Answer all

Consider a birth and death process with birth rates ); = (i + 1)2, i 2 0, and death rates u; = iu, i 2 0. (a) Determine the expected time to go from state 0 to state 4. (b) Determine the expected time to go from state 2 to state 5. (c) Determine the variances in parts (a) and (b).Problem 2: 15 points = [5 + 5 + 5] Consider a birth-and-death process, X - {X(t) : t 2 0}, with instant rates Xx = A and uk = / per hour. You are given information about the expected departure times from states 0 and 1 as follows. . E [So |X(0) = 0] = 1 hour . E[Si |X(0) = 1] = 12 minutes 1. Determine parameters A and / 2. Derive the limiting distribution, A = lim P[X(() - *] for any * 2 0 3. Find the limiting expectation, lim E[X(t)] SolutionNotation and Important Facts In this assignment, birth and death processes are presented. Please, read Sections 6.3 - 6.4 from textbook and Notes, part 6. General Techniques To determine the stationary distribution for a birth-and-death process with infinitesimal parameters, {xx : k 20} and {uk : k 2 1}, use the procedure described in the textbooks, . Set 60 = 1 and then evaluate k-1 HK i=1 for k 2 1. . If the series, A = ER, Or, converges (that is A