MATH 4431/ 6604 Probability Models

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MATH 4431/ 6604 Probability Models

Fall 2021

Assignment 2/ Oct. 8

Solve all problems 1-5 in part A (each problem is worth 10 points); in part B) read the article and conduct stochastic simulations (25 points). Show your complete work: Solutions without adequate accompanying work will receive no marks. Assignments must be submitted on Crowdmark on the link provided in eClass.

Due date: Oct. 22, 11:59PM

Part A)

1) a) Consider a workshop that has five machines and one repairperson. The machines work independently from one another: A machine fails after a time that is exponentially distributed with parameter μ =0.2 per hour. If a machine is down it is repaired in an Expo(λ) distributed amount of time, where λ = 0.5 per day. At most one machine can be under repair at any given time. Calculate the fraction of time that the repairperson is idle/ unoccupied in the long run.

b) Consider another workshop with two machines that operate independently from one another for Expo(μ) distributed times, each. A single repair facility exists with repair times that are Expo(λ) distributed.

b-i) What is the proportion of time in the long run that the probability that no machine is operating for four different designs: D1) μ = λ = 1; D2) μ = 3, λ = 1; D3) μ = 1; λ = 3; and D4) μ = λ = 2?

b-ii) Assume now that at most of the two machines can operate at any given time. What is the long-term proportion of time that no machine is working if μ = λ = 1?

2) Customers arrive at a service station according to a Poisson process with rate λ = 0.1 per min. Assume that service starts as soon as the third customer enters the queue.

    a) Calculate the expected time until the customer service starts at this station.

    b) Calculate the probability that no service is provided during the first hour.

3) Customers arrive at a service station according to a Poisson process with rate λ = 3 per hour. Denote by Nt the number of customers arriving until time t. Let Wn be the waiting time until the nth customer arrives, n = 1,2, … . Calculate the following conditional probabilities and conditional expectations, for any time points 

    a)

    b)

4) Consider the two-state discrete-time Markov chain with transition matrix

Suppose that is a Poisson process with rate λ > 0. Let a continuous-time stochastic process be given by

    a) Show that is a two-state birth-death Markov chain and determine its generator matrix Q.

    b) For the return time calculate the mean return time of the process to state 0,

    c) For the return time calculate the mean return time of the process to state 1,

5) Exercise 14.5 Textbook

Part B) Reading assignment and simulation (1a, 1b, 1c: 5 points each; 2) 10 points)

Access the article:

DJ Higham: Modeling and simulating chemical reactions, SIAM Review (2008), 50: 347-368.

1) Conduct stochastic simulations and generate graphs Figs 1-5 as in the article for the given Michaelis-Menten system with

    a) the set of kinetic parameter c(1), c(2), c(3) as in the text, and

    b) for the change in the unbinding/ reverse parameter c(2)=k2 to 1e-3.

    c) Compare a) and b) and interpret your results.

2) Conduct a stochastic simulation by adapting the Matlab code in the article for the following Lotka- Volterra system:

Denote by and the prey and predator population, respectively.

R1: prey birth; R2: predation

R3: prey death; initial populations sizes:

Graph and summarize the results of your stochastic simulation.