MMarking Streets to Improve Parking Density
Chao Xu ∗ Steven Skiena † September 18, 2018
Abstract
Street parking spots for automobiles are a scarce commodity in most urban environments.The heterogeneity of car sizes makes it inefficient to rigidly define fixed-sized spots. Instead,unmarked streets in cities like New York leave placement decisions to individual drivers, whohave no direct incentive to maximize street utilization.In this paper, we explore the effectiveness of two different behavioral interventions designedto encourage better parking, namely (1) educational campaigns to encourage drivers to “kissthe bumper” and reduce the distance between themselves and their neighbors, or (2) paintingappropriately-spaced markings on the street and urging drivers to “hit the line”. Through anal-ysis and simulation, we establish that the greatest densities are achieved when lines are paintedto create spots roughly twice the length of average-sized cars. Kiss-the-bumper campaigns arein principle more effective than hit-the-line for equal degrees of compliance, although we believethat the visual cues of painted lines induce better parking behavior.
Alternate side of the street parking is a unique but important New York City institution. Most citystreets are assigned two intervals per week (typically 1.5 hours each) during which a particular sideof the street must be vacated to allow for street cleaning. Drivers double park on the other side ofthe street during this time window, waiting for the moment when the street cleaner passes or theperiod expires, at which point they must quickly move their cars to newly clean and once-againlegal spots. Alternate side of the street parking defines the rhythm of life for many city residents[17] by mandating regular actions in order to maintain a car in the city.The effective number of New York City parking spots depend heavily on the dynamics ofalternate side of the street parking, since all cars park simultaneously but generally depart atdistinct times. Thus it is rare for two adjacent spots to open simultaneously after the streetconfiguration is frozen at the end of the forbidden interval. This means that any extra space leftbetween the hastily-parked cars cannot be reclaimed until all the vehicles move again during thenext parking switch. New York streets do not contain any laws, painted lines or boundary markingsto guide positioning or restrict the space between neighboring cars after this transition. Since it isusually easier for the driver to leave ample room between cars, much of the potential parking areais wasted. ∗ [email protected], Department of Computer Science, University of Illinois at Urbana-Champaign. Supportedin part by NSF REU supplement IIS-1128741. † Corresponding author. [email protected], Department of Computer Science, Stony Brook University. Sup-ported in part by NSF grants IIS-1017181 and DBI-1060572. a r X i v : . [ phy s i c s . s o c - ph ] M a r it-the-lineKiss-the-bumperRandom parking Figure 1: Representative street landscapes for random/R´enyi (top), kiss-the-bumper (center), andhit-the-line (bottom) for a street of length l = 20 with the optimal line spacing of k = 2, for α values of 0, 0 .
5, and 0 . • Optimizing Line Spacing – Through analysis and simulation, we establish that the greatestdensities are achieved when lines are painted to create spots roughly twice the length ofaverage-sized cars. • Relative Effectiveness of Interventions – Kiss-the-bumper campaigns are in principle moreeffective than hit-the-line for equal degrees of compliance, although we believe that the visualcues of painted lines will induce greater driver compliance.We believe that our results have genuine implications for improving the parking density of NewYork streets, and have begun preliminary efforts to interest city transit officials in a pilot study totest these ideas. More details appear in the conclusions (Section 7) of this paper.
We have not identified any substantial literature analyzing the mechanisms or dynamics of urban street parking. Municipal codes typically specify the required dimensions of on-street parking, butdo not describe the rationale behind such a selection. Representative is the City of Bowling Green,Kentucky [6], which enforces an on-street parking layout that divide the street into even parkingslots of size 6.7m to 7.9m, designed to leave at least 2.4m of maneuver space. Different citieshave different regulations for on-street parking, although all must meet the American DisabilityAct (ADA) standards for accessibility [13]. Design guides to building multistory parking garagestructures include [1, 5] Other studies seek to optimize the revenue from parking facilities [16].Perhaps the most related paper in terms of our research is Cassady and Korbza [4], who studythe performance of two different driver search strategies in minimizing (1) walking time, (2) drivingtime, and (3) combined time in reaching their destination in a typical parking lot layout. Like us,they are interested in analyzing the impact of drivers parking strategies. However, we are concernedwith maximizing the utilization of scarce street parking in an urban setting.2here has been considerable theoretical work on one particular model of street parking. TheR´enyi parking problem [15] concerns the following random process. Unit-length cars select unoc-cupied positions on the real line uniformly at random, starting from an initially empty interval oflength l . The process continues until no vacant unit-length interval remains.Each car parked in an open interval splits the interval into two smaller ones, each independentfrom the other. Therefore this random process can be analyzed by a recursive formulae. Let f ( l )denote the expected number of cars parked on a street of length l . Then f ( l ) is determined by thefollowing delay differential equation: f ( l ) = 1 + 1 l − (cid:90) l − ( f ( x ) + f ( l − x − dx = 1 + 2 l − (cid:90) l − f ( x ) dx (1)with the base cases f ( l ) = 0 for l ∈ [0 ,
1) [8].The parking density is defined as f ( l ) /l , with the R´enyi parking constant giving the limit ofdensity as l → ∞ . Numerical calculations show thatlim l →∞ f ( l ) /l ≈ . l , exact values of f ( l ) are known, including f (1) = 1, f (2) = 1 / f (3) = 2 /
3, and f (4) = (11 − /
3. The variance in this process has been determined [8, 12].Recently, the expected number of gaps of certain size was analyzed for a discrete version of theparking problem [7].The problem has been generalized into higher dimensions. Here the goal is to “park” unit n -cubes into a larger n -dimensional cube. Results in two dimensions include [2, 14]. Higher-dimensional versions are widely studied in statistical physics as the random sequential adsorptionproblem. It is investigated as the end configuration after molecules attaching itself to a surface. See[3] for a survey of the field. The limit of the parking density in n -dimensions is a difficult problem,and even the two-dimensional version remains open [9].These investigations suggest a new class of bin packing or knapsack problems [10, 11] where theheuristic employed are not under central control, but instead the heuristic is selected at random toposition each additional element. Painted street markings are the traditional way to enforce parking behavior. Each slot mustaccommodate the largest-sized vehicle on the road (the GMC Savana van, 224 inches long), leavingconsiderable wasted space when small vehicles (e.g. the Nissan Cube, 158 inches) are parked. Thelongest cars on the road approach twice the length of the shortest, making maximally-generousspots too wasteful for the city to enforce.Instead, we consider lines as optional guide markings to help drivers make better decisions onwhere to place their car. These lines serve the same role as etching images of flies in men’s urinals:providing a target to hit that encourages better behavior. Schiphol Airport in Amsterdam reportsa 80% reduction in spillage as a result of this design [18].In this paper, we will study three basic parking strategies (illustrated in Figure 1) where ob-serving street markings is optional: 3
Random parking – Consistent with the R´enyi model, the driver selects a location uniformly atrandom over all unoccupied locations on the given street. We believe this is a fairly accuratemodel of what happens in practice during the phase transition from double parking at theinstant the other side of the street becomes legal again. • Kiss the bumper – Here a driver selects a location uniformly at random and moves up to leaveminimum space between the neighboring car. Uniformly observed, this strategy providesoptimal street utilization, but the parallel maneuvering during the shift makes it impossibleto enforce such behavior. • Hit the line – Here a driver selects an unoccupied street marking (line) uniformly at randomand lines up immediately behind it. In the event that all lines are occupied, the kiss thebumper strategy is employed.We consider the situation when the population employs a pair of these basic strategies, typicallythe random/R´enyi strategy and a more sophisticated algorithm. Let α denote that fraction of thepopulation employing the more sophisticated strategy, i.e. the “good” drivers. The expecteddensity of cars parked on the street depends upon α . For α = 0, all drivers are random/R´enyidrivers, so the achievable density will be 0 . α = 1, all drivers behave in a socially-mindedway, and the resulting density approaches the optimal value of 1 . f ( l ) denote the expected number of cars that will be parked on a street of length l where the α -fraction of good drivers employ the kiss-the-bumper strategy. Then f ( l ) = 1 + αf ( l −
1) + (1 − α ) 2 l − (cid:90) l − f ( x ) dx (2)Alternately, let f k ( l, t ) denote the expected number of cars that will be parked on a street oflength l where an α -fraction of good drivers employ hit-the-line. We assume the street is paintedwith lines k units apart, such the initial line is t units away from the origin. Then f k ( l, t ) = 1 + αf ( l −
1) + (1 − α ) 1 l − (cid:90) l − f k ( x, min( x, t )) + g k ( l, x, t ) dx (3) g k ( l, x, t ) = f k ( l − x − , t − x −
1) if x + 1 ≤ tf k ( l − x − , k − (( x + 1 − t ) − k (cid:98) x +1 − tk (cid:99) )) if t < x + 1 ≤ t + k (cid:98) l − tk (cid:99) f k ( l − x − , l − x −
1) if t + k (cid:98) l − tk (cid:99) < x + 1 (4) f k ( l, t ) = f ( l ) = 0 if 0 ≤ l < P a r k i ng den s i t y α k=2kiss-the-bumper Figure 2: Parking density of kiss-the-bumper and hit-the-line as a function of α , for the optimalline spacing of k = 2 and l = 20. We now consider the problem of spacing painted lines so as to induce the greatest expected parkingdensity as a function of α . We assume that street markings occur regularly every k units over thelength l street. For unit-length cars, and compliance fraction α = 1, clearly k = 1. However, wewill show that this is not the case when there is lower compliance, and the (1 − α ) fraction of “bad”drivers employ the R´enyi strategy.In particular, through simulation we compare the expected density as a function of k and α .We also compare it to the benchmark kiss the bumper strategy for the same value of α . For eachstrategy, the simulations were run on l = 20 with 100,000 repetitions on 200 evenly spaced datapoints of α from 0 to 1. Each data point for a specific α represent the average of the parking densityover all repetitions.Our primary result is shown in Figure 2. First observe that the parking density of both thekiss-the-bumper and hit-the-line strategies range from the R´enyi parking constant to an optimalpacking as the compliance constant α ranges from zero to one. The density increases in a non-linearfashion for both strategies, with densities of 0.82 achieved by α = 0 .
5. This is only about 25% ofthe total density improvement with α = 1, so relatively high compliance rates must be achieved tosubstantially increased density.The behavior of the two strategies are similar but not identical. Kiss-the-bumper outperformshit-the-line until a compliance threshold of approximately α = 0 . k , the number of car widths betweenpainted lines. Surprisingly, the best performance is achieved when painting lines between every other spot, (i.e. k = 2), as opposed to the standard practice of delimiting the precise boundaryof every car. The reason becomes clear in hindsight. Two “good” drivers parked on consecutivelines for k = 2 will create a unit-length pocket between them, achieving optimal density even if5 P a r k i ng den s i t y α k=1k=2k=3k=5 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0 0.2 0.4 0.6 0.8 1 P a r k i ng den s i t y α k=1k=2k=3k=5 Figure 3: Effect of secondary strategies as a function of line separation: kiss the bumper (left) vs.R´enyi random selection (right) for line separations of 1, 2, 3, and 5 car lengths.eventually taken up by a normally uncooperative driver.This mechanism also explains why the relative advantage of painting lines accrues primarily forlarge values of α : enough lines must be hit to force such tight pockets. Unit-length pockets are alsooccasionally created for other painting patterns (i.e. k (cid:54) = 2), but the probability of creating themproves too small to compete with the benchmark strategy.Larger gaps between painted lines necessarily implies fewer lines available to hit. Once theseare exhausted, the “good” drivers must shift to an alternate strategy. Figure 3 contrasts thedensities achieved with kiss-the-bumper as a secondary strategy (left) vs. random/R´enyi (right).The interactions between R´enyi and painted lines can be disastrous: worse than no lines at all for k = 3. There the good drivers tend create pockets of two spaces, one of which becomes permanentlylost to a R´enyi driver.In general, much of the increase in parking density from line painting comes from kiss-the-bumper as the secondary strategy. The painted lines primarily help by creating unit-length spaces.Let f ( x ) denote the number of cars a open space of length x eventually contains. After a bumper-kiss event, f ( x ) = 1 + f ( x − y , yields f ( x ) =1 + f ( y ) + f ( x − y − The previous section posits drivers who exhibit no preferences when faced with the Nirvana ofmultiple open spots on the street. This utility function accurately models the behavior in the shiftfrom double-parking to clean streets, since the double-parked cars are uniformly distributed alongthe street. It also seems relevant to the case where drivers have random destinations on the streetafter they leave their vehicle.However, in many situations drivers are likely to pounce on the first available space they see asthey enter the street. With the new model, normal drivers will pick a random position in the firstopen space. The driver’s behavior is now limited to the first open pocket (i.e. the pocket closest to x = 0) of sufficient size to hold their unit-length vehicle. We again assume that a (1 − α ) fractionof drivers employ the R´enyi strategy, while the more civic-minded α fraction pursue one of threebeneficial approaches: 6 P a r k i ng den s i t y α k=2kiss-the-bumper (a) Hit-the-line vs. Kiss-the-bumper P a r k i ng den s i t y α k=2kiss-the-bumper (b) Crave-the-line vs. Hit-the-line Figure 4: Parking densities for hit-the-line, crave-the-line, and kiss-the bumper restricted to thefirst-available pocket. • Kiss the Bumper – Such drivers park directly behind the top car delimiting the first openpocket. • Hit the Line – Such drivers park on the first line they see in the first open pocket, if oneexists. Otherwise, they employ kiss-the-bumper. • Crave the Line – These drivers go the extra length to park at the first line they see, evenif it’s not in the first open pocket. They kiss-the-bumper if they don’t find any unoccupiedlines. Intuitively, a sequence of k crave-the-line drivers will create a sequence of k − k − α = 0 . The results reported thus far all assume unit-length vehicles, thus ignoring the natural diversityof car sizes from compacts to SUVs. But this non-uniform size distribution is an important factorgoverning urban street parking dynamics. Certainly smaller cars have an easier time finding aparking in the city than big ones, which goes a long way towards explaining the popularity ofhalf-size Smart cars in urban environments.Variable car lengths require us to consider whether our optimization criteria is to maximize thenumber of cars resting on the street, or the fraction of the total street length being utilized byparked cars. The former could best be done by only allowing half-sized cars to park, while the laterwould reserve a spot for the largest car which could fit in it. For the purposes of this paper, we7 N u m be r o f c a r s pa r k ed α k=1k=2k=1-dk=1+dkiss-the-bumper (a) Any pocket N u m be r o f c a r s pa r k ed α k=1k=2k=1-dk=1+dkiss-the-bumper (b) First pocket N u m be r o f c a r s pa r k ed α k=1k=2k=1-dk=1+dkiss-the-bumper (c) Crave the line Figure 5: Parking densities for car lengths drawn uniformly from [0.75,1.25].seek to maximize the number of parked vehicles without explicitly favoring small cars over largeones. Thus cars are served at random as per their length-frequency distribution, but the processcontinues until no more empty spots for minimum-length cars remain.In this section, we study two different models of non-uniform car lengths. Section 6.1 considersa model where cars are uniformly distributed over a size range. Section 6.2 uses car sales data andrepresentative Manhattan streets to strive for more accurate modeling.
To capture the variance of car lengths, we consider car lengths as generated by a uniform distributionin the range [1 − d, d ], for some value of d . The previously studied case of unit-length carscorresponds to d = 0.We evaluated four gap sizes for the line markings for each of the three pocket-selection modelspreviously described (random pocket, first pocket, and crave-the-line). The gap sizes we study are1 − d , 1 + d , 1 or 2.The results for d = 0 .
25 are shown in Figure 5, with the results for d = 0 .
125 and d = 0 . k = 2 gap length generally outperforms all other line markings, althoughthere are small windows where other spacings dominate for high α in hitting or craving the line.Making the gap larger or smaller than an integral size of the car did not perform well. Indeed thetight gap spacing of 1 − d results in strictly fewer parked cars as compliance increases. The optimal line spacing depends on the distribution of the car lengths driven by local residents.To better capture the dynamics of real world parking, we used new car sales figures as a proxy forthe length distribution in use on today’s roads. In particular, we compiled the January 2010 salesfor each current model from GM, Ford, Honda, Toyota, Nissan, Chrysler, and Kia. These sevencompanies together control 85% of the automobile market share in North America. Figure 6 presentsthe relative frequencies of cars by body length. It is strikingly irregular, and not particularly wellapproximated by the uniform distribution employed in the previous section.We performed simulation experiments on two typical sizes of Manhattan streets. The Broadway-to-Amsterdam and Amsterdam-to-Columbus blocks of West 92nd Street measure in at 340 feet and8 F r equen cy Car Length (in)
Figure 6: North American car sales in 2010 as a function of body length.
12 13 14 15 16 17 18 19 250 300 350 400 450 500 N u m be r o f c a r s pa r k ed α α =0.25 α =0.5 α =.75 α =1 (a) Short block
30 32 34 36 38 40 42 44 46 48 250 300 350 400 450 500 N u m be r o f c a r s pa r k ed α α =0.25 α =0.5 α =.75 α =1 (b) Long block Figure 7: Simulation result for real world data: the number of parked cars as a function of thedistance between painted lines.831 feet, respectively. We assume that adjacent parked cars must be at least one foot (12 inches)apart from each other.Figure 7 presents our results on the expected number of cars parked on both short and longblocks as a function of the gap between painted lines for four values of α . The optimal gapbetween painted lines was determined to be 385/390 inches on short/long blocks, respectively.This density oscillates depending upon how this gap compares to the length of a typical car. Aspreviously observed, optimal density is achieved when the gap is roughly twice the average carlength. Making the gap too short relative to this length significantly degrade performance, wellbelow that of unmarked streets. The magnitude of this oscillatory behavior increases strictly with α . Figure 8 gives an alternate view of this data, showing the average number of cars parked onboth short and long blocks as a function of α for optimally-sized gaps. The number of cars increasescontinually with greater compliance, with greater improvement per ∆ α as α increases.9 N u m be r o f c a r s pa r k ed α hit-the-linekiss-the-bumper (a) Short block
38 39 40 41 42 43 44 45 46 47 48 0 0.2 0.4 0.6 0.8 1 N u m be r o f c a r s pa r k ed α hit-the-linekiss-the-bumper (b) Long block Figure 8: Simulation result for real world data: the number of parked cars as a function of thecompliance rate α . We have studied the impact of two behavioral interventions (encouraging kiss-the-bumper or hit-the-line parking driving) to improve space utilization in city streets. Generally kiss-the-bumperprovides better utilization for a given level of compliance, however the absence of visual cues makesit difficult and expensive to alter current habits. If painting guide lines can increase compliance( α ) by 0 .
1, then painting guide lines two car-widths apart is the right strategy.With the proper street markings, one can generally squeeze one extra car into the shorter streeton average at α = 0 .
5, and the same on the longer street at the lower compliance of α = 0 . We wish to thank the Algorithms Reading Group at Stony Brook for helpful discussions.
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