hw21 - p = 0 . 8, = 0 . 3 and = 0 . 4. (iii) (2 pts.)...

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ENEE 324 ASSIGNMENT 21 Due Thu 12/03 Problem 21A Consider a service model where each day (e.g., between 9 AM and 5 PM), a server processes no more than two jobs. The server’s buffer can hold up to two jobs. Jobs arrive during the night (e.g., between 12 AM and 6 AM) and are accepted into the buffer on a first-come, first-served basis; jobs not accepted into the buffer are routed to other servers. Specifically: The processor processes each job in the buffer with probability p ; and if there are two jobs in the buffer, these are processed independently of each other. Jobs not processed are kept in the buffer overnight. The arrival of new jobs is independent of all processor functions, and the number U n of arrivals on Day n (again, between 12 AM and 6 AM) is an IID sequence such that P [ U n = 0] = α , P [ U n = 1] = β and P [ U n 2] = 1 - α - β Let X n be the number of jobs in the buffer at 5:01 PM on Day n . (i) (1 pt.) What is the state space S in this case? (ii) (9 pts.) Determine the transition probabilities P [ X n +1 = j | X n = i ] for every ( i,j ) ∈ S 2 . In what follows, assume
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Unformatted text preview: p = 0 . 8, = 0 . 3 and = 0 . 4. (iii) (2 pts.) Display the transition probability matrix for { X n } . (iv) (4 pts.) Determine P [ X n +2 = 1 | X n = 1]. (v) (4 pts.) Determine P [ X n +2 = 0 | X n = 2]. Problem 21B Consider the Markov chain { X n } with state space { 1 , 2 , 3 , 4 , 5 , 6 } and P = 1 / 3 1 / 6 1 / 2 1 / 6 5 / 6 3 / 8 5 / 8 1 / 4 1 / 4 1 / 2 2 / 3 1 / 3 3 / 4 1 / 4 (i) (5 pts.) Draw the state transition graph. Classify each state as recurrent or transient and identify all classes. (ii) (2 pts.) What is the conditional probability that state 5 occurs before state 3, given that X = 1? (iii) (7 pts.) Determine the stationary distribution of each recurrent class. (iv) (6 pts.) Determine the limiting values of P [ X n = 4 | X = 3] , P [ X n = 4 | X = 2] and P [ X n = 4 | X = 1] as n ....
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