# Ergodicity Bounds and Limiting Characteristics for a Modified Prendiville Model

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## Abstract

**:**

## 1. Introduction

- In the first approach, we examine the standard way to eliminate the state with the minimum number and then apply the diagonal transformation. Previously, this type of transformation (using a triangular matrix) was considered in [17]. This approach allows one to obtain exhaustive estimates of the rate of convergence to the limiting mode in the space ${l}_{1}$.
- Using the second approach, we pass from the forward Kolmogorov system to a system of the form $\frac{d\mathbf{z}}{dt}=W\left(t\right)\mathbf{z}\left(t\right)$ for $t\ge 0,$ (see [18]). This method was previously applied to the study of one class of supercomputer systems in [3] (for nonstationary models, simple methods for solving the forward Kolmogorov system do not give good results).
- We also briefly review some issues related to proportional intensities and perturbation bounds.
- Numerical examples are considered, in which estimates of the rate of convergence to the limit mode are obtained and the limit characteristics themselves are constructed.

## 2. Model Description and Basic Notions

- $k\to k+1$ with intensity function ${\lambda}_{k}\left(t\right)$ for $k=0,1,\dots ,N-1$,
- $k\to k-1$ with intensity function ${\mu}_{k}\left(t\right)$ for $k=1,2,\dots ,N$,
- $-1\to k$ with intensity function ${\gamma}_{k}\left(t\right)$ for $k=0,1,\dots ,N$,
- $k\to -2$ with intensity function ${\zeta}_{k}\left(t\right)$ for $k=0,1,\dots ,N$,
- $-2\to -1$ with intensity function $Q\left(t\right)$.

## 3. First Approach

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

## 4. Second Approach

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Example**

**1.**

- The mathematical expectation of the number of customers in the system $E(t,k)$ for $t\in [0,4]$ and different initial conditions $X\left(0\right)=0$ and $X\left(0\right)=200$.
- The probability of states $F,R$ for $t\in [0,4]$ and $X\left(0\right)=200$.
- The probability $F,R$ and several operating states ${p}_{0}\left(t\right),{p}_{1}\left(t\right)$, at $t\in [0,4]$ and different initial conditions $X\left(0\right)=0$ (solid lines) and $X\left(0\right)=200$ (dashed lines).

- The mathematical expectation of the number of customers in the system $E(t,k)$ for $t\in [0,4]$ and different initial conditions $X\left(0\right)=0$ and $X\left(0\right)=10$.
- The probability of states $F,R$ for $t\in [0,4]$ and $X\left(0\right)=10$.
- The probability $F,R$ and several operating states ${p}_{0}\left(t\right),{p}_{1}\left(t\right)$, at $t\in [0,4]$ and different initial conditions $X\left(0\right)=0$ (solid lines) and $X\left(0\right)=10$ (dashed lines).

**Example**

**2.**

- (a)
- $N=200$
- The mathematical expectation of the number of customers in the system $E(t,k)$ for $t\in [0,4]$ and different initial conditions $X\left(0\right)=0$ and $X\left(0\right)=200$.
- The probability ${p}_{50}\left(t\right)$ for $t\in [0,4]$ and different initial conditions $X\left(0\right)=0$ (solid line) and $X\left(0\right)=200$ (dashed line).
- The probability ${p}_{50}\left(t\right),{p}_{51}\left(t\right)$ for $t\in [0,4]$ and different initial conditions $X\left(0\right)=0$ (solid lines) and $X\left(0\right)=200$ (dashed lines).
- The probability ${p}_{50}\left(t\right),{p}_{51}\left(t\right),{p}_{52}\left(t\right)$ for $t\in [0,4]$ and different initial conditions $X\left(0\right)=0$ (solid lines) and $X\left(0\right)=200$ (dashed lines).

- (b)
- $N=10$
- The mathematical expectation of the number of customers in the system $E(t,k)$ for $t\in [0,4]$ and different initial conditions $X\left(0\right)=0$ and $X\left(0\right)=10$.
- The probability ${p}_{5}\left(t\right)$ for $t\in [0,4]$ and different initial conditions $X\left(0\right)=0$ (solid line) and $X\left(0\right)=10$ (dashed line).
- The probability ${p}_{5}\left(t\right),{p}_{6}\left(t\right)$ for $t\in [0,4]$ and different initial conditions $X\left(0\right)=0$ (solid lines) and $X\left(0\right)=10$ (dashed lines).
- The probability ${p}_{5}\left(t\right),{p}_{6}\left(t\right),{p}_{7}\left(t\right)$ for $t\in [0,4]$ and different initial conditions $X\left(0\right)=0$ (solid lines) and $X\left(0\right)=10$ (dashed lines).

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Probability $F,R$ for $t\in [0,4]$ and initial conditions $X\left(0\right)$ = 200 and $X\left(0\right)$ = 200, respectively.

**Figure 4.**Probability ${p}_{-2}\left(t\right),{p}_{-1}\left(t\right),{p}_{0}\left(t\right),{p}_{1}\left(t\right)$ for $t\in [0,4]$; this figure shows the rate of convergence.

**Figure 7.**Probability ${p}_{-2}\left(t\right),{p}_{-1}\left(t\right),{p}_{0}\left(t\right),{p}_{1}\left(t\right)$ for $t\in [0,4]$; this figure shows the rate of convergence for $N=10$.

**Figure 9.**Probability ${p}_{50}\left(t\right),{p}_{51}\left(t\right)$ for $t\in [0,4]$; this figure shows the rate of convergence.

**Figure 10.**Probability ${p}_{50}\left(t\right),{p}_{51}\left(t\right)$ for $t\in [0,4]$; this figure shows the rate of convergence.

**Figure 11.**Probability ${p}_{50}\left(t\right),{p}_{51}\left(t\right),{p}_{52}\left(t\right)$ for $t\in [0,4]$; this figure shows the rate of convergence.

**Figure 13.**Probability ${p}_{5}\left(t\right)$ for $t\in [0,4]$; this figure shows the rate of convergence for $N=10$.

**Figure 14.**Probability ${p}_{5}\left(t\right),{p}_{6}\left(t\right)$ for $t\in [0,4]$; this figure shows the rate of convergence for $N=10$.

**Figure 15.**Probability ${p}_{5}\left(t\right),{p}_{6}\left(t\right),{p}_{7}\left(t\right)$ for $t\in [0,4]$; this figure shows the rate of convergence for $N=10$.

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**MDPI and ACS Style**

Usov, I.; Satin, Y.; Zeifman, A.; Korolev, V.
Ergodicity Bounds and Limiting Characteristics for a Modified Prendiville Model. *Mathematics* **2022**, *10*, 4401.
https://doi.org/10.3390/math10234401

**AMA Style**

Usov I, Satin Y, Zeifman A, Korolev V.
Ergodicity Bounds and Limiting Characteristics for a Modified Prendiville Model. *Mathematics*. 2022; 10(23):4401.
https://doi.org/10.3390/math10234401

**Chicago/Turabian Style**

Usov, Ilya, Yacov Satin, Alexander Zeifman, and Victor Korolev.
2022. "Ergodicity Bounds and Limiting Characteristics for a Modified Prendiville Model" *Mathematics* 10, no. 23: 4401.
https://doi.org/10.3390/math10234401