Question

Find \(\operatorname{Pr}\\{N(t) \geq k\\}\) in a renewal process having lifetime density $$ f(x)=\left\\{\begin{array}{lll} \rho e^{-\rho(x-\delta)}, & \text { for } & x>\delta \\ 0, & \text { for } & x \leq \delta \end{array}\right. $$ where \(\delta>0\) is fixed.

Short answer

The probability that at least \(k\) renewals occur by time \(t\) in the given renewal process is: $$ \operatorname{Pr}\{N(t) \geq k\} = 1 - \sum_{n=0}^{k-1} \int_{0}^t \operatorname{Pr}\{N(u)=n\} f(t-u) du $$ where \(f(x)\) is the lifetime density function and \(F(x)\) is its corresponding cumulative distribution function.

Step-by-step solution

Determine the cdf of the given lifetime density.

To find the cumulative distribution function (cdf) of the given density function, we need to integrate it. The cdf \(F(x)\) is given by: $$ F(x) = \int f(x) dx $$ Using the given definition of the lifetime density function, we find that when \(x>\delta\): $$ F(x) = \int_{\delta}^x \rho e^{-\rho(x-\delta)} dx. $$ And, for \(x\leq\delta\), \(F(x)=0\).

Calculate the cdf for \(x>\delta\).

We will now perform the integration for \(x>\delta\): \[ F(x) = \rho \int_{\delta}^x e^{-\rho (x-\delta)} dx \\ = -\left[e^{-\rho(x-\delta)}\right]_{\delta}^x \\ = -(e^{-\rho(x-\delta)} - e^0) \\ = 1 - e^{-\rho(x-\delta)}. \] So, the complete cdf is: $$ F(x)=\left\\{\begin{array}{lll} 1 - e^{-\rho(x-\delta)}, & \text { for } & x>\delta \\\ 0, & \text { for } & x \leq \delta \end{array}\right. $$

Calculate the probability that a single renewal occurs by time \(t\).

We need to find the probability that a single renewal occurs by time \(t\). This probability can be found using the cdf: $$ \operatorname{Pr}\{N(t)=1\} = F(t) = \left\\{ \begin{array}{lll} 1-e^{-\rho(t-\delta)}, & \text { for } & t>\delta \\\ 0, & \text { for } & t \leq \delta \end{array}\right. $$

Find the probability that exactly \(k\) renewals occur by time \(t\).

We will use the Renewal Equation to find the probability that exactly \(k\) renewals occur by time \(t\). The Renewal Equation is given by: $$ \operatorname{Pr}\{N(t)=k\}=\int_{0}^t \operatorname{Pr}\{N(u)=k-1\} f(t-u) du. $$

Find the probability that at least \(k\) renewals occur by time \(t\).

We will now use the probabilities \(\operatorname{Pr}\{N(t)=k\}\) calculated in Step 4 to find the probability that at least \(k\) renewals occur by time \(t\). The required probability is given by: $$ \operatorname{Pr}\{N(t) \geq k\} = 1 - \sum_{n=0}^{k-1} \operatorname{Pr}\{N(t)=n\}. $$ In this step, we need to use the Renewal Equation and cdf calculated in the previous steps to find the probability for each value of \(n\) and sum them up to find the required probability.

Key concepts

Understanding Lifetime Density in Renewal Processes

In the context of renewal processes, the term lifetime density refers to a probability density function that describes the time between consecutive renewals or events. Specifically, it denotes the likelihood of a renewal event occurring at any given point within a specified time frame.

Like the example in the original exercise, lifetime density functions can take on various forms, but they often involve exponential distributions, especially when describing the time until the next event in memoryless processes, such as in certain types of equipment failures or service times. The notation used, as in the step-by-step solution, is a function denoted by f(x), and in our specific case, it's described as:
\begin{align*}f(x) &= \begin{cases}\rho e^{-\rho(x-\delta)}, & \text{for } x > \delta \0, & \text{for } x \leq \delta \end{cases}\end{align*}
Here, \(\rho\) represents the rate parameter, and \(\delta\) serves as a delay or a shift in the commencement of the probability density function. The exponential part of the function corresponds to memoryless property often associated with certain types of renewal processes, meaning the chance of an event occurring is the same, no matter how much time has already passed, provided it exceeds \(\delta\).

To grasp this fundamental concept, it's important to understand that the 'lifetime' in this context refers to the duration between events in the process. The density function thus provides essential information about the timing and frequency of these events.

The Role of the Cumulative Distribution Function in Renewal Processes

The cumulative distribution function (CDF), denoted as F(x), is a central concept in probability theory and statistics. For a renewal process, the CDF indicates the probability that the time between consecutive renewal events is less than or equal to a certain value, x.

In the solution to the exercise, to find the CDF, we integrate the lifetime density function over the interval from the smallest possible value to x. For the given exercise, the function looks like this:
\begin{align*}F(x) &= \begin{cases}1 - e^{-\rho(x-\delta)}, & \text{for } x > \delta \0, & \text{for } x \leq \delta \end{cases}\end{align*}
The CDF is a non-decreasing function that starts at zero and approaches one as x increases. For x values smaller than or equal to \(\delta\), the CDF remains at zero, which signifies that no renewals can occur before time \(\delta\). For values greater than \(\delta\), the CDF increases and depicts the accumulating probability of a renewal event as time extends past the threshold of \(\delta\).

The CDF is crucial in calculating probabilities regarding the time of events in a renewal process. By knowing the CDF, one can determine the likelihood of an event occurring within any given time frame, as well as use it within the Renewal Equation to solve for more complex probabilities.

Applying the Renewal Equation

The Renewal Equation is a powerful tool used to determine the probability of a certain number of renewals occurring within a particular timeframe in a renewal process. It is a fundamental concept for understanding complex stochastic processes.

The Renewal Equation has the following form:\begin{align*}\operatorname{Pr}\{N(t) = k\} &= \int_{0}^{t} \operatorname{Pr}\{N(u) = k-1\} f(t-u) du\end{align*}
This equation recursively defines the probability that exactly k renewals have occurred by time t. To solve for \(\operatorname{Pr}\{N(t) = k\}\), one must know the probabilities for fewer renewals and the lifetime density function. The integral combines these probabilities by 'summing up' the chances across all potential previous renewal times, weighted by how likely the next renewal would follow based on the lifetime density.

Moreover, the Renewal Equation allows for the calculation of the probability that at least k renewals occur, which is often found by subtracting the sum of the probabilities of having fewer than k renewals from one. In simpler terms, it helps to answer questions such as 'What is the chance that at least a certain number of events have happened by a specific point in time?'.

Understanding and applying the Renewal Equation is essential not only in theoretical study but also in practical scenarios, such as predicting when a machine will need maintenance, estimating customer return patterns, or analyzing the reliability of systems over time.