Analysis of a MAP/M/1/N Queue with Periodic and Non-Periodic Piecewise Constant Input Rate

Abstract

This paper considers a queuing system with a finite buffer, a constant main $MAP$ flow, and an additional periodic (non-periodic) Poisson piecewise-constant flow of customers. The eigenvalues of the probability translation matrix of a Kolmogorov system with constant input intensities is analyzed. The definition of the transition mode time based on the analysis of the probability translation matrix determinant is introduced for the first time. An analytical solution to the Kolmogorov equation system for a queuing system with piecewise constant arrival and service intensities is found, the solutions for a queuing system with periodic arrival and service intensities are analyzed, and numerical calculations illustrating this approach are presented.

1. Introduction

The mathematical tools of queuing theory are widely used to calculate the performance of telecommunication systems. In most cases, the operation of such systems has been studied in a stationary mode [1,2,3]. However, the stationary performance metrics of queuing systems can differ significantly from similar metrics in the transient mode [4].

At the same time, transient modes are often observed in telecommunication systems. These modes occur during the installation or reboot of networks. Transient processes can occur as a result of random traffic jumps caused by an abrupt change in the information transmission route. This situation is typical, for example, of self-organizing wireless networks and of wired networks with dynamic routing in the case of network node failure or network reboot [5]. Traffic jumps occur in software-defined networks as well, as a result of frequent changes in information transmission channels in network controllers [5,6]. An important practical problem is the consideration of periodic input flows. One reason for periodic traffic jumps is a network attack [7]. In cases of periodic impact from undesired traffic, periodic changes in input flow are observed.

The pioneering works analyzing queuing systems with periodic changes in the input flow were published in the 1970s [8,9]. Those papers used an approximate analytical method to study the queuing system $M/M/s$ with the intensity of the input flow, which varies according to a sinusoidal law. In [10], the expressions of the stationary probabilities of the system $M_t/M/\infty$ with sinusoidal arrival intensity are obtained. The works in [11,12] carry out a similar study for the systems $M_t/M/1$ and $G_t/G_t/1$, while [13,14] analyze the stability of non-homogeneous Markov processes with continuous time, describing the operation of the main classes of queuing systems with the sinusoidal Poisson input flow.

In recent years, researchers have published works describing queuing systems with jumping parameters for the information flow. For example, Ref. [15] considers a queuing system with one server, finite storage, and exponential service with the parameter $\mu$. The queuing system input receives a doubly stochastic Poisson flow, the intensity of which, $\lambda \left(t\right)$, is a jump process. A matrix and functional-analytical method for studying the stationary and non-stationary characteristics of the queuing system is proposed, and a proof of the existence and uniqueness of the stationary mode and a stabilization of the non-stationary mode of the queuing system are provided. In addition, the authors discuss the results of their numerical analysis. The work in [16] studies the stationary and transient modes of the queuing system $M_S/G/\infty$, in which the process of incoming customers is a Cox process; the incoming intensity sharply increases at random times and then returns to lower values. Expressions are found for the number of customers in the queue in the stationary and transient modes. The work in [17] studies the probabilities of states and the average number of customers in the queue of the system $G/G/c/c+K$, with a stepwise change in the intensity of receiving customers in the transitional and stationary modes. In [18], the authors present an analytical method for the analysis of inhomogeneous continuous Markov processes with piecewise constant transition intensities, including the case with periodically changing transition intensities. These authors presented analytical expressions describing for the first time both the transient and stationary modes of a random process.

Despite the great attention recently paid to the transient mode of queuing systems, most works to date [15,16,17,18,19,20,21] have studied queuing systems with a Poisson input flow, with only a small portion [15,18,20] presenting an analytical analysis of such systems. There is only one work, by the authors of the present paper, studying the $MAP/M/1/N$ system in transient mode [4].

This paper, therefore, considers the $MAP/M/1/N$ system, in which the input flow is a superposition of a Poisson flow with constant intensity $\lambda$ and a flow with jump intensity $\overline{\lambda }\left(t\right)$. In Section 2, a system of Kolmogorov equations is constructed for the case under consideration. This paper studies the probability translation matrix of a queuing system with an input $MAP$ flow and constant arrival and service intensities (Section 3). Here, the concept of transition mode time is introduced based on the study of the determinant of the translation matrix, in contrast to the transition mode time, which is based on the study of the eigenvalues of the matrix of coefficients of the Kolmogorov equations [4]. The concept introduced here is more general (averaged) than the transient time introduced in [4], as it takes into account the time dependence of all eigenvalues of the translation matrix. Such a study is of great importance for further descriptions of queuing systems with an abrupt change in the input flow. The theorem on solutions to the Kolmogorov system of equations is formulated in Section 4. Thus, the solution to the Kolmogorov system of equations is presented in an exact analytical form. Moreover, this solution does not involve a complex operation requiring the Jordan forms of the coefficient matrices of the Kolmogorov system of equations to be found; this section further describes the main properties of these solutions. A queuing system with a main Poisson flow and an additional flow with a periodically varying intensity is studied in Section 5. The eigenvalue theorem for a queuing system with a periodic input $MAP$ flow is formulated and the nature of the dependence of state probabilities on time is described. Numerical calculations in accordance with the presented method are provided in Section 6. The presented numerical results confirm the correctness and practical applicability of the developed analytical method. Note that this paper, as in [4], considers the case of simple roots of the characteristic equation of the coefficient matrix of the Kolmogorov system of equations.

2. System of Kolmogorov Equations for Queuing Systems with an Additional Piecewise Constant Flow of Customers

Let us consider a queuing system $MAP/M/1/N$ with a finite buffer and time-dependent intensities of the input customer flow.

The customer flow into the system is a Markov arrival flow. This flow is controlled by the irreducible non-periodic Markov chain ${\nu }_t$, $t\ge 0$ with continuous time and the finite state space $\{0,1,\dots ,M-1\}$.

The arrival of customers within the $MAP$ flow is described as follows: the Markov chain ${\nu }_t$, $t\ge 0$ is in a state $\nu$ during a time with an exponential distribution described by the parameter ${\lambda }_{\nu }$, $\nu =\overline{(0,M-1)}$; after this time, the process ${\nu }_t$, $t\ge 0$ with probability $p^{\left(1\right)}(\nu ,{\nu }^{\prime })$ passes into a state ${\nu }^{\prime }$ generating a customer, and the process passes into the state ${\nu }^{\prime }$ with probability $p^{\left(0\right)}(\nu ,{\nu }^{\prime })$ without generating a customer. It is assumed that transition from a state $\nu$ to the same state without generating a customer is impossible, i.e., $p^{\left(0\right)}(\nu ,\nu )=0$.

The probabilities $p^{\left(k\right)}(\nu ,{\nu }^{\prime })$, $k=\overline{0,1}$ satisfy the normalization condition

$$\sum _{\begin{array}{c}{\nu }^{\prime }=0\\ {\nu }^{\prime }\ne \nu \end{array}}^{M-1}{p}^{\left(0\right)}(\nu ,{\nu }^{\prime })+\sum _{{\nu }^{\prime }=0}^{M-1}{p}^{\left(1\right)}(\nu ,{\nu }^{\prime })=1,\nu =\overline{0,M-1}$$

(1)

Usually, the $MAP$ flow is described by the $M\times M$-matrices ${\mathbf{D}}_0$ and ${\mathbf{D}}_1$:

$${\mathbf{D}}_0=\left(\begin{array}{cccc}-{\lambda }_0& {\lambda }_0{p}_{0,1}^{\left(0\right)}& ...& {\lambda }_0{p}_{0,M-1}^{\left(0\right)}\\ {\lambda }_1{p}_{1,0}^{\left(0\right)}& -{\lambda }_1& ...& {\lambda }_1{p}_{1,M-1}^{\left(0\right)}\\ ...& ...& ...& ...\\ {\lambda }_{M-1}{p}_{M-1,0}^{\left(0\right)}& {\lambda }_{M-1}{p}_{M-1,1}^{\left(0\right)}& ...& -{\lambda }_{M-1}\end{array}\right)$$

(2)

$${\mathbf{D}}_1=\left(\begin{array}{cccc}{\lambda }_0{p}_{0,0}^{\left(1\right)}& {\lambda }_0{p}_{0,1}^{\left(1\right)}& ...& {\lambda }_0{p}_{0,M-1}^{\left(1\right)}\\ {\lambda }_1{p}_{1,0}^{\left(1\right)}& {\lambda }_1{p}_{1,1}^{\left(1\right)}& ...& {\lambda }_1{p}_{1,M-1}^{\left(1\right)}\\ ...& ...& ...& ...\\ {\lambda }_{M-1}{p}_{M-1,0}^{\left(1\right)}& {\lambda }_{M-1}{p}_{M-1,1}^{\left(1\right)}& ...& {\lambda }_{M-1}{p}_{M-1,M-1}^{\left(1\right)}\end{array}\right)$$

(3)

The average arrival rate (intensity) of customers within the $MAP$ flow is determined by the equality

$$\lambda =\theta {\mathbf{D}}_1\mathbf{e},$$

(4)

where $\theta$ is the row-vector of the stationary distribution of the control process ${\nu }_t$, $t\ge 0$ and $\mathbf{e}$ is a column-vector consisting of units only.

The vector $\theta$ can be found as the single solution to the system of linear algebraic equations

$$\begin{array}{c}\hfill \theta \left({\mathbf{D}}_0+{\mathbf{D}}_1\right)=\mathbf{0},\\ \hfill \theta \mathbf{e}=1.\end{array}$$

(5)

In contrast to the simplest flow, the $MAP$ flow is non-stationary and correlated; in the general case, adjacent time intervals between the instants of customer arrival are dependent. The $MAP$ flow correlation coefficient is calculated using formula:

$$c_{cor}=({\lambda }^{-1}\theta {\left(-{\mathbf{D}}_0\right)}^{-1}{\mathbf{D}}_1{\left(-{\mathbf{D}}_{\mathbf{0}}\right)}^{-1}\mathbf{e}-{\lambda }^{-2})/v,$$

(6)

where v is the dispersion of the lengths of the intervals between the moments of receiving customers:

$$v=\frac{2\lambda \theta {\left(-{\mathbf{D}}_0\right)}^{-1}\mathbf{e}-1}{{\lambda }^2}.$$

(7)

More detailed information about the $MAP$ flow and its properties as well as the proof of Formula (4) are presented in papers by Neuts, M. [22] and Lucantoni, D. et al. [23], as well as in monograph [1].

The service time of a customer within the system is exponentially distributed by the parameter $\mu$. The system buffer is limited by N, and customers arriving in a system in which all waiting places are occupied will be lost. For this system, it can be assumed that it receives an additional flow independent of the $MAP$ flow, that is, the simplest flow of customers with a piecewise-constant parameter $\overline{\lambda }\left(t\right)$, which depends on time (Figure 1).

$$\overline{\lambda }\left(t\right)=\left\{\begin{array}{cc}{\overline{\lambda }}_1,& 0<t<t_1\\ {\overline{\lambda }}_2,& t_1<t<t_2\\ ...& ...\\ {\overline{\lambda }}_G,& t_{G-1}<t<t_G\\ {\overline{\lambda }}_{G+1},& t>t_G\end{array}\right.$$

(8)

where $t_G>0$, G is the number of jumps.

This additional flow may be anomalous traffic resulting from a network attack or it may be an additional information flow that can be interpreted depending on the application of this model. The customers of this flow do not have priority and are indistinguishable from customers from the main $MAP$ flow in the service process.

It is known [1] that the Poisson input flow is a particular case of a $MAP$. In this case, the matrices describing the $MAP$ have a dimension of $1\times 1$ and are defined as

$${\overline{\mathbf{D}}}_0=\left(-\overline{\lambda }\left(t\right)\right),{\overline{\mathbf{D}}}_1=\left(\overline{\lambda }\left(t\right)\right),t\ge 0$$

(9)

It is known [1] that the superposition (overlay) of two $MAP$s is itself a $MAP$, thus, the total flow of customers into the system is a time-dependent $MAP$ with matrices

$${\tilde{\mathbf{D}}}_0\left(t\right)={\mathbf{D}}_0-\overline{\lambda }\left(t\right){\mathbf{I}}_M,{\tilde{\mathbf{D}}}_1\left(t\right)={\mathbf{D}}_1+\overline{\lambda }\left(t\right){\mathbf{I}}_M,t\ge 0$$

(10)

where ${\mathbf{I}}_M$ is the identity matrix of size M.

Following [1], the system of Kolmogorov equations for $K=N·M$ states of the considered $MAP/M/1/N$ system in matrix form can be written as

$$\frac{d}{dt}\overrightarrow{\mathbf{P}}\left(t\right)=\mathbf{A}\left(t\right)\overrightarrow{\mathbf{P}}\left(t\right)$$

(11)

$$\mathbf{A}\left(t\right)=\left(\begin{array}{cccc}{\mathbf{D}}_0^T-\overline{\lambda }\left(t\right){\mathbf{I}}_M& \mu {\mathbf{I}}_M& ...& 0\\ {\mathbf{D}}_1^T+\overline{\lambda }\left(t\right){\mathbf{I}}_M& {\mathbf{D}}_0^T-\overline{\lambda }\left(t\right){\mathbf{I}}_M-\mu {\mathbf{I}}_M& ...& 0\\ ...& ...& ...& ...\\ 0& 0& ...& {\mathbf{D}}_0^T+{\mathbf{D}}_1^T-\mu {\mathbf{I}}_M\end{array}\right)$$

(12)

where $\mathbf{A}\left(t\right)$ is an infinitesimal generator of a Markov chain with continuous time describing the state of the system at time t, $\overrightarrow{P}\left(t\right)$=${\left({\overrightarrow{P}}_0\left(t\right),{\overrightarrow{P}}_1\left(t\right),\dots ,{\overrightarrow{P}}_{N+1}\left(t\right)\right)}^T$, T is the transposition operator, ${\overrightarrow{P}}_k\left(t\right)$, and $k=\overline{0,N+1}$ is the macrostate probability column-vector $S_k$ (the M states of $S_k^{\left(0\right)}\dots S_k^{(M-1)}$ of the serving and buffer, determined by the dimension of the matrices (2) and (3) [1,4]).

3. Probability Translation Matrix of a Queueing System with Constant MAP Flows

The Kolmogorov Equation (11) for the state probabilities of a queuing system with $MAP$ flows are the linear homogeneous system of differential equations with piecewise-constant coefficients, thus, the probability translation matrix $\mathbf{L}(\Delta t)$ on each interval with constant coefficients can be constructed using the theory presented in [4]. This matrix relates state probabilities at time $t_0+\Delta t$ to these probabilities at $t_0=0$:

$$\overrightarrow{P}(t_0+\Delta t)=\mathbf{L}(\Delta t)\overrightarrow{P}\left(t_0\right).$$

(13)

Note that for a system with constant parameters, such a matrix has been found in an analytical form in [4]. In this case, the matrix $\mathbf{A}\left(t\right)=\mathbf{A}$ in (12) is independent of the time due to the lack of the additional flow with the rate $\overline{\lambda }\left(t\right)$. The problem in the present work is first of all to carry out the additional study of its main characteristics, such as its determinant and eigenvalues. This study is necessary for further description of the behavior of a queuing system with piecewise constant parameters. It will be shown that the eigenvalues and eigenvectors of the translation matrices of both systems with constant parameters and systems with piecewise-constant parameters completely describe the nature of the change in the probabilities of the system states in the transition mode, and the value of the determinant of the probability translation matrix determines the time of the transition mode. Note that the translation matrix of system state probabilities relates these probabilities at time t to the probabilities at time $t_0$.

Lemma 1.

All modules of the eigenvalues ${\gamma }_i$, $i=\overline{0,K-1}$ of the probability translation matrix $\mathbf{L}\left(\Delta t\right)$ defined by (13) for the interval $\Delta t$ of a queuing system with an input $MAP$ flow, constant arrival intensities λ, $\overline{\lambda }$, and service intensity μ are within the interval $\left(0,1\right)$ and satisfy the condition

$$\begin{array}{cc}\hfill & {\gamma }_0=1,\hfill \\ \hfill & 0<\left|{\gamma }_i\right|<1,i=\overline{1,K-1}.\hfill \end{array}$$

(14)

Proof. 

In accordance with definition [4], the probability translation matrix of a queuing system with $MAP$ flows and constant arrival intensities $\lambda$, $\overline{\lambda }$ and service intensity $\mu$ is an exponential of the coefficient matrix $\mathbf{A}$ of the original system of Kolmogorov differential Equation (11):

$$\mathbf{L}\left(\Delta t\right)=exp\left(\Delta t\mathbf{A}\right)={\mathbf{T}}^{-1}exp\left(\Delta t\mathbf{J}\right)\mathbf{T}$$

(15)

Here, $\mathbf{J}$ is the Jordan form of matrix $\mathbf{A}$ and $\mathbf{T}$ is the matrix composed of the eigenvectors of matrix $\mathbf{A}$. As $\mathbf{L}\left(\Delta t\right)$ and $exp\left(\Delta t\mathbf{J}\right)$ are similar matrices, the eigenvalues of $\mathbf{L}\left(\Delta t\right)$ are exponents of the eigenvalues of matrix $\mathbf{A}$. In this case, one of the eigenvalues of matrix $\mathbf{A}$ is equal to zero, and the remaining eigenvalues of matrix $\mathbf{A}$ have a negative real part. Therefore, for the eigenvalues of the translation matrix $\mathbf{L}\left(\Delta t\right)$, we have ${\gamma }_0=1$, $0<\left|{\gamma }_i\right|<1,i=\overline{1,K-1}$. □

From a practical point of view, this means that in a queuing system with a constant incoming $MAP$ flow, an exponential service time, and a buffer of limited capacity, there is always a stationary mode. Indeed, if ${\mathbf{U}}_i$ is an eigenvector corresponding to the eigenvalue ${\gamma }_i$ then ${\mathbf{U}}_0={\gamma }_0{\mathbf{U}}_0$, as ${\gamma }_0=1$, and $|{\gamma }_i|{\mathbf{U}}_i<{\mathbf{U}}_i$, as $0<|{\gamma }_i|<1$ for all $i=\overline{1,K-1}$. Thus, the probabilities of the process states in the transient mode are damped and tend to certain limit values at $t\to \infty$.

Lemma 2.

The probability translation matrix $\mathbf{L}\left(\Delta t\right)$ of a queuing system on an interval $\Delta t$ with an input $MAP$ flow and constant arrival and service intensities is stochastic.

Proof. 

In accordance with the definition of the probability translation matrix of a queuing system with $MAP$ flows and constants $\lambda$ and $\mu$ [4],

$${\mathit{P}}_{\mathit{i}}\left(\Delta t\right)=\sum _{j=0}^{K-1}{L}_{ij}\left(\Delta t\right){\mathit{P}}_{\mathit{j}}\left(0\right),i=\overline{0,N·M-1}$$

(16)

where $L_{ij}\left(\Delta t\right)$ are the elements of the probability translation matrix $\mathbf{L}\left(\Delta t\right)$ on the interval $\Delta t$ and $P_i\left(\Delta t\right)$ are the probabilities of the states. Then, if $P_0\left(0\right)=1$ and $P_j\left(0\right)=0$, $i=\overline{1,K-1}$; then, $P_i\left(\Delta t\right)=L_{i0}\left(\Delta t\right)$ for any values of $\Delta t$. This implies that $0\le L_{i0}\left(\Delta t\right)\le 1$ for any values of $\Delta t$, as $0\le P_i\Delta t\le 1$.

Similarly, it can be shown that for all $L_{ij}\left(\Delta t\right)$, $i,j=\overline{0,K-1}$ and the relation $0\le L_{ij}\left(\Delta t\right)\le 1$ holds for any $\Delta t$. In this case, for any $\Delta t$, the equality holds:

$$\sum _{i=0}^{K-1}{P}_i\left(\Delta t\right)=1$$

(17)

Substituting (16) into (17), we can obtain

$$\sum _{i=0}^{K-1}\sum _{j=0}^{K-1}{\mathit{L}}_{\mathit{ij}}\left(\Delta t\right){\mathit{P}}_{\mathit{j}}\left(0\right)=1$$

(18)

As equality (18) must hold for any $P_j\left(0\right)$, the necessary and sufficient condition is

$$\sum _{i=0}^{K-1}{\mathit{L}}_{\mathit{ij}}\left(\Delta t\right)=1,j=\overline{0,K-1}$$

(19)

Thus, the matrix $\mathbf{L}\left(\Delta t\right)$ is stochastic. □

Let us study the problem of the transient mode time by analyzing the determinant of the probability translation matrix $\mathbf{L}\left(\Delta t\right)$ for time $\Delta t$. The determinant of the probability translation matrix of a queuing system with a time-dependent input $MAP$ flow, described by (12) and (13) in accordance with the Ostrogradsky–Liouville theorem [24], is equal to

$$\begin{array}{c}{\displaystyle det\left(\mathbf{L}\left(t\right)\right)=exp\left[{\int }_0^t\mathrm{Sp}\left(\mathbf{A}\left(\tau \right)\right)d\tau \right]=}\\ \\ {\displaystyle =exp\left[\left(\left(1-N\right)\sum _{i=0}^{M-2}{\lambda }_i-\sum _{i=0}^{M-1}{\lambda }_i(1-p_{ii}^{\left(1\right)})-\right.\right.}\\ {\displaystyle -\left.\left.\mathit{N}·\mathit{M}·\mu -\frac{N-1}{t}{\int }_0^t\sum _{i=0}^{M-1}\overline{\lambda }\left(\tau \right)d\tau \right)t\right]}\end{array}$$

(20)

where $\mathrm{Sp}\left(\mathbf{A}\left(t\right)\right)$ is the trace of the coefficient matrix (13) of the Kolmogorov equation system of (12). Therefore, expression (20) is obtained by substituting the diagonal elements of the matrix in (13) for the expression of the Ostrogradsky–Liouville theorem. Now, denoting

$$\alpha \left(t\right)=\left(N-1\right)\sum _{i=0}^{M-2}{\lambda }_i+\sum _{i=0}^{M-1}{\lambda }_i\left(1-p_{ii}^{\left(1\right)}\right)+N·M·\mu +\frac{N-1}{t}{\int }_0^t\sum _{i=0}^{M-1}\overline{\lambda }\left(\tau \right)d\tau$$

(21)

we can write

$$det\left(\mathbf{L}\left(t\right)\right)=exp\left[-\alpha \left(t\right)t\right]$$

(22)

instead of (20). It can be seen from (2) that $\alpha \left(t\right)>0,t\in [0,\infty )$; therefore, $det\left(\mathbf{L}\left(t\right)\right)$ always decreases as t increases.

Now let us consider a particular (yet very important) case of a system with constant input and service intensities. For a queuing system where $\left(\overline{\lambda }=const\right.$ and $\left.\mu =const\right)$, it follows from (21) and (22) that

$$det\left(\mathbf{L}\left(t\right)\right)=exp\left(-\alpha t\right)$$

(23)

where

$$\alpha =\left(N-1\right)\sum _{i=0}^{M-2}{\lambda }_i+\sum _{i=0}^{M-1}{\lambda }_i\left(1-p_{ii}^{\left(1\right)}\right)+N·M·\mu +\left(N-1\right)M\overline{\lambda }$$

(24)

It follows from (23) and (24) that $det\left(\mathbf{L}\left(t\right)\right)=1$ for t=0 and $det\left(\mathbf{L}\left(t\right)\right)=0$ for $t\to \infty$, otherwise $0<det\left(\mathbf{L}\left(t\right)\right)<1$. This means that no single eigenvalue of the translation matrix of a system with constant parameters is equal to zero for any finite t, as this determinant is equal to the product of the eigenvalues of the probability translation matrix.

In [4], the transition mode time has been defined as the reciprocal of the smallest eigenvalue of matrix $\mathbf{A}\left(t\right)$ with a certain coefficient. Here, we determine this time based on the analysis of the determinant of the probability translation matrix (23). Indeed, this determinant is the product of the eigenvalues of its matrix [21]:

$$det\left(\mathbf{L}\left(t\right)\right)=\prod _{i=1}^K{\gamma }_i=exp\left(-\alpha t\right)$$

(25)

Then, the value

$$\beta =\sqrt[K]{\alpha }={\left[\left(N-1\right)\sum _{i=0}^{M-2}{\lambda }_i+\sum _{i=0}^{M-1}{\lambda }_i\left(1-p_{ii}^{\left(1\right)}\right)+N·M·\mu +\left(N-1\right)M\overline{\lambda }\right]}^{1/K}$$

(26)

is the average characteristic of a queuing system with constant intensities of arrival and service flows in the transient mode. Indeed, the determinant of the probability translation matrix is equal to the product of its eigenvalues, each of which, in combination with an eigenvector, describes an independent process of probability translation in the transient mode. Thus, the root of the product of these numbers characterizes the average decay rate of the transient process. Moreover, the damping is the geometric mean of the damping of all eigenvalues.

Definition 1.

The value $\beta =\sqrt[K]{\alpha }$, defined by expression (26), is called the decay rate of a transient process in a queuing system, and characterizes the decay rate of this process.

Definition 2.

The time constant of the transient mode is the time during which the value $exp\left(-\beta t\right)$, characterizing the transient mode, will decrease by e times, i.e., $exp\left(-\beta \tau \right)=exp(-1)$:

$$\tau =1/\beta$$

(27)

Strictly speaking, the transient mode is infinite in time. Indeed, $exp\left(-\beta \tau \right)\to$0 only as $t\to \infty$. In practice, one can assume that the transitional mode has ended if $exp\left(-\beta \tau \right)$ does not exceed a certain small value. Then, based on the analysis of the probability translation matrix, one can determine the time of the transition mode by analogy with [4].

Definition 3.

The transition time is the time from the beginning of the transition mode $t_0$ to the moment $t_0+{\tau }_{tr}$, when the condition $exp\left[-\beta \left(t_0+{\tau }_{tr}\right)\right]/exp\left(-\beta t_0\right)\le \epsilon$ (where ε is an infinite small value) will be satisfied.

In accordance with Definition 3, the transition time is

$${\tau }_{tr}=b/\beta$$

(28)

where $b=ln\left(1/\epsilon \right)$. Further numerical calculations show that it is reasonable to choose $b=3÷5$.

From the authors’ point of view, this definition is more general than that in [4], as it averages the behavior of all eigenvectors ${\mathbf{U}}_i$,$i=\overline{0,K-1}$ corresponding to the eigenvalues ${\gamma }_i$,$i=\overline{0,K-1}$, the linear combination of which completely describes the behavior of the system.

4. Queuing System MAP/M/1/N with a Jump in the Arrival and Service Intensities

In this section, the analytical solution to the Kolmogorov system of equations is found for a piecewise constant change in the arrival intensity $\overline{\lambda }$, and the behavior features of the considered queuing system based on this solution are described. The solution is based on finding the probability translation matrix. First, we formulate the theorem concerning the probability translation matrix of the Kolmogorov system of equations with piecewise constant coefficients.

Theorem 1

(On the solution to the Kolmogorov system of equations with piecewise constant coefficients). The probability translation matrix of a queuing system with correlated flows and piecewise constant parameters at time t is found as a product of interval translation matrices with constant parameters:

$$\mathbf{L}\left(t\right)={\mathbf{L}}_K\left(t-t_{K-1}\right)\prod _{i=G-1}^1{\mathbf{L}}_i\left(\Delta t_i\right)$$

(29)

where G is the number of intervals with constant parameters.

Proof. 

It has been shown in [4] that the probability translation matrix allows the probabilities of the system states with constant parameters at any time to be found as

$$\mathbf{P}\left(t\right)={\mathbf{L}}_1\left(t\right)\mathbf{P}\left(t_0\right),t>=t_0$$

(30)

Then, at time $t_1$ (the moment of the first jump), the probabilities of the states are

$$\mathbf{P}\left(t_1\right)={\mathbf{L}}_1\left(\Delta t_1\right)\mathbf{P}\left(t_0\right)$$

(31)

where $\Delta t_1=t_1-t_0$. Taking into account the fact that the state probabilities cannot change abruptly, for the second interval we can obtain

$$\mathbf{P}\left(t\right)={\mathbf{L}}_2\left(t-t_1\right)\mathbf{P}\left(t_1\right)={\mathbf{L}}_{\mathbf{2}}\left(t-t_1\right){\mathbf{L}}_{\mathbf{1}}\left(\Delta t_1\right)\mathbf{P}\left(t_0\right)$$

(32)

At time $t_2$ (the moment of the second jump), the state probabilities are defined as

$$\mathbf{P}\left(t_2\right)={\mathbf{L}}_2\left(\Delta t_2\right){\mathbf{L}}_{\mathbf{1}}\left(\Delta t_1\right)\mathbf{P}\left(t_0\right)$$

(33)

where $\Delta t_2=t_2-t_1$. For the third interval, we can obtain

$$\begin{array}{c}\hfill \mathbf{P}\left(t_3\right)={\mathbf{L}}_3\left(t-t_2\right){\mathbf{L}}_{\mathbf{2}}\left(\Delta t_2\right){\mathbf{L}}_1\left(\Delta t_1\right)\mathbf{P}\left(t_0\right)\\ \hfill \mathbf{P}\left(t_3\right)={\mathbf{L}}_3\left(\Delta t_3\right){\mathbf{L}}_{\mathbf{2}}\left(\Delta t_2\right){\mathbf{L}}_1\left(\Delta t_1\right)\mathbf{P}\left(t_0\right)\end{array}$$

(34)

Similarly, for the G-th interval, we can obtain

$$\begin{array}{c}\hfill \mathbf{P}\left(t\right)={\mathbf{L}}_G\left(t-t_{G-1}\right){\mathbf{L}}_{G-1}\left(\Delta t_{G-1}\right)...{\mathbf{L}}_{\mathbf{2}}\left(\Delta t_2\right){\mathbf{L}}_1\left(\Delta t_1\right)\mathbf{P}\left(t_0\right)=\\ \\ \hfill {\displaystyle ={\mathbf{L}}_G\left(t-t_{G-1}\right)\prod _{i=G-1}^1{\mathbf{L}}_i\left(\Delta t_i\right)\mathbf{P}\left(t_0\right)}\end{array}$$

(35)

where $\Delta t_i=t_i-t_{i-1}$. Thus, for the G-th interval the translation matrix of the state probabilities of a system with correlated flows is determined by expression (27). □

Note that the following main properties in the behavior of a queuing system with correlated flows in the transient mode follow from Theorem 1.

Property 1.

Let $t_1$ and $t_2$ be the moments of time at which two jumps in the intensity of arrival or service occur sequentially, with $0<t_1<t_2$. Let ${\tau }_{tr1}$ and ${\tau }_{tr2}$ be the duration of the transition modes after the first and second jumps, respectively. If the arrival or service intensity jump occurs during the transition after the previous jump $(t_2-t_1<{\tau }_{tr1})$, then the total transition time is ${\tau }_{tr}=t_1+{\tau }_{tr2}$. If the second jump occurs during the stationary mode $(t_2-t_1>{\tau }_{tr1})$, then both transient processes are independent of each other.

Property 2.

If the duration of the arrival or service intensity jump is less than the time constant in the transient mode, then the intensity jump does not affect the system behavior. If the duration of the jump is more than three time constants in the transition mode, then the jump in intensity leads to a jump in the probabilities of states.

Indeed, for a time equal to the time constant, the values of the state probabilities decrease by a factor of e [4]. During this time, the system does not have time to go into stationary mode. If the jump time is more than $(3÷5)\tau$, then, in [4], it is shown that the system has time to switch to a stationary mode and it should be taken into account in the analysis. Indeed, in [4] it is theoretically and numerically shown that the transient mode lasts on the order of $(3÷5)\tau$.

Property 3.

Let the system have a finite number of jumps at time $t_1,t_2,\dots ,t_{G-1}$. The stationary state of the queuing system after all jumps in arrival and service intensities is determined by its parameters, $\lambda \left(t\right)$, $\mu \left(t\right)$, after the last jump $(t>t_{G-1})$. However, the nature of the transition mode is determined by its previous states.

Indeed, based on (27), this can be written as

$${\displaystyle \mathbf{P}\left(t\right)={\mathbf{L}}_G\left(t-t_{G-1}\right)\mathbf{P}\left(t_{G-1}\right)}$$

(36)

where

$$\mathbf{P}\left(t_{G-1}\right)=\left(\prod _{i=G-1}^1{\mathbf{L}}_i\left(\Delta t_i\right)\right)\mathbf{P}\left(t_0\right)$$

(37)

Thus, the probability of state $\mathbf{P}\left(t_{N-1}\right)$ is the initial condition for the last jump. However, the probabilities of the system states in the stationary mode are determined only by the parameters of this system, and the initial conditions in the solution to the Kolmogorov system of equations do not affect the probabilities in the stationary mode (at $t\to \infty$) [1].

Now, let us consider the queuing system $MAP/M/1/N$ with an additional piecewise constant input flow based on the results obtained above.

5. Queuing System MAP/M/1/N with a Periodic Jump in the Arrival and Service Intensities

Let us consider a periodic abrupt change in the arrival intensity of the Poisson flow:

$$\overline{\lambda }\left(t\right)=\left\{\begin{array}{cc}{\overline{\lambda }}_1,& nT<t<t_1+nT\\ {\overline{\lambda }}_2,& t_1+nT<t<t_2+nT\\ ...& ...\\ {\overline{\lambda }}_G,& t_{G-1}+nT<t<(n+1)T\end{array}\right.$$

(38)

Here, T is the period of change of parameters, $n\in \left\{0\bigcup \mathbf{N}\right\}$ and $\mathbf{N}$ is the set of natural numbers.

Obviously, the translation matrix for the period completely describes the state of the system, as it allows us to obtain the values of the state probabilities at any moment of the period. The behavior of the periodic system, in accordance with the theory of differential equations with periodic Lyapunov coefficients [24], is described based on the analysis of the eigenvalues and eigenvectors of this matrix.

Theorem 2

(On the eigenvalues of the probability translation matrix of a periodic $MAP$ system). The modules of eigenvalues ${\gamma }_i,i=\overline{0,K-1}$ of the probability translation matrix $\mathbf{L}\left(T\right)$ over period T of a queuing system with correlated periodic arrival and service intensities lie in interval $\left(0;1\right]$, and one of these eigenvalues is always equal to one:

$${\gamma }_0=1;0<\left|{\gamma }_i\right|<1,i=\overline{1,K-1}$$

(39)

Proof. 

It follows from (13) that the determinant of the coefficient matrix is equal to zero for any $\lambda \left(t\right)$ and $\mu$. This means that one of the eigenvalues of this matrix is zero even if $\lambda \left(t\right)$ is a piecewise constant function of the form (38). Taking into account (14), it can be argued that one of the eigenvalues of the translation matrix is equal to one. Based on Lemma 2, translation matrices on intervals with constant parameters are stochastic. Therefore, the product (27) of these matrices is stochastic. However, the eigenvalues of the stochastic matrix always lie within the interval $\left(0;1\right]$ [25]. □

The further description of the behavior of a queuing system with periodic $\lambda \left(t\right)$ is based on studying the eigenvalues of the probability translation matrix over the period. In accordance with Lyapunov’s stability theory, solutions to a differential equation with periodic coefficients [24] are stable (not increasing in time) if all characteristic numbers of the translation matrix (multipliers) are less than or equal to one in absolute value. If at least one eigenvalue of this matrix is greater than unity in absolute value, then the solution is unstable (increasing in time). In addition, if the eigenvalues are real then the transient process is exponential, while if they are complex, the transient process is undulating [24]. It should be noted here that the period translation matrix found by the authors is not a monodromy matrix [24], as its order is one higher than its rank.

Corollary 1

(On the nature of the probabilities of states of an MAP system with periodic intensities). The state probabilities of a queuing system with $MAP$ flows and periodic $\lambda \left(t\right)$ and $\mu \left(t\right)$ are periodic functions with a period equal to the period of change in the arrival and service intensities.

Proof. 

As on the basis of Theorem 2 we have ${\gamma }_0=1$; $0<|{\gamma }_i|<1$, $i=\overline{1,K-1}$, then in accordance with Lyapunov’s theorem on the stability of periodic systems, the solutions to system (13) are Lyapunov stable. In this case, the system cannot be asymptotically stable in of Lyapunov sense, as one of the eigenvalues of the probability translation matrix is always equal to one. Thus, the corollary is proven. □

Obviously, for periodically changing state probabilities, one can find the average value of the probability of the i-th state as

$${\displaystyle P_{avi}=\frac{1}{T}{\int }_{t_0}^{t_0+T}{P}_i\left(t\right)dt,i=\overline{0,K-1}}$$

(40)

Such averaging makes sense if the period of change of probabilities is less than $3\tau$; otherwise, each of the transient modes should be considered as an independent one.

6. Numerical Example

In this section, the numerical study of a queuing system with a main constant $MAP$ flow and an additional jumpy Poisson flow for the case $N=3$, $M=3$ is carried out. Let $MAP$ describe the three types of traffics with intensities ${\lambda }_0=100$ packets/s, ${\lambda }_1=110$ packets/s, and ${\lambda }_2=120$ packets/s. The Poisson flow characterises anomalous traffic with intensities ${\overline{\lambda }}_1=10$ packets/s (before the jump) for the first interval and ${\overline{\lambda }}_2=200$ packets/s (after the jump) for the second interval. The service intensity on both intervals is $\mu =800$ packets/s.

These parameters characterise real data flows in wireless broadband networks used in unmanned aerial vehicles [26]. The probability matrices $p^{\left(0\right)}$ and $p^{\left(1\right)}$ for both intervals are

$${\mathbf{P}}^{\left(0\right)}=\left(\begin{array}{ccc}0& p_{01}^{\left(0\right)}& p_{02}^{\left(0\right)}\\ p_{10}^{\left(0\right)}& 0& p_{12}^{\left(0\right)}\\ p_{20}^{\left(0\right)}& p_{21}^{\left(0\right)}& 0\end{array}\right)=\left(\begin{array}{ccc}0& 0.15& 0.12\\ 0.03& 0& 0.3\\ 0.1& 0.04& 0\end{array}\right)$$

(41)

$${\mathbf{P}}^{\left(1\right)}=\left(\begin{array}{ccc}{p}_{00}^{\left(1\right)}& p_{01}^{\left(1\right)}& p_{02}^{\left(1\right)}\\ p_{10}^{\left(1\right)}& p_{11}^{\left(1\right)}& p_{12}^{\left(1\right)}\\ p_{20}^{\left(1\right)}& p_{21}^{\left(1\right)}& p_{22}^{\left(1\right)}\end{array}\right)=\left(\begin{array}{ccc}0.11& 0.17& 0.45\\ 0.15& 0.02& 0.5\\ 0.13& 0.12& 0.61\end{array}\right)$$

(42)

The dependencies of state probabilities on time are shown in Figure 2. Here, the jump in arrival intensities occurs at time $t_{jump}=0.05$ s. For the first and second intervals, $\lambda <\mu$, $\overline{{\lambda }_1}<\mu$, and $\overline{{\lambda }_2}<\mu$ are chosen ($\lambda$ is the average arrival intensity of $MAP$). At the time of the jump, the transition mode that occurs in the system as a result, for example, of its installation and reboot, has not yet ended and the total time of the transition mode, according to Property 1 is ${\tau }_{tr}=t_{jump}+t_{tr2}=0.005+0.01=0.015$ s, where $t_{tr2}$ is the time transition mode in the second interval. It can be seen from the calculation results (Figure 2) that the sum of the probabilities of all states is equal to one for any t (thin blue line), which confirms the correctness of the calculations. The red curve corresponds to the probability of a free state of the buffer $\left(P^{\left(0\right)}\left(t\right)=P_1^{\left(0\right)}\left(t\right)+P_2^{\left(0\right)}\left(t\right)+P_3^{\left(0\right)}\left(t\right)\right)$, the blue curve corresponds to the loss probability $\left(P^{\left(2\right)}\left(t\right)=p_1^{\left(2\right)}\left(t\right)+p_2^{\left(2\right)}\left(t\right)+p_3^{\left(2\right)}\left(t\right)\right)$, and the green curve corresponds to one order in the buffer $\left(p^{\left(1\right)}\left(t\right)=p_1^{\left(1\right)}\left(t\right)+p_2^{\left(1\right)}\left(t\right)+p_3^{\left(1\right)}\left(t\right)\right)$. After the end of all transient processes, the probabilities of the system states in the stationary mode take the following values: ${\pi }^{\left(0\right)}=0.67$; ${\pi }^{\left(1\right)}=0.25$; ${\pi }^{\left(2\right)}=0.08$, which fully correspond to the results based on [1].

Figure 3 shows similar calculations for another value of the intensity of the additional flow: ${\overline{\lambda }}_1=10$ packets/s before the jump and ${\overline{\lambda }}_2=700$ packets/s after the jump. Thus, after the jump, ${\overline{\lambda }}_2+\lambda \approx \mu$. For a given ratio of the arrival and service intensities, the probabilities of states in the stationary mode are approximately equal to ${\pi }^{\left(0\right)}={\pi }^{\left(1\right)}={\pi }^{\left(2\right)}=0.333$, which corresponds to the theoretical results based on the results in [1]. In this case, the transition mode time in the second interval decreased, amounting to ${\tau }_{tr2}=0.009$ s and ${\tau }_{tr}=t+t_{tr2}=0.005+0.009=0.014$ s.

Figure 4 shows the calculations of state probabilities for ${\overline{\lambda }}_1=10$ packets/s and $\overline{{\lambda }_2}=900$ packets/s, i.e., for ${\overline{\lambda }}_2+\lambda >\mu$. For this ratio, ${\pi }^{\left(0\right)}=0.26$; ${\pi }^{\left(1\right)}=0.33$; ${\pi }^{\left(2\right)}=0.41$. The total time of the transition mode was ${\tau }_{tr}=t+t_{tr2}=0.005+0.005=0.01$ s. Thus, with an increase in the intensity of the additional flow in the second interval, the jumps in the probabilities of the system states increase and the time of the transition mode decreases.

Next, let us consider a periodic change in the arrival intensity of the Poisson flow. Each period includes two intervals with constant intensities of $MAP$ and Poisson flows. The dependencies of the modules of the eigenvalues of the probability translation matrix on the duration of the second interval of the period are shown in Figure 5 and Figure 6. Here, the duration of the first interval is $t_1=0.1$ s. The calculation results show that eight of the nine eigenvalues are less than one in absolute value and tend to zero with the duration of the second interval $t_2\to \infty$, and one is equal to one for any values of $t_1$ and $t_2$. This means that the process is periodic in the system. Those solutions to the Kolmogorov system of equations are Lyapunov stable.

Figure 7, Figure 8 and Figure 9 show the dependencies of the probabilities of states with a periodic abrupt change in the arrival intensity. In the first case (Figure 7), the arrival intensity at the first interval of period ${\overline{\lambda }}_1=10$ packets/s, the intensity at the second interval of period ${\overline{\lambda }}_2=200$ packets/s, the time from the beginning of the arrival of the $MAP$ flow to the arrival of an additional Poisson flow $t_0=0.1$ s, the duration of the first interval of period $t_1=0.015$ s, and the duration of the second interval of period $t_2=0.07$ s. It can be seen from the results of the calculation that during time $t_2$ the system manages to switch to the stationary mode, with an abrupt decrease in intensity, and does not have time to switch to the stationary mode during time $t_1$, with an increase in input intensity. In this case, the averaging of the state probabilities is impossible, and these probabilities must be considered separately for each interval. In the second case (Figure 8), the period of change in the arrival intensity is much less than the time of the transition mode $(t_1=0.015$ s, $t_2=0.015$ s, and the values of the Poisson flows are ${\overline{\lambda }}_1=10$ packets/s and ${\overline{\lambda }}_2=50$ packets/s. In this case, it is possible to average the probabilities of the system states according to (40). Thus, the average value of the probability that the buffer is free is 0.84. In the third case (Figure 9), the times of the first and second intervals of the period are equal to $t_1=0.015$ s, $t_2=0.015$ s, respectively, and the arrival intensity ${\overline{\lambda }}_1=10$ packets/s and ${\overline{\lambda }}_2=500$ packets/s, respectively. In this case, the averaging of the probabilities of states is impossible despite the small period of change in the intensity of the input flow.

Numerical calculations show that in all three cases of periodic change in the intensity of the input Poisson flow, the change in the state probabilities has a periodic undamped character, which fully corresponds to the theoretical results of this paper.

7. Conclusions

This article studied the behavior of a queuing system with a constant $MAP$ flow and a piecewise constant Poisson input flow. To do this, the study of the eigenvalues and the determinant of the translation matrix of the queuing system state probabilities with constant $MAP$ flows was first carried out. A definition of the transition mode time based on the analysis of the determinant of the translation matrix was introduced and the cases of abrupt and periodic abrupt changes in the arrival and service intensities were considered. For these cases, analytical solutions to the corresponding systems of Kolmogorov equations were found and the features of the behavior of the considered queuing system described based on the obtained solutions. A theorem on the eigenvalues of the translation matrix with periodic input flows was formulated and a numerical analysis of the $MAP/M/1/N$ system was carried out using the example of an $MAP/M/1/2$ system with three states of the $MAP$ flow in transient and stationary modes.