#
On Positive Recurrence of the M_{n}/GI/1/∞ Model

## Abstract

**:**

## 1. Introduction

**positive recurrence**of the model ${M}_{n}/G/1/\infty $ under certain assumptions. Intensity of service is assumed only partially at zero (as a lower left derivative value at zero of the “integrated intensity”); in addition, an integral type condition on the “integrated intensity” over intervals of some length is assumed.

## 2. The Setting and Main Results

#### 2.1. Definition of the Process

#### 2.2. Some Notation

- 1.
- The notation ${\mathsf{P}}_{X}={\mathsf{P}}_{n,x}$ for the probability and ${\mathsf{E}}_{X}={\mathsf{E}}_{n,x}$ for the expectation will be used. Both correspond to the initial value $X=(n,x)$ of the Markov process under consideration. We highlight that this is a standard notation from the theory of homogeneous Markov processes.
- 2.
- For a possibly discontinuous distribution function F, or for its integrated intensity H, integrals written like ${\int}_{t}^{{t}^{\prime}}\dots dF\left(s\right)$ will be understood as integrals$${\int}_{t}^{{t}^{\prime}}\dots dF\left(s\right):={\int}_{(t,{t}^{\prime}]}\dots dF\left(s\right),$$
- (3)
- The following convention will be used, $dH\left(x\right)=0$ if $(n,x)=(0,0)$.

#### 2.3. Main Result for ${M}_{n}/GI/1/\infty $

**Theorem**

**1.**

**Remark**

**1.**

## 3. Lemmata

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

## 4. Proof of Theorem 1

**0.**- The proof will be split into several steps. We shall consider the embedded Markov chain, namely, the process ${X}_{t}$ at times $t=0,1,\dots $, and it will be shown that this process hits some suitable compact around “zero state” $(0,0)$ in time which admits a finite expectation. From this property, the main result will follow. The reader is warned that after this first hit the definition of the embedded Markov chain will change, as further times may become random and possibly non-integer, see step 4 of this proof.

**1.**- Let us choose $\epsilon >0$ so that$$\epsilon <\frac{1}{2}-\frac{(1+\mathsf{\Lambda})}{r}$$

**2.**

**3.**- Now, once the bound for the expected value of $\tau $ is established, we are ready to explain the details of how to obtain a bound for ${\mathsf{E}}_{n,x}{\tau}_{0}$. The rest of the proof is devoted to this implication, with the last sentences related to the corollary about the invariant measure and convergence to it.

**4.**- Consider now the process X started at time $\tau $ from state $({n}_{\tau},{x}_{\tau})$ with ${x}_{\tau}\le 1$ and ${n}_{\tau}\le M$.

**5.**- Denote$${d}^{k}:={\chi}^{k}-{\tau}^{k},\phantom{\rule{1.em}{0ex}}{\delta}^{k}:={\tau}^{k+1}-{\chi}^{k}.$$

**6.**- We have,$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\mathsf{E}}_{{X}_{0}}\left({\tau}^{1}+{d}^{1}+\sum _{i=2}^{k+1}({\delta}^{i-1}+\underset{\le T}{\underbrace{{d}^{i}}})\right)1({\chi}^{k}<{\tau}_{0})1({\chi}^{k+1}\ge {\tau}_{0})\hfill \\ \\ \hfill \phantom{\rule{1.em}{0ex}}& ={\mathsf{E}}_{{X}_{0}}\left({\tau}^{1}+{d}^{1}+\sum _{i=1}^{k}({\delta}^{i-1}+\underset{\le T}{\underbrace{{d}^{i}}})\right)\prod _{j=1}^{k}1({\chi}^{j}<{\tau}_{0})\hfill \\ \\ \hfill \phantom{\rule{1.em}{0ex}}& +{\mathsf{E}}_{{X}_{0}}({\delta}^{k}+\underset{\le T}{\underbrace{{d}^{k+1}}})\prod _{j=1}^{k}1({\chi}^{j}<{\tau}_{0}).\hfill \end{array}$$

## 5. Two Examples

**Example**

**1.**

**Example**

**2.**

## 6. Discussion

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Veretennikov, A.
On Positive Recurrence of the *M*_{n}/*GI*/1/*∞* Model. *Mathematics* **2023**, *11*, 4514.
https://doi.org/10.3390/math11214514

**AMA Style**

Veretennikov A.
On Positive Recurrence of the *M*_{n}/*GI*/1/*∞* Model. *Mathematics*. 2023; 11(21):4514.
https://doi.org/10.3390/math11214514

**Chicago/Turabian Style**

Veretennikov, Alexander.
2023. "On Positive Recurrence of the *M*_{n}/*GI*/1/*∞* Model" *Mathematics* 11, no. 21: 4514.
https://doi.org/10.3390/math11214514