**4. Multiserver Loss Systems**

Consider *n*-server queuing system *M*|*M*|*n*|0 with a Poisson input flow of intensity *nλ* and exponentially distributed service times having intensity *μ* on all *n* servers, *ρ* = *λ*/*μ*. This system can be considered as combining *n* single-server systems with input flow intensities *λ* (see Figure 2).

$$\begin{array}{rcl} \stackrel{\lambda}{\longrightarrow} & \stackrel{\mu}{\odot} & \longrightarrow & \stackrel{\mu}{\longrightarrow} \\ \stackrel{\lambda}{\longrightarrow} & \stackrel{\nu}{\longrightarrow} & \stackrel{n\lambda}{\longrightarrow} \\ \stackrel{\lambda}{\lambda} & \stackrel{\mu}{\longrightarrow} & \stackrel{\nu}{\longrightarrow} \end{array} \longrightarrow \begin{array}{rcl} \stackrel{\mu}{\longrightarrow} \\ \stackrel{\nu\lambda}{\longrightarrow} & \stackrel{\nu}{\longrightarrow} \\ \stackrel{\nu}{\longrightarrow} & \stackrel{\mu}{\longrightarrow} \end{array} \longrightarrow$$

**Figure 2.** *<sup>n</sup>* isolated *<sup>M</sup>*|*M*|1|0 systems (**left**), aggregated *<sup>M</sup>*|*M*|*n*|0 system (**right**). *bn*.

The number of customers in the system *M*|*M*|*n*|0 describes the process of death and birth with the intensities of birth and death *λn*(*k*) = *nλ*, 0 ≤ *k* < *n*, *μn*(*k*) = *kμ*, 0 < *k* ≤ *n*.

Let us denote *Pn*(*ρ*) the stationary probability of failure in the system *An* for a given *ρ*. It is not difficult to establish that *P*1(1) = 1/2. However, the combined system *An* satisfies new relation, which characterizes the synergistic effect of such a combination.

**Theorem 5.** *The following limit ratio is true: Pn*(1) ∼ 6 2 *<sup>π</sup>n*, *<sup>n</sup>* <sup>→</sup> <sup>∞</sup>.

**Proof.** Let *δ* > 0, consider the function *f*(*x*) = 1 − *x* − exp(−(1 + *δ*)*x*). The *f*(*x*) function satisfying the condition: *f*(0) = 0, *f*(1) < 0, and such that the inequalities

$$f'(x) > 0, \ 0 < x < \frac{\ln(1+\delta)}{1+\delta}, \ f'(x) < 0, \ \frac{\ln(1+\delta)}{1+\delta} < x \le 1$$

hold. Therefore, on the half interval [0, 1) there exists a single *x*(*δ*), satisfying the condition *f*(*x*(*δ*)) = 0 and such that the inequalities 1 − *x* ≥ exp(−(1 + *δ*)*x*), 0 ≤ *x* ≤ *x*(*δ*) < 1 hold. Let *pn*(*k*) = lim *<sup>t</sup>*→<sup>∞</sup> *<sup>P</sup>*(*xn*(*t*) = *<sup>k</sup>*), 0 <sup>≤</sup> *<sup>k</sup>* <sup>≤</sup> *<sup>n</sup>*, then in force [16] [Chapter 2, § 1]

$$p\_n(n-1) = p\_n(n)\frac{\mu}{\lambda}\frac{n}{n},\ p\_n(n-2) = p\_n(n)\left(\frac{\mu}{\lambda}\right)^2\frac{n(n-1)}{n^2},\dots$$

Therefore, the stationary blocking probability in virtue of the integral theorems of recovery and the law of large numbers for the recovery process [1] [Chapter 9, § 4, 5] satisfies the equality

$$P\_n(\rho) = p\_n(n) = \left(\sum\_{k=0}^n \rho^{-k} \prod\_{j=0}^{k-1} \left(1 - \frac{j}{n}\right)\right)^{-1},\tag{3}$$

where ∏−<sup>1</sup> *<sup>j</sup>*=<sup>0</sup> equals 1. From Formula (3), we obtain the inequality

$$P\_n^{-1}(1) \ge \sum\_{0 \le k \le n \ge (\delta)} \prod\_{j=0}^{k-1} \left( 1 - \frac{j}{n} \right) \ge \sum\_{0 \le k \le n \ge (\delta)} \prod\_{j=0}^{k-1} \exp\left( - (1 + \delta)j/n \right) \ge 0$$

$$\ge \sum\_{1 \le k \le n \ge (\delta)} \exp\left( - (1 + \delta)k^2/2n \right).$$

This implies that

$$P\_n^{-1}(1) \ge \int\_1^{n\ge(\delta)} e^{-(1+\delta)\mathbf{x}^2/2n} d\mathbf{x} = \sqrt{\frac{n}{1+\delta}} \int\_{\sqrt{\frac{1+\delta}{n}}}^{\mathbf{x}(\delta)\sqrt{n(1+\delta)}} e^{-y^2/2} dy \,\mathrm{d}y$$

consequently

$$P\_n(1)\sqrt{n} \le (1+\delta) \left( \int\_{\sqrt{\frac{1+\delta}{n}}}^{\chi(\delta)\sqrt{n(1+\delta)}} e^{-y^2/2} dy \right)^{-1} \to (1+\delta)\sqrt{\frac{2}{\pi}}, \ n \to \infty.$$

and so lim sup *n*→∞ *Pn*(1) 6*πn* <sup>2</sup> <sup>≤</sup> <sup>1</sup> <sup>+</sup> *<sup>δ</sup>*.

Using Formula (3) and the inequality 1 − *x* ≤ exp(−*x*), 0 ≤ *x* ≤ 1, we obtain

$$P\_n^{-1}(1) \le \sum\_{1 \le k \le n} e^{-k(k-1)/2n} \le \sum\_{1 \le k \le n} e^{-(k-1)^2/2n} \le \int\_0^\infty e^{-x^2/2n} dx\_n$$

thus it follows that 1 ≤ lim inf *<sup>n</sup>*→<sup>∞</sup> *Pn*(1) 6*πn* <sup>2</sup> . Obtained above inequalities for upper and lower limits lead to the statement of Theorem 5.

**Remark 1.** *In aggregated M*|*M*|*n*|0 *system at ρ* < 1 *following relations are valid [18]:*

$$e^{-n\ln^2\rho/2}\sqrt{\frac{2}{\pi n}}\sqrt{\frac{\rho}{8}} \preceq P\_n(\rho) \preceq (e^{-n\ln^2\rho/2})^{(\rho-1)/\ln\rho}\sqrt{\frac{2}{\pi n}}\sqrt{\frac{\ln\rho}{\rho-1}}.\tag{4}$$

*And if <sup>ρ</sup>* <sup>=</sup> *<sup>ρ</sup>*(*n*) = <sup>1</sup> <sup>−</sup> *<sup>n</sup>*−*γ*, *<sup>γ</sup>* <sup>&</sup>gt; 0, *then Theorem 5 gives*

$$
\frac{1}{2}\sqrt{\frac{1}{\pi n}} \preceq P\_n(\rho) \preceq \sqrt{\frac{2}{\pi n}}, \ \gamma \ge \frac{1}{2}.
$$

$$
\frac{1}{2}\sqrt{\frac{1}{\pi n}} \preceq P\_n(\rho) \exp\left(\frac{n^{1-2\gamma}}{2}\right) \preceq \sqrt{\frac{2}{\pi n}}, \ \gamma < \frac{1}{2}.
$$

Similar results were obtained for Erlang's loss function in [27,28] but in a more complex way.

**Remark 2.** *In aggregated M*|*M*|*n*|∞ *system following relations are valid [29] for An—stationary mean waiting time and Bn—stationary mean queue length:*

*(1) If <sup>ρ</sup>* <sup>&</sup>lt; 1, *then for some c* <sup>&</sup>lt; <sup>∞</sup>, *<sup>q</sup>* <sup>&</sup>lt; <sup>1</sup> *the relation holds An* <sup>≤</sup> *c qn*, *<sup>n</sup>* <sup>≥</sup> 1. *(2) If <sup>ρ</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>n</sup>*−*α*, 0 <sup>&</sup>lt; *<sup>α</sup>* <sup>&</sup>lt; <sup>∞</sup>, *then for n* <sup>→</sup> <sup>∞</sup>

$$A\_{\mathfrak{n}} \to \begin{cases} 0, & \mathfrak{n} < 1, \\ 1/\mu, & \mathfrak{n} = 1, \\ \infty, & \mathfrak{n} > 1. \end{cases} \quad B\_{\mathfrak{n}} \to \begin{cases} 0, & \mathfrak{n} < 1/2, \\ \infty, & \mathfrak{n} \ge 1/2. \end{cases}$$

Suppose that we have *m* independently functioning *nk*-server queuing systems with Poisson input flows of intensity *λk*, *k* = 1, ... , *m*. In the *k*-th system, the customer of the input flow is served exponentially distributed time simultaneously on *ck* channels with intensity *μk*. Let *lk* = *nk*/*ck* be a natural number and the equality *ρ<sup>k</sup>* = *λk*/(*lkμk*) = 1 holds.

We combine *n* copies of each of the *nk*-server systems under consideration, denoting *Pk <sup>n</sup>* stationary probability of failure in each of the combined systems, *k* = 1, ... , *m*. Using Theorem 5, it is not difficult to obtain the following limit relations

$$P\_n^1 \sim \sqrt{\frac{2}{\pi n l\_1}}, \dots, P\_n^m \sim \sqrt{\frac{2}{\pi n l\_m}}, n \to \infty.$$

This solution allows us to distribute the total number of *n*(*n*<sup>1</sup> + ... + *nm*) servers between flows so that the failure probabilities of customers of different flows tend to zero with the growth of a large parameter *n*. To solve this problem, one could use the exact multiplicative formula obtained in [17], but this would lead to significantly more complex calculations.

#### **5. Parameter Estimation in the Logistics Growth Model**

The recurrent model of logistic growth

$$\mathbf{x}\_0 = a, \ x\_{n+1} = b \mathbf{x}\_n (1 - \mathbf{x}\_n), \ n = 0, 1, \dots \tag{5}$$

where the parameters *a*, *b* satisfy the conditions 0 < *a* < 1, 1 < *b* < 4, attracts increased attention from biologists and physicists. For this model, both practically and theoretically, it is important to evaluate the parameter *b* based on inaccurate observations. Due to the nonlinearity of the recurrence relation (5), the least squares method applied to the estimation of the parameter *b* seems somewhat unnatural, which is confirmed by numerous computational experiments that give quite large errors. It seems more natural to apply such qualitative properties of the sequence, as the existence of its limit cycle or limit distribution [30] depending on the value of *b* in combination with the method of probability metrics [11].

Consider an additive model for introducing errors in observations *yn* = *xn* + *εn*, *n* = 1, ... Here, *εn*, *n* = 1, ... , is a sequence of i.i.d.r.v.'s having a distribution with mean zero and variance *σ*2. We introduce the following notation

$$X\_n = \sum\_{i=0}^{n-1} \frac{\varkappa\_i}{n'} \, \, Y\_n = \sum\_{i=0}^{n-1} \frac{y\_i}{n'} \, \, \, X'\_n = \sum\_{i=0}^{n-1} \frac{\varkappa\_i^2}{n'} \, \, \, Y'\_n = \sum\_{i=0}^{n-1} \frac{y\_i^2}{n} \, \, \, \,$$

Using the results of [30], it is possible to establish that for the deterministic sequence *xn*, *n* = 1, . . . , with a given *b* there are limits

$$\lim\_{n \to \infty} X\_n = \overline{x} \quad \lim\_{n \to \infty} X'\_n = \overline{x^2}. \tag{6}$$

Indeed, say that the sequence *xn*, *n* = 1, ..., has a limit cycle *x*(1), ..., *x*(*q*) of length *q* ≥ 1, if lim *<sup>k</sup>*→<sup>∞</sup> *xqk*<sup>+</sup>*<sup>j</sup>* <sup>=</sup> *<sup>x</sup>*(*j*) , *<sup>j</sup>* <sup>=</sup> 1, ..., *<sup>q</sup>*. Denote *<sup>x</sup>* <sup>=</sup> <sup>1</sup> *q q* ∑ *j*=1 *x*(*j*) , *<sup>x</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> *q q* ∑ *j*=1 [*x*(*j*) ] 2, then we

have

$$X\_{Nq} = \frac{1}{Nq} \sum\_{i=1}^{Nq} x\_i \to \overline{x}\_\prime \ X'\_{Nq} = \frac{1}{Nq} \sum\_{i=1}^{Nq} x\_i^2 \to \overline{x^2}\_\prime \ N \to \infty.$$

so Formula (6) is true in the case, when the sequence *xn*, *n* = 1, . . . , has limit cycle.

Let *p*(*dx*) be a probability measure on the *σ*-algebra of Lebesgue-measurable subsets of the segment [0, 1]. Let us say that *p*(*dx*) is the limiting distribution of the sequence *xn*, *n* = 1, ..., if for any Lebesgue-measurable set *<sup>C</sup>* <sup>⊆</sup> [0, 1] the equality holds lim*n*→<sup>∞</sup> *k*(*C*, *n*) *<sup>n</sup>* <sup>=</sup> *C p*(*dx*) = *p*(*C*), where *k*(*C*, *n*) is the number of *xi* satisfying the inclusion *xi* ∈ *C*, *i* = <sup>1</sup> <sup>1</sup>

1, ..., *n*. Then, we define *x* = 0 *xp*(*dx*), *x*<sup>2</sup> = 0 *x*<sup>2</sup> *p*(*dx*) and prove Formula (6) as follows.

Let us take an arbitrary *δ* > 0 and put *m* = & 2 *δ* ' + 1, *γ* = *δ* 2 + & 2 *δ* '−1 . Divide the half interval [0, 1) into disjoint segments

$$\mathbb{C}\_1 = \left[0, \frac{1}{m}\right), \; \mathbb{C}\_2 = \left[\frac{1}{m}, \frac{2}{m}\right), \dots, \; \mathbb{C}\_m = \left[\frac{m-1}{m}, 1\right).$$

Choose *<sup>N</sup>*(*δ*) so that for any *<sup>n</sup>* <sup>&</sup>gt; *<sup>N</sup>*(*δ*) we have & *<sup>k</sup>*(*Cj*, *<sup>n</sup>*) *n* ' ≤ *γ*, *j* = 1, ... , *m*. It is sufficiently simple to prove for *n* ≥ *N*(*δ*) the following inequalities

$$
\frac{1}{m} \cdot \sum\_{\substack{x\_i \in \mathcal{C}\_j, \ i = 1, \dots, m}} x\_i \le \frac{j}{m} \cdot \frac{k(\mathcal{C}\_{j'} n)}{n} \le \frac{j}{m} (p(\mathcal{C}\_j) + \gamma) \le \frac{j}{m}
$$

$$
\leq \frac{\gamma j}{m} + \int\_{\mathcal{C}\_j} \left( x + \frac{1}{m} \right) p(d\mathbf{x}) = \frac{\gamma j}{m} + \frac{p(\mathcal{C}\_j)}{m} + \int\_{\mathcal{C}\_j} \mathbf{x} p(d\mathbf{x}).
$$

Summing these inequalities by *j* = 1, ..., *m*, and using the equality for *m* we get for *n* ≥ *N*(*δ*) the inequality

$$X\_n \le \frac{\gamma(m+1)}{2} + \frac{1}{m} + \overline{x} = \overline{x} + \delta.$$

Analogously it is possible to obtain

$$\frac{1}{n} \cdot \sum\_{\substack{\mathbf{x}\_i \in \mathbf{C}\_j, i=1,\dots,n}} \mathbf{x}\_i \ge \frac{j-1}{m} \cdot \frac{k(\mathbf{C}\_j, n)}{n} \ge \frac{j-1}{m} (p(\mathbf{C}\_j) - \gamma) \ge$$

$$\frac{1}{n} \ge \int\_{\mathbf{C}\_j} \left(\mathbf{x} - \frac{1}{m}\right) p(d\mathbf{x}) - \frac{\gamma(j-1)}{m} = \int\_{\mathbf{C}\_j} \mathbf{x} p(d\mathbf{x}) - \frac{1}{m} - \frac{\gamma(m-1)}{2} \ge \overline{\mathbf{x}} - \delta,$$

consequently *Xn* → *x*, *n* → ∞. Similarly we have the relation *X <sup>n</sup>* → *x*2, *n* → ∞, so Formula (6) is true in the case, when the sequence *xn*, *n* = 1, ... , has limit distribution also.

Note that formally the limits *x*, *x*<sup>2</sup> may depend on the initial state *x*0. However, in the logistics growth model there is no such dependence.

We will evaluate the parameter *b* in two stages. First, we express *b* in terms of the path averages: *b* = *x*/(*x* − *x*2). Using the ratio

$$EY\_n = \frac{1}{n} \sum\_{i=0}^{n-1} E(\mathbf{x}\_i + \boldsymbol{\varepsilon}\_i) = X\_n \to \overline{\mathbf{x}}\_{\prime}$$

$$EY\_n^{\prime} = \frac{1}{n} \sum\_{i=0}^{n-1} E(\mathbf{x}\_i + \boldsymbol{\varepsilon}\_i)^2 = \frac{1}{n} \sum\_{i=0}^{n-1} (\mathbf{x}\_i^2 + \sigma^2) = X\_n^{\prime} + \sigma^2 \to \overline{\mathbf{x}^2} + \sigma^2, \ n \to \infty,$$

let us estimate the parameter *b* by the formula

$$b\_{\mathfrak{n}} = \frac{EY\_{\mathfrak{n}}}{EY\_{\mathfrak{n}} - \left(EY\_{\mathfrak{n}}' - \sigma^2\right)} \to b\_{\mathfrak{n}} \text{ } n \to \infty.$$

As a result, the parameter *b* is evaluated by the formula ˆ *bn* <sup>=</sup> *Yn Yn* − (*Y <sup>n</sup>* − *σ*2) . The convergence in probability ˆ *bn* → *b*, *n* → ∞, follows from the relations

$$VarY\_n = \frac{1}{n^2} \sum\_{i=0}^{n-1} Var(\mathbf{x}\_i + \boldsymbol{\varepsilon}\_i) = \frac{1}{n^2} \sum\_{i=0}^{n-1} Var\boldsymbol{\varepsilon}\_i = \frac{\sigma^2}{n} \to 0;$$

$$\operatorname{Var}\mathbf{Y}\_n' = \frac{1}{n^2} \sum\_{i=0}^{n-1} \operatorname{Var}(\mathbf{x}\_i + \varepsilon\_i)^2 = \frac{1}{n^2} \sum\_{i=0}^{n-1} \operatorname{Var}(2\mathbf{x}\_i \varepsilon\_i + \varepsilon\_i^2) \le \frac{4}{n} (4\sigma^2 + \sigma^4) \to 0, \ n \to \infty.$$

The following is an illustrative example of estimating parameter *b* for a logistic growth model. Calculations of *bn* were performed for the case *<sup>x</sup>*<sup>0</sup> <sup>=</sup> 0.75; *<sup>a</sup>* <sup>=</sup> 0.5; *<sup>b</sup>* <sup>=</sup> 3 at *n* = 1000 (see Figure 3). An additive model of introducing errors was considered under the assumption that *εi*, *i* = 0, .., *n* − 1, have a uniform distribution on the segment [−1/4, 1/4].

**Figure 3.** Frequency histogram for *bn*.

This method can be applied to the estimation of the parameters of the Rikker model (see, for example, in [22]). Here, the Rikker model is described by recurrent relation

$$\mathbf{x}\_0 = 1, \ \mathbf{x}\_{n+1} = a\mathbf{x}\_n \exp(-b\mathbf{x}\_n), \ a, b > 0,$$

and observations are following: *yn* = *xn* exp(*εn*), where *ε<sup>n</sup>* has normal distribution with zero mean and known variation, *n* ≥ 0. Another application of described method is the finite-difference approximation of the system of Lorentz differential equations (see, for example, in [31]), etc.

#### **6. Discussion**

All the problems of system analysis considered in this paper are based on the choice of changes in the structure of the system, the efficiency indicator, and the computational algorithm with an assessment of its complexity. In some cases, it is possible to replace the NP-problem with a fairly simple computational procedure, abandoning the high accuracy of the resulting solution in favor of a significant change in the performance indicator. Apparently, such problems require a certain proportion between the accuracy and efficiency of the resulting solution.

The proposed approach to the study of synergistic effects in complex systems can be applied to the construction of queuing systems with a large load and a small queue, to backup systems with recovery, to insurance models and other stochastic systems. It allows you to explore and find the main parameters in such popular technologies in applications as powder metallurgy, 3-D printing, fast mixing of fuel in engines, etc. The emphasis on economical, but not highly accurate calculations, makes it possible at the initial stage to correctly select the main parameters of the analyzed systems before performing more detailed and accurate calculations. This property of the proposed approach to the analysis of complex systems can be used in programs of digital economy, smart city, etc., when at the initial stage of the study it is important to determine the main indicators of the effectiveness of a complex system.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data sharing is not applicable to this article.

**Acknowledgments:** The author thanks Marina Osipova for her help in the design of the work.

**Conflicts of Interest:** The authors declare no conflict of interest.
