**5. Examples**

First note that many examples of perturbation bounds for queueing systems have been considered in [11,12,14,44,46,66,67].

Here, to compare both approaches, we will mostly deal with the queueing system *Mt*|*Mt*|*N*|*N* with losses and 1-periodic intensities. In the preceding papers on this model, other problems were considered. For example, in [68], the asymptotics of the rate of convergence to the stationary mode as *N* → <sup>∞</sup>, was studied, whereas the paper [69] dealt with the asymptotics of the convergence parameter under various limit relations between the intensities and the dimensionality of the model. In [66,67], perturbation bounds were considered under additional assumptions.

Let *N* ≥ 1 be the number of servers in the system. Assume that the customers arrival intensity *λ*(*t*) and the service intensity of a server *μ*(*t*) are 1-periodic nonnegative functions integrable on the interval [0, 1]. Therefore, the number of customers in the system (queue length) *X*(*t*) is a finite Markov chain of Class 1, that is, a BDP with the intensities *<sup>λ</sup>k*−<sup>1</sup>(*t*) = *<sup>λ</sup>*(*t*), *μk*(*t*) = *kμ*(*t*) for *k* = 1, . . . , *N*.

It should be especially noted that the process *X*(*t*) is weakly ergodic (obviously exponentially and uniformly ergodic, since the intensities are periodic and the state space is finite) if and only if

$$\int\_{0}^{1} \left(\lambda(t) + \mu(t)\right) \, dt > 0 \tag{49}$$

(see, e.g., [70]).

For definiteness, assume that 1 0 *μ*(*t*) *dt* > 0. Apply the approach described in Theorems 3 and 4. Let all *dk*= 1. Therefore,

$$B^{\*\*}(t) = \begin{pmatrix} -\left(\lambda + \mu\right) & \mu & 0 & \cdots & 0\\ \lambda & -\left(\lambda + 2\mu\right) & 2\mu & \cdots & 0\\ & \ddots & \ddots & \ddots & \ddots & \ddots\\ & 0 & \cdots & \cdots & \lambda & -\left(\lambda + N\mu\right) \end{pmatrix} \tag{50}$$

and in Equation (34) we have *αi* (*t*) = *μ*(*t*) for all *i*; hence, *α* (*t*) = *μ*(*t*).

Therefore, Theorem 3 yields the estimate

$$\|\mathbf{p}^\*(t) - \mathbf{p}^{\*\*}(t)\|\_{1D} \le e^{-\int\_s^t \mu(\tau) \, d\tau} \|\mathbf{p}^\*(s) - \mathbf{p}^{\*\*}(s)\|\_{1D}.\tag{51}$$

To find the constants in the estimates, let *μ*∗ = 1 0 *μ*(*τ*) *dτ* and consider

$$\int\_{0}^{t} \mu(\tau) \, d\tau = \mu^\* t + \int\_{0}^{\{t\}} \left(\mu(\tau) - \mu^\*\right) \, d\tau. \tag{52}$$

Find the bound for the second summand in Equation (52). Assuming *u* = {*t*}, we obtain

$$\left| \int\_{0}^{u} \left( \mu(\tau) - \mu^{\*} \right) \, d\tau \right| \le K^{\*} = \sup\_{\mu \in [0, 1]} \int\_{0}^{u} \left( \mu(\tau) - \mu^{\*} \right) \, d\tau. \tag{53}$$

Therefore,

$$\left\| e^{-\int\_{s}^{t} \mu(\tau) \, d\tau} \right\| \le e^{K^\*} e^{-\mu^\*(t-s)}.\tag{54}$$

Therefore, for the queueing system *Mt*|*Mt*|*N*|*<sup>N</sup>*, the conditions of Theorem 5 and Corollary 2

$$d = 1, \quad M = M^\* = e^{K^\*}, \quad a = a^\* = \mu^\*, \quad W = \frac{1}{N}. \tag{55}$$

These statements imply the following perturbation bounds:

$$\limsup\_{t \to \infty} \|\mathbf{p}(t) - \mathbf{\bar{p}}(t)\| \le \frac{4e^{K^\*} \left(\varepsilon^{K^\*} \left|\mathfrak{B} - \mathfrak{B}\right| \mathfrak{f} + \mu^\* \left|\mathfrak{f} - \overline{\mathfrak{f}}\right|\right)}{\mu^\* \left(\mu^\* - \varepsilon^{K^\*} \left|\mathfrak{B} - \bar{\mathfrak{B}}\right|\right)}\tag{56}$$

for the vector od=f state probabilities, and

$$\limsup\_{t \to \infty} |\phi(t) - \bar{\phi}(t)| \le \frac{N e^{K^\*} \left( e^{K^\*} \left| \mathfrak{B} - \mathfrak{B} \right| \mathfrak{f} + \mu^\* \left| \mathfrak{f} - \bar{\mathfrak{f}} \right| \right)}{\mu^\* \left( \mu^\* - e^{K^\*} \left| \mathfrak{B} - \bar{\mathfrak{B}} \right| \right)},\tag{57}$$

for limit expectations.

Moreover, for these bounds to be consistent, additional information is required concerning the form of the perturbed intensity matrix. The simplest bounds can be obtained, if it is assumed that the perturbed Markov chain is also a BDP with the same state space and the birth and death intensities *<sup>λ</sup>k*−<sup>1</sup>(*t*) and *μk*(*t*), respectively. Therefore, if the birth and death intensities themselves do not exceed *ε* for almost all *t* ≥ 0, then |f − ¯f| ≤ *ε* and |B − B¯ | ≤ 5*<sup>ε</sup>*, so that the bounds expressed by Equations (56) and (57) have the form

$$\limsup\_{t \to \infty} \left||\mathbf{p}(t) - \mathbf{\bar{p}}(t)\right|| \le \frac{4e^{K^\*} \left(5Lx^{K^\*} + \mu^\*\right)\varepsilon}{\mu^\* \left(\mu^\* - 5\varepsilon e^{K^\*}\right)}\tag{58}$$

for the vectors of state probabilities, and

$$\limsup\_{t \to \infty} |\phi(t) - \bar{\phi}(t)| \le \frac{4Ne^{K^\*} \left(5Lx^{K^\*} + \mu^\*\right)\varepsilon}{\mu^\* \left(\mu^\* - 5\varepsilon x^{K^\*}\right)}\tag{59}$$

for the limit expectations.

On the other hand, Theorem 7 can be applied as well. To construct the bounds for the corresponding parameters, Equation (24) and the fact that *D*1 = *N* is exploited. Therefore, Theorem 7 is valid for the queueing system *Mt*|*Mt*|*N*|*N* with the following values of the parameters:

$$c = c^\* = 4Ne^{K^\*}, \quad b = b^\* = \mu^\*. \tag{60}$$

According to this theorem, we obtain the estimate

$$\limsup\_{t \to \infty} \|\mathbf{p}(t) - \mathbf{p}(t)\| \le \frac{(1 + K^\* + \log(2N))\,\varepsilon}{\mu^\*}.\tag{61}$$

Moreover, the Markov chains *X*(*t*) and *X*¯(*t*) have limit expectations and

$$|\phi(t) - \bar{\phi}(t)| \le \frac{N\left(1 + K^\* + \log(2N)\right)\varepsilon}{\mu^\*}.\tag{62}$$

It is worth noting that, for the estimates expressed by Equations (61) and (62) to hold, only the condition of the smallness of perturbations is required, and *no* additional information concerning the structure of the intensity matrix is required.

Thus, in the example with the finite state space under consideration, uniform bounds turn out to be more exact.

Now consider a more special example. Let *N* = 299, *λ*(*t*) = 200(1 + sin <sup>2</sup>*πω<sup>t</sup>*), *μ*(*t*) = 1.

In Figures 1–5, there are plots of the expected number of customers in the system for some of most probable states with *ω* = 1; in Figures 6 and 7, there are plots of the expected number of customers with *ω* = 0.5.

**Figure 1.** Example 1. The mean *<sup>E</sup>*(*<sup>t</sup>*, 0) and *<sup>E</sup>*(*<sup>t</sup>*, *N*) for the original process *t* ∈ [0, <sup>19</sup>], *ω* = 1.

**Figure 2.** Example 1. The perturbation bounds for the limit expectation *<sup>E</sup>*(*<sup>t</sup>*, <sup>0</sup>), *t* ∈ [19, <sup>20</sup>], *ω* = 1.

**Figure 3.** Example 1. The perturbation bounds for the "limit" probability Pr(*X*(*t*) = <sup>190</sup>), *t* ∈ [19, <sup>20</sup>], *ω* = 1.

**Figure 4.** Example 1. The perturbation bounds for the "limit" probability Pr(*X*(*t*) = <sup>200</sup>), *t* ∈ [19, <sup>20</sup>], *ω* = 1.

**Figure 5.** Example 1. The perturbation bounds for the "limit" probability Pr(*X*(*t*) = <sup>210</sup>), *t* ∈ [19, <sup>20</sup>], *ω* = 1.

**Figure 6.** Example 1. The expectations *<sup>E</sup>*(*<sup>t</sup>*, 0) and *<sup>E</sup>*(*<sup>t</sup>*, *N*) for the original process *t* ∈ [0, <sup>18</sup>], *ω* = 0.5.

**Figure 7.** Example 1. The perturbation bounds for the limit expectation *<sup>E</sup>*(*<sup>t</sup>*, <sup>0</sup>), *t* ∈ [18, <sup>20</sup>], *ω* = 0.5.

On the other hand, as has already been noted, for the Markov chains of Classes 1–4 with countable state space, no uniform bounds could be constructed.

Consider the construction of bounds on the example of a rather simple model, which, however, does not belong to the most well-studied Class 1 (that is, which is not a BDP).

Let a queueing system be given in which the customers can appear separately or in pairs with the corresponding intensities *<sup>a</sup>*1(*t*) = *λ*(*t*) and *<sup>a</sup>*2(*t*) = 0.5*<sup>λ</sup>*(*t*), but are served one by one on one of two servers with constant intensities *μk*(*t*) = min(*k*, <sup>2</sup>)*μ*, where *λ*(*t*) is a 1-periodic function integrable on the interval [0, 1]. Therefore, the number of customers in this system belongs to Class 2, and the corresponding matrix *B*∗∗(*t*) has the form

$$B^{\*\*}(t) = \begin{pmatrix} a\_{11} & \frac{d\_1}{d\_2}\mu & 0 & \cdots & 0\\ \frac{d\_2}{d\_1}\lambda & a\_{22} & \frac{d\_2}{d\_3}2\mu & \cdots & 0\\ \frac{d\_3}{d\_1}0.5\lambda & \frac{d\_3}{d\_2}\lambda & a\_{33} & \frac{d\_3}{d\_4}2\mu & \cdots\\ 0 & \ddots & \ddots & \ddots & \ddots\\ & \ddots & \ddots & \ddots & \ddots & \ddots \end{pmatrix} \tag{63}$$

where *<sup>a</sup>*11(*t*) = − (1.5*λ*(*t*) + *μ*), *akk*(*t*) = − (1.5*λ*(*t*) + <sup>2</sup>*μ*), if *k* ≥ 2. This matrix is essentially nonnegative, such that, in the expression for the logarithmic norm, the signs of the absolute value can be omitted. Let *d*1 = 1, *dk*+<sup>1</sup> = *δdk*, and *δ* > 1. For this purpose, consider the expressions from Equation (34). We have

$$\mathfrak{a}\_1(t) = \mu - \lambda(t) \left( 0.5\delta^2 + \delta - 1.5 \right),$$

$$\mathfrak{a}\_2(t) = \mu \left( 2 - \delta^{-1} \right) - \lambda(t) \left( 0.5\delta^2 + \delta - 1.5 \right),$$

$$\mathfrak{a}\_k(t) = 2\mu \left( 1 - \delta^{-1} \right) - \lambda(t) \left( 0.5\delta^2 + \delta - 1.5 \right), \ k \ge 3.$$

Therefore, for *δ* ≤ 2, we obtain

$$\begin{split} a\left(t\right) = \inf\_{i\geq 1} a\_i\left(t\right) &= 2\mu \left(1 - \delta^{-1}\right) - \lambda\left(t\right) \left(0.5\delta^2 + \delta - 1.5\right) = \\ &= \left(\delta - 1\right) \left(\frac{2\mu}{\delta} - 0.5\lambda\left(t\right)\left(\delta + 3\right)\right), \end{split} \tag{64}$$

and the condition

$$a^\* = \int\_0^1 \left(\delta - 1\right) \left(\frac{2\mu}{\delta} - 0.5\lambda(t)\left(\delta + 3\right)\right) dt = \frac{\delta - 1}{2} \left(\frac{4\mu}{\delta} - \lambda^\*\left(\delta + 3\right)\right) > 0\tag{65}$$

will a fortiori hold if *μ* > *λ*∗ with a corresponding choice of *δ* ∈ (1, 2].

The further reasoning is almost the same as in the preceding example: instead of Equation (54), we obtain

$$e^{-\int\_s^t a(\tau) \, d\tau} \le e^{K^\*} e^{-a^\*(t-s)}\tag{66}$$

where now

$$K^\* = \sup\_{\mathfrak{u} \in [0,1]} \int\_0^\mathfrak{u} \left( \mathfrak{a}(\tau) - \mathfrak{a}^\* \right) d\tau. \tag{67}$$

Hence, the conditions of Theorem 5 and Corollary 2 for the number of customers in the system under consideration hold for

$$d = 1, \quad M = M^\* = e^{K^\*}, \quad a = a^\* = a^\*, \quad \mathcal{W} = \inf\_{k \ge 1} \frac{\delta^{k-1}}{k}. \tag{68}$$

To construct meaningful perturbation bounds, it is necessarily required to have additional information concerning the form of the perturbed intensity matrix. Therefore, Example 1 in Section 2 shows that, if a possibility of the arrival of an arbitrary number of customers ("mass arrival" in the terminology of [58]) to an empty queue is assumed, then an arbitrarily small (in the uniform norm) perturbation of the intensity matrix can "spoil" all the characteristics of the process. For example, satisfactory bounds can be constructed if we know that the intensity matrix of the perturbed system has the same form; that is, the customers can appear either separately or in pairs and are served one by one. Therefore, if the perturbations of the intensities themselves do not exceed *ε* for almost all *t* ≥ 0, then |f − ¯f| ≤ 5*ε* and |B − B¯ | ≤ 5*<sup>ε</sup>*, such that, instead of Equations (56) and (57), we obtain

$$\limsup\_{t \to \infty} \|\mathbf{p}(t) - \mathbf{\bar{p}}(t)\| \le \frac{20e^{K^\*}\varepsilon \left(Le^{K^\*} + a^\*\right)}{a^\* \left(a^\* - 20\varepsilon e^{K^\*}\right)}\tag{69}$$

for the vectors of state probabilities and

$$\limsup\_{t \to \infty} |\phi(t) - \bar{\phi}(t)| \le \frac{20e^{K^\*}\varepsilon \left(Le^{K^\*} + a^\*\right)}{a^\* \mathcal{W}\left(a^\* - 20\varepsilon e^{K^\*}\right)}\tag{70}$$

for the limit expectations.

> For example, let *λ*(*t*) = 1 + sin 2*πt*, *μ*(*t*) = 3, and *δ* = 2. Therefore, we have

$$a(t) = \mu - 2.5\lambda(t), \quad a^\* = 0.5, \quad W = 1. \tag{71}$$

Furthermore, we follow the method described in [71,72] in detail. Namely, we choose the dimensionality of the truncated process (300 in our case), the interval on which the desired accuracy is achieved ([0, 100]) in the example under consideration) and the limit interval itself (here it is [100, 101]).

Figures 8–13 expose the plots of the expected number of customers in the system and some of the most probable states.

**Figure 8.** Example 2. The expectations *<sup>E</sup>*(*<sup>t</sup>*, 0) and *<sup>E</sup>*(*<sup>t</sup>*, 299) for the original process *t* ∈ [0, <sup>100</sup>].

**Figure 9.** Example 2. The perturbation bounds for the limit expectation *<sup>E</sup>*(*<sup>t</sup>*, <sup>0</sup>), *t* ∈ [100, <sup>101</sup>].

**Figure 10.** Example 2. The probabilities of the empty queue for *X*(0) = 0 and *X*(0) = 299 for the original process *t* ∈ [0, <sup>100</sup>].

**Figure 11.** Example 2. The perturbation bounds for the "limit" probability Pr(*X*(*t*) = <sup>0</sup>), *t* ∈ [100, <sup>101</sup>].

**Figure 12.** Example 2. The perturbation bounds for the "limit" probability Pr(*X*(*t*) = <sup>1</sup>), *t* ∈ [100, <sup>101</sup>].

**Figure 13.** Example 2. The perturbation bounds for the "limit" probability Pr(*X*(*t*) = <sup>2</sup>), *t* ∈ [100, <sup>101</sup>].

**Author Contributions:** Conceptualization, A.Z and V.K.; Software, Y.S.; Investigation, A.Z.(Sections 1–5), V.K. (Sections 1–3) and Y.S. (Sections 4–5); Writing—original draft preparation, A.Z., V.K. and Y.S.; Writing—review and editing, A.Z., V.K. and Y.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** Sections 1–3 were written by Korolev and Zeifman under the support of the Russian Science Foundation, project 18-11-00155. Sections 4 and 5 were written by Satin and Zeifman under the support of the Russian Science Foundation, project 19-11-00020.

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