**Appendix D**

**Proof of Lemma 2.** Without loss of generality we suppose each *n*-mesh contains at least dim(W(F|*y*)) <sup>+</sup> 2 points. By Corollary <sup>2</sup> the solution to problem (28) can be represented in form (A1). By the condition of Lemma <sup>2</sup> there exists a set {*qs*(*n*|*y*)}*s*=1,*<sup>S</sup>* ⊆ C(*n*), such that max 1*sS* **<sup>q</sup>***s*(*y*) <sup>−</sup> *qs*(*n*|*y*)- *n*. For the vector *<sup>w</sup>*(*n*|*y*) <sup>∑</sup>*<sup>S</sup> <sup>s</sup>*=<sup>1</sup> <sup>P</sup>*s*(*y*)*w*(*qs*(*n*|*y*)|*y*) the inequalities

$$|\hat{\mathbf{w}}\_1(y) - w\_1(\epsilon\_n|y)| \leqslant \sum\_{s=1}^S \hat{\mathsf{P}}\_s(y) |w\_1(\hat{\mathsf{q}}\_s(y)|y) - w\_1(q\_s(\epsilon\_n|y)|y)| \leqslant \omega\_1(\epsilon\_n|y),$$

$$\|\|\hat{\mathbf{w}}\_2(y) - w\_2(\epsilon\_n|y)\|\| \leqslant \sum\_{s=1}^S \hat{\mathsf{P}}\_s(y) \|\|w\_1(\hat{\mathsf{q}}\_s(y)|y) - w\_1(q\_s(\epsilon\_n|y)|y)\|\| \leqslant \omega\_2(\epsilon\_n|y).$$

hold. Furthermore, the sequence of inequalities

$$\max\_{w \in \text{conv}(\mathcal{W}(\mathcal{C}(\varepsilon\_n)|y))} \mathsf{J}\_\*(w) = \mathcal{I}(y) - \min\_{w \in \text{conv}(\mathcal{W}(\mathcal{C}(\varepsilon\_n)|y))} (\widehat{\mathfrak{w}}\_1(y) - w\_1 + \|w\_2\|^2 - \|\widehat{\mathfrak{w}}\_2(y)\|^2) \geqslant 0$$

$$\begin{aligned} \mathcal{I} \geqslant \mathcal{I}(\boldsymbol{y}) - \left[ |\hat{\mathbf{w}}\_1(\boldsymbol{y}) - w\_1(\boldsymbol{\epsilon}\_n|\boldsymbol{y})| + \|\hat{\mathbf{w}}\_2(\boldsymbol{y}) - w\_2(\boldsymbol{\epsilon}\_n|\boldsymbol{y})\|^2 - 2\langle \hat{\mathbf{w}}\_2(\boldsymbol{y}), \hat{\mathbf{w}}\_2(\boldsymbol{y}) - w\_2(\boldsymbol{\epsilon}\_n|\boldsymbol{y})\rangle \right] \geqslant \mathcal{I} \\ \geqslant \mathcal{I}(\boldsymbol{y}) - \left[ \omega\_1(\boldsymbol{\epsilon}\_n|\boldsymbol{y}) + \left( \omega\_2(\boldsymbol{\epsilon}\_n|\boldsymbol{y}) + 2\frac{M}{m(\boldsymbol{y})}\mathbf{K}\_\hbar \right) \omega\_2(\boldsymbol{\epsilon}\_n|\boldsymbol{y}) \right] \end{aligned}$$

proves the convergence max*w*∈conv(W(C(*n*)|*y*)) **<sup>J</sup>**∗(*w*) ↑ J (*y*) as *<sup>n</sup>* ↓ 0.

Let **<sup>w</sup>** (*n*|*y*) **<sup>w</sup>** (*y*) as *<sup>n</sup>* <sup>↓</sup> 0. Then there exists a subsequence {*nk*}*nk*∈N, such that **<sup>w</sup>** (*nk*|*y*) <sup>→</sup> **<sup>w</sup>**(*y*) <sup>=</sup> **<sup>w</sup>** (*y*). This means that **<sup>J</sup>**∗(**w** (*y*)) = **<sup>J</sup>**∗(**w**(*y*)), which contradicts the uniqueness of the solution to the finite-dimensional dual problem (28). Lemma 2 is proven.

#### **References**


**Gurami Tsitsiashvili**

Institute for Applied Mathematics, Far Eastern Branch of Russian Academy Sciences, Radio Str. 7, IAM FEB RAS, 690041 Vladivostok, Russia; guram@iam.dvo.ru; Tel.: +89146932749

**Abstract:** In this paper, a method for detecting synergistic effects of the interaction of elements in multi-element stochastic systems of separate redundancy, multi-server queuing, and statistical estimates of nonlinear recurrent relations parameters has been developed. The detected effects are quite strong and manifest themselves even with rough estimates. This allows studying them with mathematical methods of relatively low complexity and thereby expand the set of possible applications. These methods are based on special techniques of the structural analysis of multielement stochastic models in combination with majorant asymptotic estimates of their performance indicators. They allow moving to more accurate and rather economical numerical calculations, as they indicate in which direction it is most convenient to perform these calculations.

**Keywords:** complex system; synergistic effect; performance indicator; structure change

**MSC:** 60J28
