**4. Time-Varying Single-Server Markovian System with Bulk Arrivals, Queue Skipping Policy and Catastrophes**

Consider the time-varying *M*/*M*/1 system with the intensities being periodic functions of time and the queue skipping policy as in [31] (see also [32]). Customers arrive to the system in batches according to the inhomogeneous Poisson process with the intensity *λ*(*t*). The size of an arriving batch becomes known upon its arrival to the system and is the random variable with the given probability distribution {*bn*, *n* ≥ 1}, having finite mean ¯ *b* = ∑<sup>∞</sup> *<sup>k</sup>*=<sup>1</sup> *Bk*, *Bk* <sup>=</sup> <sup>∑</sup><sup>∞</sup> *<sup>n</sup>*=*<sup>k</sup> bn*. The implemented queue skipping policy implies that whenever a batch arrives to the system its size, say *<sup>B</sup>*, is compared with the remaining total number of customers in the system, say *<sup>B</sup>*. If *<sup>B</sup>* <sup>&</sup>gt; *<sup>B</sup>*, then all customers, that are currently

in the system, are instantly removed from it, the whole batch *<sup>B</sup>* is placed in the the queue and one customer from it enters server. If *<sup>B</sup>* ≤ *<sup>B</sup>* the new batch leaves the system without having any effect on it. Whenever the server becomes free the first customer from the queue (if there is any) enters server and gets served according to the exponential distribution with the intensity *μ*(*t*). Finally the additional inhomogeneous Poisson flow of negative customers with the intensity *γ*(*t*) arrives to the system. Each negative arrival results in the removal of all customers present in the system at the time of arrival. The negative customer itself leaves the system. Since *γ*(*t*) depends on *t* it can happen that the effect of negative arrivals fades away too fast as *<sup>t</sup>* <sup>→</sup> <sup>∞</sup> (for example, if *<sup>γ</sup>*(*t*)=(<sup>1</sup> <sup>+</sup> *<sup>t</sup>*)−*n*, *<sup>n</sup>* <sup>&</sup>gt; 1). Such cases are excluded from the consideration.

Let *X*(*t*) be the total number of customers in the system at time *t*. From the system description it follows that *X*(*t*) is the CTMC with state space {0, 1, 2, ... , *b*∗}, where *b*<sup>∗</sup> is the maximum possible batch size i.e., *b*<sup>∗</sup> = max*n*≥1(*bn* > 0). Thus if the batch size distribution has infinite support then the state space is countable, otherwise it is finite.

It is straightforward to see that the transposed time-dependent generator *A*(*t*)=(*aij*(*t*))<sup>∞</sup> *<sup>i</sup>*,*j*=<sup>0</sup> for *X*(*t*) has the form

$$A(t) = \begin{pmatrix} -\lambda(t) & \mu(t) + \gamma(t) & \gamma(t) & \dots \\ \lambda(t)b\_1 - (\lambda(t)b\_2 + \mu(t) + \gamma(t)) & \mu(t) & 0 & \dots \\ \lambda(t)b\_2 & \lambda(t)b\_2 & -(\lambda(t)b\_3 + \mu(t) + \gamma(t)) & \mu(t) & \dots \\ \lambda(t)b\_3 & \lambda(t)b\_3 & \lambda(t)b\_3 & -(\lambda(t)b\_4 + \mu(t) + \gamma(t)) & \dots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix}$$

We represent the distribution of *X*(*t*) as a probability vector **p**(*t*), where **p**(*t*) = ∑*b*<sup>∗</sup> *<sup>k</sup>*=<sup>0</sup> P(*X*(*t*) = *k*)**e***<sup>k</sup>* tor all *t* ≥ 0. Given a proper **p**(0), the probabilistic dynamics of *X*(*t*) is described by the Kolmogorov forward equations *<sup>d</sup> dt***p**(*t*) = *A*(*t*)**p**(*t*), which can be rewritten in the form

$$\frac{d}{dt}\mathbf{p}(t) = A^\*(t)\mathbf{p}(t) + \mathbf{g}(t), \ t \ge 0,\tag{19}$$

.

where **g**(*t*) = (*γ*(*t*), 0, 0, . . .) *<sup>T</sup>* and *A*∗(*t*) is the matrix with the terms *a*<sup>∗</sup> *ij*(*t*) equal to

$$a\_{ij}^\*(t) = \begin{cases} a\_{0j}(t) - \gamma(t), & \text{if } i = 0, \\ a\_{ij}(t), & \text{otherwise.} \end{cases} \tag{20}$$

Due to the restrictions imposed on *γ*(*t*), we have that <sup>∞</sup> <sup>0</sup> *γ*(*t*) *dt* = ∞. Thus *X*(*t*) cannot be null ergodic irrespective of the values of *λ*(*t*) and *μ*(*t*).

**Theorem 2.** *Assume that the catastrophe intensity γ*(*t*) *is such that* <sup>∞</sup> <sup>0</sup> *γ*(*t*) *dt* = ∞*. Then the Markov chain X*(*t*) *is weakly ergodic and for any two initial conditions* **p**∗(0) *and* **p**∗∗(0) *it holds that*

$$\|\mathbf{p}^\*(t) - \mathbf{p}^{\*\*}(t)\| \le e^{-\int\_0^t \gamma(u) \, du} \|\mathbf{p}^\*(0) - \mathbf{p}^{\*\*}(0)\| \le 2e^{-\int\_0^t \gamma(u) \, du}, \ t \ge 0. \tag{21}$$

**Proof.** It is straightforward to check, that the logarithmic norm (see (3)) of the operator *A*∗(*t*) is equal to −*γ*(*t*). Denote now by *U*∗(*t*,*s*) the Cauchy operator of the Equation (19). Then the statement of the theorem follows from the inequalities -*U*∗(*t*,*s*)-<sup>≤</sup> *<sup>e</sup>* <sup>−</sup> *<sup>t</sup> <sup>s</sup> <sup>γ</sup>*(*u*) *du* and

$$\|\mathbf{p}^\*(t) - \mathbf{p}^{\*\*}(t)\| \le \|\boldsymbol{\mathcal{U}}^\*(t, 0)\| \|\mathbf{p}^\*(0) - \mathbf{p}^{\*\*}(0)\|.$$

Even though (21) is the valid ergodicity bound for *X*(*t*), it is of little help whenever the state space of *X*(*t*) is countable and one needs to perform the numerical solution of (5). This is due to the fact that the bound (21) is in the uniform operator topology, which does not allow to use the analytic frameworks (for example, [29]) for finding proper truncations of an infinite ODE system. For the latter task ergodicity bounds for *X*(*t*) in stronger (than *l*1), weighted norms are required. It can be said that with such bounds we have a weight assigned to each initial state and thus a truncation procedure becomes sensitive to the number of states. Below (in the Theorem 3) we obtain such a bound under the additional assumption, (for the definition used see [33]; appropriate test for monotone functions can be found in [Proposition 1] of [34]. Although the Theorem 2 below holds for any distribution {*bn*, *n* ≥ 1}, this assumption is essential for the Theorem 3. For distributions with tails heavier than the geometric distribution we were unable to find the conditions, which guarantee the existence of the limiting regime of queue-size process even for periodic intensities). that the batch size distribution {*bn*, *n* ≥ 1} is harmonic new better than used in expectation i.e., ∑<sup>∞</sup> *<sup>j</sup>*=*<sup>k</sup> Bj*+<sup>1</sup> <sup>≤</sup> ¯ *b* <sup>1</sup> <sup>−</sup> ¯ *b*−<sup>1</sup> *<sup>k</sup>* for all *<sup>k</sup>* <sup>≥</sup> 0.

Using the normalization condition *<sup>p</sup>*0(*t*) = <sup>1</sup> − <sup>∑</sup>*i*≥<sup>1</sup> *pi*(*t*) the forward Kolmogorov system *<sup>d</sup> dt***p**(*t*) = *A*(*t*)**p**(*t*) can be rewritten as

$$\frac{d}{dt}\mathbf{z}(t) = A^{\*\*}(t)\mathbf{z}(t) + \mathbf{f}(t), \ t \ge 0,\tag{22}$$

where

**f**(*t*) = (*λ*(*t*)*b*1, *λ*(*t*)*b*2, *λ*(*t*)*b*3, *λ*(*t*)*b*4,...) *<sup>T</sup>* and

$$A^{\*\*}(t) = \begin{pmatrix} -\left(\lambda(t) + \mu(t) + \gamma(t)\right) & \mu(t) - \lambda(t)b\_1 & -\lambda(t)b\_1 & -\lambda(t)b\_1 & \dots \\ 0 & -\left(\lambda(t)b\_2 + \mu(t) + \gamma(t)\right) & \mu(t) - \lambda(t)b\_2 & -\lambda(t)b\_2 & \dots \\ 0 & 0 & -\left(\lambda(t)B\_3 + \mu(t) + \gamma(t)\right) & \mu(t) - \lambda(t)b\_3 & \dots \\ 0 & 0 & 0 & -\left(\lambda(t)B\_4 + \mu(t) + \gamma(t)\right) & \dots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix}.\tag{23}$$

Fix *<sup>d</sup>* <sup>∈</sup> (1, 1 + (¯ *<sup>b</sup>* <sup>−</sup> <sup>1</sup>)−1] and define the increasing sequence of positive numbers {*δn*, *<sup>n</sup>* <sup>≥</sup> <sup>0</sup>} by *<sup>δ</sup><sup>n</sup>* <sup>=</sup> *<sup>d</sup>n*−1. Then instead of the matrix *<sup>B</sup>*∗∗(*t*) in (13) we have the matrix *A*˜(*t*)=(*a*˜*ij*(*t*))<sup>∞</sup> *<sup>i</sup>*,*j*=<sup>0</sup> with the following structure:

$$
\bar{A}(t) = \begin{pmatrix}
0 & - (\lambda(t)B\_2 + \mu(t) + \gamma(t)) & \frac{1}{4}\mu(t) & 0 & \dots \\
& 0 & - (\lambda(t)B\_3 + \mu(t) + \gamma(t)) & \frac{1}{4}\mu(t) & \dots \\
0 & 0 & 0 & - (\lambda(t)B\_4 + \mu(t) + \gamma(t)) & \dots \\
& \vdots & \vdots & \vdots & \ddots \\
\end{pmatrix}.\tag{24}
$$

Since the logarithmic norm (see (3)) of *A*˜(*t*) is equal to

$$\begin{aligned} -\beta^\*(t) = \sup\_i \left\{ \overline{a}\_{ii}(t) + \sum\_{j \neq i} \overline{a}\_{ji}(t) \right\} &= -\inf\_i \left\{ \gamma(t) + \left(1 - \frac{1}{d}\right) \mu(t) + \lambda(t) B\_i \right\} \\ &= -\gamma(t) - \left(1 - \frac{1}{d}\right) \mu(t), \end{aligned}$$

then from (4) we get:

$$\|\|\mathbf{z}^\*(t) - \mathbf{z}^{\*\*}(t)\|\|\_{\text{1D}} \le e^{-\int\_0^t \left(\gamma(u) + \left(1 - d^{-1}\right)\mu(u)\right) du} \|\mathbf{z}^\*(0) - \mathbf{z}^{\*\*}(0)\|\_{\text{1D}}.\tag{25}$$

Arguments similar to those used to establish the *Theorem 1* lead to the following ergodicity bounds for **p**∗(*t*) − **p**∗∗(*t*)and the conditional mean *E*(*t*, *k*):

$$\|\mathbf{p}^\*(t) - \mathbf{p}^{\*\*}(t)\| \le 4e^{-\int\_0^t \left(\gamma(u) + \left(1 - d^{-1}\right)\mu(u)\right) du} \|\mathbf{z}^\*(0) - \mathbf{z}^{\*\*}(0)\|\_{1\mathbf{D}'} \tag{26}$$

$$|E(t,k) - E(t,0)| \le \frac{1 + d^{k-1}}{W} e^{-\int\_0^t \left(\gamma(u) + \binom{1-d^{-1}}{u} \mu(u)\right) du}, \quad k \ge 1, \ W = \inf\_n \frac{d^n}{n+1}.\tag{27}$$

These results can be put together in the single theorem.

**Theorem 3.** *Assume that the distribution* {*bn*, *<sup>n</sup>* <sup>≥</sup> <sup>1</sup>} *with finite mean* ¯ *b is harmonic new better than used in expectation. Then if* <sup>∞</sup> 0 *<sup>γ</sup>*(*t*)+(<sup>1</sup> <sup>−</sup> *<sup>d</sup>*−1)*μ*(*t*) *dt* = +∞ *for some <sup>d</sup>* <sup>∈</sup> (1, 1 + (¯ *<sup>b</sup>* <sup>−</sup> <sup>1</sup>)−1]*, then the Markov chain <sup>X</sup>*(*t*) *is weakly ergodic and the ergodicity bound (26) holds.*

We close this section with the example, showing the dependence on *t* of the same two quantities — *p*0(*t*) and *E*(*t*, *k*)—considered in the Section 3. Assume here that *bk* = <sup>1</sup> 3 2 3 *k*−<sup>1</sup> , *λ*(*t*) = 9(1 + sin 2*πt*), *μ*(*t*) = 8(1 + cos 2*πt*) and *γ*(*t*) = 1, i.e., the catastrophe intensity is constant and the mean size ¯ *b* of an arriving batch is equal to 3. It can be checked that *d* = <sup>3</sup> <sup>2</sup> satisfies the conditions of the *Theorem 3*. Then from (26) and (27) we get the upper bounds

$$\|\|\mathbf{p}^\*(t) - \mathbf{p}^{\*\*}(t)\|\| \le 4e^{-\frac{5}{3}t} \|\mathbf{z}^\*(0) - \mathbf{z}^{\*\*}(0)\|\_{1D\_\prime} \tag{28}$$

$$|E(t,k) - E(t,0)| \le \frac{1 + \left(\frac{3}{2}\right)^{k-1}}{\frac{9}{8}} e^{-\frac{5}{3}t}, \ k \ge 0. \tag{29}$$

In Figure 5 it is depicted how *p*0(*t*) behaves as *t* increases and Figure 6 shows its limiting value. If *<sup>t</sup>* <sup>≥</sup> 60 then the right part of (28) does not exceed 3 · <sup>10</sup><sup>−</sup>2, i.e., starting from the instant *t* = 60 = *t* ∗ the system "forgets" its initial state and the distribution of *X*(*t*) for *t* > *t* ∗ can be regarded as limiting. Moreover, since the limiting distribution of *X*(*t*) is periodic, it is sufficient to solve (numerically, (it must be noticed that since *bk* > 0 for all *k*, the system of ODEs contains infinite number of equations. Thus in order to solve it numerically one has to truncate it. We perform this truncation according to the method in [30])). the system of ODEs only in the interval [0, *t* ∗ + *T*], where *T* is the smallest common multiple of the periods of *λ*(*t*) and *μ*(*t*) i.e., *T* = 1. The probability distribution of *X*(*t*) in the interval [*t* ∗, *t* <sup>∗</sup> <sup>+</sup> *<sup>T</sup>*] is the estimate (with error not greater than 3 · <sup>10</sup>−<sup>2</sup> in *l*1-norm) of the limiting probability distribution of *X*(*t*). The upper bound on the rate of convergence of the conditional mean number of customers in the system *E*(*t*, *k*) is given in (29). If *t* ≥ *t* ∗ then the right part does not exceed 0, 3, i.e., starting from the instant *t* = *t* ∗ the system "forgets" its initial state and the value of *E*(*t*, *k*) can be regarded as the limiting value of the mean number of customers with the error not greater than 0, 3. The rate of convergence of *E*(*t*, *k*) and the behaviour of its limiting value can be seen in Figures 7 and 8. As in the previous numerical example, the obtained upper bounds are not tight: the system enters the periodic limiting regime before the instant *t* = *t* ∗.

**Figure 5.** Rate of convergence of the empty system probability *p*0(*t*) in the interval [0, 60] given two different initial conditions: *p*0(0) = 1 (**red line**), *p*250(0) = 1 (**blue line**).

**Figure 6.** Limiting probability *p*0(*t*) of the empty queue given two different initial conditions: *p*0(0) = 1 (**red line**), *p*250(0) = 1 (**blue line**).

**Figure 7.** Rate of convergence of the conditional mean *E*(*t*, *k*) number of customers in the system in the interval [0, 60]: *E*(*t*, 0) (**red line**), *E*(*t*, 250) (**blue line**).

**Figure 8.** Limiting conditional mean *E*(*t*, *k*) number of customers in the system: *E*(*t*, 0) (**red line**), *E*(*t*, 250) (**blue line**).
