/*! 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 53 Write a program to approximate \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 53

Write a program to approximate \(m(t)\) for the interarrival distribution \(F * G\), where \(F\) is exponential with mean 1 and \(G\) is exponential with mean \(3 .\)

Short Answer

Expert verified
To approximate \(m(t)\) for the interarrival distribution \(F*G\), where \(F\) is exponential with mean 1 and \(G\) is exponential with mean 3, first import numpy and scipy.stats libraries in Python. Next, define the exponential distribution functions \(F\) and \(G\) with their respective parameters. Then, create a convolution function, \(F*G(x)\), using \(F\) and \(G\). Afterward, write a function to approximate the value of \(m(t)\) using the convolution function \(F*G(x)\). Finally, use the approximating function to find the value of \(m(t)\) for different values of \(t\). For example, to find the value of \(m(5)\), call the function as `approximate_mt(5, F, G)`.

Step by step solution

01

Import Required Libraries

Start by importing the required libraries, numpy and scipy.stats, in python to perform the calculations and write a program.
02

Define Exponential Distributions

Define the exponential distribution functions F and G with their respective parameters (means), using the scipy.stats package. ```python import numpy as np import scipy.stats F = scipy.stats.expon(scale=1) # Mean = 1 G = scipy.stats.expon(scale=3) # Mean = 3 ```
03

Define Convolution Function

Define the convolution function, \(F*G(x)\), using the F and G functions. ```python def convolution_FG(x, F, G, n_terms=50): t_values = np.linspace(0, x, n_terms+1) delta_t = t_values[1] - t_values[0] values = F.pdf(t_values[:-1]) * G.pdf(x - t_values[:-1]) * delta_t return np.sum(values) ```
04

Approximate m(t) using \(F*G(x)\)

Now, write the function to approximate the value of \(m(t)\) using the convolution function \(F*G(x)\). ```python def approximate_mt(t, F, G): if t <= 0: return 0 return (1 - G.cdf(t)) * convolution_FG(t, F, G) ```
05

Use the approximating function

Use the approximating function to find the value of \(m(t)\) for different values of t. For example, if you want to find the value of \(m(5)\), simply call the function: ```python t = 5 result = approximate_mt(t, F, G) print(f"m({t}) is approximately {result:.4f}") ``` This program can be easily used to approximate \(m(t)\) for the interarrival distribution \(F*G\) where F and G are exponential distributions with given parameters (mean 1 and mean 3, respectively).

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Distribution
The exponential distribution is a continuous probability distribution used to model the time between events in a process where events occur continuously and independently at a constant average rate.

### Key Features- **Memoryless Property**: The probability of an event occurring in the next instant is always the same, regardless of how much time has elapsed. - **Parameter (Rate or Mean)**: For a given rate \( \lambda \), the mean of the distribution is \( \frac{1}{\lambda} \). In the exercise, \( F \) has a mean of 1 (\( \lambda = 1 \)) and \( G \) has a mean of 3 (\( \lambda = \frac{1}{3} \)).### Applications- Commonly used to model **interarrival times** in queueing theory where events like phone calls to a call center occur at a constant average rate.- Useful in fields such as reliability engineering and survival analysis.

Understanding the exponential distribution is crucial in identifying the characteristics of random processes that follow this pattern.
Convolution of Distributions
Convolution refers to the operation used to determine the probability distribution of the sum of two independent random variables. This technique is particularly useful when dealing with independent random variables that follow different distributions, as in the case of two exponential distributions. ### Importance in Probability - When two interarrival times follow exponential distributions with different means, convolution helps to determine the overall distribution function. - The result of the convolution of two exponential distributions is a **gamma distribution** if both have the same rate, or a **hyperexponential distribution** if they have different rates, as in this exercise. ### In Practice - In programming, as shown in the solution, the convolution is computationally approximated using discrete summation. - This allows for the approximation of the combined effect of independent stochastic (random) processes. Understanding convolution is essential for modeling compound processes and identifying the resulting distribution function in complex scenarios like network traffic flows.
Interarrival Times
Interarrival times refer to the duration between consecutive events in a stochastic process, often modeled using exponential distributions.

### Significance- Key in the study of queueing theory and network traffic, where understanding the time between packet arrivals can help in optimizing and designing efficient systems.- An important metric for evaluating the performance and dynamics of systems that handle random events.### Mathematical Expression- Consider two independent interarrival times modeled by distributions \( F \) and \( G \). The resulting distribution of their sum can be determined using convolution. - Represents how long a system stays idle between events and can be used to evaluate resource utilization.### Real-World Applications- Widely used in telecommunications, where the focus is on analyzing and predicting call times and message arrivals. - Understanding and modeling interarrival times is critical in improving system performance and reliability.Grasping the concept of interarrival times helps in predicting the behavior of systems subject to random events, shaping strategies for their management and optimization.

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

Let \(\left\\{N_{1}(t), t \geqslant 0\right\\}\) and \(\left[N_{2}(t), t \geqslant 0\right\\}\) be independent renewal processes. Let \(N(t)=\) \(N_{1}(t)+N_{2}(t)\) (a) Are the interarrival times of \(\\{N(t), t \geqslant 0\\}\) independent? (b) Are they identically distributed? (c) Is \(\\{N(t), t \geqslant 0\\}\) a renewal process?

Compute the renewal function when the interarrival distribution \(F\) is such that $$ 1-F(t)=p e^{-\mu_{1} t}+(1-p) e^{-\mu_{2 t} t} $$

Wald's equation can also be proved by using renewal reward processes. Let \(N\) be a stopping time for the sequence of independent and identically distributed random variables \(X_{i}, i \geqslant 1\) (a) Let \(N_{1}=N\). Argue that the sequence of random variables \(X_{N_{1}+1}, X_{N_{1}+2}, \ldots\) is independent of \(X_{1}, \ldots, X_{N}\) and has the same distribution as the original sequence \(X_{i}, i \geqslant 1\) Now treat \(X_{N_{1}+1}, X_{N_{1}+2}, \ldots\) as a new sequence, and define a stopping time \(\mathrm{N}_{2}\) for this sequence that is defined exactly as \(\mathrm{N}_{1}\) is on the original sequence. (For instance, if \(N_{1}=\min \left(n: X_{n}>0\right\\}\), then \(\left.N_{2}=\min \left[n: X_{N_{1}+n}>0\right\\} .\right)\) Similarly, define a stopping time \(N_{3}\) on the sequence \(X_{N_{1}+N_{2}+1}, X_{N_{1}+N_{2}+2}, \ldots\) that is identically defined on this sequence as \(N_{1}\) is on the original sequence, and so on. (b) Is the reward process in which \(X_{i}\) is the reward earned during period \(i\) a renewal Ireward process? If so, what is the length of the successive cycles? (c) Derive an expression for the average reward per unit time. (d) Use the strong law of large numbers to derive a second expression for the average reward per unit time. (e) Conclude Wald's equation.

Consider a single-server bank for which customers arrive in accordance with a Poisson process with rate \(\lambda .\) If a customer will enter the bank only if the server is free when he arrives, and if the service time of a customer has the distribution \(G\), then what proportion of time is the server busy?

Consider a train station to which customers arrive in accordance with a Poisson process having rate \(\lambda\). A train is summoned whenever there are \(N\) customers waiting in the station, but it takes \(K\) units of time for the train to arrive at the station. When it arrives, it picks up all waiting customers. Assuming that the train station incurs a cost at a rate of \(n c\) per unit time whenever there are \(n\) customers present, find the long-run average cost.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

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.