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Copyright © Peter Glynn All rights reserved. MS&E 221 Spring 2021 Problem Session on Markov Jump Process MS&E 221 May 25, 2021 Question 2.1 (Light) : Let N 1 = ( N 1 ( t ) : t 0) be a Poisson process with rate λ 1 and N 2 = ( N 2 ( t ) : t 0) be a Poisson process with rate λ 2 . Suppose that a light can be switched from dim to bright and from bright to dim, and it is initially dim. Whenever an event from N 1 occurs when the light is dim, the light switches to bright; whenever an event from N 1 occurs when the light is bright, it stays bright; whenever an event from N 2 occurs when the light is dim, it stays dim; whenever an event from N 2 occurs when the light is bright, it switches to dim. (a) Model the situation using a 2-state Markov jump process X = ( X ( t ) : t 0). (b) What is the long-run proportion of time when the light is bright?

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