[7] Suppose a car rental agency has three locations in NY: Downtown location (labeled D), Uptown location (labeled U) and a Brooklyn location (labeled B). The agency has a group of delivery drivers to serve all three locations. The agency's statistician has determined the following: Of the calls to the Downtown location, 30% are delivered in Downtown area, 30% are delivered in the Uptown, and 40% are delivered in Brooklyn. . Of the calls to the Uptown location, 40% are delivered in Downtown area, 40% are delivered in the Uptown, and 20% are delivered in Brooklyn. Of the calls to the Brooklyn location, 50% are delivered in Downtown area, 30% are delivered in the Uptown, and 20% are delivered in Brooklyn. After making a delivery, a driver goes to the nearest location to make the next delivery. This way, the location of a specific driver is determined only by his or her previous location. a) Write down the state vector and transition matrix. b) Is Theorem 3 applicable to this transition matrix? c) On average (i.e. in the long run), what percentage of cars stay in Downtown? hint. In this example, the powers of P matrix converges to the desired form pretty quickly so you can even compute these powers manually or use Excel use definition or of stationarity as well)
[7] Suppose a car rental agency has three locations in NY: Downtown location (labeled D), Uptown location (labeled U) and a Brooklyn location (labeled B). The agency has a group of delivery drivers to serve all three locations. The agency's statistician has determined the following: Of the calls to the Downtown location, 30% are delivered in Downtown area, 30% are delivered in the Uptown, and 40% are delivered in Brooklyn. . Of the calls to the Uptown location, 40% are delivered in Downtown area, 40% are delivered in the Uptown, and 20% are delivered in Brooklyn. Of the calls to the Brooklyn location, 50% are delivered in Downtown area, 30% are delivered in the Uptown, and 20% are delivered in Brooklyn. After making a delivery, a driver goes to the nearest location to make the next delivery. This way, the location of a specific driver is determined only by his or her previous location. a) Write down the state vector and transition matrix. b) Is Theorem 3 applicable to this transition matrix? c) On average (i.e. in the long run), what percentage of cars stay in Downtown? hint. In this example, the powers of P matrix converges to the desired form pretty quickly so you can even compute these powers manually or use Excel use definition or of stationarity as well)
A First Course in Probability (10th Edition)
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Chapter1: Combinatorial Analysis
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Show full answers and steps to this exercise. Please explain how you get to the answers using Markov Chain Theorem, especially part b) using theorem 3 of Markov Chain
![[7] Suppose a car rental agency has three locations in NY: Downtown location (labeled
D), Uptown location (labeled U) and a Brooklyn location (labeled B). The agency
has a group of delivery drivers to serve all three locations. The agency's statistician
has determined the following:
Of the calls to the Downtown location, 30% are delivered in Downtown area,
30% are delivered in the Uptown, and 40% are delivered in Brooklyn.
. Of the calls to the Uptown location, 40% are delivered in Downtown area,
40% are delivered in the Uptown, and 20% are delivered in Brooklyn.
. Of the calls to the Brooklyn location, 50% are delivered in Downtown area,
30% are delivered in the Uptown, and 20% are delivered in Brooklyn.
After making a delivery, a driver goes to the nearest location to make the next
delivery. This way, the location of a specific driver is determined only by his or
her previous location.
a) Write down the state vector and transition matrix.
b) Is Theorem 3 applicable to this transition matrix?
c) On average (i.e. in the long run), what percentage of cars stay in Downtown?
hint: In this example, the powers of P matrix converges to the desired form
pretty quickly so you can even compute these powers manually or use Excel
use definition
or
of stationarity as well)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa3574180-5be7-46af-9f18-c1fbb9ec682c%2F8b1bdd73-4f01-428a-b48e-8f06b642fa30%2F6vxb5dr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:[7] Suppose a car rental agency has three locations in NY: Downtown location (labeled
D), Uptown location (labeled U) and a Brooklyn location (labeled B). The agency
has a group of delivery drivers to serve all three locations. The agency's statistician
has determined the following:
Of the calls to the Downtown location, 30% are delivered in Downtown area,
30% are delivered in the Uptown, and 40% are delivered in Brooklyn.
. Of the calls to the Uptown location, 40% are delivered in Downtown area,
40% are delivered in the Uptown, and 20% are delivered in Brooklyn.
. Of the calls to the Brooklyn location, 50% are delivered in Downtown area,
30% are delivered in the Uptown, and 20% are delivered in Brooklyn.
After making a delivery, a driver goes to the nearest location to make the next
delivery. This way, the location of a specific driver is determined only by his or
her previous location.
a) Write down the state vector and transition matrix.
b) Is Theorem 3 applicable to this transition matrix?
c) On average (i.e. in the long run), what percentage of cars stay in Downtown?
hint: In this example, the powers of P matrix converges to the desired form
pretty quickly so you can even compute these powers manually or use Excel
use definition
or
of stationarity as well)
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