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A MISO model for power consumption in virtualized servers

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Abstract

Energy efficiency is always a concern in hosting servers. When any new development is added to a host server, the power consumption of the host server must be theoretically and empirically re-evaluated. Because of the ongoing development trends in computing systems at the hardware, software and middleware levels, deriving a direct mathematical model for quantifying the power consumption of a host server is difficult. Therefore, a system identification is used to construct the power consumption model for virtualized hosting servers. To date, three types of system identifications have been used in the literature for defining the power consumption model: the first-principles, the black-box and the gray-box identification approaches. To the best of our knowledge, the majority of these approaches are apparently used to model the power consumption in a single-input single-output (SISO) system model, in which a hardware component is reconfigured to meet the power budget target. In this paper, to accommodate the ongoing development trends in computing systems, we propose a multi-input single-output (MISO) model for modeling the power consumption of virtualized hosting servers. We use the black-box system identification method, and we utilize the Auto-Regressive eXogenous (ARX) mathematical model to construct the MISO power model. We compare our MISO power model with the SISO power model that is used in existing state-of-the-art works. Empirically, our MISO power model exhibits higher accuracy than the existing SISO power model in predicting power consumption. Using our model, we can achieve approximately 98 % accuracy in predicting the power consumption of virtualized hosting servers.

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Correspondence to Fawaz Al-Hazemi.

Appendix: System dynamics of MISO power model

Appendix: System dynamics of MISO power model

Any estimated model should be capable of predicting the dynamic behavior of a system, and our MISO model should have this prediction capability. Let us recall Eq. 8 and consider the 1st-order model and the 2nd-order input delay as the following Eq. 17

$$\begin{aligned}&\Delta p(k)=a \cdot \Delta p(k-1) + b_{f1} \cdot \Delta f(k-1) \nonumber \\&\quad +\, b_{f2} \cdot \Delta f(k-2)+ b_{c1} \cdot \Delta c(k-1) + b_{c2} \cdot \Delta c(k-2).\nonumber \\ \end{aligned}$$
(17)

Additionally, let us assume that we know the initial condition \(\Delta p(1)\) and the inputs of the previous two steps \(\Delta f(1)\), \(\Delta f(0)\), \(\Delta c(1)\), \(\Delta c(0)\). Then, we could derive the solution to Eq. 17 as follows. The first prediction of \(\Delta p(2)\) is

$$\begin{aligned} \Delta p(2)&= a \cdot \Delta p(1) + b_{f1} \cdot \Delta f(1) + b_{f2} \cdot \Delta f(0)\nonumber \\&+\,\, b_{c1} \cdot \Delta c(1) + b_{c2} \cdot \Delta c(0), \end{aligned}$$
(18)

and the second prediction \(\Delta p(3)\) is

$$\begin{aligned} \Delta p(3)&= a \cdot \Delta p(2) + b_{f1} \cdot \Delta f(2) + b_{f2} \cdot \Delta f(1)\nonumber \\&+\,\, b_{c1} \cdot \Delta c(2) + b_{c2} \cdot \Delta c(1). \end{aligned}$$
(19)

We can re-write Eq. 19 after substituting Eq. 18, as follows

$$\begin{aligned}&\Delta p(3)=a \cdot [a \cdot \Delta p(1) + b_{f1} \cdot \Delta f(1) + b_{f2} \cdot \Delta f(0)\nonumber \\&+\,\,b_{c1} \cdot \Delta c(1) + b_{c2} \cdot \Delta c(0)] +\;b_{f1} \cdot \Delta f(2) + b_{f2} \cdot \Delta f(1)\nonumber \\&+\,\, b_{c1} \cdot \Delta c(2) + b_{c2} \cdot \Delta c(1). \end{aligned}$$
(20)

Or

$$\begin{aligned}&\Delta p(3)=a^2 \cdot \Delta p(1) + a \cdot b_{f1} \cdot \Delta f(1) + a \cdot b_{f2} \cdot \Delta f(0)\nonumber \\&\quad +\,\, a \cdot b_{c1} \cdot \Delta c(1) + a \cdot b_{c2} \cdot \Delta c(0) + b_{f1} \cdot \Delta f(2) \nonumber \\&\quad +\,\, b_{f2} \cdot \Delta f(1)+ b_{c1} \cdot \Delta c(2) + b_{c2} \cdot \Delta c(1). \end{aligned}$$
(21)

The third prediction \(\Delta p(4)\) is

$$\begin{aligned}&\Delta p(4)=a \cdot \Delta p(3) + b_{f1} \cdot \Delta f(3) + b_{f2} \cdot \Delta f(2)\nonumber \\&+\,\, b_{c1} \cdot \Delta c(3) + b_{c2} \cdot \Delta c(2). \end{aligned}$$
(22)

Similarly, we can re-write Eq. 22 after substituting Eq. 21, as follows

$$\begin{aligned}&\Delta p(4)=a \cdot [a^2 \cdot \Delta p(1) + a \cdot b_{f1} \cdot \Delta f(1) \nonumber \\&\quad +\,\, a \cdot b_{f2} \cdot \Delta f(0)+ a \cdot b_{c1} \cdot \Delta c(1) + a \cdot b_{c2} \cdot \Delta c(0) \nonumber \\&\quad +\,\, b_{f1} \cdot \Delta f(2) + b_{f2} \cdot \Delta f(1)+ b_{c1} \cdot \Delta c(2)\nonumber \\&\quad +\,\, b_{c2} \cdot \Delta c(1)] + b_{f1} \cdot \Delta f(3) + b_{f2} \cdot \Delta f(2)\nonumber \\&\quad + \,\, b_{c1} \cdot \Delta c(3) + b_{c2} \cdot \Delta c(2) \end{aligned}$$
(23)

Or

$$\begin{aligned}&\Delta p(4)=a^3 \cdot \Delta p(1) + a^2 \cdot b_{f1} \cdot \Delta f(1) + a^2 \cdot b_{f2} \cdot \Delta f(0)\nonumber \\&\quad + \,\, a^2 \cdot b_{c1} \cdot \Delta c(1) + a^2 \cdot b_{c2} \cdot \Delta c(0) + a \cdot b_{f1} \cdot \Delta f(2) \nonumber \\&\quad + \,\,a \cdot b_{f2} \cdot \Delta f(1)+ a \cdot b_{c1} \cdot \Delta c(2) + a \cdot b_{c2} \cdot \Delta c(1) \nonumber \\&\quad +\,\, b_{f1} \cdot \Delta f(3) + b_{f2} \cdot \Delta f(2)+ b_{c1} \cdot \Delta c(3)\nonumber \\&\quad +\,\, b_{c2} \cdot \Delta c(2) . \end{aligned}$$
(24)

Then, we can obtain the following solution

$$\begin{aligned}&\Delta p(k)=a^{k-1} \cdot \Delta p(1) + \sum _{i=1}^{k-1} {a^{k-1-i} \cdot b_{f1} \cdot \Delta f(i)} \nonumber \\&\quad + \sum _{i=1}^{k-1} {a^{k-1-i} \cdot b_{f2} \cdot \Delta f(i-1)} + \sum _{i=1}^{k-1} {a^{k-1-i} \cdot b_{c1} \cdot \Delta c(i)} \nonumber \\&\quad + \sum _{i=1}^{k-1} {a^{k-1-i} \cdot b_{c2} \cdot \Delta c(i-1)}, \end{aligned}$$
(25)

where \(k\) is greater than 1 (\(k>1\)), and \(a\), \(b_{f1}\), \(b_{f2}\), \(b_{c1}\), and \(b_{c2}\) are estimated parameters.

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Al-Hazemi, F., Peng, Y. & Youn, CH. A MISO model for power consumption in virtualized servers. Cluster Comput 18, 847–863 (2015). https://doi.org/10.1007/s10586-015-0436-x

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