91Ó°ÊÓ

Consider a side road connecting to a major highway at a stop sign. According to a study by D. R. Drew, \({ }^{44}\) the average delay \(D\), in seconds, for a car waiting at the stop sign to enter the highway is given by $$ D=\frac{e^{q T}-1-q T}{q}, $$ where \(q\) is the flow rate, or the number of cars per second passing the stop sign on the highway, and \(T\) is the critical headway, or the minimum length of time in seconds between cars on the highway that will allow for safe entry. We assume that the critical headway is \(T=5\) seconds. a. What is the average delay time if the flow rate is 500 cars per hour ( \(0.14\) car per second)? b. The service rate \(s\) for a stop sign is the number of cars per second that can leave the stop sign. It is related to the delay by $$ s=D^{-1} . $$ Use function composition to represent the service rate as a function of flow rate. Reminder: \((a / b)^{-1}=b / a\). c. What flow rate will permit a stop sign service rate of 5 cars per minute ( \(0.083\) car per second)?

Step by step solution

01

Identify Known Values

For part (a), the given values are the flow rate of cars, \( q = 0.14 \) cars per second and the critical headway, \( T = 5 \) seconds.

02

Calculate Average Delay Time

Substitute the values \( q = 0.14 \) and \( T = 5 \) into the delay formula. This gives us \( D = \frac{e^{0.14 \times 5} - 1 - 0.14 \times 5}{0.14} \). Calculate this to find the average delay time, \( D \).

03

Calculate Exponential and Substitute

First compute the exponential term \( e^{0.7} \), then substitute back to find \[ D = \frac{e^{0.7} - 1 - 0.7}{0.14} \approx \frac{1.6645 - 1 - 0.7}{0.14} \approx \frac{0.6645}{0.14} \approx 4.746 \] seconds.

04

Express Service Rate in terms of Flow Rate

Since the service rate \( s = D^{-1} \), and \[ D = \frac{e^{qT} - 1 - qT}{q} \]. We can express \( s \) as \[ s = \left(\frac{e^{qT} - 1 - qT}{q}\right)^{-1} = \frac{q}{e^{qT} - 1 - qT} \].

05

Solve for Flow Rate Given Service Rate

For part (c), set \( s = 0.083 \) and solve for \( q \). The service rate equation is \( 0.083 = \frac{q}{e^{5q} - 1 - 5q} \). Rearrange to find \( q \), giving \( 0.083(e^{5q} - 1 - 5q) = q \).

06

Approximate with Trial and Error

Since an analytical solution is difficult, use trial and error with values of \( q \) around 0.1 to find \( q \) that satisfies the equation \( 0.083(e^{5q} - 1 - 5q) = q \).\ Through trials, \( q \approx 0.11 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Flow Rate

Flow rate is a critical component of traffic flow management, particularly at intersections where minor roads meet major highways. Simply put, flow rate refers to the number of vehicles passing a particular point per second. In the context of our problem, it measures how many cars pass by the stop sign on the highway. Typically, flow rates are measured in cars per hour, yet for precision, especially in calculations like our average delay time, it is converted to cars per second. For instance, a flow rate of 500 cars per hour translates to approximately 0.14 cars per second. This conversion involves dividing the number of cars per hour by 3600, the total number of seconds in an hour. Understanding the flow rate allows us to gauge how busy a road is, and it plays a significant role in calculating potential delays at intersections.

The Role of Critical Headway

Critical headway is another vital concept for traffic engineers. It refers to the minimum time interval between vehicles on the main road that allows a car from a side road to safely enter. For instance, if the critical headway is set at 5 seconds, a driver will need at least that amount of time between two consecutive cars on the highway to merge safely. This headway ensures that there is adequate time for a vehicle to accelerate and merge without causing disruptions or accidents. Incorporating the critical headway into the average delay formula helps determine how often a car from the side road will have the opportunity to enter the major road smoothly. If the time between traffic on the highway is consistently less than this value, delays are inevitable.

Decoding Service Rate

The service rate at a stop sign describes the efficiency or capacity of the intersection in terms of vehicles passing through. It is measured in cars per second and inversely related to the average delay.To express the service rate in terms of flow rate, we first need to calculate the average delay, denoted as \( D \). The service rate \( s \) is then the reciprocal of the average delay (i.e., \( s = D^{-1} \)). Therefore, when we have a function for \( D \), like in our exercise, we can find \( s \) by taking the inverse.The relationship can be shown as: \[ s = \frac{q}{e^{qT} - 1 - qT} \]. This tells us how many cars can actually leave the stop sign based on the traffic flow on the highway and the given critical headway.

Exponential Function in Traffic Analysis

Exponential functions often appear in traffic calculations, as they help model growth and decay processes. In our exercise, the term \( e^{qT} \) is part of the average delay formula, indicating the influence of both flow rate \( q \) and critical headway \( T \) on delay.The exponential component reflects how quickly potential delay times ramp up as the flow rate increases or as the time between cars (headway) becomes smaller. Essentially, the faster the flow and the tighter the headways, the more dramatically the delay increases.To calculate delay, the exponential function needs to be evaluated precisely, emphasizing its sensitivity to small changes in input. Understanding its role can improve decisions on traffic management and alleviate congestion.