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A monkey climbs a ladder with n bars, so they jump to the next bar at rate A but fall at the bottom of the ladder independently at rate μ. Once at the top, the monkey can only fall with rate μ. a. Consider the state space S = {0, 1,..., n} with the bottom to be at 0 and the top of the ladder at n. For n = 4 write the matrix Q (with parameters qij) of the CTMC associated with the position of the monkey. b. In the general case, find the fraction of time the monkey spends at the bottom of the ladder, as a function of a = X/(X + μ).

A monkey climbs a ladder with n bars, so they jump to the next bar at rate A but fall at the bottom of the ladder independently at rate μ. Once at the top, the monkey can only fall with rate μ. a. Consider the state space S = {0, 1,..., n} with the bottom to be at 0 and the top of the ladder at n. For n = 4 write the matrix Q (with parameters qij) of the CTMC associated with the position of the monkey. b. In the general case, find the fraction of time the monkey spends at the bottom of the ladder, as a function of a = X/(X + μ).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.7: The Inverse Of A Matrix
Problem 30E
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Problem 1 A monkey climbs a ladder with n bars, so they jump to the next bar at rate X but fall at the bottom of the ladder independently at rate μ. Once at the top, the monkey can only fall with rate . a. Consider the state space S = {0, 1,..., n} with the bottom to be at 0 and the top of the ladder at n. For n = 4 write the matrix Q (with parameters qij) of the CTMC associated with the position of the monkey. b. In the general case, find the fraction of time the monkey spends at the bottom of the ladder, as a function of a = X/(X + μ).
Transcribed Image Text:Problem 1 A monkey climbs a ladder with n bars, so they jump to the next bar at rate X but fall at the bottom of the ladder independently at rate μ. Once at the top, the monkey can only fall with rate . a. Consider the state space S = {0, 1,..., n} with the bottom to be at 0 and the top of the ladder at n. For n = 4 write the matrix Q (with parameters qij) of the CTMC associated with the position of the monkey. b. In the general case, find the fraction of time the monkey spends at the bottom of the ladder, as a function of a = X/(X + μ).
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