A toll tunnel has decided to experiment with the use of a debit card for the collection of tolls. Initially, only one lane will be used. Cars are estimated to arrive at this experimental lane at the rate of 800 per hour. It will take exactly 2.5 seconds to verify the debit card.

In how much time would you expect the customer to wait in line, pay with the debit card, and leave?

Note: Round your intermediate calculations to 3 decimal places and final answer to 1 decimal place.



How many cars would you expect to see in the system?

Note: Round your intermediate calculations to 3 decimal places and final answer to 2 decimal places.



This is a classic queueing theory problem that can be modeled using the **M/M/1** queuing system, which is a system with a single server (one lane) and Poisson arrivals.

### Notations:
- \(\lambda\): Arrival rate (cars per second)
- \(\mu\): Service rate (cars per second)

#### Given:
- Cars arrive at a rate of 800 per hour, so:
  \[
  \lambda = \frac{800}{3600} \text{ cars per second} = 0.222 \text{ cars per second}
  \]
  
- The service time (time to verify the debit card) is 2.5 seconds, so the service rate is:
  \[
  \mu = \frac{1}{2.5} \text{ cars per second} = 0.4 \text{ cars per second}
  \]

### 1. Expected time in the system (\(W\)):

In an M/M/1 queue, the total time spent in the system (waiting time + service time) is given by:

\[
W = \frac{1}{\mu - \lambda}
\]

Substitute the values for \(\mu\) and \(\lambda\):

\[
W = \frac{1}{0.4 - 0.222} = \frac{1}{0.178} = 5.618 \text{ seconds}
\]

Thus, the expected time in the system (rounded to 1 decimal place) is:
\[
W = 5.6 \text{ seconds}
\]

### 2. Expected number of cars in the system (\(L\)):

The average number of cars in the system is given by the formula:

\[
L = \frac{\lambda}{\mu - \lambda}
\]

Substitute the values for \(\lambda\) and \(\mu\):

\[
L = \frac{0.222}{0.4 - 0.222} = \frac{0.222}{0.178} = 1.247
\]

Thus, the expected number of cars in the system (rounded to 2 decimal places) is:
\[
L = 1.25 \text{ cars}
\]

### Summary:
- The expected time a customer spends in the system (waiting + service time) is **5.6 seconds**.
- The expected number of cars in the system is **1.25 cars**.

Would you like more details or have any other questions? 

Here are 5 related questions you might consider:
1. How would the wait time change if more lanes were added for debit card verification?
2. What would happen to the system if the arrival rate increased to 1000 cars per hour?
3. How can we decrease the waiting time for customers in such a system?
4. What is the probability that a car will have to wait in line before being served?
5. How would introducing a priority lane for fast transactions affect the system?

**Tip:** Queueing theory helps optimize systems with multiple users and limited resources, such as toll booths, customer service centers, or traffic lights.

A toll tunnel has decided to experiment with the use of a debit card for the collection of tolls. Initially, only one lane will be used. Cars are estimated to arrive at this experimental lane at the rate of 800 per hour. It will take exactly 2.5 seconds to verify the debit card. In how much time would you expect the customer to wait in line, pay with the debit card, and leave? How many cars would you expect to see in the system?

Solve a queueing theory problem involving a toll tunnel that uses a debit card system for payments. Learn how to calculate the expected wait time (5.6 seconds) and the number of cars in the system (1.25 cars) using M/M/1 queueing model with Poisson arrivals and debit card verification taking 2.5 seconds per car. This is an advanced problem suitable for operations research students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation