Question

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

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

Import Required Libraries

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

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 ```

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) ```

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) ```

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).