Consider a Markov chain {an = 0, 1, - - -} on the state space S = {0, 1, 2}. Suppose that the Markov chain has the transition matrix 011 P=100 it' 1....
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IMG-20210403-WA0024.jpgIMG-20210403-WA0021.jpgIMG-20210330-WA0018.jpgIMG-20210330-WA0017.jpgIdentify a problem in an IT-related system where statistical methods could be used to analyze, evaluate, and recommend performance improvements for the system.


Summarize the problem, and discuss how you would use the discrete mathematics, probability, and statistical methods from this course to analyze the associated data, present the information, and draw inferences to develop a solution that would improve system performance. Explain your reasoning.

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Problem 2: 10 points Consider a birth-and-death process, X = {X(t) : t 2 0} , associated with the service line that consists of N = 10 servers. When all 10 servers are occupied, the new request is refused and not coming into the service line. As there are k < 10 ongoing service requests, the next incoming request is going to occupy either of the vacant servers. This situation is described with the possible state space, S = {0, 1, 2, ... , 10} and the infinitesimal transition probabilities (as A 4 0) are described as follows: P [X(t+A) - X(t) = 1 X(t) =k] = >k =3. (10 -k) for 0 < < < 10, and Ak =0 for k > 10, and P [X(t + 4) - X(t) = -1\X(t) = k] =/k = 2 . k for 0 < k < 10. 1. Derive the stationary distribution of the number of ongoing service requests, lim P [X(t) = k] , 1-too for 0 < k < 10. 2. Determine the limiting probability of the idle state, I = 0, that is: lim P [X(t) = 0] 3. Determine the limiting probability of having all servers occupied: lim P [X(t) = 10] 1-+ 00

Starting with the R Punkting for the simulation of the junegration - death process, modify it to include "birth's in addition to immigrations and death, Use your modified function to simulate a bug realisation Prom the birth-immigration- death process where the birth, death , and immigration rates are all one, starting from an initial condition of zero.

Consider a Markov chain {Km 11 = O, 1, - - -} on the state space S = {0,1,2}. Suppose that the Markov chain has the transition matrix A A .2. 10 10 10 P = A :4. A 10 10 10 i .2. .4_ 1o 10 10 1. Show that the Markov chain has a unique stationary mass. 2. Let h denote the stationary mass of the Markov chain. Find h(x) for all a: E S. 3. Show that the Markov chain has the steady state mass. 4. Let h" denote the steady state mass of the Markov chain. Find h*(x) for all a: E S.

Consider a Markov chain {an = 0, 1, - - -} on the state space S = {0, 1, 2}. Suppose that the Markov chain has the transition matrix 011 P=100 it" 1. Show that the state space is irreducible. 2. Show that the Markov chain is periodic. Find the period of the Markov chain. 3. Let h denote a stationary mass of the Markov chain. Find h(:r) for all a: E S.

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