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Research Project Convex Optimization CVX Matlab Latex

Publication and Source code

Introduction

Data centers have been increasingly deployed for cloud computing. Due to space limitation, servers are installed compactly on racks. The increased of computational power causes high energy cost on both computing and cooling. As a result, the operation of data center becomes extremely expensive. How we can increase energy efficiency of data centers becomes a more urgent problem in recently built large-scale data centers.

Based on our investigation, a significant proportion of energy consumption in a data center is from AC systems and computation systems. We propose that energy consumption can be optimized by assigning different workloads to different machines.

Different from previous studies. Our study is highlighted in these innovations:
  1. Mixed-integer convex optimization is applied for energy modeling.
  2. The scheduling problem is established on Dynamic Voltage and Frequency Scaling (DVFS) enabled CPUs.
  3. In system model, we consider heterogeneous heat correlation among servers.
  4. Different from existing works, which consider energy consumption of computation and AC individually, our optimization includes both factors.
  5. A thermal interference model is built on high-fidelity cross coefficient matrix.

Method

We set objective function as minimize the power consumption from both AC and computation. The objective function is related to the tasks scheduling \( X \) and AC output temperature \( T^{sup} \): \( \min_{( X, T^{sup} )} P^{cmp} + P^{AC}\).

The constraints are composed by 4 parts:
  1. Tasks scheduling model
  2. Heat flow model
  3. Server power model
  4. AC power model
All these models except the server power model, they can be described by linear equations. For the server power model, we describe the power consumption \( P^{cmp}_{i} \) as cube function of processor's frequency \( f_i \): \( P^{cmp}_{i} (f_i) = \beta_1 + \beta_2 f^3_i \), different from previous studies that assume a linear correlation between power consumption \( P^{cmp}_i \) and frequency \( \mu_i \): \( P^{cmp}_i(\mu_i) = \beta_1 + \beta_3 \mu_i \).

Performance Evaluation

We model a data center layout using computational fluid dynamics (CFD) simulation, where 25 servers are deployed:
Then we convert the CFD model into thermal interference model described by a matrix: To verify our assumption that power consumption is cube function of frequency of modern CPU, we measure an actual power consumption of a desktop CPU, by setting it to 3 different working statuses:
  1. DVFS enabled status
  2. DVFS disabled (non-DVFS) status
  3. P-state status
Then we compare them with theoretical curve fitting results as shown in the following figure. The above experiment result shows that the the cube function can accurately describe the power consumption in practice. We generate tasks with random computation resource requirement between 400MHz to 500MHz. Comparing the optimal result with other methods, we can get these observations:
  1. The DVFS-enabled data center is more energy efficient than non-DVFS data center.
  2. The naive coolest-first scheduling does not achieve this privilege.
  3. For DVFS case, in region (0 to 80%load) all three proposed optimization achieve similar energy efficiency.
  4. With the increase of load, AC energy consumption becomes a dominant factor.
  5. At 100% load, no policy is better than others.

Conclusion

In the study, the proposed optimization accounts for the non-linear computation power of DVFS-enabled servers and the heterogeneous thermal correlation among servers, and can be approximately solved in polynomial time. The simulation results verified the effectiveness of our optimization, especially when the data center is heavily loaded.