Agent-Based Modeling for Transportation Planning: A Method for Estimating Parking Search Time Based on Demand and Supply Nir Fulman, Itzhak Benenson Department of Geography and Human Environment, Porter School of the Environment and Earth Sciences, Tel Aviv University nirfulma@post.tau.ac.il; bennya@post.tau.ac.il Abstract Levy et al., 2013], while simulation modeling makes it pos- sible to estimate parking search time and the distance be- We estimate parking cruising time curves - the tween a driver’s place of overnight parking and destination probability Pi() of longer than  parking search for [Levy et al., 2013; Levy and Benenson, 2015]. Note that destination Ni located within an area with hetero- simulation models of cruising for parking include car fol- geneous demand and supply. To do that, we esti- lowing effects [Levy and Benenson, 2015; Arnott and Wil- mate cruising time curves for an area of homoge- liams, 2017], but we are not aware of analytical models that neous demand and supply and then average these account for this phenomenon. curves based on (1) a model of parking search be- Importantly, the analytical and simulation approaches re- havior established in a serious parking game; and sult in qualitatively different estimates of cruising time, as (2) a “Maximally Dense” parking pattern obtained dependent on the occupation rate. According to [Levy et al., for the case where drivers possess full knowledge 2013] the average search time in a homogeneous grid-like of the available parking spots and are able to park city area remains low in analytical models, even when the at the spot closest to their destination that is vacant occupation rate is very high, ca. 98%, whereas simulation at the moment they start searching for parking. We studies of cruising time for the same area result in essential- verify the proposed methods by comparing their ly higher estimates starting from ca. 90% occupation (Fig- outcomes to the cruising time curves obtained in an ure 1). agent-based model of parking search in a city. As a practical example, we construct a map of cruising time for the Israeli city of Bat Yam. We demon- strate that despite low (0.65) overall demand-to- supply ratio in Bat Yam, high demand-to-supply ratio in the center of the city may result in longer than 10 minutes parking search there. We discuss the application of the proposed approach for urban planning. 1 Introduction: Demand-to-Supply Ratio as a Major Determinant of Parking Search Time Long parking search time is a perpetual problem of every Figure 1: Cruising time as dependent on occupancy rate in analyti- big city, and quantitative estimation of parking cruising time cal and simulation models of [Levy et al., 2013]. is a long-standing challenge for transportation research. Given only a moment’s thought, the inherent reason for this problem is that demand D exceeds supply S and the demand According to [Levy et al., 2013], the reason for the gap in to supply ratio R = D/S > 1. A greater level of detail is nec- Figure 1 is the primarily clustered distribution of vacant essary to estimate parking search time for a designated area, parking places that inevitably emerges in a parking model and should include vehicles arrival and departure rates in the with stochastic arrivals and departures. area, parking occupation rate, spatial distributions of the High occupation rate and above 100% demand-to-supply departing drivers, and of destinations of the arriving drivers. ratio are characteristic of the central part of every large city. The analytical study of cruising for parking can be per- At the same time, the spatial patterns of demand and supply formed with stochastic or deterministic models [Arnott and there are always heterogeneous and the level of heterogenei- Rowse 1999; 2013; Anderson and de Palma, 2004; 2013; ty is dictated by the city: the demand for parking is defined by the size and use of the buildings, while the supply is de- fined by the parking capacity of street links and off-street lots, as well as parking permissions and prices. In this paper we demonstrate that this heterogeneity has far-reaching con- sequences and local mismatch results in the emergence of essentially larger areas where drivers have to cruise for longer. We investigate this idea in depth with an agent- based model of parking search, and present a fast and effi- cient algorithm for estimating parking search time based on the patterns of parking demand and supply. The output of the algorithm is a map of cruising time that is validated with the help of the simulation model. As a practical example we Figure 2: Torus 20x20 grid city (a); zoom to a city block (b). construct the map of cruising time for the Israeli city of Bat Yam, with a population of 120,000, and discuss the applica- tion of the proposed approach for urban management and Street links and junctions are stored as GIS layers, with planning. the demand being an attribute of a junction, and the number of parking spots an attribute of a link. Model experiments are performed on a 20x20 grid with N = 400 destinations 2 The PARKGRID Agent-Based Model of (junctions), L = 800 links and P = L*40 = N*80 = 32,000 Parking Search parking places. We do not consider off-street parking lots in Cruising is the collective outcome of individual drivers’ the current version of this model. parking search. In what follows we investigate the problem of parking search with the spatially explicit agent-based 2.2 PARKGRID Basic Assumptions PARKGRID model that is based on the knowledge of park- PARKGRID agents - drivers are explicitly considered from ing search behavior obtained in a serious parking search the moment they reach their destination and start cruising game [Benenson et al., 2015]. PARKGRID is a stand-alone for parking; whereas drivers en route to their destinations C# application and can be freely downloaded from are ignored. While cruising, a model driver either finds a https://www.researchgame.net/profile/Nir_Fulman. vacant parking spot and parks, or leaves the system after a PARKGRID continues the tradition of PARKAGENT long unsuccessful cruise. We assume that a driver cruises at [Levy et al., 2013; Levy and Benenson, 2015] and is a GIS- a constant speed of 12 km/h [Carrese et al., 2004] that is, it based application that is based on the layers of streets, desti- takes a driver 30 seconds to traverse a 100m link. We thus nations, and parking places. In this paper, we consider an consider 30 sec as a model time unit - tick. At each model abstract grid city for estimating basic dependencies, and tick, the list of cruising and due-to-depart drivers is con- then apply our results to a real city. structed, randomly re-ordered, and each driver acts in its turn. 2.1 Urban Street Network in PARKGRID Driver Types, Arrivals, Departures PARKGRID simulates on-street parking in an abstract grid Each model driver c is assigned a destination Ni; c appears city where the street network is represented by two-way at Ni and starts its parking search driving along a randomly links Li and junctions Nj (Figure 2). The length of a street chosen link that is incident to Ni. Each driver is also as- link is 100 m. To avoid boundary effects, the grid is folded signed a parking time, the distribution of which is uniform into a torus - the right ends of its rightmost links in Figure on the [TPmin, TPmax] time interval. Drivers that aim at Ni are 2a are connected to the leftmost junctions and the top ends generated according to a Poisson process with a per-hour of the top links are connected to the junctions at the bottom. average λi that depends on whether a driver is an employee In this way each junction has exactly four incident links. For or a visitor to the destination, and is proportional to the des- further simplicity, we set drivers’ destinations at the junc- tination’s Ni’s demand Di. The car vacates the spot after the tions. parking time is over. In the current version of the model, we assume that driv- We consider two types of drivers: employees who park in ers’ destinations are located at the junctions and each desti- the city and do not leave until the end of the simulation nation junction Ni is characterized by its demand Di that can (TPmin = 8 hours); and visitors with TPmin = 1 hour, TPmax = vary between buildings. Each link contains 20 parking plac- 2 hours. Employees arrive to the city in the morning, and es of 5m length on each of its sides, 40 parking spots alto- their arrival time is uniformly distributed on the time inter- gether. This entails the ratio of the total number of destina- val [9:00, 10:00]. Visitors arrive to the city and leave it be- tions to the total number of curb parking spots equal to R city tween 9:00 and 16:00. The simulation starts at 9:00 with an = 80. empty city and stops at 16:00. d < 100, 100 ≤ d < 200 200 ≤ d < 300 d ≤ 300 < 400 d ≥ 400 Decision at a previous junction Closer Further Closer Further Closer Further Closer Further Closer Further Closer 0.00 1.00 0.65 0.35 0.85 0.15 0.90 0.10 Irrelevant Further Irrelevant 0.00 1.00 0.80 0.20 0.85 0.15 1.00 0.00 Table 1: Probability to choose a link that takes a driver closer to/further away from a destination, as depending on a distance d between a junction and a destination and the decision made at the previous junction [Benenson et al., 2015]. previous time step, then its search time on this link is esti- Drivers’ Cruising Behavior mated as 30/(f + 1) sec. The parking search behavior that we implement in the mod- Maximum cruising time M in all investigated scenarios is el is based on the results of the PARKGAME serious game set to M = 20 min; during this time a model driver traverses [Benenson et al., 2015] and is formalized as a biased, to- 40 street links and 1,600 parking places. We assume that wards destination, random walk [O’Sullivan and Perry, drivers that fail to find curb parking during the maximum 2013]. Visually, a driver cruises around the destination until search time, park at a “distant off-street parking lot” that finding a free, on street parking spot, repeatedly approach- always has vacant spots. We ignore them when estimating ing the destination and driving further away from it (Figure average cruising time. 3a). Drivers’ turn decisions at junctions depend on two pa- rameters: the distance between the junction and destination, and the decision taken at the previous junction – to approach 3 Model Study the destination or drive further away from it. The probabili- ties to turn closer to/further away from the destination, as 3.1 Homogeneous Demand and Supply Patterns dependent on the distance to destination and the decision In the basic scenario we consider a homogeneous city in made at the previous junction, were based on more than 200 which the average number of drivers who aim at destination PARKGAME game sessions with 35 participants (Table 1). Ni is Di = q*Rcity, q < 1. Note that q is an average over the Given a driver’s destination Ni, the biased towards destina- city occupation rate in this case. tion random walk model of parking search determines the Let a fraction e of drivers who arrive to the city in the driver’s search neighborhood U(Ni) and, for each link l ∈ morning be employees who stay there until the end of the U(Ni), the probability wl of traversing this link during a pe- day. For a city with N destinations this means that riod of search (Figure 3b). PARKGAME experiments e*q*Rcity*N drivers arrive, uniformly, to the city between demonstrate that these probabilities do not depend on a 9:00 and 10:00, search for parking, park (if successful) and driver’s characteristics (risk-taker or risk avoider) and park- the car stays at the parking spot until 16:00, the end of the ing occupation rate around the destination. model day. The rest of the drivers that arrive throughout the day and depart the same day, are visitors whose parking time is uniformly distributed on the [1, 2] (hours) time in- terval. The average parking time of a visitor is thus 1.5 hours and to compensate visitors’ departures, we assume that visitors’ arrival rate λ is (1 – e)*q*Rcity*N per 1.5 hour that is, λ = ((1- e)/1.5)*q*Rcity*N per hour. All drivers in the basic scenario employ the biased ran- dom walk search tactic, with the parameters presented in Table 1. We start with investigating the dependency of park- ing search time in a city with a homogeneous distribution of demand and supply, that is, Di, λi and [Ti,min, Ti,max] are iden- tical for every destination, and estimate the probability p(q, τ) to find parking in time less than τ (“cruising time curve”), as dependent on q. We then extend these results to the case a b of heterogeneous demand. Figure 3: Driver’s parking search as a biased random walk. Typical 3.2 Homogeneous Scenario Outcomes search path (a); U(Ni) and probabilities to visit links in it (b). We start with the case of relatively low demand, q = 0.85 and e = 0.85. That is, the average number of employees that In the model, a driver parks on the first street link that is aim at each destination equals to e*q*Rcity = 0.85*0.85*80 = not fully occupied. If, during a 30 sec iteration, a driver 57.8 cars, while the visitors arrive during the whole day at parks on a traversed link that had f free parking places at the an average rate ((1-e)/1.5)*q*Rcity = 0.1*0.85*80 = 6.8 cars/hour/destination. a a b Figure 5: Model output for the basic scenario of homogeneous b demand for q = 0.85, e = 0.85. Percentage of time the link is fully occupied (a), cruising time curve (b). Figure 4: PARKGRID basic scenario, q = 0.85, e = 0.85. Dynam- ics of the total arrivals and departures (a), and the fraction of occu- pied spots (b). Figure 6: Cruising time curves (fraction of driving cruising for a time t before finding a vacant parking) for q varying between 0.950 – 0.995 and e = 0.85 As presented in Figure 4, the average occupation rate in each destination Ni  H the demand is set equal to Di = (q + the city stabilizes, as expected, at q = 0.85 towards 11:00 α)*Rcity, while for destinations in L, Di = (q – α)*Rcity. and from then on remains steady, fluctuating around 0.85 Figure 8 presents the case of α = 0.5 and H and L as 5x5 with the STD of 0.0025. In what follows, we consider the neighborhoods. The demand Di of every destination in H is steady period 11:00 - 16:00 only. equal to Di = (q + α)*Rcity = (0.85 + 0.5)*80 = 108 and, to As should be expected, a street link’s occupation rates are compensate, the destination’s demand in L is equal to Di = symmetrically distributed around 85% average, with an (q – α)*Rcity = (0.85 – 0.5)*80 = 28. For the rest of the des- STD = 1%. On average, a link is fully occupied during ca. 7 tinations we preserve the demand Di = q*Rcity = 0.85*80 = minutes per hour that is, 13% of the time (Figure 5a). High 68. parking availability results in an average cruising time of 17 seconds, with only 12% of drivers cruising for longer than 30 seconds, a consequence of not finding a vacant spot along the first link after the destination (Figure 5b). With an increase in q, the expected search time becomes longer and longer (Figure 6). The cruising time curves in Figure 6 enable estimating the average search time as dependent on the occupation rate and Figure 7 merges between Levy et al [2013] outcomes presented in Figure 1 and the PARKGRID estimates of the average cruising time as dependent on occupation. Figure 8: Demand patterns for the heterogeneous scenario, q = 0.85, e = 0.85, R = 80, α = 0.5. Figure 7: PARKGRID search time vs Levy et al [2013] results. The effects of the H and L neighborhoods on the city parking pattern are different. The capacity of the links inside As can be seen, the PARKGRID average search time is H is insufficient for absorbing all drivers who aim to park higher than obtained in Levy et al [2013] analytical model, there and some of them eventually park beyond H, increas- while lower than the estimates obtained in simulations for ing parking occupation in H’s surroundings. The L neigh- the occupation rates below ~99.5% and higher for higher borhood hardly influences the parking pattern, because the average occupation rates. Several explanations can be pro- demand there is far below parking capacity. posed: Levy et al [2013] (1) artificially preserved a constant To reflect the effect of spillovers generated by drivers number of drivers in the system, substituting one driver that who aim to park at H, we apply the PARKFIT algorithm leaves the system by one driver that enters it; (2) accounted (Levy and Benenson [2015]) that, based on the PARKGRID for the parking search on the way to the destination; and (3) demand and supply patterns, generates a Maximally Dense accounted for the car-following and the time that it takes a (MD) parking occupation pattern. driver to occupy a vacant spot. In PARKGRID, the arrival and departure processes are independent, parking search 3.4 PARKFIT Algorithm and Maximally Dense starts after a destination is reached, and car-following and Parking Pattern the time that it takes a driver to occupy a spot are ignored. The PARKFIT algorithm aims at estimating the parking pattern in a “city of autonomous vehicles”. Its major as- 3.3 The Case of Heterogeneous Demand sumption is that each car “knows” the vacant parking place To investigate the consequences of heterogeneous demand, that is closest to its destination when starting its parking we consider a city with two neighborhoods H and L, where search, “books” it when entering the system, and drives the demand differs from the average over the city. We as- there directly, meanwhile the spot cannot be occupied by sume that in the neighborhood H the demand is higher than other drivers. q*Rcity, while the demand in L is lower and adjusted to the Let k, k = 1, 2, 3, …, K be destinations of the Dk demand, demand in H, to preserve the overall q*Rcity. Formally, for and drivers (i.e., autonomous cars) know distances between each of the parking spots in the area and their destinations. The steps of the PARKFIT algorithm are as follows: the border of this extension, where the links’ occupation rate (1) Build a list L of all pairs (the is lower than 100%. Drivers whose destinations are not length of this list is D1 + D2 + D3 + … + DK) and randomly within H and its MD extension will cruise over a neighbor- reorder it. This list defines the order of drivers’ arrival to the hood with an average or even lower than average occupation area rate. (2) Loop by drivers in L. For each driver consider the The average occupation rate ri,ave over the driver’s search parking spot closest to its destination that is vacant at a neighborhood U(Ni) can be estimated as moment of the driver’s arrival to the system and assign it to the driver. ri,ave = lU(Ni){wl*rl} (1) (3) Randomly release spots in respect to the departure Where U(Ni) is the driver’s random walk search neigh- rate per time period that, on average, is necessary to traverse borhood, wl is the probability of traversing each link the link. lU(Ni) during its search as presented in Figure 3b, and rl is In the areas where destinations’ demand is lower than the the average occupation rate of link l in the maximally dense parking supply around, as in the neighborhood L, PARKFIT pattern. algorithm generates patches of 100% occupation around each destination with intervals of vacant spots between patches. In cases where destinations’ demand is higher than the supply nearby, as in the neighborhood H, PARKFIT spreads the excess demand beyond the area of the high- demand destinations (Figure 9). In both cases, the occupa- tion rate of the link within highly occupied patches is equal to 100% minus departure rate per the time unit necessary for traversing the link (30 seconds for the grid city that we con- sider). α=0 α = 0.5 a b Figure 9. MD patterns generated by PARKFIT for q = 0.85, Rcity = 80, α = 0 (a) and α = 0.5 (b). 3.5 Cruising Time as Dependent on Local De- mand-to-Supply Ratio A driver’s search conditions are very different depending on whether its destination Ni is located within H, L or over the Figure 10: Directly simulated vs obtained according to (2) cruis- rest of the area. The success of a driver’s parking search is ing time curves for three selected destinations within and outside defined by the overlap of the search neighborhood U(Ni) H. and the MD pattern. For drivers with destinations Ni that are Consequently, we can roughly estimate the cruising time close to the center of H, the only chance to park is to occupy curve Pi(τ) for the destination Ni located within the hetero- a spot that is freed by a departing driver. Drivers whose des- geneous neighborhood (according to the demand and sup- tination is close to the boundary of the MD-extension of H ply) U(Ni) based on ri,ave. The simplest approximation is as have a significantly higher chance to find a free spot beyond follows: Pi,1(τ) = p(ri,ave, τ) (2) 4 Predicting Cruising Time in Bat Yam Instead of (2) that is based on the average occupation rate As a practical example, we estimate the time of residents’ (1), we can directly average cruising time curves p(rl, τ) that search for overnight parking in the Israeli city of Bat Yam. are characteristic of the links of the MD pattern: Pi,2(τ) = lU(Ni){wl*p(rl, τ)} (3) 4.1 Parking Demand and Supply in Bat Yam Our estimates are based on the demand and supply data of We have verified approximation (2) by comparing cruis- 2010, when Bat Yam’s population was ca. 130,000, total car ing time curves Pi(τ) that are estimated directly in simula- ownership 35,000, and the total number of residential build- tions and Pi,1(τ) for locations within and outside (yet close ings 3,300 with 51,000 apartments. These data as well as to) H, and for which U(Ni) neighborhoods are heterogene- layers of streets, off-street parking facilities and buildings ous. We employed the weights wl as presented in Figure 3b were supplied to us by the Bat Yam municipality. Residen- and cruising time curves for the homogeneous case as pre- tial buildings in Bat Yam provide their tenants a total of sented in Figure 6. Figure 10 presents Pi,1(τ) and directly 17,500 dedicated parking places that should be excluded estimated Pi(τ) curves, and the fit is very good. from the demand and supply data. We associate destinations Figure 11 presents the results of systematic comparison of the overnight parking with residential buildings and esti- between aggregate properties of the Pi(τ) and its Pi,1(τ) ap- mate the demand for parking in each as (35,000 – 17,500) / proximation for all 400 destinations Ni of the demand pat- 51,000 = 0.34 times the number of apartments (i.e. house- tern presented in Figure 8. As can be seen, direct and ap- holds) in the building. proximate estimates of the average search times as well as Parking supply data is based on two GIS layers - a layer the probability to cruise for over 3 minutes strongly corre- of streets and a layer of off-street parking facilities. Based late with R2 ~ 0.95. on the layer of streets, 27,000 spots for curb parking were constructed automatically, 5 meters apart on both sides of two-way street links, and on the right side of one-way links, with a necessary gap from the junction. In addition, 1,500 spots are available for the city’s residents in its parking lots and in the evening Bat Yam residents can park at these spots free of charge. The average overnight demand/supply ratio is thus very low (35,000 – 17,500) / (27,000 + 1,500) ≈ 0.61 car/parking spot. However, the distribution of demand and the road net- work that characterizes the supply in Bat Yam are both highly heterogeneous, and the demand in the center of Bat Yam is high and significantly exceeds the supply there (Figure 12). a 4.2 Maps of Parking Search Time in Bat Yam We estimate cruising time in Bat-Yam assuming, as above, an 85:15 ratio between the numbers of parking resi- dents and visitors that come to visit Bat-Yam residents in the evening. Starting from the building-based estimates of demand, and the link and lot-based estimates of supply, we have (1) transferred buildings’ demand to the nearest junc- tion; (2) established Bat Yam MD pattern for 85:15 ratio of residents and visitors (Figure 13a); and then (3) estimated the cruising time curve for every destination applying for- mula (3), assuming that a driver’s area of search and cruis- ing behavior is the same as revealed in the experiments pre- sented in Figure 3b and Table 1. Figure 13b presents esti- mates of the average cruising time in Bat Yam, while Figure b 14 presents the probability to cruise for parking for more than a certain time as dependent on driver’s destination. Figure 11: Pi,1(τ) estimates according to (2) versus direct estimates with the PARKGRID. Average search times (a); Probability to cruise longer than 3 minutes (b). 2.5 minutes 10 minutes Figure 14. Bat-Yam maps of the probability to cruise longer than a b 2.5 and 10 minutes. 5 Conclusions The surplus of demand over supply is critical for parking search in the city. We investigate the dependence of the parking search time on the local demand-to-supply ratio and propose an algorithm for estimating search time based on static demand and supply patterns. We apply our model to the Israeli city of Bat Yam and show that despite a low, ca. 61%, average demand to supply ratio, spatial heterogeneity of the demand and supply pat- terns results in lengthy parking searches for a significant fraction of drivers. The proposed method can be applied to every city where the patterns of parking demand and supply are known at a c resolution of buildings, roads and parking lots. We consider Figure 12: Bat Yam: Parking demand by buildings (a), road net- our approach as a fast and efficient approximation for direct work (b) Demand/Supply ratio by Transport Analysis Zones (c). estimates of the cruising time, which can also be obtained in a dynamic agent-based model simulation, such as PARKAGENT [Levy et al., 2013]. This approach can be applied to any city of arbitrary size. It should be stressed that the perspectives of agent-based modeling of human-driven systems such as parking, critical- ly depend on our knowledge of agents’ behavior. In this respect, we consider serious games as a method to account for the dynamic nature of the system that is missed in the stated and revealed preferences surveys. In the same time, the conditions of serious games are fully controlled by the researcher and can be used to create situations that cannot be observed. To the best of our knowledge, the biased ran- dom walk search tactic that is revealed in the game and em- ployed in the PARKGRID model is the first example of a successful merge between a serious game and a parking agent-based model. From a practical point of view, the estimates of parking a b search times presented in this paper should serve as initial information for an urban planner who aims at assessing the Figure 13. Bat-Yam MD (a) and average cruising time (b) maps. consequences of construction of, for example, a new office, commercial or residential building. If parking supply in the area is insufficient for the planned demand, a planner can choose to increase supply by adding parking lots. Our meth- od can be applied for predicting the decrease in the search time and its spatio-temporal extent. The policy maker can also attempt to decrease demand by rigid limitations on vehicular entrance to a designated area to certain groups of drivers, or by increasing parking prices, or even introducing flexible prices that are adapted to the expected demand for parking [SFMTA, 2016]. Incorpora- tion of drivers’ reactions to prices when cruising for parking is thus the extension of our approach, and the first step in this direction is presented in [Fulman and Benenson, 2018]. 6 References [Anderson and de Palma, 2004] Simon P. Anderson and André De Palma. Parking in the city. Papers in Regional Science, 86(4): 621–632, 2004. [Arnott and Rowse, 1999] Richard Arnott and John Rowse. Modeling parking. Journal of urban economics, 45(1): 97-124, 1999. 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