**5. Time-Varying Markovian Bulk-Arrival and Bulk-Service System with State-Dependent Control**

In the recent paper [35] the authors considered the Markovian bulk-arrival and bulkservice system with the general state-dependent control (see also [35–39]). The total number

*X*(*t*) of customers at time *t* in that system constitutes CTMC with state space {0, 1, 2, ... }. Its generator *Q*(*t*)=(*qij*(*t*))<sup>∞</sup> *<sup>i</sup>*,*j*=<sup>0</sup> has quite a specific structure:

$$q\_{ij} = \begin{cases} h\_{ij\prime} & \text{if } 0 \le i \le k - 1, \ j \ge 0, \\ b\_{i-j+k\prime} & \text{if } i \ge k, \ j \ge i - k, \\ 0, & \text{otherwise}, \end{cases} \tag{30}$$

where *k* ≥ 1 is the fixed integer. For further explanations and the motivation behind such structure of *Q*(*t*) we refer the reader to [Section 1] in [35]. The purpose of this section is to show that for at least one particular case of this system, even when the intensities are time-dependent, one can obtain the upper bounds for the rate of convergence using the method based on the logarithmic norm. Specifically, we take the example, (in the example of [Section 7] in [35] the entries of the intensity matrix *<sup>Q</sup>*(*t*) are: *hi*,*i*−<sup>1</sup> = *<sup>μ</sup>*, *hi*,*i*+<sup>1</sup> = *<sup>λ</sup>*, *hi*,*<sup>i</sup>* = −(*λ* + *μ*), *b*<sup>0</sup> = *a*, *bk*<sup>+</sup><sup>1</sup> = *b*, *bk* = −(*a* + *b*) and *k* = 3). from the Section 7 of [35], with the exception that all the transition intensities are time-dependent i.e., *bi* = *λ*(*t*) and *ai* = *μ*(*t*) and are both nonnegative locally integrable on [0, ∞). Then the transposed generator *A*(*t*)=(*aij*(*t*))<sup>∞</sup> *<sup>i</sup>*,*j*=<sup>0</sup> = *<sup>Q</sup>T*(*t*) of *<sup>X</sup>*(*t*) has the form

$$A(t) = \begin{pmatrix} -\lambda(t) & \mu(t) & 0 & \mu(t) & 0 & 0 & \dots \\ \lambda(t) & -(\lambda(t) + \mu(t)) & \mu(t) & 0 & \mu(t) & 0 & \dots \\ 0 & \lambda(t) & -(\lambda(t) + \mu(t)) & 0 & 0 & \mu(t) & \dots \\ 0 & 0 & \lambda(t) & -(\lambda(t) + \mu(t)) & 0 & 0 & \dots \\ 0 & 0 & 0 & \lambda(t) & -(\lambda(t) + \mu(t)) & 0 & \dots \\ 0 & 0 & 0 & 0 & \lambda(t) & -(\lambda(t) + \mu(t)) & \dots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix} \tag{31}$$

Denote the distribution of *X*(*t*) by **p**(*t*) i.e., **p**(*t*)=(*p*0(*t*), *p*1(*t*),...)<sup>T</sup> = ∑<sup>∞</sup> *<sup>k</sup>*=<sup>0</sup> P (*X*(*t*) = *k*)**e***<sup>k</sup>* (as above, **e***<sup>k</sup>* denotes the *k*th unit basis vector). The ergodicity bound for *X*(*t*) in the null ergodic case is given below in the *Theorem 4*.

**Theorem 4.** *If* <sup>∞</sup> 0 *<sup>λ</sup>*(*t*)(<sup>1</sup> <sup>−</sup> *<sup>σ</sup>*) + *<sup>μ</sup>*(*t*)(<sup>1</sup> <sup>−</sup> *<sup>σ</sup>*−3) *dt* = +∞ *for some σ* ∈ (0, 1)*, then the Markov chain X*(*t*) *is null ergodic,*

$$\sum\_{i=0}^{\infty} \sigma^i p\_i(t) \le e^{-\int\_0^t \left(\lambda(u) + \mu(u) - \sigma \lambda(u) - \sigma^{-3} \mu(u)\right) du} \sum\_{i=0}^{\infty} \sigma^i p\_i(0), \ t \ge 0,\tag{32}$$

*and for any n* ≥ 0 *and N* ≥ 0 *the following inequality holds:*

$$\mathbb{P}(X(t) > n | X(0) = N) \ge 1 - \sigma^{N-n} e^{-\int\_0^t \left(\lambda(u) + \mu(u) - \sigma \lambda(u) - \sigma^{-3} \mu(u)\right) du}.\tag{33}$$

**Proof.** Fix *σ* > 0 and define the decreasing sequence of positive numbers {*δn*, *n* ≥ 0} by *δ<sup>n</sup>* = *σn*. Put **p**˜(*t*) = Λ**p**(*t*), where Λ = *diag*(*δ*0, *δ*1,...). Then we have (6). Denote by <sup>−</sup>*α*˜ *<sup>k</sup>*(*t*) the sum of all elements in the *<sup>k</sup>*th column of *<sup>A</sup>*˜(*t*) i.e.

$$\begin{aligned} \bar{\mathfrak{a}}\_0(t) &= (1 - \sigma)\lambda(t), \\ \bar{\mathfrak{a}}\_k(t) &= (1 - \sigma) \left( \lambda(t) + \mu(t) - \sigma^{-1}\mu(t) \right), \ k = 1, 2, \\ \bar{\mathfrak{a}}\_k(t) &= \underbrace{\lambda(t) + \mu(t) - \sigma\lambda(t) - \sigma^{-3}\mu(t)}\_{=\mathfrak{F}(t)}, k \ge 3. \end{aligned}$$

If 0 < *σ* < 1 then *α*˜ <sup>0</sup>(*t*) ≥ *β*(*t*), *α*˜ <sup>1</sup>(*t*) ≥ *β*(*t*) and *α*˜ <sup>2</sup>(*t*) ≥ *β*(*t*), and thus (32) and (33) follow from (4) and (8) respectively.

The ergodicity bound in the weakly ergodic case, state below in the Theorem 5, is obtained by analogy with the Theorem 1. Define an increasing sequence of positive

numbers {*δn*, *n* ≥ 0}. Then the matrix *B*∗∗(*t*) built from the matrix *A*(*t*), in the same way as it is done in the Section 3, has the form:

$$B^{\ast \ast}(t) = \begin{pmatrix} -(\lambda(t) + \mu(t)) & \mu(t)\frac{\delta\_1}{2} & -\mu(t)\frac{\delta\_1}{3} & \mu(t)\frac{\delta\_1}{4} & 0 & 0 & \dots \\ \lambda(t)\frac{\delta\_1}{\delta\_1} & -(\lambda(t) + \mu(t)) & 0 & 0 & \mu(t)\frac{\delta\_2}{5} & 0 & \dots \\ 0 & \lambda(t)\frac{\delta\_2}{2} & -(\lambda(t) + \mu(t)) & 0 & 0 & \mu(t)\frac{\delta\_2}{5} & \dots \\ 0 & 0 & \lambda(t)\frac{\delta\_4}{25} & -(\lambda(t) + \mu(t)) & 0 & 0 & \dots \\ 0 & 0 & 0 & \lambda(t)\frac{\delta\_3}{34} & -(\lambda(t) + \mu(t)) & 0 & \dots \\ 0 & 0 & 0 & 0 & \lambda(t)\frac{\delta\_4}{5} & -(\lambda(t) + \mu(t)) & \dots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix} \tag{34}$$

Denote by −*α*˜ *<sup>k</sup>*(*t*) the sum of all elements in the *k*th column of *B*∗∗(*t*) i.e.,

$$\begin{aligned} \boldsymbol{\mu}\_{1}(t) &= \boldsymbol{\lambda}(t) + \boldsymbol{\mu}(t) - \boldsymbol{\lambda}(t)\frac{\delta\_{2}}{\delta\_{1}}, \\ \boldsymbol{\mu}\_{2}(t) &= \boldsymbol{\lambda}(t) + \boldsymbol{\mu}(t) - \boldsymbol{\lambda}(t)\frac{\delta\_{3}}{\delta\_{2}} - \boldsymbol{\mu}(t)\frac{\delta\_{1}}{\delta\_{2}}, \\ \boldsymbol{\mu}\_{3}(t) &= \boldsymbol{\lambda}(t) + \boldsymbol{\mu}(t) - \boldsymbol{\lambda}(t)\frac{\delta\_{4}}{\delta\_{3}} - \boldsymbol{\mu}(t)\frac{\delta\_{1}}{\delta\_{3}}, \\ \boldsymbol{\mu}\_{k}(t) &= \boldsymbol{\lambda}(t) + \boldsymbol{\mu}(t) - \boldsymbol{\lambda}(t)\frac{\delta\_{k+1}}{\delta\_{k}} - \boldsymbol{\mu}(t)\frac{\delta\_{k-3}}{\delta\_{k}}, \ k \ge 4. \end{aligned}$$

Since the logarithmic norm of *B*∗∗(*t*) is equal to −*β*(*t*) = − min(min1≤*k*≤<sup>3</sup> *<sup>α</sup>k*(*t*), inf*k*≥<sup>4</sup> *<sup>α</sup>k*(*t*)), we can apply (4) to (13) and (15) with *<sup>δ</sup>k*+<sup>1</sup> = *σδk*, *k* ≥ 5.

**Theorem 5.** *If* <sup>∞</sup> 0 *<sup>λ</sup>*(*t*)(<sup>1</sup> <sup>−</sup> *<sup>σ</sup>*) + *<sup>μ</sup>*(*t*)(<sup>1</sup> <sup>−</sup> *<sup>σ</sup>*−3) *dt* = +∞ *for some σ* > 0*, then the Markov chain X*(*t*) *is weakly ergodic and the ergodicity bound (15) holds.*

As the numerical example we again consider the periodic case: *λ*(*t*) = 3(1 + sin *πt*) and *μ*(*t*) = 4(1 + cos 2*πt*). By direct inspection it can be checked that the sequence {*δn*, *<sup>n</sup>* <sup>≥</sup> <sup>1</sup>}, defined by *<sup>δ</sup><sup>n</sup>* <sup>=</sup> <sup>10</sup> 9 *n*−1 , leads to *β*(*t*) = *α*2(*t*). Thus the conditions of the *Theorem 5* are fulfilled with *σ* = <sup>10</sup> <sup>9</sup> . The pre-limiting and the limiting values of the same quantities as in the two previous examples—*p*0(*t*) and *E*(*t*, *k*)—are shown in Figures 9–12.

**Figure 9.** Rate of convergence of the empty system probability *p*0(*t*) in the interval [0, 45] given two different initial conditions: *p*0(0) = 1 (**red line**), *p*250(0) = 1 (**blue line**).

**Figure 10.** Limiting probability *p*0(*t*) of the empty queue given two different initial conditions: *p*0(0) = 1 (**red line**), *p*300(0) = 1 (*blue line*).

**Figure 11.** Rate of convergence of the conditional mean *E*(*t*, *k*) number of customers in the system in the interval [0, 45]: *E*(*t*, 0) (**red line**), *E*(*t*, 300) (**blue line**).

**Figure 12.** Limiting conditional mean *E*(*t*, *k*) number of customers in the system: *E*(*t*, 0) (**red line**), *E*(*t*, 300) (**blue line**).
