## Abstract

In this paper, we are interested in stochastic fluid flow model with upward jumps at level zero and instantaneous phase transitions at jumps epochs. This mathematical model is governed by a non-homogeneous differential system with specific boundary conditions. We derive three algorithms for computing explicitly the fluid level distribution. We study the general case of the model, the case where the phase process conserves its state when the fluid jumps, and a special case where the phase process transits to a deterministic fixed state at jumps epochs. Numerical results are carried out to illustrate the numerical stability and the advantages in computational time for the proposed approaches.

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## Acknowledgements

The authors are grateful to the anonymous referee for helpful comments and suggestions that led to substantial improvements of the manuscript.

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## Appendix

### Appendix

In order to prove Theorem 4, we need the following lemma.

### Lemma 1

If a square matrix *C* of order *n* is such that

then *C* is invertible and \(C^{-1} \ge 0\).

### Proof

We prove this lemma in two steps. Firstly, we prove the following: for all \(x \in \mathbb {R}^n\), such that \(C x \ge 0\) we have necessarily \(x\ge 0\). Secondly, we prove the required result.

Let \(x\in \mathbb {R}^n\) be a vector such that \(y=Cx \ge 0\) and let *m* be the index between 0 and *n* such that \(x_m:= \min _{1\mathop{\le} i\mathop{\le} n} x_i\). We have:

Since \(x_k\ge x_m\), for all *k* and using the hypothesis on *C*, \(\vert c_{m,k}\vert =-c_{m,k} \ge 0\), for all \(k\ne m\), it comes that:

Hence, we obtain:

Since *A* is a strict diagonal dominant matrix we have:

As \(y_m\ge 0\), we get \(x_m \ge 0\). Since \(x_k \ge x_m\), it follows that \(x_k\ge 0\), for all *k* which means that \(x\ge 0\). We conclude that for all \(x \in \mathbb {R}^n\) satisfying \(C x \ge 0\) we have necessarily \(x\ge 0\).

Now, let us prove that the matrix *C* is invertible. By using the finite dimensional rank theorem, the matrix *C* is invertible if and only if its kernel is reduced to the zero vector. For this purpose, let \(x\in \mathbb {R}^n\) be a vector such that \(Cx = 0\), we have \(C x\ge 0\) and \(-Cx= C(-x) \ge 0\). According to the last result we obtain that \(x\ge 0\) and \(-x\ge 0\) which implies necessarily that \(x=0\), then we get that *C* is invertible. It remains to show that \(C^{-1} \ge 0\). Since *C* is invertible, let \(\pmb {f}_j^*\) be the \(j^{th}\) column vector of the matrix \(C^{-1}\). Since \(CC^{-1}=I\), we obtain \(C \pmb {f}_j^*=\pmb {e}_j^*\), where \(\pmb {e}_j^*\) is the \(j^{th}\) column vector of the canonical basis of \(\mathbb {R}^n\). It is clear that \(\pmb {e}_j^*\ge 0\), which means that \(C\pmb {f}_j^*\ge 0\), so according to the first step of the proof we get \(\pmb {f}_j^*\ge 0\) and therefore \(C^{-1}\ge 0\). \(\square\)

### Proof of Theorem 2

We consider the row vector \(\pmb {F}(x)=(\pmb {F}_{\widehat{S}}(x),\pmb {F}_{S_0}(x))\), where \(\pmb {F}_{\widehat{S}}(x)=(F_i(x), i \in \widehat{S})\) and \(\pmb {F}_{S_0}(x)=(F_i(x), i \in S_0)\). So, the two differential systems related to \(S_0\) are as follows:

Since the submatrix \((-A_{S_0 S_0})\) satisfies the hypothesis of Lemma 1, \((-A_{S_0 S_0 })\) is invertible and then we get:

As soon as \(\pmb {F}_{\widehat{S}}(x)\) is determined, \(\pmb {F}_{S_0}(x)\) will be determined and therefore the solution \(\pmb {F}(x)=(\pmb {F}_{\widehat{S}}(x),\pmb {F}_{S_0}(x))\) will be given in a unique way. To seek the expression of \(\pmb {F}_{\widehat{S}}(x)\), we consider the following two differential systems related to \(\widehat{S}\):

By replacing \(\pmb {F}_{S_0}(x)\) by its expression (11), we obtain the following Problem:

where

So, in order to prove the uniqueness of \(\pmb {F}_{\widehat{S}}(x)\), we check that the conditions of Problem \(( \widehat{\mathcal {P}})\) are similar to Problem \(({\mathcal {P}})\). In others words, we must show that: \(\widehat{A}\) is an irreducible infinitesimal generator, \(D_{\widehat{S} \widehat{S}}\) is an invertible diagonal matrix, \(\widehat{Q}\) is a stochastic matrix and \(\sum _{i\, \in\, \widehat{S}} \widehat{\pi _i} d_i<0\), where \(\widehat{\pmb {\pi }}\) is the stationary distribution related to \(\widehat{A}\). In fact, we have that *A* is irreducible, then \(\widehat{A}\) is an irreducible infinitesimal generator and that its steady-state vector \(\widehat{\pmb {\pi }}\) is:

The stability condition associated to the Problem \(( \widehat{\mathcal {P}})\) is \(\sum _{i\, \in\, \widehat{S}} \widehat{\pi }_i d_i< 0\), which is indeed satisfied since \(\sum _{i\, \in\, \widehat{S}} \widehat{\pi }_i d_i=\frac{1}{1- \pmb {\pi }_{S_0} \pmb {1}_{S_0}^*} \sum _{i\, \in\, S} \pi _i d_i\). It is obvious that \(D_{\widehat{S} \widehat{S}}\) is an invertible diagonal matrix. Now, it remains to show that \(\widehat{Q}\) is a stochastic matrix. In fact, we have:

On the one hand, since *Q* is a stochastic matrix, we have \(Q_{\widehat{S}\widehat{S}}\ge 0\) and \(Q_{\widehat{S} S_0} \ge 0\). Moreover, the non-diagonal coefficients of the matrix *A* are all nonnegative which gives that \(A_{S_0 \widehat{S}}\ge 0\). Taking into account the fact that \((-A_{S_0 S_0 })\) satisfies all the hypothesis of Lemma 1 we obtain immediately that \((-A_{S_0 S_0 })^{-1}\ge 0\). Then \(\widehat{Q} _{i,j}\ge 0\), for all \(i,j \in \widehat{S}\). On the other hand, we have:

Thus, we get that \(\widehat{Q}\) is a stochastic matrix. In another side, \(\pmb {F}(0) \pmb {1}^*=1\), implies \(\pmb {F}_{\widehat{S}}(0) \pmb {1}_{\widehat{S}}^*\mathop{+} \pmb {F}_{S_0}(0) \pmb {1}_{S_0}^*=1\). Exploiting Identity (11) at point \(x=0\) and taking into account the condition “\(F_i(0)=0,\ i\in S_+\)”, we obtain:

By applying the same development made in the proof of Theorem 1 to Problem \(( \widehat{\mathcal{P}})\), the uniqueness of \(\pmb {F}_{\widehat{S}}(x)\) holds and it is given by the following expression:

where \(\pmb {F}_{\widehat{S}}(0)\) and \(\pmb {F}_{S_-}^{\prime }(0)\) are calculated in a unique way as solution of the following linear system:

where \(\widehat{ L}=D_{\widehat{S}\widehat{S}}( \widehat{ Q}-I)\), \(\widehat{M}=\widehat{A} D_{\widehat{S}\widehat{S}}^{-1}\), \(\widehat{B}=D_{\widehat{S} \widehat{S}} \widehat{Q} D_{\widehat{S} \widehat{S}}^{-1}\), \(\widehat{\Delta }\) is the diagonal matrix composed by all the eigenvalues of \(\widehat{M}\), \(\widehat{P}\) is the matrix composed of the corresponding eigenvectors and \(\widehat{K}(q)=\mathrm { diag}\left( \frac{e^{\lambda _i q}-1}{\lambda _i}, i \in \widehat{S }\right)\).

Once \(\pmb {F}_{\widehat{S}}(x)\) is calculated, \(\pmb {F}_{S_0}(x)\) will be computed through the following expression:

Finally, we conclude that \(\pmb {F}(x)=(\pmb {F}_{\widehat{S}}(x),\pmb {F}_{S_0}(x))\) is the unique solution of the Problem \((\mathcal {P})\) and this completes the proof.

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Nabli, H., Abdallah, I. Stochastic Fluid Models with Upward Jumps and Phase Transitions.
*Methodol Comput Appl Probab* **25**, 7 (2023). https://doi.org/10.1007/s11009-023-09982-5

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DOI: https://doi.org/10.1007/s11009-023-09982-5