The airport branch of a car rental company maintains a fleet...

The airport branch of a car rental company maintains a fleet of 50 SUVs. The interarrival time between requests for an SUV is 2.4 hours, on average, with a standard deviation of 2.4 hours. There is no indication of a systematic arrival pattern over the course of a day. Assume that, if all S UVs are rented, customers are willing to wait until there is an SUV available. An SUV is rented, on average, for 3 days, with a standard deviation of I day. a. What is the average number of SUVs parked in the company's lot? b. Through a marketing survey, the company has discovered that if it reduces its daily rental price of $$\$80$$ by $$\$25$$, the average demand would increase to 12 rental requests per day and the average rental duration will become 4 days. Is this price decrease warranted? Provide an analysis! c. What is the average time a customer has to wait to rent an SUV? Please use the initial parameters rather than the infonnation in (b). d. How would the waiting time change if the company decides to limit all SUV rentals to exactly 4 days? Assume that if such a restriction is imposed, the average interarrival time will increase to 3 hours, with the standard deviation changing to 3 hours.

The airport branch of a car rental company maintains a fleet of 50 SUVs. The interarrival time between requests for an SUV is 2.4 hours, on average, with a
standard deviation of 2.4 hours. There is no indication of a systematic arrival pattern over the course of a day. Assume that, if all S UVs are rented, customers are willing to wait until there is an SUV available. An SUV is rented, on average, for 3 days, with a standard deviation of I day.
a. What is the average number of SUVs parked in the company's lot?
b. Through a marketing survey, the company has discovered that if it reduces its daily rental price of $$\$80$$ by $$\$25$$, the average demand would increase to 12 rental requests per day and the average rental duration will become 4 days. Is this price decrease warranted? Provide an analysis!
c. What is the average time a customer has to wait to rent an SUV? Please use the initial parameters rather than the infonnation in (b).
d. How would the waiting time change if the company decides to limit all SUV rentals to exactly 4 days? Assume that if such a restriction is imposed, the average interarrival time will increase to 3 hours, with the standard deviation changing to 3 hours.

Key Concepts

Sensitivity and Impact Analysis

Sensitivity analysis evaluates how changes in input parameters, like arrival rates or service durations, affect overall system performance. This approach is instrumental in decision-making, allowing practitioners to assess the potential impacts of operational changes, pricing strategies, or policy modifications on system metrics like waiting times and utilization.

System Utilization and Inventory Management

System utilization refers to the proportion of time that service resources are actively in use. In contexts like inventory management, understanding utilization helps determine the balance between supply and demand. This analysis is critical for making informed decisions about resource allocation, ensuring that inventory levels are optimized to meet fluctuating customer demand.

Variability (Coefficient of Variation)

Variability, commonly measured by the coefficient of variation (the ratio of the standard deviation to the mean), reflects the degree of unpredictability in interarrival and service time distributions. High variability can lead to longer wait times and more pronounced queue buildups, making it essential to account for this factor in system design and analysis.

Queueing Model Selection

Selecting an appropriate queueing model (such as M/M/1, M/G/1, or G/G/1) is central to accurately representing and analyzing a service system’s dynamics. The choice of model depends on the specific characteristics of the arrival and service time distributions, as well as the number of service channels, and is crucial for obtaining valid performance estimates and operational insights.

Interarrival Time Distribution

The interarrival time distribution characterizes the time intervals between successive customer or request arrivals. Understanding this distribution is crucial because it helps determine the arrival rate and variability in the system, which directly impacts waiting times and the overall performance of queueing systems.

Little's Law

Little's Law is a fundamental theorem in queueing theory that relates the average number of items in a system (L) to the average arrival rate (?) and the average time an item spends in the system (W) via the equation L = ?W. It is widely used to analyze various operational systems by providing a straightforward way to estimate congestion and waiting times.

Queueing Theory

Queueing theory involves the study and modeling of waiting lines and service processes. It encompasses the analysis of arrival patterns, service mechanisms, and system performance metrics like wait times, queue lengths, and resource utilization. This field provides the mathematical frameworks needed to analyze and predict system behaviors in situations where resources are limited relative to demand.

Service Time Distribution

The service time distribution details the statistical behavior of the time required to complete a service. This concept is vital because variability in service times affects system performance metrics, including average waiting times and resource utilization. It helps in modeling and predicting how long entities remain in the system, especially when the service duration is variable.

Transcript

00:01 In this problem, we are asked to compare two different roots.

00:04 We're trying to figure out if root a really is five minutes faster than root b.

00:09 We're given data on both routes.

00:12 Route a, our in, the number of trials that we took, they said we tested it 20 times.

00:21 We found that the average length of time that root a took, or the mean of root a was 40.

00:27 And we found that the standard deviation for root a was three.

00:33 For root b, we also tested it 20 times.

00:36 So our n for root b is 20.

00:41 We found that the mean of all of those trials was 43 minutes.

00:49 And we found that the standard deviation for root b was two minutes.

00:54 So on first glance, it does look like root a is a little bit quicker.

00:59 But we need to construct a confidence interval to be sure.

01:03 So for a confidence interval, we are ultimately going to be using the equation.

01:08 Mean of a minus the mean of b plus or minus.

01:15 So if you look at this, this is just the difference between a and b.

01:19 Plus or minus are t statistic based on the degrees of freedom times standard error.

01:27 Now i know that this is a lot, but this actually isn't too.

01:29 Bad once we just start to plug in our values into our formulas.

01:34 To find the standard error when comparing two means, we need to take the square root of the standard deviation of the first squared divided by n of the first plus the standard deviation of the second squared divided by n of the second.

01:58 And then we're going to take the square root of all of that.

02:01 So if we were to plug in our values here, that gives us three squared over 20 plus two squared over 20, which reduces down to the square root of 13 over 20.

02:18 That's just 9 plus 4.

02:21 So 13 over 20, which equals 0 .8062.

02:26 About i rounded, but that's about our answer there.

02:31 To find the degrees of freedom by hand is quite complex.

02:36 So your book suggests that you use your calculator or the computer to calculate this...

Please give Ace some feedback