An Optimization-Based Scheme for Efficient Virtual Machine Placement

Abstract

According to the important methodology of convex optimization theory, the energy-efficient and scalability problems of modern data centers are studied. Then a novel virtual machine (VM) placement scheme is proposed for solving these problems in large scale. Firstly, by referring the definition of VM placement fairness and utility function, the basic algorithm of VM placement which fulfills server constraints of physical machines is discussed. Then, we abstract the VM placement as an optimization problem which considers the inherent dependencies and traffic between VMs. By given the structural differences of recently proposed data center architectures, we further investigate a comparative analysis on the impact of the network architectures, server constraints and application dependencies on the potential performance gain of optimization-based VM placement. Comparing with the existing schemes, the performance improvements are illustrated from multiple perspectives, such as reducing the number of physical machines deployment, decreasing communication cost between VMs, improving energy-efficient and scalability of data centers.

Appendix

1.1 Appendix A: Proof of Theorem 1

Proof

The constraints of $\left( {\varvec{P}}_1\right)$ are linear and the constraint domain is convex set. In addition,

$$\begin{aligned} \frac{\partial ^{2}U_p }{\partial x_{pv}^2 \left( t \right) }<0 \hbox { and } \frac{\partial ^{2}U_p }{\partial x_{pv} \left( t \right) \partial x_{pw} \left( t \right) }<0 \end{aligned}$$

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are satisfied. Thus, the Hesse matrix of the objective function is negative definition. That means the objection function is concave but not strictly concave with respect to

$$\begin{aligned} x\left( t \right) =\left( {x_{pv} \left( t \right) ,p\in P,v\in V} \right) . \end{aligned}$$

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According to [27], $\left( {\varvec{P}}_1\right)$ is a convex programming problem, it is easy to conclude that the optimal solution exists but not unique. Assuming that $y_1 \left( t \right)$ and $y_2 \left( t \right)$ are two different solutions for the total load requirements. Let $\alpha$ be any real number which ranges from 0 to 1. Then, since $U_p \left( {y_p \left( t \right) } \right) =w_p \log y_p \left( t \right)$ is the objective function, when $y_1 \left( t \right) \ne y_2 \left( t \right)$, clearly:

$$\begin{aligned} U_p \left( {\alpha y_1 \left( t \right) +\left( {1-\alpha } \right) y_2 \left( t \right) } \right) >\alpha U_p \left( {y_1 \left( t \right) } \right) +\left( {1-\alpha } \right) U_p \left( {y_2 \left( t \right) } \right) . \end{aligned}$$

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The objection function is strictly concave with respect to $y=\left( {y_p \left( t \right) ,p\in P} \right)$, the unique optimal solution exists for total load requirements. $\square$

1.2 Appendix B: Proof of Theorem 2

Proof

Based on the fact that ${\varvec{0}}\in R, R\ne \varnothing$. Assuming $x_i \left( t \right) \ge 0,i=1,2,\ldots ,n, R$ is not empty bounded set. Due to the constraints of $\left( {\varvec{P}}_2\right)$ are linear, it is ensured that $R$ is a closed set. For the case that the objective function given by Eq. (29) is linear and continuous function, $\left( {\varvec{P}}_{\varvec{2}}\right)$ is equivalent to seek the maximum values problem upon the non-empty, bounded closed set. Therefore, the unique optimal solution exists, namely $R^{*}\ne \varnothing$. $\square$

1.3 Appendix C: The Expressions of Communication Distance for Different Topologies

In traditional Tree topologies, we assign $n_0$ as the output number of the access switch, and $n_1$ as the output number of the aggregation switch, then

$$\begin{aligned} Distance_{kl}^{Tree} =\left\{ {\begin{array}{ll} 0&{}\quad { if}\;k=l \\ 1&{}\quad { if}\;\left\lfloor {\frac{k-1}{n_0 }} \right\rfloor =\left\lfloor {\frac{l-1}{n_0 }} \right\rfloor \\ 3&{}\quad { if}\;\left\lfloor {\frac{k-1}{n_0 }} \right\rfloor \ne \left\lfloor {\frac{l-1}{n_0 }} \right\rfloor \wedge \left\lfloor {\frac{k-1}{n_0 n_1 }} \right\rfloor =\left\lfloor {\frac{l-1}{n_0 n_1 }} \right\rfloor \\ 5&{}\quad { if}\;\left\lfloor {\frac{k-1}{n_0 n_1 }} \right\rfloor \ne \left\lfloor {\frac{l-1}{n_0 n_1 }} \right\rfloor \\ \end{array}} \right. . \end{aligned}$$

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For the VL2 topologies, in order to achieve variable load balancing, assume that all traffic from the access switch will go out through the core switch for further forwarding, setting $n_0$ as the output number of the access switch, then

$$\begin{aligned} Distance_{kl}^{VL2} =\left\{ {\begin{array}{ll} 0&{}\quad { if}\;k=l \\ 1&{}\quad { if}\;\left\lfloor {\frac{k-1}{n_0 }} \right\rfloor =\left\lfloor {\frac{l-1}{n_0 }} \right\rfloor \\ 5&{}\quad { if}\;\left\lfloor {\frac{k-1}{n_0 n_1 }} \right\rfloor \ne \left\lfloor {\frac{l-1}{n_0 n_1 }} \right\rfloor \\ \end{array}} \right. . \end{aligned}$$

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In Fat-Tree architecture, communication distance is the function of $n_{port}$, the total number of ports per switch, then:

$$\begin{aligned} Distance_{kl}^{Fat-Tree} \!=\!\left\{ {\begin{array}{ll} 0&{}\quad { if}\;k\!=l\! \\ 1&{}\quad { if}\;\left\lfloor {\frac{\!2(k\!-\!1)}{n_{port} }} \right\rfloor \!=\!\left\lfloor {\frac{2(l\!-\!1)}{n_{port} }} \right\rfloor \\ 3&{}\quad { if}\;\left\lfloor {\frac{\!2(k\!-\!1)}{n_{port} }} \right\rfloor \ne \left\lfloor {\frac{2(l\!-\!1)}{n_{port} }} \right\rfloor \wedge \left\lfloor {\frac{\!4(k\!-\!1)}{n_{port}^2 }} \right\rfloor \!=\!\left\lfloor {\frac{\!4(l\!-\!1)}{n_{port}^2 }} \right\rfloor \\ 5&{}\quad { if}\;\left\lfloor {\frac{\!4(k\!-\!1)}{n_{port}^2 }} \right\rfloor \ne \left\lfloor {\frac{\!4(l\!-\!1)}{n_{port}^2 }} \right\rfloor \\ \end{array}} \right. . \qquad \end{aligned}$$

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In BCube framework, communication distance is the Hamming distance function of server address, then:

$$\begin{aligned} Distance_{kl}^{BCube} =\left\{ {\begin{array}{ll} 0&{}\quad { if}\;k=l \\ 2\;hamming\left( {addr\left( k \right) ,addr\left( l \right) } \right) -1&{}\quad { if}\;k\ne l \\ \end{array}} \right. . \end{aligned}$$

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