Polynomial least squares workload prediction

Method description

This technique makes the prediction by using a linear model to perform a best dth _{order polynomial fit, because the prediction length L for this application}

is short and ranges usually from 1 to 9 samples.

The polynomial fit is performed by minimizing the error within the ob- served window of temperatures, by using the following function:

kw − ˇAxk2

2 (4.18)

where wt contains the frequency requirements ∀t = 1...N , N is the length

of the observation window of historical data. Vector xt ² <d+1 and matrix

ˇ

Figure 4.10: Structure of matrix ˇA.

Equation 4.18 can be solved as a least squares minimization problem to de- rive vector x. The prediction on the future workload requirement is performed by assuming that the linear model just derived will hold for the next L data samples.

Assuming this assumption hold, the future workload requirement is given by following equation:

wt= ˇAxt, ∀ N ≤ t ≤ N + L (4.19)

where wt ² <p for t > N is the predicted workload requirement at time t. We

tested the predictor on the benchmarks described in the experimental setup section of this thesis and we achieved good accuracies for short-term forecasts (L ranging from 1 to 9).

Design methodology

The structure of matrix ˇA is shown in Figure 4.10. As it can be noted, the first row of matrix ˇA considers only the first component of vector x since the first entry of the row is the only nonzero element. The second row considers all the history contained in vector x, all with the same weight. The third row is a weight function that considers elements of vector x according to a quadratic weight. All rows of matrix ˇA follow the same trend including the last row that weight entries of vector x with a dth _{polynomial function.}

Values of N (the length of the observation window) and d (the order of the polynomial fit) that provide the best prediction depend on the workload re- quirement (task arrival process) statistical properties. These kind of processes are usually non-stationary and depend on the interaction between the user and the MPSoC itself. For the aforementioned reasons, to chose these parameters we suggest to use empirical studies.

4.4

Summary

This chapter has introduced the concepts to model the workload of an MP- SoC system. Moreover some considerations have been made about the system energy models used in this thesis.

The first section has introduced the concept that minimizing power sav- ing, improving performance and increase chip reliability are three important tradeoffs facing MPSoC designers. A theoretical quantification of the energy efficiency is also provided.

The second section has described the power and the workload models used in this thesis. A description of the way the task arrival process is abstracted has also been presented. Finally the relation between the working frequencies of the cores composing the MPSoC and its power dissipation has been described. The third section has provided a theoretical and experimental analysis about the effects that workload prediction has on the performance of any thermal management technique. Two workload estimation techniques are also presented.

Policies for Thermal

Control with Air Cooling

5

This chapter we analyze and explore the use of four families of control tech- niques for thermal management of multi-processors system on chip (MPSoC). In particular, we aim at achieving an online smooth thermal control action that minimizes the performance loss as well as the computational and hard- ware overhead of embedding a thermal management system inside the multi processor system on chip. The optimization problem considers the thermal pro- file of the system, its evolution over time and current time-varying workload requirements. This problem is formulated as a finite-horizon optimal control problem. Thus, we implement the policies on a hardware simulation platform and we collect data of the system. A detailed comparison and analysis is also reported.

Figure 5.1: Diagram of a generic DVFS-based thermal management system

5.1

Introduction

The structure of state-of-the-art DVFS-based thermal management systems is reported in the diagram of Figure 5.1.

The thermal management policy regulator monitors the multi processors system on chip (MPSoC). Without loss of generality, we assume this MPSOC to be partitioned into p islands (or subsystems), each with independent fre- quency and voltage settings. For our purposes, we consider frequencies as inputs to the system, since we are abstracting away the computation. Vector fτ ² <p represents the value of the clock frequencies at time τ . The frequency

value of input i at time τ is denoted by (fτ)i. Input i ranges from 1 to p.

The regulator sets working frequencies fτ +1 according to a specific policy.

The frequency setting the regulator does at time τ is performed by taking into account the current frequency setting fτ, temperature measurements ˜tτ

coming from on-die thermal sensors and a workload requirement coming from the scheduler wτ ² <p. For each functional unit i = 1..p, the workload is

defined as the minimum value of the clock frequency that the functional unit should have in order to execute the required tasks within the specified system constraints. The regulator provides a frequency assignment that minimizes the difference between the required and the achieved workload.

The author of this thesis proposed previously various policies for thermal management [96], [95], [99], [100]. These policies were developed and analyzed independently. In this thesis, we provide a comprehensive comparison of four families of thermal management methods for multi processors system on chip. Thus, this chapter provides the analytical and experimental basis for selecting among four similar, but different, control methods.

These families are closely related and some are based on model predictive control (MPC) [2], which has received much attention lately. These policies aim at achieving an online smooth thermal control action that minimizes the performance loss as well as the computational and hardware overhead of em- bedding a thermal management system inside the MPSoC. Control models and

Figure 5.2: UltraSPARK T1 processor, die photograph by courtesy of SUN [49]

policies differ according to both the way details are included in the problem formulation and the way the solution is computed.

5.2

2D-MPSoC case study