91Ó°ÊÓ

A queue, or line, of traffic can form when a feeder road meets a main road with a high volume of traffic. In this exercise we assume that gaps in traffic on the main road (allowing cars to enter from the feeder road) appear randomly and that cars on the feeder road arrive at the intersection randomly also. We let \(s\) be the average rate at which a gap in traffic appears and \(a\) the average arrival rate, both per minute. Assume that \(s\) is greater than \(a\). a. The average length \(L\) of a queue is given by the rational function $$ L=\frac{a^{2}}{s(s-a)} . $$ Explain what happens to average queue length if arrival rate and gap rate are nearly the same. b. The average waiting time \(w\) (the time in minutes spent in the queue) is given by $$ w=\frac{a}{s(s-a)} . $$ If the gap rate is 3 per minute, what arrival rate will result in a driver's average waiting time of 2 minutes?

Step by step solution

01

Understanding Queue Length Equation

The average queue length is given by \( L = \frac{a^2}{s(s-a)} \). When the arrival rate \( a \) approaches the gap rate \( s \), the denominator \( s-a \) becomes very small, which causes \( L \) to become very large. This leads to a significant increase in the average queue length.

02

Analyzing Waiting Time Equation

The waiting time \( w \) is given by \( w = \frac{a}{s(s-a)} \). We need to determine \( a \) for \( w = 2 \) when \( s = 3 \).

03

Substitute Known Values

Substitute \( w = 2 \) and \( s = 3 \) into the waiting time formula: \[ 2 = \frac{a}{3(3-a)} \].

04

Solve for Arrival Rate

First, multiply both sides by \( 3(3-a) \): \[ 2 \cdot 3(3-a) = a \].Next, simplify and rearrange the equation:\[ 6(3-a) = a \]\[ 18 - 6a = a \]Combine like terms:\[ 18 = 7a \]Finally, solve for \( a \):\[ a = \frac{18}{7} \approx 2.57 \] (cars per minute).

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Key Concepts

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

Traffic Flow

Traffic flow refers to the movement of vehicles or pedestrians through intersections and roadways. In queue theory, especially where feeder roads meet busy highways, traffic flow can considerably influence the formation of queues. This concept is particularly important in understanding how vehicles merge into main traffic streams.

When considering traffic flow in terms of probability, both vehicles arriving at an intersection and the gaps available in the main traffic flow behave randomly. The feeder road vehicles attempt to enter traffic whenever there is a sufficient gap. If these gaps are infrequent or short, queues can quickly build up.

• Gap rate \( s \): This is the rate at which gaps occur in the main road traffic. More frequent gaps mean more chances for cars from a feeder road to join the traffic, thus optimizing traffic flow.
• Arrival rate \( a \): This is the frequency of cars arriving at the intersection from the feeder road. When this rate is higher than the gap rate, it results in congestion and longer queues.

Understanding these two rates helps in designing better intersection controls and traffic lights, aiming to balance the flow and reduce congestion.

Rational Functions

In the context of traffic, a rational function is utilized to understand how queue length and waiting time react to changes in arrival and gap rates. A rational function is simply a ratio of two polynomials. In this case, the functions take into account how many cars arrive and how often gaps occur on the main road.

The formula for the average queue length \( L \) is a classic example: \[ L = \frac{a^2}{s(s-a)} \] This illustrates a rational function because it divides two polynomials of arrival rate \( a \) and gap rate \( s \).

• If \( a \) (arrival rate) increases relative to \( s \) (gap rate), the denominator \( s-a \) shrinks, which drastically increases \( L \), implying longer queues.
• The form of these equations helps visualize how sensitive the system is to small changes, especially as the arrival rate nears the gap rate.

Rational functions are crucial in predicting and optimizing systems prone to congestion, such as traffic flow, by showing potential issues before they occur.

Waiting Time Calculation

Waiting time is an essential metric in traffic management, representing the average duration vehicles spend in a queue before merging into the main road. Calculating this time helps in assessing and improving the efficiency of traffic systems. In our exercise, the average waiting time \( w \) is given by:\[ w = \frac{a}{s(s-a)} \]
The formula helps traffic engineers determine how changes in arrival and gap rates impact waiting times. This is especially useful in planning signal timings or road expansions.

• To find the arrival rate \( a \) that corresponds to a specific waiting time, such as 2 minutes when \( s = 3 \), you plug these values into the equation to solve for \( a \).
• This results in setting up the equation: \[ 2 = \frac{a}{3(3-a)} \] Solving it can help find the critical rate \( a \) where significant waiting is reduced.

In this problem, the solution showed that an arrival rate of approximately \( 2.57 \) cars per minute leads to the desired average waiting time of 2 minutes. This shows the delicate balance required in traffic management to minimize delays.