Question

Consider a renewal process with underlying distribution function \(F(x)\). Let \(\mathbb{W}\) be the time when the interval duration from the preceding renewal event first exceeds \(\xi>0\) (a fixed constant). Determine an integral equation satisfied by $$ V(t)=\operatorname{Pr}\\{W \leq t\\} $$ Calculate \(E[W] .\) (Assume an event occurs at time \(t=0 .\) )

Short answer

The short answer to the problem is: To find an integral equation satisfied by \(V(t)\), we derive the following equation: \[V(t) = F(\xi) \frac{V(t)}{F(\xi)} + (1-F(\xi)) \int_{\xi}^{\infty} \frac{V(t-x)}{1-F(\xi)} f(x) dx,\] where \(F(x)\) is the underlying distribution function of inter-arrival times, and \(f(x) = F'(x)\) is its probability density function. The expected value of \(W\) can be computed by evaluating the following integral with the given underlying distribution functions: \[E[W] = \int_{0}^{\infty} \left[ F(\xi) \frac{V(t)}{F(\xi)} + (1-F(\xi)) \int_{\xi}^{\infty} \frac{V(t-x)}{1-F(\xi)} f(x) dx \right] dt.\]

Step-by-step solution

Define the renewal process and the underlying distribution function F(x)

A renewal process is a sequence of events occurring in time, such that the inter-arrival times between successive events are independent and identically distributed random variables. In this problem, the underlying distribution function of inter-arrival times is given by \(F(x)\). The distribution function \(F(x)\) gives the probability that the inter-arrival time is less than or equal to x, i.e., \(F(x) = \operatorname{Pr}\left\{X \leq x\right\}\), where \(X\) denotes a random variable for the inter-arrival time.

Write down the definition of the probability function V(t)

In this problem, we have a fixed constant \(\xi > 0\) and we need to find the probability function \(V(t)\), which represents the probability that the interval duration from the preceding renewal event first exceeds \(\xi\) by time \(t\). In other words, \(V(t) = \operatorname{Pr}\{W \leq t\}\).

Derive an integral equation for V(t)

To derive an integral equation for \(V(t)\), let us consider the probability that the interval duration from the preceding renewal event first exceeds \(\xi\) just before time \(t\). Since the inter-arrival times are independent, we can write the following: \begin{align*} V(t) &= \operatorname{Pr}\{ W \le t \} \\ &= \operatorname{Pr}\{ W \le t | X \le \xi \} \operatorname{Pr}\{ X \le \xi \} + \operatorname{Pr}\{ W \le t | X > \xi \} \operatorname{Pr}\{ X > \xi \} \\ &= F(\xi) \operatorname{Pr}\{W \leq t | X \leq \xi \} + (1-F(\xi)) \operatorname{Pr}\{W \leq t | X>\xi \}, \end{align*} where \(X\) is a random variable representing the inter-arrival time. Now, let's analyze the two conditional probabilities on the right-hand side of the equation. If \(X \leq \xi\), it means that the interval duration from the preceding renewal event exceeds \(\xi\) within the given interval, so the probability becomes \(\operatorname{Pr}\{W \leq t | X \leq \xi \} = \frac{V(t)}{F(\xi)}\). If \(X > \xi\), it means that the interval duration from the preceding renewal event exceeds \(\xi\) after time \(\xi\), so we can write the probability as \(\operatorname{Pr}\{W \leq t | X>\xi \} = \operatorname{Pr}\{W-\xi \leq t-X | X>\xi \}\). Using the total probability theorem, we can write the following integral equation: \[ \operatorname{Pr}\{W-\xi \leq t-X | X>\xi \} = \int_{\xi}^{\infty} \frac{V(t-x)}{1-F(\xi)} f(x) dx, \] where \(f(x) = F'(x)\) is the probability density function of \(X\). Substituting this back into the equation for \(V(t)\), we obtain: \[ V(t) = F(\xi) \frac{V(t)}{F(\xi)} + (1-F(\xi)) \int_{\xi}^{\infty} \frac{V(t-x)}{1-F(\xi)} f(x) dx. \]

Compute the expected value of W

With the derived equation for \(V(t)\), we can compute the expected value of \(W\) using the following formula: \[ E[W] = \int_{0}^{\infty} V(t) dt. \] Now, substitute the derived integral equation for \(V(t)\) into this formula: \[ E[W] = \int_{0}^{\infty} \left[ F(\xi) \frac{V(t)}{F(\xi)} + (1-F(\xi)) \int_{\xi}^{\infty} \frac{V(t-x)}{1-F(\xi)} f(x) dx \right] dt. \] To compute the expected value of \(W\), you would need to know the distribution function \(F(x)\) and its probability density function \(f(x)\). Once these functions are given, you can evaluate the integral to obtain the expected value of \(W\).

Key concepts

Renewal Theory

Renewal theory is a branch of probability theory that deals with the occurrence of events over time, where each event marks a renewal of the process. It is often used to model scenarios like the breakdown and repair of machines or the arrival of customers at a store.

In a renewal process, times between consecutive events are considered to be independent and identically distributed random variables with a certain distribution function. One of the fundamental questions in renewal theory is understanding how often renewals occur over a long period. This is characterized by the renewal function, which counts the number of renewals up to time t. Renewal theory helps in calculating vital quantities like the expected number of renewals or the waiting time until the next renewal, which is directly related to the exercise at hand.

Analyses within renewal theory often involve integral equations, which are equations containing an unknown function under an integral sign. Solving these equations is essential for obtaining the distribution function that dictates the renewal process, as demonstrated in the step by step solution provided for this exercise.

Distribution Function

The distribution function, also known as the cumulative distribution function (CDF), is a significant concept in statistics and probability theory. It describes the probability that a random variable X will take on a value less than or equal to x. Mathematically, it is expressed as F(x) = Pr{X ≤ x}.

This CDF rises monotonically from 0 to 1, and it encompasses all the possible outcomes of the random variable. As seen in the original exercise, the distribution function F(x) is pivotal in defining the process and deriving the probability function V(t) in the context of the renewal process. The shape of the CDF can tell us a lot about the nature of the random variable, such as where its values are concentrated and how spread out they are.

Expected Value

The expected value, or mean, of a random variable is a measure of its central tendency, giving us an average outcome if an experiment is repeated many times. For a continuous random variable, it is calculated using the formula E[X] = ∫ x f(x) dx over all possible values of x, where f(x) is the probability density function associated with the random variable X.

In the context of renewal theory, the expected value could represent the average time to the next renewal or the average duration between renewals. For instance, the expected value E[W] sought in the exercise is essentially the average time when the interval duration from the preceding renewal first exceeds a certain threshold ξ. This measure is very useful because it gives us a single summary statistic that characterizes the average behavior of the renewal process over time.

Integral Equation

An integral equation is a relationship involving an unknown function, its derivatives, and integrals. Solving an integral equation is often key in applications involving differential equations, renewal theory, and physics.

In our exercise, the integral equation emerges as a way to express V(t), the probability that the time W from a preceding event first exceeds ξ by time t. The equation is built upon combining the renewal process’s distribution function F(x) and probability density function f(x) to model the behavior of the process. These equations are usually solved by applying analytical methods or numerical techniques for more complex cases where an analytical solution is not straightforward.

Probability Density Function

The probability density function (PDF) is another fundamental concept in probability theory and statistics. It is a function that describes the likelihood of a continuous random variable to take on a particular value. Unlike the distribution function which provides cumulative probabilities, the PDF gives the probability per unit interval and is defined such that the area under the curve of the PDF over a range of possible values equals the probability of the events within that range.

In mathematical notation, if F(x) is the distribution function for a random variable X, then f(x) = F'(x) is the corresponding PDF for X. In the situation presented in our renewal process exercise, the PDF helps derive the more complex integral equation form of V(t) necessary to solve for the expected value E[W]. PDFs are essential for modeling the underlying behavior of random variables in a wide variety of disciplines, from engineering to finance.