffic system for a smaller geographic scale network, despite its simplicity in data requirements. Based 11 on scarce aggregated data, this model can be used to analyze how on-street and garage parking policies can affect the 12 traffic performance; and how the traffic performance can affect the decision to use on-street or garage parking.

13
Data Availability

14
The data used to support the findings of this study are available from the corresponding author upon request.

16
The authors declare that there is no conflict of interest regarding the publication of this paper.