e with mean 1/μ to process a job. Arriving jobs join a single line and are processed by the next availa he system at time t. What must be true of μ for X to be a positive recurrent Markov chain? μ >1/2 H>2 ≥ 1/3 >1/3 None of the other choices.
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- ') Consider an M/M/S/S queueing system where customers arrive from a fixed population base of S. This system can be modelled as a Markov chain with the following parameters: Xk= (5 -kJ) Ma = kM Draw the state-transition diagram and show that the probability of an empty system is. (1-e. Hence find the expected number of customers in the system.How many states have a nonsimplified Markov chain for a system consistingof n components? Assume that each component has two states: operationaland failed.Consider the illustration: Suppose that a group of robots is traversing this maze. At each step, each robot will choose a path and move along it, where it is equally likely to select each available path and cannot choose to stay where it is. (At the end of each step, each robot will be in one of the four numbered rooms.) Part (a): Construct the appropriate transition matrix for the Markov chain modeling this scenario. Part (b): Find the steady state probability vector.
- Consider the Markov chain with three states,S={1,2,3}, that has the following transition matrix P= Draw the state transition diagram for this chain. If we know P(X1=1) =P(X1=2) =1/4, find P(X1=3, X2=2,X3=1)For systems with n components, the number of states in a non-simplified Markov chain is as follows: Each part may be in one of two states: working or broken.Jobs arrive at processing station A according to a Poisson process with rate 1 per minute. If the single server is busy, they wait for the single server to become available. After service, with probability 1/2 the job departs the system; otherwise it moves on to station B, which operates like station A except that after service all jobs depart the system. The service times are independent and exponentially distributed with mean 1/2 minute at each of the stations. What is the average number of jobs at station B (waiting or in service)? Type your answer.
- 3. Consider the Markov chain with three states, S={1,2,3}, that has the following transition matrix P= a. Draw the state transition diagram for this chain. (10 marks) b. If we know P(Xi=1)=P(X1=2)=1/4, find P(X1=3, X2=2,X3=1) (10 marks)The following is an illustration of the one-step transition probabilities for a renewal Markov chain with no recurrence.In the Erdös-Rényi random network model, suppose N=101 and p=1/20, that is, there are 101 vertices, and every pair of vertices has a probability of 1/20 of being connected by an edge. For the network model given what is the probability that a network generated with those parameters has exactly 400 edges? No need to give the decimal value, the mathematical expression will suffice
- Computer Science Consider the demand and supply system p = ad +bdqd +ud p = as +bsqs +us with the equilibrium condition qd = qs = q. The parameters bd and bs are −2 and 1.5, respectively. Find the parameters ad and as are such that q = 5 and p = 10. Throughout this question use N = 100, and set the random seed to 14022022. The variable ud has a standard deviation of 3. All randomly generated variables have a mean of zero and are normally distributed unless something else is specified. All Monte Carlo studies should be done with 10,000 repetitions. Part a. Illustrate the supply and demand curves in a graph, together with a sample of simulated price and quantity data. Provide an additional graph where you have included the OLS estimated line of demand equation above. Are your estimates close to the true values of ad and bd ? Solve the question in Python language on Jupyter notebook.2.2 Let X. X₁.... be a Markov chain with transition matrix 1 2 3 10 1/2 1/2) 2 1 0 0 3 1/3 1/3 1/3) and initial distribution a = (1/2.0, 1/2). Find the following: (a) P(X₂ = 1|X₁ = 3) (b) P(X₁ = 3, X₂ = 1) - (c) P(X₁ = 31X₂ = 1) (d) P(Xo = 1|X₁ = 3.X4 = 1.X₂ = 2)Let A be a random variable with unified distribution between -2, 2 ( A~U[-2,2] ) and b is a constant. The stochastic process X(t) is defined as : X(t) = At + b Find the expected value and autocorrelation of X(t).