2. Stochastic model

In this paper, we consider a large-scale storage system that consists of N servers in a communication network. Let F_N be the total number of files that can be stored in these servers. It will be assumed that F_N is finite. The file storage system operates as follows: As long as the storage capacity is not exceeded new files can be admitted and files with one copy can be duplicated.

For $i\in\{1,2\}$, $X_i^N (t)$ denotes the number of files with i copies present in the network at time t and $(X_0(t))$ denotes the number of files lost for good. Let $(m^N(t))$ be the number of free places in the network at time $t\geq 0$. The sequence of the processes $(m^N(t))$ is defined on $\bar{\mathbb{N}}=\mathbb{N}\cup \{+\infty\}$ and is given by

(2)\begin{equation} m^N(t)=F_{N}-2X_{2}^N (t)-X_{1}^N (t) \end{equation}

The file duplication and admitting policies can be described as follows : conditionally on $(X_{1}^{N} (t), X_2^N(t))=(x_1,x_2)$ with $x_1 \gt 0$ and $ 2x_2+x_1 \lt F_{N}$, a file with one copy gets an additional copy with rate $\frac{\lambda N}{x_1}$. If $m^N(t)\geq 2$, new files can be stored with rate $\xi N$. Copies of files disappear independently at rate µ. If the last replica of a given file is lost before being repaired, the file is then definitively lost.

All events are supposed to occur after an exponentially distributed time. The admitting, failure, and duplication processes are then independent Poisson processes. The process ${X}^{N}(t)=(X_{1}^{N} (t),X_{2}^{N} (t))$ is then a Markov process on the state space

\begin{equation*} {\mathcal{D}}^N=\{(x_1,x_2)\in{\mathbb{N}}^2\;|\; 2x_2+x_1\leq {F}_{N} \} \end{equation*}

For $(x_1,x_2) \in {\mathbb{N}}^{2}$ the ${\mathcal{Q}}$-matrix $Q^{N} = (q^{N}(.,.))$ of $({X}^{N}(t))$ is given by

(3)\begin{equation} (x_1,x_2) \longrightarrow (x_1,x_2)+ \left\{ \begin{array}{ll} (0,1) \ \ \xi N \mathbb{1}_{\lbrace x_1 + 2 x_2 \lt F_{N} -1 \rbrace} \\ (1,-1)\ \ 2 \mu x_2\\ (-1,1) \ \ \lambda N \mathbb{1}_{\lbrace x_1 \gt 0, x_1 + 2 x_2 \lt F_{N} \rbrace}\\ (-1,0) \ \ \mu x_1 \end{array} \right. \end{equation}

2.1. Stochastic differential equations

The evolution equations associated to the Markov processes $(X_{0}^{N}(t))$, $(X_{1}^{N}(t))$ and $(X_{2}^{N}(t))$ are given by:

(4)\begin{equation} X_{0}^{N}(t) = X_{0}^N(0)+ \overset{+ \infty}{\underset{i = 1}{\sum }} \int_{0}^{t} \mathbb{1}_{\lbrace i \leq X_{1}^{N}(u^{-}) \rbrace} {\mathcal{N}}_{\mu,i}(du). \end{equation}

(5)\begin{eqnarray} X_{1}^{N}(t)& = & X_{1}^{N}(0) - \int_{0}^{t} \mathbb{1}_{\lbrace X_{1}^{N}(u^{-}) \gt 0,2X_2^N(u^-)+X_1^N(u^-) \lt F_{N} \rbrace} {\mathcal{N}}_{\lambda N}(du) \end{eqnarray}

\begin{eqnarray*} \nonumber &-& \overset{+ \infty}{\underset{i = 1}{\sum }} \int_{0}^{t} \mathbb{1}_{\lbrace i \leq X_{1}^{N}(u^{-}) \rbrace} {\mathcal{N}}_{\mu,i}(du) \end{eqnarray*}

\begin{eqnarray*} \nonumber &+ &\overset{+ \infty}{\underset{i = 1}{\sum }} \int_{0}^{t} \mathbb{1}_{\lbrace i \leq X_{2}^{N}(u^{-}) \rbrace} {\mathcal{N}}_{2 \mu,i}(du). \end{eqnarray*}

(6)\begin{eqnarray} X_{2}^{N}(t) &=& X_{2}^{N}(0) + \int_{0}^{t}\mathbb{1}_{\lbrace 2X_2^N(u^-)+X_1^N(u^-) \lt F_{N}-1 \rbrace}{\mathcal{N}}_{\xi N}(du) \end{eqnarray}

\begin{eqnarray*} \nonumber &-&\overset{+ \infty}{\underset{i = 1}{\sum }} \int_{0}^{t} \mathbb{1}_{\lbrace i \leq X_{2}^{N}(u^{-}) \rbrace} {\mathcal{N}}_{2 \mu,i}(du) \end{eqnarray*}

\begin{eqnarray*} \nonumber &+&\int_{0}^{t} \mathbb{1}_{\lbrace X_{1}^{N}(u^{-}) \gt 0, 2X_2^N(u^-)+X_1^N(u^-) \lt F_{N} \rbrace}{\mathcal{N}}_{\lambda N}(du) \end{eqnarray*}

where $({\mathcal{N}}_{\alpha,i})$ denotes an i.i.d sequence of Poisson processes with parameter α. All the sequences of Poisson processes are assumed to be independent. And $x(u^-)=\lim\limits_{\substack{s\to u \\ s \lt u}}x(s) $

The equations (5) and (6) can be rewritten as

(7)\begin{align} X_{1}^{N}(t) &= X_{1}^{N}(0)+ {M}_{1}^{N}(t) -\mu \int_{0}^{t} X_{1}^{N}(u) du +2\mu \int_{0}^{t} X_2^{N}(u) du \end{align}

\begin{align*} \nonumber & -\lambda N \int_{0}^{t} \mathbb{1}_{\lbrace X_{1}^{N}(u^{-}) \gt 0,2X_2^N(u^-)+X_1^N(u^-) \lt F_{N} \rbrace} du \end{align*}

(8)\begin{align} X_2^N(t)&=X_2^N(0)+ {M}_{2}^{N}(t)-2\mu \int_{0}^{t} X_2^{N}(u) du \end{align}

\begin{align*} \nonumber &+ \xi N \int_{0}^{t} \mathbb{1}_{\{2X_2^N(u^-)+X_1^N(u^-) \lt F_{N}-1\}} du\\ \nonumber & +\lambda N \int_{0}^{t}\mathbb{1}_{\lbrace X_{1}^{N}(u^{-}) \gt 0,2X_2^N(u^-)+X_1^N(u^-) \lt F_{N} \rbrace}du \nonumber \end{align*}

where $({M}_1^N(t))$ and $({M}_2^N(t))$ are martingales associated to Markov processes $(X_{1}^{N}(t))$ and $(X_{2}^{N}(t))$ (see[Reference Robert13] pp 348) given by :

(9)\begin{equation} \begin{aligned} {M}_{1}^{N}(t) &= \overset{+ \infty}{\underset{i = 1}{\sum }} \int_{0}^{t} \mathsf{1}_{\lbrace i \leq X_{2}^{N}(u^{-}) \rbrace} [ {\mathcal{N}}_{2 \mu,i}(du) -2 \mu du] \\ &- \int_{0}^{t} \mathbb{1}_{\{X_1^N(u) \gt 0,2X_2^N(u)+X_1^N(u) \lt F_{N}\}} [ {\mathcal{N}}_{\lambda N}(du) - \lambda N du ]\\ &- \overset{+ \infty}{\underset{i = 1}{\sum }} \int_{0}^{t} \mathsf{1}_{\lbrace i \leq X_{1}^{N}(u^{-}) \rbrace} [ {\mathcal{N}}_{\mu,i}(du) - \mu du] \end{aligned} \end{equation}

(10)\begin{equation} \begin{aligned} {M}_{2}^{N}(t) &= \int_{0}^{t} \mathbb{1}_{\lbrace 2X_2^N(u)+X_1^N(u) \lt F_{N}-1\rbrace}[ {\mathcal{N}}_{\xi N}(du) - \xi N du ] \\ &+\int_{0}^{t} \mathbb{1}_{\{X_1^N(u) \gt 0,2X_2^N(u)+X_1^N(u) \lt F_{N}\}} [ {\mathcal{N}}_{\lambda N}(du) - \lambda N du ] -\\ &-\overset{+ \infty}{\underset{i = 1}{\sum }} \int_{0}^{t} \mathbb{1}_{\lbrace i \leq X_{2}^{N}(u^{-}) \rbrace} [ {\mathcal{N}}_{2 \mu,i}(du) - 2 \mu du] \end{aligned} \end{equation}

The predictable increasing processes associated to the martingales $({M}_{1}^{N}(t))$ and $({M}_{2}^{N}(t))$ are, respectively, given by

(11)\begin{equation}\begin{aligned} \langle {M}_{1}^{N}\rangle(t) &= 2 \mu \int_{0}^{t} X_{2}^{N}(u) du + \mu \int_{0}^{t} X_{1}^{N}(u) du \\ &+ \lambda N \int_{0}^{t}\mathbb{1}_{\lbrace X_{1}^{N}(u) \gt 0,\; 2X_2^N(u)+X_1^N(u) \lt F_{N} \rbrace} du \end{aligned} \end{equation}

(12)\begin{equation} \begin{aligned} \langle {M}_{2}^{N}\rangle(t)& = \xi N \int_{0}^{t} \mathbb{1}_{\lbrace 2X_2^N(u)+X_1^N(u) \lt F_{N}-1 \rbrace} du +2 \mu \int_{0}^{t} X_{2}^{N}(u) du\\ & + \lambda N \int_{0}^{t}\mathbb{1}_{\lbrace X_{1}^{N}(u) \gt 0,\;2X_2^N(u)+X_1^N(u) \lt F_{N} \rbrace} du \end{aligned} \end{equation}

3. Fluid equations and Skorokhod problem

Let ${\mathcal{S}}$ be the convex domain in $ {\mathbb{R}}^2$ given by

\begin{equation*}{\mathcal{S}}=\{(x_1,x_2)\in{\mathbb{R}}^2 |x_1\geq 0,x_2\geq 0, 2x_2+x_1\leq {\bar{\beta}}\}\end{equation*}

and $\mathcal{D}({\mathbb{R}}_+,{\mathbb{R}}^2)$ the space of ${\mathbb{R}}^2$-valued right continuous functions on ${\mathbb{R}}_+$ with left limits. Let ${\mathcal{M}}_{m,n}(\mathbb{R})$ be the space of m × n matrices over $\mathbb{R}$.

In this paper, we consider the following Skorokhod problem in the convex domain ${\mathcal{S}}$. Let $\theta\in{\mathcal{M}}_{2,1}(\mathbb{R})$, $A\in{\mathcal{M}}_{2,2}(\mathbb{R})$ and $ R\in{\mathcal{M}}_{2,2}(\mathbb{R})$. Let ν be the measure on $[0,+\infty[\times \bar{\mathbb{N}}$ satisfying $\nu([0,t]\times \bar{\mathbb{N}})=t$ for all $t\geq 0$.

Definition 3.1. The couple of functions $z\in\mathcal{D}({\mathbb{R}}_+,{\mathbb{R}}^2)$ and $y\in\mathcal{D}({\mathbb{R}}_+,{\mathbb{R}}^2)$ with $z(0)\in {\mathcal{S}}$, is called the solution of the Skorokhod problem associated with the data $(\theta,\nu,A,R,{\mathcal{S}})$ and the function

(13)\begin{equation} x(t)= z(0)+t\theta+\mathcal{V}(t,\Gamma)+\int_{0}^t Az(s)ds \end{equation}

where for Γ in a σ-algebra $\mathcal{B}(\bar{\mathbb{N}})$

\begin{equation*}\mathcal{V}(t,\Gamma)=\begin{pmatrix} 0\\\nu(t,\Gamma)\end{pmatrix}\end{equation*}

if the three following conditions hold :

1. (1)

   (14)\begin{equation} z(t)=z(0)+t\theta+\mathcal{V}(t,\Gamma)+\int_{0}^t Az(s)ds+Ry(t) \end{equation}

2. (2) $z(t)\in {\mathcal{S}}$ for all $t\geq 0$

3. (3) for $i=1,2$ the component y_i of the function y are non-decreasing functions with $y_i(0)=0$, and for $t\geq 0$

   (15)\begin{align} &y_1(t)=\int_{0}^{t}\mathbb{1}_{\{z_1(s)=0\}} dy_1(s) \end{align}

   (16)\begin{align} &y_2(t)=\int_{0}^{t}\mathbb{1}_{\{z_1 \gt 0,z_1(s)+2z_2(s)={\bar{\beta}}\}} dy_2(s) \end{align}

If $z\in\mathcal{D}({\mathbb{R}}_+,{\mathbb{R}}^d)$ and $y\in\mathcal{D}({\mathbb{R}}_+,{\mathbb{R}}^d)$ with $z(0)\in {\mathcal{S}}$ is a solution of the above Skorokhod problem then the function $z=(z(t))$ has the following properties. First z behaves on the interior of the set S like a solution of the following ordinary differential equation

(17)\begin{equation} x(t)=x(0)+\theta t+\mathcal{V}(t,\Gamma)+\int_0^t Ax(s)ds \end{equation}

And second, z is reflected instantaneously at the boundaries $(\partial{\mathcal{S}})_1=\{x_1=0\}$ and $(\partial{\mathcal{S}})_2=\{x_1+2x_2={\bar{\beta}}\}$ of the set ${\mathcal{S}}$. The direction of the reflection on the boundary $(\partial{\mathcal{S}})_1$ is the first column vector of the reflection matrix R and the direction of reflection on $(\partial{\mathcal{S}})_2$ is the second column vector the matrix R. See for example, Tanaka[Reference Tanaka15].

3.1. Fluid equations

If $({X}^N(t))$ is a sequence of processes, one defines the renormalized sequence of processes of $({X}^N(t))$ by

\begin{equation*} {\bar{X}}^N(t)\stackrel{def}{=} \frac{{X}^{N}(t)}{N},\; \text{for}\; t\geq 0\end{equation*}

From equations (2), (7), (8) one gets the fluid stochastic differential equations associated with the sequence of processes $({\bar{X}}_1^N(t))$ and $({\bar{X}}_2^N(t))$

(18)\begin{equation} \begin{aligned} {\bar{X}}_{1}^{N}(t) = & {\bar{X}}_{1}^{N}(0)+ {\bar{M}}_{1}^{N}(t)-\lambda t-\mu \int_{0}^{t} {\bar{X}}_{1}^{N}(u) du \\ &\quad +2\mu \int_{0}^{t} {\bar{X}}_2^{N}(u) du + \lambda \int_{0}^{t} \mathbb{1}_{\lbrace {\bar{X}}_{1}^{N}(u) \gt 0, m^N(u)=0\rbrace} du \\ &\qquad+\lambda \int_{0}^{t} \mathbb{1}_{\lbrace {\bar{X}}_{1}^{N}(u)=0\rbrace} du \end{aligned}\end{equation}

(19)\begin{equation} \begin{aligned} {\bar{X}}_2^N(t) =& {\bar{X}}_2^N(0) + {\bar{M}}_{2}^{N}(t)+(\lambda+\xi)t-2\mu \int_{0}^{t} {\bar{X}}_2^{N}(u) du\\ &\quad -\xi \int_{0}^{t} \mathbb{1}_{\{m^N(u)\leq 1\}} du -\lambda \int_{0}^{t} \mathbb{1}_{\lbrace {\bar{X}}_{1}^{N}(u) \gt 0 ,m^N(u)=0\rbrace} du\\ &\qquad- \lambda \int_{0}^{t} \mathbb{1}_{\lbrace {\bar{X}}_{1}^{N}(u)= 0\rbrace} du \end{aligned} \end{equation}

The process $(m^N(t))$ evolves on a very rapid time-scale compared with the process ${\bar{X}}^N(t)\stackrel{def}{=}( {\bar{X}}_1^N(t), {\bar{X}}_2^N(t))$. One can see that, while the velocity of the process $({\overline{X}}^N(t))$ is of the order O(1), the velocity of the process $(m^N(t))$ is much faster than $({\overline{X}}^N(t))$ and is of the order O(N).

We consider as in Hunt and Kurtz [Reference Hunt and Kurtz5] the random measure ν ^N on $[0,+\infty[\times \bar{\mathbb{N}}$ defined by

(20)\begin{equation} \nu^N((0,t)\times \Gamma)=\int_0^t\mathbb{1}_{\{m^N(u)\in\Gamma\}}du \end{equation}

for all $t\in [0,+\infty[$ and Γ in a σ-algebra $\mathcal{B}(\bar{\mathbb{N}})$. Note that the measure ν ^N satisfies the condition $\nu^N((0,t)\times \bar{\mathbb{N}})=t$. There is a subsequence of the sequence $(\nu^N)$ that converges in distribution to random measure ν satisfying $\nu((0,t)\times ~\bar{\mathbb{N}})=t$. (see Hunt and Kurtz [Reference Hunt and Kurtz5] for more details). In terms of the random measure ν ^N equations (18), (19) becomes

(21)\begin{equation} \begin{aligned} {\bar{X}}_{1}^{N}(t) = & {\bar{X}}_{1}^{N}(0)+ {\bar{M}}_{1}^{N}(t)-\lambda t-\mu \int_{0}^{t} {\bar{X}}_{1}^{N}(u) du \\ &\quad +2\mu \int_{0}^{t} {\bar{X}}_2^{N}(u) du + \lambda \int_{0}^{t} \mathbb{1}_{\lbrace {\bar{X}}_{1}^{N}(u) \gt 0, m^N(u)=0\rbrace} du \\ &\qquad+\lambda \int_{0}^{t} \mathbb{1}_{\lbrace {\bar{X}}_{1}^{N}(u)=0\rbrace} du \end{aligned} \end{equation}

(22)\begin{equation} \begin{aligned} {\bar{X}}_2^N(t) =& {\bar{X}}_2^N(0) + {\bar{M}}_{2}^{N}(t)+(\lambda+\xi)t-2\mu \int_{0}^{t} {\bar{X}}_2^{N}(u) du\\ &\quad -\xi \nu^N([0,t]\times\{0,1\}) -\lambda \int_{0}^{t} \mathbb{1}_{\lbrace {\bar{X}}_{1}^{N}(u) \gt 0 ,m^N(u)=0\rbrace} du\\ &\qquad- \lambda \int_{0}^{t} \mathbb{1}_{\lbrace {\bar{X}}_{1}^{N}(u)= 0\rbrace} du \end{aligned} \end{equation}

The above equations can be rewritten in the matrix form as follows:

(23)\begin{equation} \begin{aligned} {\bar{X}}^N(t)&={\bar{X}}^N(0)+{\bar{M}}^N(t)+t\bar{\theta}-\xi\mathcal{V}^N(t,\{0,1\})\\ &\qquad+ \int_{0}^t A{\bar{X}}^N(s)ds+RY^N(t) \end{aligned} \end{equation}

where

\begin{equation*} {\bar{X}}^N(t)=\begin{pmatrix}{\bar{X}}_{1}^{N}(t)\\ {\bar{X}}_{2}^N(t)\end{pmatrix}\; {\bar{M}}^N(t)=\begin{pmatrix} \frac{{M}_{1}^{N}(t)}{N}\\ \frac{{M}_{2}^{N}(t)}{N} \end{pmatrix}\end{equation*}

\begin{equation*} \bar{\theta}=\begin{pmatrix} -\lambda\\\xi +\lambda\end{pmatrix}, \; A=\begin{pmatrix} -\mu & 2\mu \\ 0 & -2\mu \end{pmatrix},\;\; R=\begin{pmatrix} \lambda & \lambda\\ -\lambda & -\lambda\end{pmatrix}\end{equation*}

\begin{equation*}\mathcal{V}^N(t,\{0,1\})=\begin{pmatrix} \nonumber 0 \\ \nu^N([0,t] \times \{0,1\})\end{pmatrix}\end{equation*}

\begin{equation*} Y^N(t)=\begin{pmatrix} \int_{0}^{t} \mathbb{1}_{\{{\bar{X}}_1^N(u)= 0\}}du\\ \int_{0}^{t} \mathbb{1}_{\lbrace {\bar{X}}_{1}^{N}(u) \gt 0 ,m^N(u)=0\rbrace} du\end{pmatrix}\end{equation*}

As illustrated in Figure 1 the couple of processes $( {\bar{X}}^N(t) )$ and $(Y^N (t))$ can be interpreted as the solution of the Skorokhod problem associated with data $(\bar{\theta} ,\nu^N, A,R,{\mathcal{S}})$ and

(24)\begin{equation} {\bar{V}}^N(t)={\bar{X}}^N(0)+{\bar{M}}^N(t)+t\bar{\theta}-\xi\mathcal{V}^N(t,\Gamma)+ \int_{0}^t A{\bar{X}}^N(s)ds \end{equation}

The following graphic illustrates the simulation for the process $ ( {\overline{X}}_1(t), {\overline{X}}_2(t))$, and it has been shown that the process $ ( {\overline{X}}_1(t))$ is well reflected at the boundary ${(\partial{\mathcal{S}})}_1$ and the process $ ( {\overline{X}}_1(t) + 2 ( {\overline{X}}_2(t)) )$ is reflected at the boundary $(\partial{\mathcal{S}})_2$.

Figure 1. Simulation of the process $({\overline{X}}_1^N(t), {\overline{X}}_2^N(t))$ in the convex ${\mathcal{S}}$

In the next theorem, we prove the relative compactness of the sequence of processes $\left({\overline{X}}^{N}(.),Y^{N}(.), {\mathcal{V}}^N(.)\right)$ in $\mathcal{D}({\mathbb{R}}_+,{\mathbb{R}}^2)\times {\mathcal{M}}_{1}({\mathbb{R}}_+\times\bar{\mathbb{N}})$. Where ${\mathcal{M}}_{1}({\mathbb{R}}_+\times\bar{\mathbb{N}})$ is the space of Radon measures on ${\mathbb{R}}_+\times\bar{\mathbb{N}}$.

Theorem 3.2 Suppose that

\begin{eqnarray*} \nonumber \underset{N \rightarrow + \infty}{lim} ( {\overline{X}}_1^N(0), {\overline{X}}_2^N(0)) = (x_1, x_2) \in {\mathcal{S}}, \end{eqnarray*}

the sequence $\left({\overline{X}}^{N}(.),Y^{N}(.), \nu^N(.)\right)$ is then relatively compact in $\mathcal{D}({\mathbb{R}}_+,{\mathbb{R}}^3)$ and the limit $\left( x(.) ,y(.), \nu(.)\right)$ of any convergent subsequence satisfies:

(25)\begin{equation} \begin{aligned} x_1(t) &= x_1 - \lambda t-\mu \int_{0}^t x_1(s)ds + 2\mu \int_{0}^t x_2(s) ds \\ &+ \lambda \int_{[0,t] \times \mathbb{N}} \mathbb{1}_{\lbrace x_1(s) \gt 0 \rbrace} \mathbb{1}_{\lbrace 0 \rbrace} (u) \nu (ds \times du) +\lambda y_1(t) \end{aligned} \end{equation}

(26)\begin{equation} \begin{aligned} x_2(t) &= x_2 + (\lambda+\xi) t -2 \mu \int_{0}^t x_2(s) ds- \xi \nu ([0,t] \times \{0,1\}) \\ & - \lambda \int_{[0,t] \times \mathbb{N}} \mathbb{1}_{\lbrace x_1(s) \gt 0 \rbrace} \mathbb{1}_{\lbrace 0 \rbrace} (u) \nu (ds \times du) - \lambda y_1(t) \end{aligned} \end{equation}

where the function y₁ is a non-decreasing function with $y_1(0)=0$, and for $t\geq 0$

\begin{equation*}\begin{aligned} y_1(t)=\int_{0}^{t}\mathbb{1}_{\{x_1(s)=0\}} dy_1(s) \end{aligned} \end{equation*}

Lemma 3.3. The sequences of processes $\left(\frac{{M}_{1}^{N}(t)}{N}\right)_{t\geq0}$ and $\left(\frac{{M}_{2}^{N}(t)}{N}\right)_{t\geq0}$ converge in distribution to 0 uniformly on compact sets.

Proof. Doob’s inequalities show that, for ϵ > 0 and $t \geq 0$

\begin{equation*} \mathbb{P}\left( \underset{0 \leq s\leq t }{sup}\frac{{M}_{i}^{N}(s)}{N} \geq \epsilon \right)\leq \frac{1}{\epsilon^{2} N^{2}} \mathbb{E} ( \langle {M}_{i}^{N}\rangle(t))\end{equation*}

From equations (11), (12) one gets

\begin{equation*} \begin{aligned} \mathbb{E} ( \langle {M}_{1}^{N}\rangle(t))&\leq \mu {F}_{N}+\lambda Nt\\ \mathbb{E} ( \langle {M}_{2}^{N}\rangle(t))&\leq \mu {F}_{N}+(\lambda+\xi) Nt\\ \end{aligned} \end{equation*}

Then from (1) the sequences of processes $\left(\frac{{M}_{1}^{N}(t)}{N}\right)_{t\geq0}$ and $\left(\frac{{M}_{2}^{N}(t)}{N}\right)_{t\geq0}$ converge in distribution to 0 uniformly on any bounded time interval.

Proof. proof of Theorem 3.2

First, we prove the relative compactness of the process

\begin{equation*}{\bar{X}}^N(t)=\begin{pmatrix}{\bar{X}}_{1}^{N}(t)\\ {\bar{X}}_{2}^N(t)\end{pmatrix} \end{equation*}

For this we prove separately that $({\overline{X}}_1^{N}(t))$ and $({\overline{X}}_2^{N}(t))$ are tight.

For $T \gt 0, \delta \gt 0$ we denote by $\omega_{g}^T(\delta)$ the modulus of continuity of the function g on $[0,T]$:

(28)\begin{equation} \omega_{g}^T(\delta)=\sup_{0\leq s\leq t\leq T,|t-s|\leq \delta}\vert g(t)-g(s)\vert \end{equation}

The equation (21) shows that the processes $({\overline{X}}_1^{N}(t),Y_1^N(t))$ with $Y_1^N(t)=\lambda\int_{0}^{t} \mathbb{1}_{\{X_1^N(u)= 0\}} du)$ is the unique solution of the Skorokhod problem associated to the process

(29)\begin{equation}\begin{aligned} {\overline{V}}_1^N(t)& = {\overline{X}}_1^N(0) + {\overline{M}}_1^N(t) - \lambda t + \mu \int_{0}^t (2 {\overline{X}}_2^N(u) - {\overline{X}}_1^N(u) ) du \\ &+ \lambda \int_{[0,t] \times \mathbb{N}} \mathbb{1}_{\lbrace {X_1}^{N}(u) \gt 0 \rbrace}\mathbb{1}_{\lbrace 0 \rbrace} (y) \nu^N (ds \times dy) \end{aligned} \end{equation}

By using explicit representation of the solution of the Skorokhod in dimension 1, see El Karoui and Chaleyat-Maurel [Reference Karoui and Chaleyat-Maurel6], one has

\begin{equation*} \|{\overline{X}}_1^N\|_{\infty,t}\stackrel{def}{=}\sup_{0\leq s\leq t}|{\overline{X}}_1^N(s)|\leq 2 \|{\overline{V}}_1^{N}\|_{\infty,t} \end{equation*}

and

\begin{equation*}|\lambda Y_{1}^N(t)|\leq \|{\overline{V}}_1^{N}\|_{\infty,t}\end{equation*}

By equation (29), one gets that

\begin{equation*}\begin{aligned}\nonumber \|{\overline{V}}_1^{N}\|_{\infty,t}&\leq |{\overline{X}}_1^N(0))|+ 2 \lambda t+\mu \int_0^t\|{\overline{X}}_1^N\|_{\infty,s}ds\\ \nonumber &+2\mu\int_0^t\|{\overline{X}}_2^N\|_{\infty,s}ds+ \|{\overline{M}}_1^{N}\|_{\infty,t} \end{aligned} \end{equation*}

and using inequalities given above,

\begin{equation*}\begin{aligned}\nonumber \|{\overline{X}}_1^N\|_{\infty,t}&\leq 2 |{\overline{X}}_1^N(0)|+4 \lambda t+2\mu\int_0^t\|{\overline{X}}_1^N\|_{\infty,s}ds\\ \nonumber &+4\mu\int_0^t\|{\overline{X}}_2^N\|_{\infty,s}ds+ 2\|{\overline{M}}_1^{N}\|_{\infty,t}\nonumber \end{aligned} \end{equation*}

\begin{equation*}\begin{aligned}\nonumber \|{\overline{X}}_2^N\|_{\infty,t}&\leq |{\overline{X}}_1^N(0)|+|{\overline{X}}_2^N(0)|+(4\lambda+3\xi) t + \|{\overline{M}}_1^{N}\|_{\infty,t}+\|{\overline{M}}_2^{N}\|_{\infty,t}\\ \nonumber &+\mu\int_0^t\|{\overline{X}}_1^N\|_{\infty,s}ds+4\mu\int_0^t\|{\overline{X}}_2^N\|_{\infty,s}ds \end{aligned} \end{equation*}

Then

\begin{equation*}\nonumber \|{\overline{X}}_1^N\|_{\infty,t}+\|{\overline{X}}_2^N\|_{\infty,t}\leq H^N(T)+8\mu\int_0^t(\|{\overline{X}}_1^N\|_{\infty,s}+\|{\overline{X}}_2^N\|_{\infty,s})ds \end{equation*}

with

\begin{equation*}H^N(t)=3|{\overline{X}}_1^N(0)|+ |{\overline{X}}_2^N(0)|+(8\lambda+3\xi)T+3\|{\overline{M}}_1^{N}\|_{\infty,T}+\|{\overline{M}}_2^{N}\|_{\infty,T}\end{equation*}

Gronwall’s lemma gives that the relation

\begin{equation*}\|{\overline{X}}_1^N\|_{\infty,t}+\|{\overline{X}}_2^N\|_{\infty,t}\leq H^N(T)e^{8\mu t}\end{equation*}

holds for all $t\in[0,T]$. The convergence of martingales and of $|{\overline{X}}_1^N(0)|$, $|{\overline{X}}_2^N(0)|$ shows that the sequence $(H^N(T))$ converges in distribution. Consequently for ϵ > 0, there exists some C > 0 such that for $i=1,2$ and all $N\in\mathbb{N}$

\begin{equation*}\mathbb{P}\left(\|{\overline{X}}_1^N\|_{\infty,t} + \|{\overline{X}}_2^N\|_{\infty,t} \gt C\right)\leq \epsilon. \end{equation*}

If η > 0, there exists N ₁ and δ > 0 such that for all $N\geq N_1$

\begin{equation*}\delta(\lambda+4\mu C)\leq \frac{\eta}{2}\end{equation*}

and

\begin{equation*}\mathbb{P}\left(\omega_{{\overline{M}}_{1}^{N}}^T(\delta)\geq \frac{\eta}{2}\right)\leq \epsilon\end{equation*}

One gets finally

\begin{align*} \mathbb{P}\left(\omega_{{\overline{V}}_{1}^{N}}^T(\delta)\geq \eta\right)&\leq\mathbb{P}\left(2\lambda\delta+2\mu\delta(\|{\overline{X}}_1^N\|_{\infty,T}+\|{\overline{X}}_2^N\|_{\infty,T})\geq \frac{\eta}{2}\right)\\ &+\mathbb{P}\left(\omega_{{\overline{M}}_{1}^{N}}^T(\delta)\geq \frac{\eta}{2}\right)\leq 3\epsilon \end{align*}

Consequently the sequence $({\overline{V}}_{1}^{N}(t))$ is tight and by continuity of the solution of the Skorokhod problem in dimension 1 the sequences $({\overline{X}}_1^N(t))$ and $(\overline{Y}_1^{N}(t))$ are tight, see Billingsley [Reference Patrick8].

From equation ((22)) one gets for s < t :

\begin{equation*} \begin{aligned} \nonumber |{\overline{X}}_2^N(t)-{\overline{X}}_2^N(s)| &\leq (\lambda+\xi)(t-s)+2\mu\int_s^t|{\overline{X}}_2^N(u)|du+|{\overline{M}}_{2}^{N}(t)-{\overline{M}}_{2}^{N}(s)| \\ \nonumber &+ (2\lambda+\xi)(t-s)+\lambda (Y_1^N(t)-Y_1^N(s)) \end{aligned} \end{equation*}

and

(30)\begin{equation} \begin{aligned} \mathbb{P}\left(\omega_{{\overline{X}}_2^N}^T(\delta) \geq \eta\right) &\leq \mathbb{P}\left(\omega_{{\overline{M}}_{2}^{N}}^T(\delta)\geq \eta/3\right) +\mathbb{P}\left(\omega_{Y_{1}^{N}}^T(\delta)\geq \eta/3\right)\\ &+\mathbb{P}\left(2\mu\delta\|{\overline{X}}_2^N\|_{\infty,T}+\delta(2 \lambda+3 \xi) \geq \eta/3\right) \end{aligned} \end{equation}

There exists $N_1\geq 0$ such that $\delta(2\mu C-\xi)\leq\epsilon$ and

\begin{equation*}\mathbb{P}\left(\omega_{{\overline{M}}_{2}^{N}}^T(\delta)\geq \frac{\eta}{3}\right)\leq \epsilon\end{equation*}

and

\begin{equation*}\mathbb{P}\left(\lambda\omega_{\frac{{Y}_{1}^{N}}{N}}^T(\delta)\geq \frac{\eta}{3}\right)\leq \epsilon\end{equation*}

and, consequently

\begin{equation*}\mathbb{P}\left(\omega_{{\overline{X}}_2^N}^T(\delta)\geq \eta\right)\leq 3\epsilon\end{equation*}

The sequence ${\overline{X}}_2^N$ is therefore tight.

It remains to prove the relative compactness of the sequence of random measures ν ^N on $[0,+\infty[\times\bar{\mathbb{N}}$. Since for all N and all $t\geq 0$

\begin{equation*} \nu^N([0,t[\times\bar{\mathbb{N}})=t\end{equation*}

The result is given in Hunt and Kurtz [Reference Thomas17] Lemma 1.3.

Remark 3.4. The dynamical system associated to the equations given in (25) and (26) are given by

\begin{equation*} \left\{\begin{array}{cc} x_1(t)=&x_1-\lambda t-\mu\int_0^tx_1(s)ds+2\mu\int_0^tx_2(s)ds+\lambda y_1(t)\\ x_2(t)=&x_2+(\lambda+\xi) t-2\mu\int_0^tx_2(s)ds-\lambda y_1(t) \end{array} \right. \end{equation*}

The unique solution of this reflected ordinary differential equations noted $x(t) = (x_1(t), x_2(t))$ is given by

1. (1) If $ (x_1,x_2)\in{{\mathcal{S}}}_1\cup{{\mathcal{S}}}_2 $, then for all $ t \geq 0 $,

   (31)\begin{equation}\begin{aligned} \left\{\begin{array}{ll} &{x}_1(t)=\left(x_1+2x_2-\dfrac{\lambda+2\xi}{\mu}\right)e^{-\mu t}-\left(2x_2-\dfrac{\lambda+\xi}{\mu}\right)e^{-2\mu t}+\dfrac{\xi}{\mu} \\ &{x}_2(t)=\dfrac{\lambda+\xi}{2\mu}+\left(x_2-\dfrac{\lambda+\xi}{2\mu}\right)e^{-2\mu t} \end{array} \right. \end{aligned} \end{equation}

2. (2) If $ (x_1,x_2)\in{{\mathcal{S}}}_3 $, $ x_1 = 0 $, then

   (32)\begin{align} x_1(t) &= \frac{\xi}{\mu}\left(e^{-\mu (t-{\tau}_1)}-1\right)^2 \mathbb{1}_{[{\tau}_1,+\infty[}(t) \end{align}

   (33)\begin{align} x_2(t) &= \left( x_2+\xi t \right) \mathbb{1}_{[0,{\tau}_1]}(t) + \left(\frac{\lambda}{2\mu} + \frac{\xi}{2\mu}\left(1 - e^{-2\mu (t-{\tau}_1)}\right)\right)\mathbb{1}_{[{\tau}_1,+\infty[}(t) \end{align}

   (34)\begin{align} y_1(t) &= \frac{(\lambda - 2 \mu x_2) t - \mu \xi t^2}{\lambda} \mathbb{1}_{[0,{\tau}_1]}(t) + \frac{(\lambda - 2 \mu x_2)^2}{4\lambda \mu \xi }\mathbb{1}_{[{\tau}_1,+\infty)}(t) \end{align}

   where $ {\tau}_1 = \frac{\lambda - 2 \mu x_2}{2 \mu \xi} $.

With

\begin{equation*} {{\mathcal{S}}}_1= \left\{(x_1,x_2)\in S\;|\;\bigl(x_1+2x_2-\frac{\lambda +2\xi}{\mu}\bigr)\bigl(2x_2-\frac{\lambda+\xi}{\mu}\bigr)\leq 0\right\} \end{equation*}

\begin{equation*} {{\mathcal{S}}}_2= \left\{(x_1,x_2)\in S\;|\;x_1+2x_2 \gt \frac{\lambda +2\xi}{\mu}\;,\;\frac{\lambda+\xi}{\mu} \lt 2x_2\right\} \end{equation*}

\begin{equation*} {{\mathcal{S}}}_3= \left\{(x_1,x_2)\in S\;|\;x_1+2x_2 \lt \frac{\lambda +2\xi}{\mu}\;,\;\frac{\lambda+\xi}{\mu} \gt 2x_2\right\} \end{equation*}

See (4) for the explicit solution to the reflected ODE obtained above. This dynamical system admits a unique equilibrium point

\begin{equation*} \left(\frac{\xi}{\mu},\frac{\lambda+\xi}{2\mu}\right) \end{equation*}

Thus, according to the position of this equilibrium point in the convex set ${\mathcal{S}}$, three possible regimes can be considered. Let

\begin{equation*} \rho \stackrel{def}{=} \frac{\lambda+ 2 \xi}{\mu}\end{equation*}

The under-loaded regime ( $\rho \lt {\bar{\beta}}$), the critically loaded regime ( $\rho = {\bar{\beta}}$) and the overloaded regime ( $\rho \gt {\bar{\beta}}$). Each of the aforementioned regimes will be developed in detail in the next sections.

4. The under-loaded regime

Throughout this section, we assume that the condition

(35)\begin{equation} \rho \lt {\bar{\beta}} \end{equation}

holds.

In the Under-loaded regime, the equilibrium point ρ is less than ${\bar{\beta}}$, and the figure below, fig. 2, illustrates the stabilization of the process $({\overline{X}}_1(t), {\overline{X}}_2(t))$ at the equilibrium point and never reaches the boundary $(\partial{\mathcal{S}})_2$.

Figure 2. Simulation of the process $({\overline{X}}_1(t), {\overline{X}}_2(t))$ with respect to the boundary $(\partial{\mathcal{S}})_2$

Let $(X_1^N(t))$ and $(X_2^N(t))$ the processes given, respectively, by equations (7) and (8). Recall that $(X_1^N(t)+X_2^N(t))$ is the process describing the total number of files that are present in the system at time t. Let $(Z^N(t))$ be the process given by

(36)\begin{equation} Z^N(t)=\dfrac{X_1^N(t)+2X_2^N(t)-N\rho}{\sqrt{N}} \end{equation}

The Q-matrix $Q^{N} = (q^{N}(.,.))$ of the Markov process $ (X_1^N(t) ,X_2^N(t) ,Z^N(t)) $ is defined by;

For $(x_1,x_2)\in{\mathcal{D}}^N$ and $z=\dfrac{x_1+2x_2-N\rho}{\sqrt{N}}$

(37)\begin{equation} (x_1,x_2,z) \longrightarrow (x_1,x_2,z)+ \left\{ \begin{array}{ll} (0,1,\frac{2}{\sqrt{N}}) \ \ \xi N \mathbb{1}_{\lbrace z \lt \frac{{F}_{N} -1}{\sqrt{N}}-N\rho \rbrace} \\ (1,-1,-\frac{1}{\sqrt{N}})\ \ 2 \mu x_2\\ (-1,1,\frac{1}{\sqrt{N}}) \ \ \lambda N \mathbb{1}_{\lbrace x_1 \gt 0, z \lt \frac{{F}_{N}}{\sqrt{N}}-N\rho \rbrace}\\ (-1,0,-\frac{1}{\sqrt{N}}) \ \ \mu x_1 \end{array} \right. \end{equation}

and the generator of $ (X_1^N(t) ,X_2^N(t) ,Z^N(t)) $ is given by,

\begin{align*} A^N f(x_1, x_2, z) & = \xi N \mathbb{1}_{\lbrace z \lt \frac{{F}_{N} - N \rho - 1}{\sqrt{N}} \rbrace} [f(x_1 , x_2 +1, z+ \frac{2}{\sqrt{N}}) - f(x_1, x_2, z)] \\ &+ \lambda N \mathbb{1}_{\lbrace x_1 \gt 0 \ , \ z \lt \frac{{F}_{N} - N \rho}{\sqrt{N}} \rbrace} [f(x_1 - 1, x_2 + 1, z+ \frac{1}{\sqrt{N}})- f(x_1, x_2, z)] \\ & + \mu x_1 [f(x_1 - 1, x_2, z- \frac{1}{\sqrt{N}}) - f(x_1, x_2, z)] \\ &+ 2 \mu x_2 [f(x_1 + 1, x_2 - 1, z- \frac{1} {\sqrt{N}}) - f(x_1, x_2, z)] \end{align*}

For any function f depending only on the third variable z, i.e.,

\begin{equation*} f(x_1,x_2,z)=g(z)\quad \forall\;(x_1,x_2)\in\mathbb{N}^2 \quad with \quad x_1 \gt 0 \end{equation*}

for some twice differentiable function g on $\mathbb{R}$ one gets

\begin{align*} \nonumber A^N g( z) & = \xi N \mathbb{1}_{\lbrace z \lt \frac{{F}_{N} - N \rho - 1}{\sqrt{N}} \rbrace} [g( z+ \frac{2}{\sqrt{N}}) - g( z)] \\ \nonumber &+ \lambda N \mathbb{1}_{\lbrace z \lt \frac{{F}_{N} - N \rho}{\sqrt{N}} \rbrace} [g( z+ \frac{1}{\sqrt{N}})- g( z)] \\ \nonumber & + \mu(\sqrt{N}z+N\rho)[g(z- \frac{1}{\sqrt{N}}) - g( z)] \end{align*}

Remark that condition (35) implies that terms $\frac{{F}_{N} - N \rho - 1}{\sqrt{N}}$ and $\frac{{F}_{N} - N \rho}{\sqrt{N}}$ converge to $+\infty$. Thus the generator converges to

\begin{equation*} -\mu zg'(z)+(\lambda+3\xi)g''(z)\qquad z\in\mathbb{R} \end{equation*}

when $N\rightarrow +\infty$, which is the generator of an Ornstein–Uhlenbeck process with variance converges to $\frac{\lambda+3\xi}{\mu}$. By results given in Ethier and Kurtz [Reference Ethier and Kurtz2] one can see that for some positive constant α the process $(X_1^N(t)+2X_2^N(t))$ lives in $[N\rho-\alpha N, N\rho+\alpha N]\subset [0,N{\bar{\beta}}]$ and the probability of saturation of the system is therefore small. In the under-loaded regime one can suppose that the capacity of the system is infinite, i.e., ${F}_{N}=+\infty$. In this case, the complete study of the process $(X_1^N(t),X_2^N(t))$ is made in the article El Kharroubi and El Masmari [Reference Kharroubi and Masmari7].

5. The overloaded regime

Throughout this section, we assume that the condition

(38)\begin{equation} \rho \gt {\bar{\beta}} \end{equation}

holds.

In the Overloaded regime, the equilibrium point ρ exceeds $ {\bar{\beta}} $, and the figure below, fig. 3, illustrates that the process $ ({\overline{X}}_1(t), {\overline{X}}_2(t)) $ being constrained by the boundary $ (\partial{\mathcal{S}})_2 $ and never reaching the equilibrium point.

Figure 3. Simulation of the process $({\overline{X}}_1(t), {\overline{X}}_2(t))$ with respect to the boundary $(\partial{\mathcal{S}})_2$

the Q-matrix $Q^{N} = (q^{N}(.,.))$ and the generator of the Markov process $ ({X_1}^{N}(t/N), {X_2}^{N}(t/N) ,m^N(t/N)) $ are given by,

\begin{equation*} \nonumber \left\{ \begin{array}{ll} q^{N}((x_1,x_2, m),(x_1 - 1, x_2, m+1)= \frac{1}{N} \mu x_1\\ q^{N}((x_1,x_2, m),(x_1 +1, x_2-1, m+ 1)= \frac{2}{N} \mu x_2 \\ q^{N}((x_1,x_2, m),(x_1 -1, x_2 + 1, m- 1)= \lambda \mathbb{1}_{\lbrace x_1 \gt 0 \ , \ m\geq 1 \rbrace} \\ q^{N}((x_1,x_2, m),(x_1 , x_2 +1, m- 2)= \xi \mathbb{1}_{\lbrace m\geq 2 \rbrace} \end{array} \right. \end{equation*}

\begin{align*} A_N f(x_1, x_2, m) & = \frac{1}{N} \mu x_1 [f(x_1 - 1, x_2, m+1) - f(x_1, x_2, m)] \\ &+ \frac{2}{N} \mu x_2 [f(x_1 +1, x_2-1, m+ 1)- f(x_1, x_2, m)] \\ & + \lambda \mathbb{1}_{\lbrace x_1 \gt 0 \ , \ {F}_{N} - (2 x_2 + x_1) \geq 1 \rbrace} [f(x_1 -1, x_2 + 1, m- 1) - f(x_1, x_2, m)] \\ &+ \xi \mathbb{1}_{\lbrace {F}_{N} - (2 x_2 + x_1) \geq 2 \rbrace} [f(x_1 , x_2 +1, m- 2) - f(x_1, x_2, m)] \end{align*}

For any function f depending only on the third variable m, i.e.,

\begin{equation*} f(x_1,x_2,m)=g(m)\quad \forall\;(x_1,x_2)\in\mathbb{N}^2 \end{equation*}

for some function g on $\mathbb{N}$ one gets

\begin{equation*}\begin{aligned}\nonumber A_N g( m) & = \mu\frac{{F}_{N}-m}{N}\left(g(m+1) - g(m)\right) \\ \nonumber &+ \lambda \mathbb{1}_{\lbrace x_1 \gt 0 \ , m \geq 1 \rbrace} [g( m- 1) - g(m)] \\ \nonumber &+ \xi \mathbb{1}_{\lbrace m \geq 2 \rbrace} [g(m- 2) - g( m)] \end{aligned} \end{equation*}

This generator converges to

\begin{equation*}\begin{aligned}\nonumber Ag( m) & = \mu{\bar{\beta}}\left(g(m+1) - g(m)\right) \\ \nonumber &+ \lambda \mathbb{1}_{\lbrace x_1 \gt 0, m \geq 1 \rbrace} [g( m- 1) - g(m)] \\ \nonumber &+ \xi \mathbb{1}_{\lbrace m \geq 2 \rbrace} [g(m- 2) - g( m)] \end{aligned} \end{equation*}

Thus, for any $x=(x_1,x_2)\in\mathbb{N}^*\times\mathbb{N}$, this is the generator of the Markov process $(m(t))$ with transitions

(39)\begin{equation} m \longrightarrow m+ \left\{ \begin{array}{ll} +1 \ \ \mu {\bar{\beta}} \\ -1 \ \ \lambda \mathbb{1}_{\{m \geq 1\}} \\ -2 \ \ \xi \mathbb{1}_{\{m \geq 2\}} \end{array} \right. \end{equation}

Proposition 5.1. Under the condition (38), the process $(m(t))$ has a unique invariant distribution π, its generating function $g(u) = {\underset{n \geq 0}{\sum }} \pi(n) u^n$ is given by, for $u\in[-1,1]$

(40)\begin{equation} g(u) = \frac{1}{-\mu {\bar{\beta}}+ (\lambda+\xi) u +\xi} \left[(\lambda u+\xi (1+u)) \pi(0) +\xi (1+u) u \pi(1)\right] \end{equation}

Where $( \pi(0), \pi(1))$ are given by

(41)\begin{equation} \pi(0) = \frac{({1+y}_{*}) (\lambda+2 \xi - \mu {\bar{\beta}})}{(\lambda+2\xi)( 1+y_{*}) -2\mu{\bar{\beta}}y_{*}} \end{equation}

(42)\begin{equation} \pi(1)= \frac{-\mu {\bar{\beta}} + \lambda +2 \xi}{2 \xi} - \frac{\lambda + 2 \xi}{2 \xi } \pi(0) \end{equation}

with

\begin{equation*} y_{*} = \frac{(\lambda+\xi)- \sqrt{(\lambda+\xi)^2 + 4 \xi \mu {\bar{\beta}}}}{2 \mu {\bar{\beta}}} \end{equation*}

Proof. The existence and uniqueness of the stationary distribution is a simple consequence of Foster’s criterion. See Proposition 8.14 of Robert [Reference Robert13]. For $u \in [-1,1]$, define

\begin{equation*}g(u) = {\underset{n \geq 0}{\sum }} \pi(n) u^n\end{equation*}

The equilibrium equation

(43)\begin{equation} \begin{aligned} &\sum_{m=0}^{+\infty}[\mu{\bar{\beta}}\left(f(m+1)-f(m)\right)+\lambda\mathbb{1}_{\lbrace x_1 \gt 0 \ , m \geq 1 \rbrace}\left(f(m-1)-f(m)\right)\\ &\qquad+\xi\mathbb{1}_{\lbrace m \geq 2 \rbrace}\left(f(m-2)-f(m)\right)] \pi(m)=0 \end{aligned} \end{equation}

for $f(m)=u^m$, gives the following relation

\begin{equation*} g(u) (\mu {\bar{\beta}} u^2 (u-1) + \lambda (u-u^2) +\xi (1-u^2)) = \lambda (u-u^2) \pi(0) + \xi (1-u^2) ( \pi(0) + u \pi(1)) \end{equation*}

Let

\begin{equation*}P(u)\stackrel{def}{=} - \mu {\bar{\beta}} u^2 +(\lambda +\xi) u +\xi \end{equation*}

then we have

(44)\begin{equation} P(u) g(u) = ((\lambda+\xi) u+ \xi )\pi (0) + \xi (1+u) u \pi(1)) \end{equation}

Note that $P(-1)=-( \mu {\bar{\beta}}+\lambda) \lt 0$, $P(0)=\xi$ and $P(1)=-\mu {\bar{\beta}}+\lambda+2\xi \gt 0$ by Condition (38). The function P(u) has a unique root in $[-1,1]$ and it is necessarily $y_*$.

We have therefore that $y_*$ is a root of the RHS of the Relation (44), hence

\begin{equation*} \mu{\bar{\beta}}y_*^2\pi(0)+\xi y_*(1+y_*)\pi(1)=0 \end{equation*}

and the relation $g(1)=1$ gives the additional identity

\begin{equation*} \frac{\lambda+2\xi}{2 \xi} \pi(0)+\pi(1) =\frac{\lambda + 2 \xi - \mu {\bar{\beta}}}{2 \xi} \end{equation*}

The proposition is proved.

5.1. Fluid limits

Our aim in this section is to identify the limit of the renormalized processes $({\bar{X}}_1^N(t))$ and $({\bar{X}}_2^N(t))$ given, respectively, by equations (18) and (19). We assume that

(45)\begin{equation} \underset{N \rightarrow + \infty}{\text{lim}} \left( {\overline{X}}_1^N(0), {\overline{X}}_2^N(0) \right) = \left( x_1, x_2 \right) \end{equation}

and we successively study the cases where $(x_1,x_2)$ is chosen inside the set ${\mathcal{S}}$ and the case where $(x_1,x_2)$ lies on the boundary $(\partial {\mathcal{S}})_2$.

5.1.1. Starting from the interior of $ {\mathcal{S}}$

Let $T_1^N$ be the hitting time

\begin{equation*}T_1^N=\inf\{t \gt 0\;|\; m^N(t)\in\{0,1\}\}\end{equation*}

Note that before time $T_1^N$ the Markov process $(X_1^N(t),X_2^N(t))$ coincides with the Markov process describing the storage process with infinite capacity ( ${F}_{N}=+\infty$).

The Proposition 5.4 proves the convergence in distribution of the hitting time $T_1^N$. The proof of this result is inspired by the study of $M/M/N/N$ queue ( see Robert [Reference Robert13] and Fricker, Robert, and Tibi [Reference Fricker, Robert and Tibi4]). Let $\phi_c^N$ be the function on ${\mathbb{R}}^+$ defined by

\begin{equation*}\phi_c(t) = ce^{\mu t} (\rho + \frac{c \xi }{2 \mu} e^{\mu t})\end{equation*}

for $c \in {\mathbb{R}}^*$, $N\in\mathbb{N}^*$.

Lemma 5.2. Let $v=(1,2)$. The function

\begin{align*} {g}_{c} : &(t,w) \in {\mathbb{R}}^+ \times \mathbb{N}^*\times\mathbb{N} \rightarrow (1+c e^{\mu t} )^ {v\cdot w} e^{- N\phi_c(t)}\\ &\text{where}\qquad v\cdot w=w_1+2w_2 \end{align*}

is space-time harmonic with respect to the Q-matrix Q given in (4) with ${F}_{N}=+\infty$. In other words

\begin{equation*}\frac{\partial {g}_{c}}{\partial t} (t,w) + Q ({g}_{c}) (t,w) =0,\; \text{for all} \ t \in {\mathbb{R}}^+ \; \text{and for all} \; w \in \mathbb{N}^*\times\mathbb{N}\end{equation*}

Proof. For $t\in{\mathbb{R}}_+$ and $w\in\mathbb{N}^*\times\mathbb{N}$

\begin{equation*}\nonumber \begin{aligned} \frac{\partial {g}_{c}}{\partial t} (t,x) & =e^{- \phi_c (t)}ce^{\mu t} \biggl[ v\cdot w \mu(1+c e^{\mu t})^ {v\cdot w-1}\\ & \quad- \biggl( \lambda N + \xi N(2+ c e^{\mu t})\biggr) (1+c e^{\mu t})^{v\cdot w}\biggr] \end{aligned} \end{equation*}

on other hand

\begin{equation*}Q({g}_{c})(t,w)=Q({g}_{c}(t,.))(w)\end{equation*}

is given by

\begin{equation*}\nonumber \begin{aligned} Q({g}_{c})(t,w)& =\lambda N \biggl[(1+c e^{\mu t})^{v\cdot w+1} e^{- \phi_c (t)} - (1+c e^{\mu t})^{v\cdot w} e^{- \phi_c (t)}\biggr] \\ &\qquad + \mu {v\cdot w} \biggl[(1+c e^{\mu t})^{v\cdot w-1} e^{- \phi_c (t)} - (1+c e^{\mu t})^{v\cdot w} e^{- \phi_c (t)}]\\ & \quad\quad\quad+ \xi N \biggl[(1+c e^{\mu t})^{v\cdot w+2} e^{- \phi_c (t)} - (1+c e^{\mu t})^{v\cdot w}e^{- \phi_c (t)}\biggr] \end{aligned} \end{equation*}

\begin{equation*}\nonumber \begin{aligned} & = e^{- \phi_c (t)} \biggl[(\lambda + \xi) N \biggl((1+c e^{\mu t})^{v\cdot w+1} - (1+c e^{\mu t})^{v\cdot w} \biggr)\\ &\qquad +\mu{v\cdot w} \biggl((1+c e^{\mu t})^{v\cdot w-1} - (1+c e^{\mu t})^{v\cdot w}\biggr )\\ & \qquad\qquad+ \xi N \biggl((1+c e^{\mu t})^{v\cdot w+2} -(1+c e^{\mu t})^{v\cdot w+1}\biggr )\biggr] \end{aligned} \end{equation*}

\begin{equation*}\nonumber \begin{aligned} & = e^{- \phi_c (t)} \biggl[- {v\cdot w}c \mu e^{\mu t} (1+c e^{\mu t})^{v\cdot w-1} \\ &\qquad +c e^{\mu t} \biggl(\lambda N + \xi N (2 + c e^{\mu t})\biggr) (1+c e^{\mu t})^{v\cdot w} \biggr]\\ &\qquad\qquad = - \frac{\partial {g}_{c}}{\partial t} (t,w) \end{aligned} \end{equation*}

Proposition 5.3.

1. (1) For $c \in {\mathbb{R}}^*$ and $N\in\mathbb{N}^*$ the process

   (46)\begin{equation} \left( {g}_{c}(t,{X}^N(t))\right) \end{equation}

   is a martingale.

2. (2) For $N\in\mathbb{N}^*$ the following processes are martingales.

   (47)\begin{equation}\left(e^{\mu t} (v\cdot {X}^N(t) - N\rho )\right) \end{equation}

   (48)\begin{equation} \left( e^{2 \mu t} \left((v\cdot {X}^N(t) - N\rho )^2 - v\cdot {X}^N(t) - N\frac{\xi}{\mu}\right)\right) \end{equation}

Proof.

1. (1) By Lemma 5.2 the function $(t,w)\rightarrow {g}_{c}(t,w)$ is space-time harmonic for the Q-matrix Q given in ((4)) with ${F}_{N}=+\infty$. Since $t \rightarrow \frac{\partial {g}_{c}}{\partial t}$ is continuous, then the process $({g}_{c}(t,{X}^N(t))$ is a local martingale (See Corollary B.5 in Robert [Reference Robert13]). Furthermore, for $t \in {\mathbb{R}}^+$,

   \begin{eqnarray*} \nonumber v\cdot {X}^N(t) \leq (2 X_2^N(0) + X_1^N(0))+ 2 {\mathcal{N}}_{\xi_N}(]0,t]) + {\mathcal{N}}_{\lambda_N}(]0,t]) \end{eqnarray*}

   one gets for $t\geq 0$,

   \begin{equation*}\mathbb{E} ( \sup_{0 \leq s \leq t }|{g}_{c}(t,{X}^N(t)|) \lt + \infty\end{equation*}

   Thus the process $({g}_{c}(t,{X}^N(t))$ is a martingale (see proposition A.7 in Robert [Reference Robert13]).

2. (2) Let Ψ be the function on ${\mathbb{R}}^+\times \mathbb{N}$ defined by

   \begin{equation*}\Psi(x,z)=(1+x)^ze^{-N\rho x}e^{-\frac{N\xi}{2\mu}x^2}\end{equation*}

   Note that

   \begin{equation*}\Psi(ce^{\mu t},v\cdot {X}^N(t))={g}_{c}(t,{X}^N(t))\end{equation*}

   and therefore $(\Psi(ce^{\mu t},v\cdot {X}^N(t))$ is a martingale. On other hand, it is well known that

   \begin{equation*}e^{-N\rho x}(1+x)^z=\sum_{n\geq 0}C_n^{N\rho}(z)\frac{{x}^n}{n!} \end{equation*}

   where $C_n^{N\rho}(z)$ is the nth Poisson-Charlier polynomial (see Chihara [Reference Theodore16]). Hence, the expansion of $\Psi(x,z)$ is given by

   (49)\begin{equation} \Psi(x,z)=\sum_{n\geq 0}\left(\sum_{k=0}^nC_{n-k}^{N\rho}(z)b_k\right)\frac{{x}^n}{n!} \end{equation}

   where $b_{2k+1}=0$ and $b_{2k}=\left(-\frac{N\xi}{2\mu}\right)^k$

Replacing in (49) x and z by $ce^{\mu t}$ and $v\cdot {X}^N(t)$, respectively, one gets that for any $n\in\mathbb{N}^*$,

\begin{equation*}\left( e^{n\mu t}(\sum_{k=0}^nC_{n-k}^{N\rho}(v\cdot {X}^N(t))b_k)\right)\end{equation*}

is a martingale. In particular for n = 1 and n = 2 one gets that the processes

\begin{equation*}\left(e^{\mu t}(v\cdot {X}^N(t)-N\rho)\right)\end{equation*}

and

\begin{equation*}\left(e^{2\mu t}(v\cdot {X}^N(t)-N\rho)^2-v\cdot {X}^N(t)-\frac{N\xi}{2\mu})\right)\end{equation*}

are martingales.

Proposition 5.4. if Conditions (38) and (45) hold with $x_1+2x_2 \lt {\bar{\beta}}$ then the hitting time $T_{1}^N$ converges in distribution to T ₀ where

(50)\begin{equation} T_0 = \frac{1}{\mu} \text{log} \left( \frac{\lambda + 2 \xi - \mu (x_1 + 2 x_2)}{\lambda + 2 \xi - \mu {\bar{\beta}}} \right) \end{equation}

Proof. We assume that Conditions (38) and (45) hold with $x_1+2x_2 \lt {\bar{\beta}}$. Doob’s optional stopping Theorem applied to the martingale given in (47) and to $T_1^N$ show that the process

\begin{equation*} \nonumber \left(e^{\mu t \wedge T_{1}^N}\left[ v\cdot {X}^{N}(t \wedge T_{1}^N)- N\rho \right]\right) \end{equation*}

is a martingale. Thus, the following equality holds

\begin{equation*} \mathbb{E} \left(e^{\mu t \wedge T_{1}^N}\left[N\rho -v\cdot {X}^{N}(t \wedge T_{1}^N) \right]\right)= N\rho-v\cdot {X}^{N}(0) \end{equation*}

Since $v\cdot {X}^{N} (t \wedge T_{1}^N) \leq {F}_{N}-1$, one gets that,

\begin{equation*} \nonumber \mathbb{E} (e^{\mu t \wedge T_{1}^N} ) \leq \frac{(\lambda + 2\xi) N - \mu v\cdot {X}^{N}(0)}{(\lambda + 2\xi) N - \mu {F}_{N}+\mu} \end{equation*}

By letting t go to infinity, monotone convergence Theorem shows that

\begin{equation*} \nonumber \mathbb{E} (e^{\mu T_{1}^N} ) \leq \frac{\lambda + 2\xi - \mu v\cdot{\bar{X}}^N(0)}{\lambda + 2\xi - \mu \frac{{F}_{N}}{N}+\frac{\mu}{N}} \end{equation*}

And that implies uniform integrability of the martingale

\begin{equation*} \nonumber \mathbb{E} (e^{\mu t \wedge T_{1}^N} (v\cdot {X}^{N} (t \wedge T_{1}^N)- \rho N)) \end{equation*}

One gets therefore the following identity

(51)\begin{equation}\mathbb{E} (e^{\mu T_1^N}) = \frac{\lambda +2 \xi -\mu v\cdot{\bar{X}}^N(0)}{\lambda + 2 \xi - \mu \frac{{\bar{F}}_N}{N}+\frac{\mu}{N} } \end{equation}

Doob’s optional stopping theorem applied again to the martingale given by (48) and to the stopping time $T_1^N$ shows that the process

\begin{equation*} \nonumber \left(e^{{2 \mu t \wedge T_{1}^N}} (v\cdot {X}^{N}(t \wedge T_{1}^N)- \rho N)^{2} - v\cdot {X}^{N}(t\wedge T_{1}^N) - \frac{\xi N}{\mu}\right) \end{equation*}

is a martingale. Since $v\cdot {X}^{N} (t \wedge T_{1}^N) \leq {F}_{N}-1$, $N{\bar{\beta}} \lt N\rho$ and $N{\bar{\beta}}= {F}_{N}$ one could then use the same arguments used above to get the following identity

(52)\begin{equation}\mathbb{E} (e^{{2 \mu T_{1}^N}}) = \frac{N (v\cdot{\bar{X}}^N(0)- \rho )^{2} - v\cdot{\bar{X}}^N(0) - \frac{\xi }{\mu}}{N(\frac{{F}_{N}-1}{N}- \rho)^{2} - \frac{{F}_{N}-1}{N} - \frac{\xi }{\mu}} \end{equation}

One then deduces that $var(e^{\mu T_{1}^N}) = O(1/N)$ and the Tchebychev inequality implies that, for ϵ > 0,

\begin{equation*}\mathbb{P} (| e^{\mu T_{1}^N} - \mathbb{E} (e^{\mu T_{1}^N} ) | \gt \epsilon) \leq \frac{var(e^{\mu T_{1}^N})}{\epsilon^{2}},\end{equation*}

Hence, using the identity given by (51), the sequence $(T_1^N)$ converges in probability to T ₀.

Theorem 5.5 If Conditions (38) and (45) hold with $x_1+2x_2 \lt {\bar{\beta}}$ and $x_2 \gt \frac{\lambda+\xi}{2\mu}$ Then for the convergence in distribution,

\begin{equation*} \nonumber \underset{N \rightarrow + \infty}{lim} ({\overline{X}}_1^N(t),{\overline{X}}_2^N(t))_{0\leq t \leq T_0} =({\bar{x}}_1(t),{\bar{x}}_2(t))_{0\leq t \leq T_0} \end{equation*}

with $({\bar{x}}_1(t), {\bar{x}}_2(t)) $ are given in (31).

Note that at time T ₀ , the fluid limit $({\bar{x}}_1(t), {\bar{x}}_2(t)) $ hits the boundary ${(\partial{\mathcal{S}})}_2$, i.e., ${\bar{x}}_1(T_0) + 2 {\bar{x}}_2(T_0) = {\bar{\beta}}$.

Proof. We assume that Conditions (38) and (45) hold with $x_1+2x_2 \lt {\bar{\beta}}$ and $x_2 \gt \frac{\lambda+\xi}{2\mu}$. By Theorem 3.2 the sequence

\begin{equation*}\left({\overline{X}}^{N}(t),Y^{N}(t), \nu^N(t)\right)\end{equation*}

is relatively compact in $\mathcal{D}({\mathbb{R}}_+,{\mathbb{R}}^3)$ and the limit $\left( x(.) ,y(.), \nu(.)\right)$ of any convergent subsequence satisfies for all $t\geq 0$:

(53)\begin{equation} \begin{aligned} x_1(t) &= x_1 - \lambda t-\mu \int_{0}^t x_1(s)ds + 2\mu \int_{0}^t x_2(s) ds \\ &+ \lambda \int_{[0,t] \times \mathbb{N}} \mathbb{1}_{\lbrace x_1(s) \gt 0 \rbrace} \mathbb{1}_{\lbrace 0 \rbrace} (u) \nu (ds \times du) +\lambda y_1(t) \end{aligned} \end{equation}

(54)\begin{equation} \begin{aligned} x_2(t) &= x_2 + (\lambda+\xi) t -2 \mu \int_{0}^t x_2(s) ds- \xi \nu ([0,t] \times \{0,1\}) \\ & - \lambda \int_{[0,t] \times \mathbb{N}} \mathbb{1}_{\lbrace x_1(s) \gt 0 \rbrace} \mathbb{1}_{\lbrace 0 \rbrace} (u) \nu (ds \times du) - \lambda y_1(t) \end{aligned} \end{equation}

The condition $x_2 \gt \frac{\lambda+\xi}{2\mu}$ implies that the function $y_1(t)=0$ for all $t\geq 0$ (see Theorem 2 in El Kharroubi and El Masmari [Reference Kharroubi and Masmari7]). Thus, it is sufficient to show that for all $t\leq T_0$

\begin{equation*}\nu ([0,t] \times \{0,1\})=0\end{equation*}

Let us first recall that,

\begin{equation*} \nu^N((0,t)\times \{0,1\})=\int_0^t\mathbb{1}_{\{m^N(u)\in\{0,1\}\}}du\end{equation*}

and that the increasing sequence of hitting times $(T_1^N)$ converges in probability to T ₀. For any $t\leq T_0$ and for any ϵ > 0

\begin{align*} \mathbb{P}\{\sup_{s\leq t}\nu^N((0,s)\times \{0,1\})\geq\epsilon\}&\leq \mathbb{P}\{\sup_{s\leq t\wedge T_1^N }\nu^N((0,s)\times \{0,1\})\geq\epsilon\}\\ \qquad &+ \mathbb{P}\{\sup_{T_1^N\leq s\leq t }\nu^N((0,s)\times \{0,1\})\geq\epsilon\} \end{align*}

The first term of the RHS of the above Inequality is equal to zero. Since for $T_1^N\leq s\leq t $

\begin{align*} \nu^N((0,s)\times \{0,1\})&=\int_0^{T_1^N}\mathbb{1}_{\{m^N(u)\in\{0,1\}\}}du+\int_{T_1^N}^s\mathbb{1}_{\{m^N(u)\in\{0,1\}\}}du\\ &\leq T_0-T_1^N \end{align*}

\begin{equation*}\mathbb{P}\{\sup_{s\leq t}\nu^N((0,s)\times \{0,1\})\geq\epsilon\}\leq \mathbb{P}\{|T^N-T_0|\geq\epsilon\} \end{equation*}

Thus,

\begin{equation*}\lim_{N\rightarrow +\infty}\mathbb{P}\{\sup_{s\leq t}\nu^N((0,s)\times \{0,1\})\geq\epsilon\}=0\end{equation*}

Application 1 : In that case, simulations have shown that if one assumes that if the system starts from the interior of the domain $ {\mathcal{S}}$, $ x_1 + 2x_2 \lt {\bar{\beta}}$, the storage system before $ T_1^N$ behaves like the system with infinite capacity, and the processes $( X_{0}^N(t))$, $( X_{1}^N(t))$ and $( X_{2}^N(t))$ in the finite capacity case are close to the processes $( X_{0}^N(t))$, $( X_{1}^N(t))$ and $( X_{2}^N(t))$ without constraints on the boundary ${F}_{N}$. And the better choice of parameters that guarantees reliability remains the same as in the infinite capacity case before T ₁,

\begin{equation*} {{\mathcal{S}}}_3= \left\{(x_1,x_2)\in S\;|\;x_1+2x_2 \lt \frac{\lambda +2\xi}{\mu}\;,\;\frac{\lambda+\xi}{\mu} \gt 2x_2\right\} \end{equation*}

The graphs below illustrate the closeness of the processes with finite capacity, represented by the color red, and those with infinite capacity, represented by the color blue in figs. 4a), 4b and 4c)

Figure 4. Comparison between the stochastic processes in the finite and infinite case before $T_{1}^N$. a) The stochastic processes $( X_{0}^N(t))$ in the finite and infinite case. b) The stochastic processes $( X_{1}^N(t))$ in the finite and infinite case. c) The stochastic processes $( X_{2}^N(t))$ in the finite and infinite case.

5.1.2. Starting from the boundary of $(\partial{\mathcal{S}})_2$ of the set ${\mathcal{S}}$

Theorem 5.6 If Conditions (38) and (45) hold with $x_1+2x_2={\bar{\beta}}$. Then for the convergence in distribution,

\begin{equation*} \underset{N \rightarrow + \infty}{lim} ({\bar{X}}_1^N(t),{\bar{X}}_2^N(t))_{t \geq 0} = (x_1(t), x_2(t))_{t \geq 0} \end{equation*}

where $(x_1(t), x_2(t))_{t \geq 0} $ is the solution of the ordinary differential equation,

(55)\begin{equation} \begin{aligned} x_1(t) &= x_1 - \lambda (1-\pi(0))t + \mu \int_{0}^t (2 x_2(u) - x_1(u)) \, du + \lambda y_1(t) \\ x_2(t) &= x_2 + \left(\frac{\mu{\bar{\beta}}}{2}+\frac{\lambda}{2}(1-\pi(0))\right) t -2 \mu \int_{0}^t x_2(u) \, du - \lambda y_1(t) \end{aligned} \end{equation}

where $\pi(0)$ is defined by Equation (41).

Proof. Our goal is to identify the measure ν in Equations (53) and (54). The Q-matrix of the Markov process $({X}^N(.), m^N(.))$ is given by,

\begin{equation*}\nonumber (x^N,m^N) \longrightarrow (x^N,m^N)+ \left\{ \begin{array}{ll} (x^N+e_2,m^N-2) \ \ \xi N \mathbb{1}_{\{m^N \geq 2\}} \\ (x^N+e_1-e_2,m^N+1)\ \ 2 \mu x_2^N\\ (x^N+e_2-e_1,m^N-1) \ \ \lambda N \mathbb{1}_{\{x_1^N \gt 0\}} \mathbb{1}_{\{m^N \geq 1\}} \\ (x^N-\frac{{e}_1}{N},m^N+1) \ \ \mu x_1^N \end{array} \right. \end{equation*}

Thus, the process

\begin{equation*}\left(f({X}^N(t), m^N(t) ) - f({X}^N(0), m^N(0) )-\int_0^t (Qf)({X}^N(s), m^N(s) )ds\right) \end{equation*}

is a martingale for all bounded function f on ${\mathbb{R}}^+\times \bar{\mathbb{N}}$. In particular the process

(56)\begin{equation} \begin{aligned} \mathcal{M}^N(t)\stackrel{def}{=}&g(m^N(t)) - g(m^N(0))\\ & - \int_{0}^t [g(m^N(s)-2)- g(m^N(s))]\xi N \mathbb{1}_{\{m^N(s) \geq 2\}}ds \\ &\quad -\int_{0}^t [g(m^N(s)+1) - g(m^N(s))] \mu(2X_2^N+X_1^N(s))ds \\ &\qquad -\int_{0}^t [g(m^N(s)-1) - g(m^N(s))] \lambda N \mathbb{1}_{\{X_1^N(s) \gt 0\}} \mathbb{1}_{\{m^N(s) \geq 1\}}ds \end{aligned} \end{equation}

is a martingale for all bounded function g on $\bar{\mathbb{N}}$. It follows from Doob’s inequality that the process $ (\frac{\mathcal{M}^N(t)}{N})$ converges in distribution to 0.

Since $2{\bar{X}}_2^N(t)+{\bar{X}}_1^N(t)=\frac{{F}_{N}}{N}-\frac{m^N(t)}{N}$, Equation (56) can be rewritten as

(57)\begin{equation} \begin{aligned} \frac{{\mathcal{M}}^N(t)}{N}=&\frac{g(m^N(t)) - g(m^N(0))}{N}\\ & - \int_{0}^t\biggl\{[g(m^N(s)-2)- g(m^N(s))]\xi \mathbb{1}_{\{m^N(s) \geq 2\}} \\ &\quad + [g(m^N(s)+1) - g(m^N(s))] \mu(\frac{{F}_{N}}{N}-\frac{m^N(t)}{N}) \\ &\qquad + [g(m^N(s)-1) - g(m^N(s))] \lambda \mathbb{1}_{\{X_1^N(s) \gt 0\}} \mathbb{1}_{\{m^N(s) \geq 1\}}\biggr\} ds \end{aligned} \end{equation}

In terms of measure $\nu^N(.)$, we may rewrite the last term on the RHS of (57) as follows :

(58)\begin{equation} \begin{aligned} & \int_{0}^t\biggl\{(g(y-2)- g(y))\xi \mathbb{1}_{\{y \geq 2\}}\\ &\quad + (g(y+1) - g(y)) \mu(\frac{{F}_{N}}{N}-\frac{y}{N})\\ &\qquad + [g(y-1) - g(y)] \lambda \mathbb{1}_{\{{\bar{X}}_1^N(s) \gt 0\}} \mathbb{1}_{\{y \geq 1\}}\biggr\}\nu^N(ds\times dy) \end{aligned} \end{equation}

which also converges to 0 since $\frac{1}{N}\left(g(m^N(t)) - g(m^N(0))\right)$ converges to 0 as $N\rightarrow+\infty$. Furthermore, by continuous mapping theorem one gets that

\begin{equation*}\begin{aligned}\nonumber \int_{[0,t]\times \mathbb{N}} \biggl\{[g(y-2)- g(y)] \xi \mathbb{1}_{\{y \geq 2\}} + [g(y+1) - g(y)] \mu {\bar{\beta}} \\ \nonumber + [g(y-1) - g(y)] \lambda \mathbb{1}_{\{x_1(s) \gt 0\}} \mathbb{1}_{\{y \geq 1\}} \biggr\} \nu(ds \times dy) =0 \end{aligned} \end{equation*}

for all $t\geq 0$.

Thus, for almost all $t \geq 0$,

\begin{equation*}\begin{aligned}\nonumber \sum_{y\in \mathbb{N}} \biggl\{[g(y-2)- g(y)] \xi \mathbb{1}_{\{y \geq 2\}} + [g(y+1) - g(y)] \mu {\bar{\beta}} \\ \nonumber + [g(y-1) - g(y)] \lambda \mathbb{1}_{\{x_1(t) \gt 0\}} \mathbb{1}_{\{y \geq 1\}} \biggr\}\nu_t (y) =0 \end{aligned} \end{equation*}

Hence, for all $t\geq 0$ such that $x_1(t) \gt 0$, the measure $\nu_t(.)=\pi$ where the measure π is invariant for the Markov process $(m(t))$ with Q-matrix given by (39). The theorem is proved.

Application 2: In the Overloaded case, if the system starts from the boundary $\partial {{\mathcal{S}}}_2$, i.e., $x_1 + 2 x_2 = {\bar{\beta}}$, and the parameters have been fixed as follows:

• - The number of nodes N = 100,

• - The duplication rate is $\lambda = 0,3$,

• - The admitting rate is ξ = 0.3,

• - The loss rate is µ = 0.01,

• - The maximal number of files to be stored in the system is $F_{max} = 7000$.

In the simulations below, it will be shown that the fluid limits $(x_0(t), x_1(t),x_2(t))$ obtained in (55) coincide with the stochastic process $(X_0^N(t), X_1^N(t),X_2^N(t))$ defined for the model. And the new coming files for storage are accepted with approximately the rate $\xi N (1 - \pi_{{\bar{\beta}}}(0))$. On the other hand, the reliability of the system is not impacted by the capacity of the system in this case.

See the graphs below, fig. 5 which clearly illustrate the striking alignment between the stochastic processes and their corresponding fluid limits. This strong correspondence underscores the critical role of the fluid limits in accurately capturing and describing the asymptotic behavior of the stochastic processes.

Figure 5. Comparison between the stochastic processes $X_0^N(t), X_1^N(t), X_2^N(t)$ and their respective fluid limits $x_0(t), x_1(t), x_2(t)$. a) The stochastic process $ (X_0^N(t))$. b) The associated fluid limit $ x_0(t)$. c) The stochastic process $ (X_1^N(t))$. d) The associated fluid limit $ x_1(t)$. e) The stochastic process $ (X_2^N(t))$. f) The associated fluid limit $ x_2(t)$.

6. The critically loaded regime

In the Critically-loaded regime, the equilibrium point ρ is identical to $ {\bar{\beta}} $, and the figure below, fig. 6, illustrates that the process $ ({\overline{X}}_1(t), {\overline{X}}_2(t)) $ being constrained by the boundary $ (\partial{\mathcal{S}})_2 $ which coincides with the equilibrium point.

Figure 6. The equilibrium point in the critically loaded regime

Throughout this section, we assume that the condition

(59)\begin{equation} \rho={\bar{\beta}} \end{equation}

holds. Let $\lbrace Z_1^N(t),Z_2^N(t),Z^N(t)\rbrace$ be the Markov process defined by

\begin{equation*}Z_1^N(t)=\sqrt{N}(\rho_1-{\bar{X}}_1^N(t)),\quad Z_2^N(t)=\sqrt{N}(\rho_2-{\bar{X}}_2^N(t))\end{equation*}

and

\begin{equation*}Z^N(t)=\sqrt{N}(\rho-{\bar{X}}_1^N(t)-2{\bar{X}}_2^N(t))=Z_1^N(t)+2Z_2^N(t)\end{equation*}

where

\begin{equation*}\rho_1=\frac{\xi}{\mu},\quad \rho_2=\frac{\lambda+\xi}{2\mu}\end{equation*}

In the following proposition we prove that the sequence of processes

\begin{equation*}\lbrace Z_1^N(t),Z_2^N(t),Z^N(t)\rbrace\end{equation*}

converges in distribution to a reflected three-dimensional Ornstein–Uhlenbeck process.

The Q-matrix $Q^{N} = (q^{N}(.,.))$ and the generator of the Markov process $\lbrace Z_1^N(t),Z_2^N(t),Z^N(t)\rbrace$

are given by:

\begin{equation*} \nonumber \left\{ \begin{array}{ll} q^{N}((z_1,z_2, z),(z_1+\frac{1}{\sqrt{N}}, z_2, z+\frac{1}{\sqrt{N}})=\mu N(\rho_1-\frac{z_1}{\sqrt{N}})\\ q^{N}((z_1,z_2, z),(z_1-\frac{1}{\sqrt{N}}, z_2+\frac{1}{\sqrt{N}}, z+\frac{1}{\sqrt{N}})= 2\mu N(\rho_2-\frac{z_2}{\sqrt{N}} ) \\ q^{N}((z_1,z_2, z),(z_1+\frac{1}{\sqrt{N}}, z_2-\frac{1}{\sqrt{N}}, z-\frac{1}{\sqrt{N}})= \lambda N \mathbb{1}_{\lbrace z_1 \lt \sqrt{N}\rho_1,\;z\geq \frac{1}{\sqrt{N}}+\sqrt{N}({\bar{\beta}}-\frac{{F}_{N}}{N}) \rbrace} \\ q^{N}((z_1,z_2, z),(z_1, z_2-\frac{1}{\sqrt{N}}, z-2\frac{1}{\sqrt{N}})= \xi N \mathbb{1}_{\lbrace z\geq \frac{2}{\sqrt{N}}+\sqrt{N}({\bar{\beta}}-\frac{{F}_{N}}{N}) \rbrace} \end{array} \right. \end{equation*}

\begin{equation*}\begin{aligned}\nonumber &A_N f(z_1, z_2, z) = \mu N(\rho_1-\frac{{z}_1}{\sqrt{N}}) [f(z_1+\frac{1}{\sqrt{N}}, z_2, z+\frac{1}{\sqrt{N}}) - f(z_1,z_2, z)] \\ \nonumber &+ 2\mu N(\rho_2-\frac{{z}_2}{\sqrt{N}} ) [f(z_1-\frac{1}{\sqrt{N}}, z_2+\frac{1}{\sqrt{N}}, z+\frac{1}{\sqrt{N}})- f(z_1,z_2, z)] \\ \nonumber & + \lambda N \mathbb{1}_{\lbrace z_1 \lt \sqrt{N}\rho_1,\;z\geq \frac{1}{\sqrt{N}}+\sqrt{N}({\bar{\beta}}-\frac{{F}_{N}}{N}) \rbrace} [f(z_1+\frac{1}{\sqrt{N}}, z_2-\frac{1}{\sqrt{N}}, z-\frac{1}{\sqrt{N}}) - f(z_1,z_2, z)] \\ \nonumber &+ \xi N \mathbb{1}_{\lbrace z\geq \frac{2}{\sqrt{N}}+\sqrt{N}({\bar{\beta}}-\frac{{F}_{N}}{N}) \rbrace} [f(z_1, z_2-\frac{1}{\sqrt{N}}, z-2\frac{1}{\sqrt{N}}) - f(z_1,z_2, z)] \end{aligned} \end{equation*}

Proposition 6.1. If f is twice differentiable on ${\mathbb{R}}^3$ and such that $\nabla f(z_1,z_2,0)=0$ then the generator converges to

(60)\begin{equation} \begin{aligned} Af(z_1,z_2,z)&=\mu (2z_2-z_1)\frac{\partial f}{\partial x_1}(z_1,z_2,z)-2\mu z_1\frac{\partial f}{\partial x_2}(z_1,z_2,z)\\ &-\mu z\frac{\partial f}{\partial {x}_{3}}(z_1,z_2,z)+(\lambda+\xi)\frac{{\partial}^2 f}{\partial x_1^2}(z_1,z_2,z)+(\lambda+\xi)\frac{{\partial}^2 f}{\partial x_2^2}(z_1,z_2,z)\\ &(\lambda+\frac{5}{2}\xi)\frac{{\partial}^2 f}{\partial {x}_{3}^2}(z_1,z_2,z)-(2\lambda+\xi)\frac{{\partial}^2 f}{\partial x_1\partial x_2}(z_1,z_2,z)-2\lambda\frac{{\partial}^2 f}{\partial x_1\partial {x}_{3}}(z_1,z_2,z)\\ &+(2\lambda+3\xi)\frac{{\partial}^2 f}{\partial x_2\partial {x}_{3}}(z_1,z_2,z) \end{aligned} \end{equation}

for z > 0 and to

(61)\begin{equation} \begin{aligned} &(\lambda+\xi)\frac{{\partial}^2 f}{\partial x_1^2}(z_1,z_2,0)+(\lambda+\xi)\frac{{\partial}^2 f}{\partial x_2^2}(z_1,z_2,0)\\ &(\lambda+\frac{5}{2}\xi)\frac{{\partial}^2 f}{\partial {x}_{3}^2}(z_1,z_2,0)-(2\lambda+\xi)\frac{{\partial}^2 f}{\partial x_1\partial x_2}(z_1,z_2,0)-2\lambda\frac{{\partial}^2 f}{\partial x_1\partial {x}_{3}}(z_1,z_2,0)\\ &+(2\lambda+3\xi)\frac{{\partial}^2 f}{\partial x_2\partial {x}_{3}}(z_1,z_2,0) \end{aligned} \end{equation}

which is the generator of the three-dimensional Ornstein–Uhlenbeck process reflected on the boundary of the half-space z > 0. One could refer to the following papers Ward and Glynn [Reference Ward and Glynn19], Ward and Glynn [Reference Ward and Glynn18], and Lidong and Chunmei [Reference Zhang and Jiang20] where the properties of the reflected Ornstein–Uhlenbeck and the associated infinitesimal generator are presented.