Estimating Parking Spot Occupancy

Transcription

1 1 Estimating Parking Spot Occupancy David M.W. Landry and Matthew R. Morin Abstract Whether or not a car occupies a given parking spot at a given time can be modeled as a random variable. By looking at a parking lot as a collection of these random variables, we create a spatial random process. In this paper, we will consider factors that affect the probability and distribution of occupancy in a parking lot. We then create a simplified model accounting for some of the more regular and important factors and simulate results with this model. I. INTRODUCTION Finding an open parking spot is a common event in many peoples lives. Sometimes the parking options are plentiful, while at other times finding a parking spot can be a painful process, especially when under strict temporal and spatial constraints. There are various factors that influence the availability of parking, and they are too numerous to mention. This article presents a simple model for predicting parking behaviour. The model only considers two factors that seem particularly important. These are time of day and distance from a point of interest. One application of this model is to minimize the amount of time it takes an individual to find a parking space. Also, insight gained from observing modeled behavior could be used to make parking lot layouts that are more time efficient. II. ANALYTICAL MODEL Various factors influence parking behavior. These include distances from parking spot to point of interest 1, the time of day, parking privileges, events, weather, and so on. This model makes assumptions to create a simplified, analytical model. The intent is to allow an individual to intuitively grasp certain trends in parking behavior. The model ignores temporal and spatial irregularities such as weather, holidays, sporting events, accidents, special privileges, etc. Instead it assumes that time of day and distance between the parking spot and the point of interest are the most important factors for parking behaviour. 1 The point of interest is the location that drivers and their passengers are trying to reach. A. Spatial and Temporal Random Process The Bernoulli distribution is useful in modeling the occupancy of a single parking spot. The probability of a parking spot occupancy, p, is a dynamic value that is a function of time of day and distance from the point of interest. A parking lot, then, is a collection of Bernoullidistributed random variables, each with different values for p. This forms our random process with diefferent for parking occupancy verses time and distance. How probability is distributed across the lot affects occupancy. An interesting problem is how to select the probability of a parking spot, given a particular time and distance, that reasonably reflect what happens in real life. In building this model, we have found two methods for generating those probabilities. B. Independent Distributions One way to select the probabilities for each parking spot is to assume two independent distributions; one for time and one for distance. By assuming independence, this scheme assumes that individuals entering a parking lot do not base their decision on where to park on the current time. These distributions are then overlayed on the factor they represent. The random variable p can then be generated for a given distance, d, and time of day, t. Let s look at these distributions separately and then combine them. 1) Spatial Distribution: The point of interest is a key parameter and usually is self evident. It could be, for example, a store entrance, a parking garage exit, or the location of a bus or train stop. Each parking spot is a certain distance from the point of interest. The spatial random process takes distance as a parameter and returns a number between zero and one. By examining aerial photos of parking lots and through trial and error, a Gaussian distribution centered at the point of interest seems to provide a reasonable approximation to reality. Overlaying the Gaussian onto distance gives { 1 p(d) = σ /2σ 2, x > 0 2π e d2 0, otherwise, (1)

2 2 Fig. 1. This graph overlays a Rayleigh distribution onto the time of day. By adjusting the variance, we can affect how broad the peak probability is and what time of day has peak probability of occupancy. Fig. 2. Changing kt affects the density of states. where d is the distance from the point of interest and σ is the standard deviation. Using a Guassian represents the idea that most people are interested in parking closer to the point of interest. 2) Temporal Distribution: Deciding how occupancy is distributed through time is less reliable than the spatial distribution. Temporal distribution of occupancy could change dramatically by venue. For example, parking for an office building or school building likely has a dramatically different temporal distribution when compared to that of a grocery store or mall. In this model, a distribution was picked to model behavior in a office parking situation. In picking this venue, we assume that there is a period at the beginning of the day where occupancy increases sharply as people arrive for work. As the day progresses, some people might leave early while others leave late, resulting in a more gradual drop in occupancy probability. During the middle of the day, there might be a dip in occupancy as people leave for lunch. The Rayleigh distribution roughly fits this trend. It increase sharply, then drops off more gradually. As shown in Figure 1, after overlaying the distribution onto time of day, adjusting the variance affects how broad high regions of probability are and shifts what hour peak probability falls on. A draw-back of this distribution is that we cannot adjust breadth of the peak and time of the peak independently. This compromises how well the temporal distribution can be adjusted based on real data for a particular parking lot. Another draw-back is that a dip in occupancy for lunch cannot be included in this distribution. Fig. 3. Altering µ shifts the location of the 0.5 probability state. 3) Joint Distribution: Passing a distance to the spatial distribution returns a number representing the probability of occupancy at that distance. Similarly, passing a time of day into the temporal distribution returns the probability of occupancy at that time. Since time and distance are assumed to be independent, it is simple to find the probability of occupancy for a particular parking spot at a given time as p(d, t) = p(d) p(t), (2) where p is the random variable for probability of occupancy, d is the distance from the point of interest, and t is the time of day. C. Fermi-Dirac Distribution The second model will use the Fermi-Dirac distribution, commonly used in solid state physics [1]. It is used

3 3 to show the probability that a certain energy state will be occupied by an electron. This has an interesting parallel to modeling parking. Replacing the energy parameter with the distance parameter, d, like so p(d) = e d µ 1 kt + 1, (3) results in a distribution that can be used to generate a probability for a particular parking space. As can be seen in Figures 2 and 3 this distribution has two parameters that can be used to adjust its shape. This first parameter, kt, affects the steepness of the slope. The second parameter, µ, shifts the mean of the distribution. By changing kt, the desnsity of occupancy can be altered. Then, by varying µ with respect to time, occupancy with respect to time can be modeled. D. Multiple Points of Interest A single parking space often has multiple points of interest. For example, with a parking lot in front of a strip mall, each store entrance becomes a point of interest. However, this means that each parking space has an independent probability of occupancy for each point of interest. For two points of interest the probability of being occupied can be expressed as, p = p 1 + p 2 p 1 p 2 (4) where p 1 and p 2 are the events where a parking spot is occupied for the first point of interest and the second point of interest respectively. This new p value is the probability that the parking space will be occupied by individuals interested in the first point of interest or the second. E. Expected Time The question that interests people parking is where to look for a parking space. Individuals tend to realize the trade-off inherent in distance from the point of interest: the further away the driver looks, the more quickly they find a spot, but the longer it takes them to walk to their destination. For a given set of parking spots we can calculate the expected time till arrival at the point of interest. Upon arrival, there are two states the set of parking spaces can be in: (1) all the spaces could be occupied or (2) at least one space could be available. The expected wait time for these two states can be represented as T o and T u, where T o is the expected time for a parking spot to become available and the driver to walk to the point of interest (given all parking in the set is occupied), and T u is the expected time to walk to point of interest if at least one spot is immediately open. There is some probability, p, that the set of parking spaces will be completely occupied. This means that the overall expected time can be represented as T total = pt o + (1 p)t u (5) With this expression for expected time, we can determine which group of parking spots should be used to minimize the time for the driver to actually reach the point of interest. This can be done for any groupings of parking spaces, including by distance from the point of interest, by row, or by any other arbitrary groupings. 1) Expected Time If All Spots Occupied: The expected time, T o, of this space is broken into two parts, the wait time until a parking spot is available and the travel time from parking spot to the point of interest. In order to model wait time, we must define a random variable to represent how long a parking space is occupied. Let X is a random variable representing the time a car occupies a parking spot. Assume N parking spots, each with the a wait time distribution of X. The time it takes for one of those parking spaces to open up can be defined as Y, where Y = min(x 1, X 2..., X N ). (6) Given this definition of Y, the distribution of Y can be derived in terms of X as F Y (x) = P {Y x} = 1 P {Y > x} = 1 P {min(x 1, X 2..., X N ) > x} = 1 P {X 1 > x, X 2 > x,..., X N > x} = 1 P {X > x} N = 1 (1 F X (x)) N, (7) f Y (x) = df Y (x) dx = Nf X (x)(1 F X (x)) N 1. (8) Given this probability distribution, the expected wait time until one parking space becomes available can be calculated as E[Y ] = xnf X (x)(1 F X (x)) N 1 dx. (9) Since the X is i.i.d., each parking space is equally likely to be the first to become available. This means that the distance the driver is expected to walk, d o, can be shown as d o = 1 N d i, (10) N i

4 4 Fig. 4. The model was used to simulate a parking lot around a retail building with one entrance. In this image, a realization of this simulation is shown. The green represents a building. The point of interest is the entrance, which is represented by the yellow pixel. The red pixels are parking spaces that are occupied. where d i is the distance from the point of interest to a particular parking space. Assuming an average walking velocity of v, the total expected time to arrive at the point of interest can be expressed as T o = E[Y ] + d o v. (11) 2) Expected time if at least one space is open: To find the expected time, T u, we must examine which parking space the driver is likely to park in. If more than one spot is open, it is assumed that the driver will park in the closest available. The probability that the n th space is the first available can be expressed as n 1 p u = (1 p n ) p i. (12) i=1 The expected distance to walk can now be written as E[d u ] = N d j p u,j, (13) j=1 which means the expected overall time is T u = E[d] v. (14) III. SIMULATION RESULTS This section demonstrates some specific applications of this model. This shows how the model can help give information concerning behaviors for specific parking layouts. Fig. 5. This demonstrates parking spaces available for a point of interest signified by the yellow pixel. Each red pixel represents a parking space. This simulates parking on a street where the point of interest is near a four way intersection. The streets in the paper will be a clockwise order starting at the street with the point of interest. They are named: street 1, street 2, street 3, street 4. A. Simulating Parking Occupancy The first example simulates the behavior for a retail building with parking distributed all around it in parking aisles as shown in Figure 4. The parking space distances from the point of interest are calculated as line of sight, except where the building obstructs a direct course. At these parking spots the shortest distance that goes around the building are used. Notice how the parking is more sparse behind the building because the walking distance is increased. The probabilites that determine occupancy are given by a spatial random process with the Fermi-Dirac equation, with µ = 20, and kt = 5. This occupancy can be simulated for any given time. B. Simulating Estimated Time to Destination The second example is for street parking around an intersection, where the point of interest is next to one of the streets, as shown in Figure 5. We used the Fermi- Dirac expression for our spatial random process with µ = 75 and kt = 17, where distance is measured as people would walk, which is along the streets. In this example each street has a different number of parking spaces with cars parked end to end. The street names as shown in Figure 5 are, going in a clockwise order starting at the street with the point of interest: street 1, street 2, street 3, and street 4. The number of parking spaces on each road are in the same order 30, 15, 25 and 5. The point of interest is next to the 4th parking space on the street 1. From this we calculate, T total, the

5 5 Future work could be done to show how the best parking locations change with different random process parameters, such as we domonstrated with the Fermi- Dirac paramter µ. It would also be interest to show how the best parking areas can change with time, or how it changes with different parking layouts. REFERENCES [1] R. Naumann, Introduction to the physics and chemistry of materials. CRC Press, [Online]. Available: http: //books.google.com/books?id=fa4paqaamaaj Fig. 6. For each street the T total value is shown verses µ. Several of the plots cross at different µ values demonstrating how minimizing T total can change depending on parameters used for the spatial and temporal random process. expected time it takes to get to the point of interest for each street. We can then determine which street would be expected to take the least time to arrive at the point of interest. To do this we made the random variable of the time that people spend in a parking spot to be uniform between 15 to 75 minutes. We also set our walking velocity to 3 mph. Assuming that a driver picks a street and then only parks on that street, the expected time till arrival at point of interest for each street are (in the same order as the streets were given previously) 9.3, 7.6, 2.3, and 20.1 minutes. So street 3 would minimize time to park and walk to point of interest. We can then compute T total for various values of µ with each street. This is shown in a plot in Figure 6. Notice how the optimal street changes as µ varies. From this we can see that the best parking decision may change depending on the distribution of a specific parking layout at different times, if µ is a function of time. This can be also true of kt, depending on the layout. Therefore understanding the spatial and temporal random process of parking for a given layout can help decision making in minimizing the drivers time. IV. CONCLUSION A parking model was developed with the goal of helping in design of parking systems as well as helping drivers make better parking decisions. The parking model was used in a couple of simple examples to demonstrate its potential utility. The model can to be extended to more complicated systems and gives some intuition for those systems.