Joint energy optimization on the server and network sides for geo-distributed data centers

Abstract

Energy optimization has become an emerging concern for cloud service providers. Existing methods focus on reducing the energy consumption of either server inside the data center or data transmission between data centers. Moreover, most of the works are based on assumptions that servers and workloads are homogeneous. This is not in accordance with the fact that modern data centers are built from various classes of servers. In this paper, we consider the joint energy optimization of intra- and inter-data center in both homo- and heterogeneous cases. We first propose an optimization model to minimize the joint energy cost of servers and network sides. To tackle the time-coupling constraint of carbon emission, we apply the Lyapunov optimization framework to transform the original problem into a well-studied queue stability problem. For better scalability of time complexity, we derive a distributed solution by using generalized benders decomposition. Then, we extend the model to deal with the situation where requests and data centers are heterogeneous as data centers are typically built from servers with different specifications. To better deal with the dynamic of the network (e.g., the occurrence of faults), we leverage a deep Q-network (DQN) and propose a fault-tolerant DQN-based solution. Finally, the simulation results show the high efficiency of our proposal in cost-saving and performance-enhancing.

Appendices

Appendix 1: Description of Equivalency between Virtual Queue and Constraint (11)

These queues own stability which is equivalent to $\lim_{T \to \infty } {\mathbb{E}}\{ X_{i} (T)/T\} = 0$. And for (15), it exists that

$$ X_{i} (t + 1) \ge X_{i} (t) - N_{i}^{\max } + (E_{i} (t) + e_{i} (t))N_{i} (t). $$

(41)

The following inequality can be obtained for the summation of the time slot from 0 to $t$ with the above inequality (41):

$$ \frac{{X_{i} (T) - X_{i} (0)}}{T} + N_{i}^{\max } \ge \frac{1}{T}\sum\limits_{t = 0}^{T - 1} {(E_{i} (t) + e_{i} (t))N_{i} (t)} . $$

(42)

Bring $X(0) = 0$ and take the two sides of inequality (42) to the limit, we can get the following inequality:

$$ \mathop {\lim }\limits_{T \to \infty } \frac{{{\mathbb{E}}\{ X_{i} (t)\} }}{T} + N_{i}^{\max } \ge \mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\sum\limits_{t = 0}^{T - 1} {(E_{i} (t) + e_{i} (t))N_{i} (t)} . $$

(43)

If the virtual queues $X_{i} (t)$ are stable, it should be satisfied that $\lim_{T \to \infty } {\mathbb{E}}\{ X_{i} (T)/\} /T = 0$ and the above inequality (43) can be converted to the carbon emission constraint (11) of problem (14). Therefore, satisfying carbon emission constraint is equivalent to the stability of virtual queues. Thus, in this way, the carbon emissions constraint (14) can be constrained into the stability of the virtual queue.

Appendix 2

Proof of Lemma 1

Similar to [11], there is a fact obviously which is that $(\max [x - y + z,0])^{2} \le x^{2} + y^{2} + z^{2} - 2x(y - z),\forall x,y,z \ge 0.$ So, for virtual queue $X_{i} (t),$ there are

$$ \begin{gathered} X_{i}^{2} (t + 1) - X_{i}^{2} (t) \le (N_{i}^{\max } )^{2} + (E_{i} (t) + e_{i} (t))^{2} N_{i}^{2} (t) \hfill \\ - 2X_{i} (t)[N_{i}^{\max } - (E_{i} (t) + e_{i} (t))N_{i} (t)]. \hfill \\ \end{gathered} $$

(44)

Based on the above inequality, we can get:

$$ \begin{gathered} \Delta ({\mathbf{X}}(t)) \le \sum\nolimits_{i = 1}^{I} {\frac{1}{2}{\mathbb{E}}\{ (N_{i}^{\max } )^{2} + (E_{i} (t) + e_{i} (t))^{2} N_{i}^{2} (t)|{\mathbf{X}}(t)\} } \hfill \\ - \sum\nolimits_{i = 1}^{I} {X_{i} (t){\mathbb{E}}\{ N_{i}^{\max } - (E_{i} (t) + e_{i} (t))N_{i} (t)|{\mathbf{X}}(t)\} } . \hfill \\ \end{gathered} $$

(45)

It is obvious that $(E_{i} (t) + e_{i} (t))N_{i} (t)$ is bounded by $\max_{i,t} (E_{i} (t) + e_{i} (t))N_{i} (t)$. So, the bound value of the express $1/2\sum\nolimits_{i = 1}^{I} {{\mathbb{E}}\{ (N_{i}^{\max } )^{2} + (E_{i} (t) + e_{i} (t))^{2} N_{i}^{2} (t)|{\mathbf{X}}(t)\} }$ is defined by $B = 1/2 \cdot (\sum\nolimits_{i = 1}^{I} {(N_{i}^{\max } )^{2} + I(\max_{i,t} (E_{i} (t) + e_{i} (t))N_{i} (t)} )^{2} )$.

The following inequality can be obtained by adding the express $H{\mathbb{E}}\{ \sum\nolimits_{i = 1}^{I} {(E_{i} (t) + e_{i} (t)) \cdot M_{i} (t)} \}$ to both sides of inequality (45):

$$ \begin{gathered} \Delta ({\mathbf{X}}(t)) + H{\mathbb{E}}\{ \sum\limits_{i = 1}^{I} {(E_{i} (t) + e_{i} (t))M_{i} (t)} |{\mathbf{X}}(t)\} \le B - \sum\limits_{i = 1}^{I} {X_{i} (t)N_{i}^{\max } } \hfill \\ + \sum\limits_{i = 1}^{I} {{\mathbb{E}}\{ (E_{i} (t) + e_{i} (t))(HM_{i} (t) + X_{i} (t)N_{i} (t))|{\mathbf{X}}(t)\} } . \hfill \\ \end{gathered} $$