On an exact solution of the rate matrix of $G∕M∕1$-type Markov process with small number of phases

Highlights

• Structured $G∕M∕1$-type Markov process with 2 phases at each level is considered.

• Explicit expression for the unknown rate matrix R is derived.

• Matrix polynomial equation of arbitrary power is transformed into linear.

• Novel approach to power saving by randomized regime switching is discussed.

Abstract

In this research paper we consider the matrix polynomial equation arising naturally in the equilibrium analysis of a structured $G∕M∕1$-type Markov process. We obtain an explicit expression for the unknown rate matrix $R$ being 2 $\times$ 2 matrix. The method is based on symbolic solution of the determinantal polynomial equation. Using Cayley–Hamilton theorem, the matrix polynomial equation for the matrix $R$ is reduced to the system of linear equations. Motivated by applications in Edge Computing by means of Internet of Things devices having tight constraints in energy consumption, we demonstrate the applicability of the method by a novel approach to energy efficiency of a single-server computing system. A new randomized regime switching scheme is proposed, which, as it is shown by means of numerical experiment, provides significant decrease of energy consumption of the system under study.

Introduction

Matrix analytic method continues to prove its applicability to a wide variety of models in the field of modern computing and communication systems. Extensively developed by M. Neuts in the celebrated work [28], the powerful method allows to study sophisticated objects: stability [[23], [31]] and performance [4] of stochastic model of a high-performance cluster, power management in datacenters [8], redundancy in high-performance systems [10], cloud computing [5], to name a few.

The system under study is usually modelled as a continuous time Markov process $\{(X\left(t\right),Y\left(t\right)),t⩾0\}$ with countable state space $E≔\{(0,j),j=1,\dots ,m_0;(i,j),i⩾1,j=1,\dots ,m\}$, where the phase variable $Y\left(t\right)$ may take one of $m$ (or $m_0$ for boundary states) values and level variable $X\left(t\right)$ may be increased/decreased at each transition. The state space $E$ can be partitioned into levels with level $n⩾1$ being the subset $\{(n,j),j=1,\dots ,m\}\subset E$. In many fields of interest, it is assumed that the level is increased by at most one, and decreased by at most $N-1$ (we focus on the case $N<\infty$) units at each transition epoch. These models belong to the so-called structured $G∕M∕1$-type Markov processes, extensively studied in [28], with the natural example of such a process being the queue length process, embedded at arrival epochs. The infinitesimal generator matrix of a structured $G∕M∕1$-type process has the following block-multidiagonal representation:

$$Q=\left(\begin{array}{ccccc}\hfill A^{0,0}\hfill & \hfill A^{0,1}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill \\ \hfill A^{1,0}\hfill & \hfill A^{1,1}\hfill & \hfill A^{\left(0\right)}\hfill & \hfill 0\hfill & \hfill \dots \hfill \\ \hfill A^{2,0}\hfill & \hfill A^{\left(2\right)}\hfill & \hfill A^{\left(1\right)}\hfill & \hfill A^{\left(0\right)}\hfill & \hfill \dots \hfill \\ \hfill A^{3,0}\hfill & \hfill A^{\left(3\right)}\hfill & \hfill A^{\left(2\right)}\hfill & \hfill A^{\left(1\right)}\hfill & \hfill \dots \hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill \\ \hfill \mathbf{0}\hfill & \hfill A^{(N+1)}\hfill & \hfill A^{\left(N\right)}\hfill & \hfill A^{(N-1)}\hfill & \hfill \dots \hfill \\ \hfill \mathbf{0}\hfill & \hfill \mathbf{0}\hfill & \hfill A^{(N+1)}\hfill & \hfill A^{\left(N\right)}\hfill & \hfill \dots \hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill \end{array}\right),$$

where $A^{\left(i\right)},i=0,\dots ,N+1$ are square matrices of order $m$, satisfying the balance equation

$$A\mathbf{1}=\mathbf{0},\text{ where }A≔\sum _{i=0}^{N+1}{A}^{\left(i\right)},$$

$\mathbf{1}$ ($\mathbf{0}$) is the vector of ones (zeros) of corresponding dimension, $A^{0,0}$ is a square matrix of order $m_0$ and $A^{i,0},A^{0,1}$ are possibly rectangular matrices. (Recall that for this type of processes the off-diagonal elements of matrix $Q$, i.e. the rates of transitions of the process, are nonnegative.)

The key component of the method is to obtain the steady-state probability vector $\pi =\left({\pi }_{i,j}\right),i,j\in E$ of the system states in the level-wise matrix-geometric form [28] (for more details on the method see [[3], [17]])

$${\pi }_k={\pi }_{k-1}R,k⩾1,$$

where ${\pi }_k=({\pi }_{k,1},\dots ,{\pi }_{k,m})$ and the matrix $R$ is the minimal nonnegative solution of a matrix polynomial equation

$$P\left(R\right)≔\sum _{i=0}^N{R}^i{A}^{\left(i\right)}=\mathbf{0}.$$

Thus, the analysis is essentially reduced to obtaining the matrix $R$. However, in general, the rate matrix $R$ is obtained by means of converging iterative procedures [[2], [11], [28]] (see also the comparison of iterative procedures [18]), and, to the best of our knowledge, the explicit solution in general is available only for a number of special cases (in particular, when matrix $A^{\left(0\right)}$ is a rank-one matrix [21], or $A^{\left(2\right)}$ is a rank-one matrix in a system with $N=2$ [[11], [15]], see also [20]).

To avoid the numerical procedure of obtaining the matrix $R$, some alternative methods are developed. The spectral decomposition-based methods were suggested in [[6], [22]] (which required eigenanalysis of a matrix polynomial), some of them utilizing special structure of the model to decrease the computational complexity [7]. In [12], the Spectral Analysis method based on Jordan canonical form is suggested to obtain a closed form analytic solution of (4). In [25] an extensive study of the Jordan form representation of $R$ (i.e. computing eigenvalues and generalized eigenvectors of $R$) is performed, and the finite memory recursions are suggested to decrease memory requirements of the algorithm. A more general discussion of Jordan canonical form method is provided in [26]. Alternatively, new method of steady-state analysis and obtaining the closed form solution by finding the roots of some (complex-variable) polynomial is developed in [5].

A particular case of $G∕M∕1$ process with $N=2$ is the so-called Quasi-Birth-Death (QBD) process, when the level variable is allowed to increase or decrease by at most one at a time. In this case the matrix polynomial equation (4) is reduced to the matrix quadratic equation

$$R^2{A}^{\left(2\right)}+R{A}^{\left(1\right)}+A^{\left(0\right)}=0,$$

first used to find a solution of a QBD process (by means of Complex Analysis-based method) in late 60’s [[9], [32]].

Recently a method of obtaining explicit solution for the rate matrix $R$ of a QBD process with a small number of phases was proposed [27], where, in particular, the 2 $\times$ 2 matrices $A^{\left(0\right)},A^{\left(2\right)}$ may be of full rank (we also note a closely related work [19], where both 2 $\times$ 2 matrices were full rank, however, upper-triangular). In the present paper we extend the results of [27] to a more general structured $G∕M∕1$-type Markov process with $N>1$ and two phases at a level.

As an example application of the proposed solution, we elaborate on the new method of randomized switching for power saving (first proposed in [27]) which we feel may be implemented in Edge Computing systems based on Internet of Things devices. By numerical experiments, we investigate the optimal configuration of the switching parameters that minimize the average energy consumption and guarantee the required quality of service. We note that the proposed randomized switching approach may be applied to systems, where cost effectiveness and service elasticity is important, such as high-performance and cloud-based computing systems [24], as well as teletraffic systems (e.g. on-demand content servers), where the operational cost (e.g. energy cost, or cloud service cost), as well as the system speed, is to be adopted to the working conditions. The approach is suitable for heavy load conditions, since the switching is performed autonomously, without any knowledge on the system state, except the type of the current event.

This research paper is organized as follows. First, we briefly describe the results of [27]. Then we present an algebraic approach on obtaining the matrix $R$. Next, we apply this approach to solve the optimization problem related to Energy–Performance tradeoff, and illustrate the approach with simulation results.