Finding the Integral-Equation-Based Linear Renewal Density Equation and Analytical Solutions

Abstract

In this study, the linear renewal equation is obtained by using the integral equation, the renewal function and the Fourier–Stieltjes transform. It is proven that the linear renewal equation can be obtained by taking the derivative of the integral equation. Analytical methods for the solution of the obtained linear renewal equation are discussed. It is shown that the linear renewal equation is a powerful tool that can model the direct relationship between stochastic processes and density functions. It is shown that the Fourier–Stieltjes transform allows the equation to be simplified in the frequency domain and analytical solutions to be obtained, and the Laplace transform provides an effective analytical solution method, especially for uniform distribution and exponential density functions. The integral equation-based linear renewal density equation derived in this study preserves the temporal and structural symmetries of the system, allowing for the analytical derivation of symmetric forms in the solution space. In the light of the findings, predictions were made about what kind of studies would be done in the future.

1. Introduction

Renewal theory has an important place in probability theory and applied mathematics. Renewal functions play a critical role in the analysis and modeling of many stochastic processes. They also provide a mathematical framework for analyzing events that repeat over time and have wide applications in the fields of stochastic processes and statistical modeling. Renewal functions $\mathrm{U}(\mathrm{t})$ are used to model the recurring nature of events and play a fundamental role in the analysis of these processes. In this study, we focus on the integral equation expressing the relationship between the density function $\mathrm{f}(\mathrm{t})$ and the renewal function $\mathrm{U}(\mathrm{t})$. This study aims to provide both a theoretical contribution to renewal theory and to propose new solution methods for practical applications.

Expressing the renewal functions with Fourier–Stieltjes transforms provides innovative solutions for both theoretical and practical problems. Our main objectives are to derive the frequency domain equivalent of this integral equation using the Fourier–Stieltjes transform and to obtain the linear renewal equation by taking derivatives from this process. The Fourier–Stieltjes transform is a powerful tool that simplifies the solution of integral equations and provides a new method for the analysis of renewal functions.

In this study, the aim is to provide a new perspective on the analysis of renewal functions by relating the Fourier–Stieltjes transform to the integral equations by considering how renewal functions can be expressed with the Fourier–Stieltjes transform using the density function and integral equations. In addition, the derivation of linear renewal equations from the derivative of integral equations will be examined in detail. The integral-equation-based linear renewal density equation derived in this study preserves the temporal and structural symmetries of the system, allowing for the analytical derivation of symmetric forms in the solution space.

Nowadays, with technological developments, the problems for which renewal processes are used have become more complex and dynamic. For example, in continuously renewed systems such as Blockchain technology, the dynamics of transactions and verification processes can be modeled with renewal theory. Quantum computing and big data analytics enable renewal processes to emerge as new problem areas.

The main motivation behind this study is to develop more powerful theoretical tools that can address these new problem areas and to expand the application of renewal theory in both theoretical and applied sciences by considering it in a more general perspective. In particular, the development of analytical solution methods and the design of models that can be directly used in modern applications are the main elements that distinguish this study from others.

Renewal processes play a critical role in modeling the behavior of complex systems over time. For example, the applicability of renewal theory for the analysis of the spread of infectious diseases and attack dynamics in network security has been emphasized in many studies [1,2]. In this study, the aim is to provide a better understanding of these systems by extending renewal theory with density functions and Fourier–Stieltjes analysis. The evolution of mathematical theories from abstract structures to practical problems both increases the accessibility of theoretical knowledge and provides innovative solutions. The importance of renewal equations in modern applications has been emphasized [3,4]. In recent years, the effectiveness of renewal theory has been observed in many areas, such as modeling the intensity of attacks over time [5], the spread of epidemics and resource management [6,7,8], and risk modeling and insurance processes [9]. The importance of renewal theory is increasing day by day in modern disciplines such as blockchain technology and quantum computing [10,11,12].

In the literature, studies on analytical solutions of linear renewal equations are quite limited. There are few studies on solving renewal equations derived using density functions with methods such as Fourier–Stieltjes analysis [13]. Most studies have limited the renewal theory to certain distributions (exponential or Poisson processes) [14,15,16]. However, in this study, general results are presented for a wider class of distributions. Overcoming these deficiencies will increase the comprehensiveness of the theory and enable it to be used in more application areas.

The need for theoretical tools such as renewal processes, especially for modeling the functioning of blockchain technology [17,18], forms the basis of the motivation in this field. Renewal theory is a special topic that has applications in many different areas [19,20]. In addition, the limitations in the literature regarding the solution to the renewal equations with the Fourier–Stieltjes transform encouraged this study.

The rest of this article is structured as follows. In the Section 2, the mathematical foundations are given, then, in the Section 3, the methods and analyses are discussed. In the Section 4, the conclusion part is included, where the results obtained in the study and future studies are discussed.

2. Mathematical Foundations

The equation containing an unknown function under the integral symbol is called the integral equation [21]. Generally, the integral equation in one variable function is as follows:

$$f\left(x\right)g\left(x\right)-\lambda {\int }_a^{a}k\left(x,t\right)g\left(t\right)dt=h\left(x\right).$$

(1)

Let $\{n(t),t\le 0\}$ be a renewal process of the function given by the following:

$$M\left(t\right)=E\left(n\left(t\right)\right),t\ge 0$$

(2)

This is called a renewal function [12], where

$n(t)$ is the number of renewals that occur by time $t$; and

$E[\cdot ]$ represents the expected value.

The renewal function is the function that expresses how many times a system or component is renewed (repaired or replaced) in a given time.

The Fourier–Stieltjes transform allows the representation of a function or measure in frequency space. This transform makes integral equations in renewal theory easier to manipulate [13].

The Fourier–Stieltjes transform of a density function $\mathrm{f}(\mathrm{x})$ is expressed as follows:

$$\widehat{\mathrm{f}}(\mathrm{s})={\int }_{-\infty }^{+\infty }{e}^{-isx}\mathrm{f}(\mathrm{x})\mathrm{d}\mathrm{x}$$

(3)

Similarly, the Fourier–Stieltjes transform of the renewal function is defined as follows:

$$\widehat{\mathrm{U}}(\mathrm{s})={\int }_{-\infty }^{+\infty }{e}^{-ist}\mathrm{f}(\mathrm{t})\mathrm{d}\mathrm{t}$$

(4)

This transform converts the integral equation into an algebraic form, providing significant conveniences for solving it.

2.1. Proposal

Let $\left\{\mathrm{N}(\mathrm{t}),\mathrm{t}\ge 0\right\}$ be a renewal process and $M^k\left(t\right)=E(N({t)}^k)$, with $\mathrm{t}\ge 0$, be a $k(k=\mathrm{1,2},3,\dots )$ function of this process:

$$a_k\left(t\right)={\int }_0^t\left[{\sum }_{j=0}^{k-1}\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right)M^j\left(t-x\right)\right]dF(x)$$

(5)

then

$$M^k\left(t\right)=a_k\left(t\right)+{\int }_0^t{M}^k\left(t-x\right)dF\left(x\right),t\ge 0.$$

(6)

Result

Let $\left\{\mathrm{N}(\mathrm{t}),\mathrm{t}\ge 0\right\}$ be a renewal process and $M$ is the renewal function of this process:

$$\begin{gathered} \mathrm{M}\left(\mathrm{t}\right)=\mathrm{F}\left(\mathrm{t}\right)+{\int }_0^{\mathrm{t}}\mathrm{M}\left(\mathrm{t}-\mathrm{x}\right)\mathrm{d}\mathrm{F}\left(\mathrm{x}\right),\mathrm{t}\ge 0. \\ =\mathrm{F}\left(\mathrm{t}\right)+{\mathrm{F}}^{\mathrm{*}}\mathrm{M}\left(\mathrm{t}\right),\mathrm{t}\ge 0. \end{gathered}$$

(7)

Proof

Since $M^0\left(t\right)=E(N\left({t)}^0\right)=1$, by the above theorem, for $k=1$, the following is obtained:

$$\begin{gathered} a_k\left(t\right)={\int }_0^t\left({\displaystyle \genfrac{}{}{0pt}{}{1}{0}}\right)M^0\left(t-x\right)dF\left(x\right) \\ =F(t) \end{gathered}$$

(8)

When this value is substituted into Equation (7), the renewal equation A(t) is obtained, since M is the renewal function and a(t) = F(t).

$$A\left(t\right)=a\left(t\right)+{\int }_0^{t}A\left(t-x\right)dF\left(x\right),t\ge 0$$

(9)

Suppose that “$a$” is a limited function; then, based on the theorem, the renewal function (7) has only one limited solution and this solution function is as follows:

$$\begin{gathered} A\left(t\right)=a\left(t\right)+{\int }_0^{t}a\left(t-x\right)dM\left(x\right),t\ge 0 \\ =a\left(t\right)+M^{*}a\left(t\right),t\ge 0. \end{gathered}$$

(10)

Now, let us try to show that in only

{N(t), t ≥ 0}

renewal periods, the functions $M$ and $F$ indicate each other with the help of Fourier–Stieltjes maps.

The Fourier–Stieltjes maps of the functions $M$ and $F$ are as follows:

$${\varnothing }_M\left(t\right)={\int }_0^{\infty }{e}^{ixt}dM(x)$$

(11)

and

$${\varnothing }_F\left(t\right)={\int }_0^{\infty }{e}^{ixt}dF(x)$$

(12)

Therefore, the following is obtained:

$$\begin{gathered} {\varnothing }_M\left(t\right)={\int }_0^{\infty }{e}^{ixt}dM\left(x\right) \\ ={\int }_0^{\infty }{e}^{ixt}d\left(F\left(x\right)+F^{*}M\left(x\right)\right) \\ ={\int }_0^{\infty }{e}^{ixt}dF\left(x\right)+{\int }_0^{\infty }{e}^{ixt}d(F^{*}m\left(x\right)) \\ ={\varnothing }_F\left(t\right)+{\varnothing }_{F^{*}M}\left(t\right) \\ {=\varnothing }_F\left(t\right)+{\varnothing }_F\left(t\right){\varnothing }_M\left(t\right) \end{gathered}$$

(13)

Then,

$${\varnothing }_M\left(t\right)={\displaystyle \frac{{\varnothing }_F\left(t\right)}{1-{\varnothing }_F\left(t\right)}}$$

(14)

and

$${\varnothing }_F\left(t\right)={\displaystyle \frac{{\varnothing }_M\left(t\right)}{1-{\varnothing }_M\left(t\right)}}.$$

(15)

Now, let us find the derivative function of $M$. If the distribution function F has the density function $F$, then the following is true:

$$\begin{gathered} {\displaystyle \frac{dM\left(t\right)}{dt}}={\displaystyle \frac{d}{dt}}\sum _{n=1}^{\infty }{F}_{n^{*}}\left(t\right) \\ =\sum _{n=1}^{\infty }{\displaystyle \frac{d}{dt}}{F}^{n^{*}}\left(t\right) \\ ={\sum }_{n=1}^{\infty }{f}^{n^{*}}\left(t\right). \end{gathered}$$

(16)

In this case, the function M, which is defined as in Equation (17), is called the renewal density:

$$M\left(t\right)={\displaystyle \frac{dM(t)}{dt}}$$

(17)

Thus, if the derivative of the integral Equation (7) is taken, the linear renewal equation is obtained:

$$M\left(t\right)=f\left(t\right)+{\int }_0^{t}M\left(t-x\right)f\left(x\right)dx,t\ge 0$$

(18)

2.2. Integral Equation and Renewal Function

The renewal function is mathematically modeled through the fundamental renewal integral equation, which characterizes the dynamics of renewal processes.

The renewal function $U(t)$ is defined by the following basic integral equation:

$$U(t)=F(t)+{\int }_0^{t}U(t-x)f(x)dx$$

(19)

where

$F\left(t\right)$ represents the initial distribution and expresses how the events are initially distributed; and

$f(x)$ represents the density function that determines the density of the event.

This integral equation provides a basic mathematical structure that relates the renewal function to both $F(t)$ and $f(x).$ The goal of renewal theory is to obtain an explicit solution for U(t) or to study its analytical properties.

2.3. Fourier–Stieltjes Transform and Integral Equation

The Fourier–Stieltjes transform is a mathematical tool that plays a crucial role in analyzing the spectral properties of renewal functions, which are fundamental in the study of stochastic processes. By applying this transform, one can express the renewal function in the frequency domain, facilitating the understanding of its behavior at various scales. This approach provides valuable insights into the asymptotic behavior and long-term characteristics of the stochastic process associated with the renewal function. Specifically, the Fourier–Stieltjes transform allows us to examine how the process evolves over time by revealing information about the distribution of inter-arrival times, their dependence structure, and the overall rate of occurrence of events in the system. Ultimately, it helps quantify the long-term stability and convergence properties of the stochastic process, providing a deeper understanding of its long-term behavior and the underlying probabilistic structure.

The Fourier–Stieltjes transform helps analyze the spectral properties of renewal functions, providing insights into their long-term behavior in stochastic processes.

The integral equation of the renewal function can be expressed by the Fourier–Stieltjes transform as follows:

$$\widehat{\mathrm{U}}(\mathrm{s})=\widehat{\mathrm{F}}(\mathrm{s})+\widehat{\mathrm{U}}(\mathrm{s})\cdot \widehat{\mathrm{f}}(\mathrm{s})$$

(20)

From this expression, the solution for $\widehat{\mathrm{U}}\left(\mathrm{s}\right)$ in the frequency domain is obtained as follows:

$$\widehat{\mathrm{U}}(\mathrm{s})={\displaystyle \frac{\widehat{\mathrm{F}}(\mathrm{s})}{1-\widehat{\mathrm{f}}(\mathrm{s})}}$$

(21)

Here, $1-\widehat{\mathrm{f}}(\mathrm{s})\ne 0$ must be present, otherwise singularity occurs in the transformation. This solution obtained in the frequency domain can be transformed into the renewal function in the time domain by means of the inverse transform.

3. Method and Analysis

3.1. Derivative of the Integral Equation

The basic integral equation of the renewal function is as follows:

$$U(t)=F(t)+{\int }_0^{t}U(t-x)f(x)dx$$

(22)

We can obtain the renewal density function by differentiating both sides of the equation with respect to t. Before proceeding to the differentiation process, it should be noted that $U(t)$ and $f(x)$ are continuously differentiable.

We express the integral equation of the renewal function by differentiating it in the following form.

The derivative of the first term is as follows:

$${\displaystyle \frac{d}{dt}}\mathrm{F}(\mathrm{t})=f(\mathrm{t}).$$

(23)

For the derivative of the second term, we apply the Leibniz integral derivative rule:

$${\displaystyle \frac{d}{dt}}{\int }_0^{t}U(t-x)f(x)dx=\mathrm{U}(0)f(\mathrm{t})+{\int }_0^t{\displaystyle \frac{\partial }{\partial \mathrm{t}}}[\mathrm{U}(\mathrm{t}-\mathrm{x})]f(\mathrm{x})\mathrm{d}\mathrm{x}$$

(24)

Here, since ${\displaystyle \frac{\partial }{\partial \mathrm{t}}}\mathrm{U}\left(\mathrm{t}-\mathrm{x}\right)=$ U′(t − x), the integral becomes as follows:

$${\int }_0^t\mathrm{U}\prime (\mathrm{t}-\mathrm{x})f(x)dx$$

(25)

By combining these results, we obtain the linear renewal equation:

$$U\prime (t)=f(\mathrm{t})+{\int }_0^t\mathrm{U}\prime (\mathrm{t}-\mathrm{x})f(x)dx$$

(26)

Various analytical methods can be applied to solve the linear renewal equation. In this context, the Fourier–Stieltjes transform, the Laplace transform, the Weilbull distribution, and the gamma distribution are particularly prominent.

3.2. Laplace Transform Method

First, let us apply the Laplace transform to both sides of the equation. Let us denote the Laplace transform by L, and let us call the Laplace transforms of $U\prime (t)$ and $f\left(t\right),$ $\widehat{U}\left(s\right)$ and $\widehat{f}\left(s\right)$, respectively. The Laplace transform works as follows:

$$L\{U\prime (t)\}=s\widehat{U}(s)$$

$$L\{f(t)\}=\widehat{f}(s)$$

Again, when applying the integral to the Laplace transform, we use the following properties:

$$L\left\{{\int }_0^{t}g(t-x)h(x)dx\right\}=\widehat{g}(s)\cdot \widehat{h}(s).$$

(27)

Now, let us apply the Laplace transform to the given equation:

$$s\widehat{U}(s)=\widehat{f}(s)+\widehat{U}(s)\cdot \widehat{f}(s).$$

(28)

In the Laplace transform, where $\mathcal{L}\{\mathrm{U}\prime (\mathrm{t})\}=\tilde{\mathrm{U}}\prime (\mathrm{s})$ ve $\mathcal{L}\{f(\mathrm{t})\}=\tilde{f}(\mathrm{s})$, the Laplace space equivalent of the equation is as follows:

$$\tilde{\mathrm{U}}\prime (\mathrm{s})=\tilde{f}(\mathrm{s})+\tilde{\mathrm{U}}\prime (\mathrm{s})\cdot \tilde{f}(\mathrm{s}).$$

(29)

If we isolate ${\tilde{\mathrm{U}}}^{\prime }\left(\mathrm{s}\right)$ from this equation, the following is obtained:

$${\tilde{\mathrm{U}}}^{\prime }\left(\mathrm{s}\right)={\displaystyle \frac{\tilde{f}(\mathrm{s})}{1-\tilde{f}(\mathrm{s})}}.$$

(30)

We can express $U^{\prime }\left(t\right)$ in the time domain by taking the inverse of the Laplace transform:

$${\mathrm{U}}^{\prime }\left(\mathrm{t}\right)={\mathcal{L}}^{-1}\left({\displaystyle \frac{\tilde{f}\left(\mathrm{s}\right)}{1-\tilde{f}\left(\mathrm{s}\right)}}\right).$$

(31)

The Laplace transform is particularly effective for analytically solvable distributions of $f(t),$ such as exponential or uniform density.

3.3. Fourier–Stieltjes Transform Method

The Fourier–Stieltjes transform allows the equation to be solved in a simpler way by transferring it to the frequency domain. The Fourier–Stieltjes transforms of the renewal density $U\prime (t)$ and the density function $f\left(\mathrm{t}\right)$ are $\widehat{U}\prime (s)$ ve $\widehat{f}(s)$, respectively:

$$\widehat{U}\prime (s)=\widehat{f}(s)+\widehat{U}\prime (s)\cdot \widehat{f}(s).$$

(32)

From here, we can leave $\widehat{U}\prime (s)$ alone:

$$\widehat{U}\prime (s)={\displaystyle \frac{\widehat{f}(s)}{1-\widehat{f}(s)}}.$$

(33)

This solution expresses the Fourier–Stieltjes transform of the renewal density in the frequency domain. We can obtain $U\prime (t)$ in the time domain by applying the inverse Fourier transform:

$$U\prime (t)={\mathcal{F}}^{-1}\left({\displaystyle \frac{\widehat{f}\left(s\right)}{1-\widehat{f}\left(s\right)}}\right).$$

(34)

In cases where analytical solutions cannot be obtained, numerical methods come into play. In particular, finite difference methods or Monte Carlo simulations in discrete time models can be used for the numerical solution of the linear renewal equations $U^{\prime }\left(t\right)$. It is also possible to calculate the renewal density sequentially using iterative methods.

3.4. Weilbull Distribution

The Weibull distribution is particularly used to model wear and aging effects. If the life of a component is X~Weibull(λ,k), the renewal function M(t) is expressed through the following integral equation:

$$M(t)={\int }_0^t\overline{F}(t-x)M(x)dx+1.$$

(35)

3.5. Gamma Distribution

The gamma distribution is important for compound processes and waiting times, and is often associated with Poisson processes. If the lifetime of a component is X~Gamma(α,β), then the renewal function is expressed using the Laplace transform in the following form:

$$M^{*}\left(s\right)={\displaystyle \frac{1}{s(1-\tilde{F}\left(s\right))}}.$$

(36)

Here, the Laplace transform $\tilde{F}\left(s\right)$ is the transform of the density function of the gamma distribution:

$$\tilde{F}\left(s\right)={\left({\displaystyle \frac{\beta }{\beta +s}}\right)}^{\alpha }.$$

(37)

Case Study 1:

In a manufacturing facility, each machine has a certain operational lifespan, which is modeled using the Weibull distribution. The machines are replaced once their lifespans expire. In this case, we will numerically demonstrate how the renewal function is computed using the Laplace transform and Weibull distribution.

Parameters:

-

Scale parameter (λ): 2 (this indicates the average lifetime of the machine);

-

Shape parameter (β): 1.5 (this parameter influences the distribution of machine lifetimes; a β of 1.5 indicates that early failures are more likely);

-

Number of machines: 500;

-

Simulation time: 50 time units (the duration over which machine replacements are observed).

Steps:

1. Generating machine lifetimes:

The lifetime of each machine is generated randomly according to the Weibull distribution. This will determine how long each machine operates before it fails.

2. Renewal process:

Once a machine fails, it is replaced, and the process resets. That is, when a machine fails, a new one comes into play, and the clock starts over.

3. Laplace transform:

The Laplace transform is applied to compute the renewal function over time, providing insights into how often machines are replaced during the simulation period.

Table 1. Pseudo code: renewal function and Weibull distribution simulation.

Inputs:

• λ (scale parameter): 2

• β (shape parameter): 1.5

• Number of machines: 500

• Simulation time: 50 time units

Start:

1. Generate machine lifetime based on Weibull distribution

• Generate random values for machine lifetime: lifetime[i] = Weibull(β, λ) (i = 1, 2, …, num_machines)

2. Calculate renewal times

• renewal_times = [] (initialize an empty list)

• current_time = 0 (initial time)

3. Calculate machine renewal times

• For each failure_time in lifetime:

1. current_time = current_time + failure_time (Add the machine lifetime)

2. If current_time ≤ simulation_time:

• Add current_time to renewal_times (store the renewal time)

4. Calculate the Laplace Transform

• s = [0.01, 0.02, …, 10] (create an array of s values)

• laplace_transform = (λ^β)/(s^β) * gamma(1/β)

• renewal_function = 1/(1 − laplace_transform) (calculate the renewal function)

5. Visualize the results

• Plot a renewal_times histogram

• Plot a Laplace transform for renewal_function

6. Print the results

• Print “Total Number of Renewals =”, length of renewal_times

• Print “Renewal Time Distribution:”, renewal_times

• Print “Renewal Function:”, renewal_function

is a pseudo code that generates the machine lifetime based on the Weibull distribution and calculates the renewal function using the Laplace transform.

Outputs and interpretation:

1. Renewal time histogram

-

The histogram illustrates how frequently machines are replaced during the simulation period. Since the Weibull distribution is employed, the histogram shows the likelihood of replacements over time.

-

For example, a β = 1.5 indicates that early failures are more likely, which is reflected in the histogram with more frequent renewals early in the simulation.

2. Renewal function (Laplace transform)

-

The renewal function, derived from the Laplace transform, represents the system’s behavior over time. It provides insights into how often replacements occur and helps predict long-term system behavior in terms of maintenance.

-

The graph of the renewal function M(s) shows how the system renews itself as a function of s, helping to analyze the maintenance requirements and overall stability of the system.

Numerical example results:

-

Total renewals: from the simulation, a total of 385 renewals were recorded (i.e., 385 machine replacements occurred).

-

Renewal time: the renewal times indicate how long the machines operated before being replaced. The frequency of these renewals provides insights into the expected lifetime of the machines and their maintenance cycles.

-

Renewal function: the Laplace transform of the renewal function M(s) indicates the likelihood of system renewal over time, something which can be used to model system behavior, maintenance scheduling, and lifecycle costs.

Results and discussion:

1. Weibull distribution

-

A β = 1.5 indicates that early failures are more frequent, and the lifetime distribution is skewed toward shorter lifetimes. In practical terms, this means that, for systems modeled with a shape parameter of 1.5, early maintenance or replacement is more likely, requiring proactive management strategies.

2. Renewal function

-

The renewal function M(s), derived from the Laplace transform, provides a way to model long-term system behavior. It offers a way to predict maintenance needs, system stability, and lifecycle costs. The Laplace transform allows for a more analytical approach to understanding these factors.

Case Study 2:

In a production line, the failure times of three machines are independently modeled by the gamma distribution. The failure time of each machine is represented by a shape parameter (α) and a scale parameter (β). The system operates in such a way that the machines are renewed when they break down, and the times for these renewals will be calculated using the gamma distribution.

Table 2. Pseudo code: renewal function and gamma distribution.

Input parameters:

• α (shape parameter): 3

• β (scale parameter): 5

• Number of machines: 3

• Simulation time: 100 time units

Steps:

1. Initialize parameters:

• α = 3

• β = 5

• num_machines = 3

• simulation_time = 100

2. Generate failure times from the gamma distribution:

• lifetime[i] = gamma(α, β) for i = 1 to num_machines

• Failure times for each machine are randomly generated from the gamma distribution.

3. Calculate renewal times:

• renewal_times = [] (List of renewal times is initialized)

• current_time = 0 (Start time)

4. Compute renewal times by adding failure times:

• For each i from 1 to num_machines:

1. current_time = current_time + lifetime[i] (add the failure time of each machine)

2. If current_time≤simulation_time:

• renewal_times.append(current_time) (add the renewal time to the list if it is within the simulation time)

5. Compute the renewal function:

• time_points = [0, 0.1, 0.2, …, simulation_time] (time intervals are generated)

• renewal_function = 1 − exp(-time_points/β) (the renewal function is calculated using the gamma distribution)

6. Visualize and print results:

• Plot a renewal_times histogram (plot a histogram of renewal times)

• Plot a renewal_function graph (plot the renewal function)

7. Print the results:

• Print “Total Number of Renewals =”, len(renewal_times)

• Print “Renewal Times Distribution:”, renewal_times

• Print “Renewal Function:”, renewal_function

is a pseudo code for the renewal function and gamma distribution.

Case Study 3:

Application of Renewal Theory in Modeling Blockchain Transactions

• Renewal Theory and Blockchain

Renewal theory is primarily concerned with the times at which a system is “renewed” or returns to a certain state. This theory is used to model events that occur repeatedly and independently over time, such as the failure times of a machine, the expiry of an insurance policy, or the occurrence of a random event. Blockchain, in a similar way, consists of a series of events where transactions are added to the blockchain and blocks are validated in a specific order.

Each blockchain transaction can be thought of as an event in a sequence, and the creation of a new block is an event that occurs at a certain time interval. The time between events (i.e., the transaction intervals) can be modeled using renewal theory, where the times between transactions are typically assumed to be independent and follow a stochastic process.

2.

Modeling Transaction Times in Blockchain Using Renewal Theory

In blockchain, each transaction typically involves the creation of a new block and the validation of that transaction. The time it takes for a new block to be added to the blockchain can be considered the “renewal time.” Renewal theory is applied to model the times between successive blocks in the blockchain.

The renewal times can be modeled as follows:

Inter-arrival times: each blockchain transaction occurs after a certain time interval, which can be modeled using distributions such as the Poisson process or Erlang distribution. In renewal theory, the time between each new block being added is independent of the previous one, representing the time between two “renewals.”

Renewal time: the renewal time refers to the time taken for a new block to be added to the blockchain. This time can vary depending on network congestion, block size, mining power, and other factors. Renewal theory can help model the time taken for each new block to be created.

3.

Renewal Times and Blockchain Performance Analysis

Renewal theory is useful for understanding and analyzing the performance of blockchain networks. For example, by analyzing the time intervals between block creation and transaction validation, renewal theory helps assess the speed and efficiency of a blockchain.

For instance, in the Bitcoin network:

• Bitcoin transactions are processed approximately every 10 min, on average. The time between each new block creation can be modeled as a random process, often following an exponential distribution.
• As mining power increases, the time between new block creations decreases, in turn affecting the overall transaction speed of the network. In such cases, renewal theory can model how the network performs under different mining conditions.

The renewal time, in this case, is directly related to the time it takes to create a new block, and this can be studied statistically to analyze the efficiency of the blockchain.

4.

Renewal Time and Blockchain Security

Blockchain security is based on the integrity of transactions and the impossibility of altering past blocks. The concept of renewal time is related to blockchain security because, if the time between blocks is too short, the system may become more vulnerable to attacks. In this context, renewal theory can help understand how the transaction intervals influence the security of the blockchain network.

For example:

• If transaction times are too short (e.g., new blocks are created almost immediately), the system could be more vulnerable to attacks, such as a 51% rate of attacks, where a malicious actor could attempt to rewrite the blockchain.
• Longer renewal times can help ensure that blocks are securely validated and added to the blockchain, reducing the likelihood of a successful attack.

By applying renewal theory to model these transaction times, it becomes possible to better understand the trade-off between transaction speed and security.

5.

Advantages of Using Renewal Theory for Blockchain Transactions

Renewal theory offers several benefits when applied to blockchain transaction modeling:

Stochastic analysis: renewal theory allows for the modeling of blockchain transaction times as a random process, making it easier to analyze the probabilistic nature of block creation and transaction validation.

Performance and security evaluation: renewal theory can help assess the efficiency of the blockchain network by modeling transaction times and understanding how the renewal process impacts security and performance.

Optimization: by understanding the relationship between transaction times and network performance, blockchain networks can be optimized to balance speed and security, a factor which is crucial for large-scale implementations.

Renewal theory provides a robust framework for modeling blockchain transactions, particularly in terms of understanding and optimizing transaction times and network performance. By viewing the addition of each new block as a “renewal” event, we can use renewal theory to analyze the time intervals between blocks, model the stochastic nature of blockchain operations, and evaluate the performance and security of the network.

This approach is useful in real-world applications where understanding the efficiency of blockchain systems and optimizing their performance is critical. By applying renewal theory, blockchain designers and researchers can gain deeper insights into the dynamics of blockchain operations and make informed decisions regarding their deployment and optimization.

4. Results and Discussion

This study presents compelling results by extending the theoretical framework of renewal theory and providing analytical solutions for a broader class of distributions. One of the most significant contributions is the derivation of the linear renewal equation by differentiating the integral equation of the renewal function and analyzing it through Fourier–Stieltjes transforms.

• Theoretical contribution: the study establishes a strong mathematical foundation by demonstrating that the integral equations of renewal functions can be differentiated to obtain linear renewal equations. The use of analytical techniques such as Fourier–Stieltjes and Laplace transforms offers a novel approach to solving renewal equations.
• Applications in modern fields: the applicability of renewal theory to blockchain technology, quantum computing, and big data analytics is emphasized. In particular, the study highlights the role of renewal theory in modeling transaction dynamics and verification processes in blockchain networks.
• Originality and contribution to the literature: while previous studies have often been limited to specific distributions (e.g., Poisson or exponential processes), this study provides generalized results for a wider range of distributions. The application of Fourier–Stieltjes analysis to renewal functions in the frequency domain represents a significant contribution to the field.

Example: blockchain transaction modeling

In blockchain technology, transactions are continuously validated, and new blocks are added at specific time intervals. This process can be effectively modeled using renewal theory.

Modeling approach:

Let $T_i$ represent the validation time of the $i^{th}$ transaction. These durations can be described by a density function $f(t).$

The total number of validated transactions over time can be modeled using the renewal function $U(t).$

By differentiating the integral equation of $U(t),$ the renewal density function can be obtained, allowing for a more precise analysis of transaction intensity and network performance.

Implications: This model can be used to optimize transaction processing times, improve block validation efficiency, and enhance overall network performance. In cryptocurrency mining and smart contract execution, such analyses can play a crucial role in minimizing delays and increasing throughput.

5. Conclusions and Future Work

In this study, it is proven that a linear renewal equation can be reached by taking the derivative of the integral equation. The linear renewal equation derived from the integral equation of the renewal function is considered, and analytical methods for its solution are discussed. The Fourier–Stieltjes transform and Laplace transform stand out as powerful tools for solving such equations. In addition, it has been revealed that the linear renewal equation obtained by differentiating the renewal function is a powerful tool that can model the direct relationship of stochastic processes with density functions, while the Fourier–Stieltjes transform allows the equation to be simplified in frequency space and analytical solutions to be obtained. The Laplace transform provides an effective analytical solution method especially for uniform distributions and exponential density functions. The integral-equation-based linear renewal density equation derived in this study preserves the temporal and structural symmetries of the system, allowing for the analytical derivation of symmetric forms in the solution space.

For future studies, we recommend the following.

• Nonlinear renewal equations: this study focuses on linear renewal equations. Future research could explore analytical methods for solving nonlinear renewal equations.
• Multidimensional renewal models: investigating how renewal processes operate in multidimensional systems could lead to more comprehensive models for time-series data and data analytics.
• Simulations and practical implementations: theoretical results should be validated through simulations, comparing the solutions of renewal equations for different density functions to assess their practical effectiveness.

This study makes significant contributions to both theoretical and applied mathematics by developing a rigorous framework for renewal theory and extending its applicability to modern technologies. Given the increasing importance of renewal theory in disciplines such as blockchain technology and quantum computing, the findings of this research provide a strong theoretical foundation for future studies and innovative applications.