/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 24 Give an efficient method for sim... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 24

Give an efficient method for simulating a nonhomogeneous Poisson process with intensity function $$ \lambda(t)=b+\frac{1}{t+a}, \quad t \geqslant 0 $$

Short Answer

Expert verified
To efficiently simulate a nonhomogeneous Poisson process with intensity function \(\lambda(t) = b + \frac{1}{t+a}\), follow these steps: 1. Calculate the maximum intensity value: \(\lambda_{max} = \lambda(0) = b + \frac{1}{a}\). 2. Simulate a homogeneous Poisson process with intensity \(\lambda_{max}\) using standard methods, such as the inter-arrival times having an exponential distribution with mean \(1/\lambda_{max}\). 3. Perform the thinning process by accepting or rejecting each point \(t_k\) in the homogeneous process, based on a randomly generated number \(u_k\), using the rule: \(u_k \leq \frac{\lambda(t_k)}{\lambda_{max}}\). 4. Collect the accepted points from the thinning process, which will represent the simulated nonhomogeneous Poisson process.

Step by step solution

01

Identify the maximum intensity function value

First, we need to find the maximum value of the given intensity function over the interval of interest. This will be used to scale our homogeneous Poisson process to ensure we accept/reject points correctly. In our case, we have: $$ \lambda(t) = b + \frac{1}{t+a}, \quad t \geqslant 0 $$ For this exercise, we will assume that \(b>0\) and \(a>0\). The maximum value of the intensity function can be found by taking the first derivative and setting it to zero to find the critical points. Then, we will evaluate the intensity function at the critical points and boundaries to determine the maximum value.
02

First derivative

To find the maximum value of the intensity function, we need to find the first derivative of the function and set it to zero to find the critical points: $$ \frac{d}{dt} \bigg( b + \frac{1}{t+a} \bigg) = -\frac{1}{(t+a)^2} $$ Since the first derivative is always negative, the intensity function is strictly decreasing. Therefore, its maximum value occurs at the lower-bound of the interval (\(t=0\)).
03

Calculate maximum intensity value

Evaluate the intensity function at the lower bound of the interval to find the maximum value: $$ \lambda_{max} = \lambda(0) = b + \frac{1}{a} $$
04

Simulate a homogeneous Poisson process

Next, we simulate a homogeneous Poisson process with intensity \(\lambda_{max}\). This can be done using standard methods, such as the inter-arrival times having an exponential distribution with mean \(1/\lambda_{max}\). Generate points \(t_1, t_2, \dots\) in the homogeneous Poisson process until the last point exceeds the time horizon of interest.
05

Thinning the homogeneous Poisson process

Now, we will thin the homogeneous Poisson process by accepting or rejecting each point \(t_k\) in the process. To do this, generate a random number \(u_k\) between 0 and 1 for each point \(t_k\). If: $$ u_k \leq \frac{\lambda(t_k)}{\lambda_{max}} $$ we accept the point \(t_k\) in the nonhomogeneous Poisson process. Otherwise, we reject it. Repeat this process for all generated points in the homogeneous Poisson process.
06

Combine accepted points

Finally, collect all the accepted points from Step 5. These accepted points represent the occurrences of events in our simulated nonhomogeneous Poisson process. The generated nonhomogeneous Poisson process with intensity function \(\lambda(t) = b + \frac{1}{t+a}\) is now successfully simulated using the thinning method.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

If \(0 \leqslant X \leqslant a\), show that (a) \(E\left[X^{2}\right] \leqslant a E[X]\) (b) \(\operatorname{Var}(X) \leqslant E[X](a-E[X])\) (c) \(\operatorname{Var}(X) \leqslant a^{2} / 4\).

The Discrete Rejection Metbod: Suppose we want to simulate \(X\) having probability mass function \(P\\{X=i\\}=P_{i}, i=1, \ldots, n\) and suppose we can easily simulate from the probability mass function \(Q_{i}, \sum_{i} Q_{i}=1, Q_{i} \geqslant 0 .\) Let \(C\) be such that \(P_{i} \leqslant C Q_{i}, i=1, \ldots, n .\) Show that the following algorithm generates the desired random variable: Step 1: Generate \(Y\) having mass function \(Q\) and \(U\) an independent random number. Step \(2:\) If \(U \leqslant P_{Y} / C Q_{Y}\), set \(X=Y .\) Otherwise return to step \(1 .\)

Let \(X_{1}, \ldots, X_{n}\) be independent exponential random variables each having rate 1 . Set $$ \begin{aligned} &W_{1}=X_{1} / n \\ &W_{i}=W_{i-1}+\frac{X_{i}}{n-i+1}, \quad i=2, \ldots, n \end{aligned} $$ Explain why \(W_{1}, \ldots, W_{n}\) has the same joint distribution as the order statistics of a sample of \(n\) exponentials each having rate 1 .

Consider the following procedure for randomly choosing a subset of size \(k\) from the numbers \(1,2, \ldots, n:\) Fix \(p\) and generate the first \(n\) time units of a renewal process whose interarrival distribution is geometric with mean \(1 / p-\) that is, \(P\\{\) interarrival time \(=k\\}=p(1-p)^{k-1}, k=1,2, \ldots .\) Suppose events occur at times \(i_{1}k\) then randomly choose (by some method) a subset of size \(k\) from \(i_{1}, \ldots, i_{m}\) and then stop. If \(m

Consider a queueing system in which each service time, independent of the past, has mean \(\mu\). Let \(W_{n}\) and \(D_{n}\) denote, respectively, the amounts of time customer \(n\) spends in the system and in queue. Hence, \(D_{n}=W_{n}-S_{n}\) where \(S_{n}\) is the service time of customer \(n\). Therefore, $$ E\left[D_{n}\right]=E\left[W_{n}\right]-\mu $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.