Umbrella Traffic Flow Analysis
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Finding the traffic flux at the major doorways was done using two different methods. | Finding the traffic flux at the major doorways was done using two different methods. | ||
| - | The first method is a Markov simulation of traffic flow around campus. A Markov matrix expresses the probability that something in a given state at some time will move to another given state at the next time. In our model, it expresses the probability that somebody leaving a certain doorway will move to another given doorway. Two matrices are needed for our model because the probabilities of where are user is going to move are different for users entering and leaving a given doorway. For example, a user exiting the upper door of Baker hall might go to the library, the UC, or some other building. A user entering the same door must exit through either the same door, the lower door at Porter hall, or one of the other doors along the length of the building. Using these movement probabilities, a model of campus was constructed and a rainy day was simulated to see how traffic moved around campus. | + | The first method is a Markov simulation of traffic flow around campus. A [http://en.wikipedia.org/wiki/Stochastic_matrix Markov matrix] expresses the probability that something in a given state at some time will move to another given state at the next time. In our model, it expresses the probability that somebody leaving a certain doorway will move to another given doorway. Two matrices are needed for our model because the probabilities of where are user is going to move are different for users entering and leaving a given doorway. For example, a user exiting the upper door of Baker hall might go to the library, the UC, or some other building. A user entering the same door must exit through either the same door, the lower door at Porter hall, or one of the other doors along the length of the building. Using these movement probabilities, a model of campus was constructed and a rainy day was simulated to see how traffic moved around campus. |
Unfortunately, an accurate model of the campus would require large amounts of data averaged over a large time span, which was not feasible for us to collect within the given time frame of this project, and may not be feasible for other clients to collect. To estimate the necessary capacity of the system without using a full model simulation, we counted traffic flow in and out of major doorways in half- to one-hour increments. This gives us point estimates of traffic flux, allowing us to see roughly which doorways would need more units. | Unfortunately, an accurate model of the campus would require large amounts of data averaged over a large time span, which was not feasible for us to collect within the given time frame of this project, and may not be feasible for other clients to collect. To estimate the necessary capacity of the system without using a full model simulation, we counted traffic flow in and out of major doorways in half- to one-hour increments. This gives us point estimates of traffic flux, allowing us to see roughly which doorways would need more units. | ||
Revision as of 17:28, 9 November 2008
Contents |
Executive Summary
Purpose
The physical construction of our device is relatively simple. It contains few components, and those components are already well-developed and commonly used. The size of the device is constrained by the umbrellas, and device weight is not a large concern since it will be stationary. Given this, we did not feel that a strictly mechanical analysis would provide us the insight that we need to develop a complete solution. In addition to the design of the hardware, the issue of implementing the system had to be addressed. This is where we chose to focus our analysis. The largest issue of implementation is determining where to place the units, and how many modular units to put at each location. Traffic flow into and out of each doorway is the major determinant of this, so we decided to find out what the traffic flow was like as a function of time at the major doorways on campus.
Approach
Finding the traffic flux at the major doorways was done using two different methods.
The first method is a Markov simulation of traffic flow around campus. A Markov matrix expresses the probability that something in a given state at some time will move to another given state at the next time. In our model, it expresses the probability that somebody leaving a certain doorway will move to another given doorway. Two matrices are needed for our model because the probabilities of where are user is going to move are different for users entering and leaving a given doorway. For example, a user exiting the upper door of Baker hall might go to the library, the UC, or some other building. A user entering the same door must exit through either the same door, the lower door at Porter hall, or one of the other doors along the length of the building. Using these movement probabilities, a model of campus was constructed and a rainy day was simulated to see how traffic moved around campus.
Unfortunately, an accurate model of the campus would require large amounts of data averaged over a large time span, which was not feasible for us to collect within the given time frame of this project, and may not be feasible for other clients to collect. To estimate the necessary capacity of the system without using a full model simulation, we counted traffic flow in and out of major doorways in half- to one-hour increments. This gives us point estimates of traffic flux, allowing us to see roughly which doorways would need more units.
Model
Assumptions
- It is constantly raining throughout the model day.
- The Markov movement probabilities are time-independent.
- The probability of taking an umbrella is equal for all users.
- Movement times are normally distributed.
- The Markov model is appropriate for modeling behavior.
- The movement probabilities are not affected by the existence of the distribution system.
Description
After determining the Markov probability matrices, we decided on an algorithm to simulate a day's worth of movement activity.
The simulation steps through each day minute by minute. At each time step, the system can first add new users. Each user in the system keeps track of its current position, its wait time at that position, whether it is inside or outside, and whether or not it has an umbrella. Next, it loops through each user. Each user's wait time is reduced by 1. When a user's wait time hits zero, it selects the next location to move to from the appropriate Markov matrix, and its wait time at that location. The wait times are selected from a normal distribution given in an Excel file. If the user was inside, and is traveling outside to get to another building, the wait time is given by the distance between the two locations; if a user was outside and is going inside, the wait time is dependent only on the location it has arrived at.
When a user leaves a building, it takes an umbrella, tracked by decrementing an array. When a user enters a building, it deposits an umbrella at the new location, by incrementing an array.
After each user has been dealt with for a given time step, the distribution of umbrellas by location is saved in a large matrix. Once all time steps have been run, the matrix is saved as an Excel file. Plotting the data in the Excel file shows which locations need to keep the largest stock of umbrellas.
