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This paper proposes to take the relationship between delay and workload into account in the power optimization of microprocessors in mobile embedded systems. Since the components outside a device continuously change their values or properties, the workload to be handled by the systems becomes dynamic and variable. This variable workload is formulated as a staircase function of the delay taken at the previous iteration in this paper and applied to the power optimization of DVFS (dynamic voltage-frequency scaling). In doing so, a graph representation of all possible workload/mode changes during the lifetime of a device, Workload Transition Graph (WTG), is proposed. Then, the power optimization problem is transformed into finding a cycle (closed walk) in WTG which minimizes the average power consumption over it. Out of the obtained optimal cycle of WTG, one can derive the optimal power management policy of the target device. It is shown that the proposed policy is valid for both continuous and discrete DVFS models. The effectiveness of the proposed power optimization policy is demonstrated with the simulation results of synthetic and real-life examples.

Today’s mobile embedded systems often interact with physical processes or external environments, referred to as Cyber-Physical Systems (CPSs). Such systems are usually modeled with interactions between the physical world and the devices [

In a class of applications, the computational workload of the embedded systems depends on the variation of the sampled input value, while the computation delay, in turn, affects the input variation of the next iteration. Usually, if it invests more time at one iteration for processing information, it would have more work to do at the next iteration. One example of such delay-workload dependency can be found in an object tracking which is frequently used in drone, surveillance camera, or augmented reality [

Such workload-delay relations can be popularly found in modern mobile embedded systems, which rely on computer vision algorithms to capture what happens in the external world. In those applications, it is typical that the

The workload-delay dependency can also be found in many different types of applications. Real-time pattern matching over event streams [

Nowadays, most modern microprocessors used in mobile embedded systems support dynamic voltage-frequency scaling (DVFS) [

The workload-delay dependency has been firstly modeled and applied to the DVFS optimization in [

This work differs from our previous work [

The workload-delay dependency is modeled in a staircase function generalizing the previous model and validated with a real-life example.

A novel data structure, Workload Transition Graph (WTG), is proposed to represent all possible operation workload/mode changes of a device.

Based on WTG, a power management policy is derived and shown to be optimal.

Bogdan and Marculescu [

Recently, Pant et al. [

A design guideline for flexible delay constraints in distributed embedded systems was proposed by Goswami et al. [

This section presents the system model assumed in this paper, which is followed by the formulation of the power optimization problem.

In this paper, we assume that a system has multiple operation modes due to DVFS feature, where the operating frequency and voltage can be modulated. For simplicity, we first assume that there are infinitely many operation modes available, among which one is chosen at each iteration. It will be shown that the proposed technique can be applied to a discrete DVFS as well in Section

The workload is defined to be a number of clock cycles elapsed to complete the given computation. We denote the number of cycles elapsed to handle the workload of the

The delay

As stated earlier, the workload is dependent upon the previous execution delay. Usually, the workload is not a continuous function of the delay variation. Rather, the changes happen in a discrete manner. Therefore, the workload at the

At the

The dynamic power consumption of CMOS circuits is

Our objective is to minimize the average power consumption of a given system as follows:

Given the modeling constant

In this section, we describe the proposed operation management policy as an answer to the problem defined in the previous section. In doing so, we first derive the condition for feasible and schedulable systems. Then, we study when the workload changes and how it affects the power dissipation. Based on that, we propose a novel graph representation that captures all possible workload transitions in the power-optimal operation. Finally, we derive the power-optimal operation policy with the given workload function

In this subsection, we examine under which condition a given system is feasible. First, the system should be schedulable within the given real-time constraint

Given the workload function

Suppose that the delay at the

Once the workload gets bigger than

Given the workload function

Then, the valid workload levels and the execution delay range during the lifetime of a given system can be formulated as below.

Given the workload function

In this subsection, we examine when a workload transition between valid workload levels possibly occurs and how it affects the system.

As presented in (

However, such workload transitions can occur only within limited ranges. Figure

Examples of workload transitions: (a) transitions to lower workload levels and (b) transitions to higher workload levels. The transitions from

The same principle is also applied to the transition from a workload level to higher ones. If the delay can be lengthened properly with a speed scaling factor within the range

A workload transition from

The essential difficulty of the presented power optimization problem lies in the fact that two conflicting forces should be handled at the same time. In order to minimize the power, on one hand, it tries to scale down the speed (thus lengthen the delay) as much as possible as described in (

Therefore, no one simple intuition can be exploited to solve the problem. Rather, we need to compare different modes in a comprehensive way. In order to be able to explore all possible execution modes and quantify their effects, we need to devise a data structure that includes elementary information on how workload transitions change the system status and power dissipation. In line with that purpose, we propose a graph representation of the workload evolution, Workload Transition Graph (WTG), which captures all possible transition scenarios of the workload changes during the lifetime of a system.

A valid workload transition from one workload level to another can be caused by any delay within the corresponding range. A transition from

In the power-optimal trace

We prove this by contradiction. Let us suppose that there exists a power-optimal trace

From Theorem

The scaling factor of a valid workload transition from

Now, we define WTG.

WTG is defined to be a graph

With these definitions, a power-optimal execution trace

Algorithm

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Let us take Figure

A workload staircase function example with six workload levels.

The corresponding WTG of the workload function

The feasible delay range to handle workload

Different initial workload levels may result in different WTGs as shown in Figures

In this subsection, we present the proposed operation policy that compromises the energy-delay tradeoff caused by the delay-workload dependency.

As stated earlier, a power-optimal execution trace

Given the power-optimal execution trace

A cycle (closed walk) of WTG is a walk whose starting and ending vertices are the same. That is, a walk

Suppose that a cycle of WTG,

An arbitrary walk of a WTG,

Now, consider a power-optimal trace

A walk example of the WTG shown in Figure

From Theorem

If the current workload level vertex is in the optimal cycle

Otherwise, take a path

Figure

Flowchart diagram of the proposed operation policy.

Whilst we assume a continuous DVFS model for ease of presentation and generality, modern microprocessors in reality have finite DVFS modes with a set of predefined operation voltages and frequencies. In this section, we show that the continuity of the model presented in (

Let us suppose that we now have a system with a discrete and finite DVFS model, where only

Given a set of feasible scaling factors

Likewise, the scaling factor of a valid transition, in Definition

Given a set of feasible scaling factors

In this section, we validate the proposed model and operation policy with experimental analysis and simulations.

It is firstly shown that the proposed workload-delay dependency is evidently observed in an object tracking application. The performance of a commonly used object tracking method [

Workload-delay dependency function

We compare the proposed power management policy with two others. The first one is ALAP, where the speed is chosen to be the slowest one with respect to the real-time constraint. The other comparison is made against a stable trace with the maximum speed as another extreme (ASAP). When the initial workload is

There are two kinds of cycles in WTG: the first one is a self-loop which implies a

(a) A synthetic example,

The average power consumption of all self-loops in the synthetic example is tied to

In principle, a stable operation mode is the case that the staircase workload function

This paper formulates the delay-workload dependency in power optimization problem of embedded systems as a staircase function of the delay taken at the previous iteration. In applying it to the power optimization of DVFS-enabled electronic devices, a novel graph representation, called WTG, is proposed for exploring all possible workload/mode changes. Then, it is shown that the power optimization problem is equivalent to finding a cycle of the graph that has the minimum average power consumption. The effectiveness of the proposed operation policy is proven by the power simulations of synthetic and real-life examples. It has been observed that staying in a low speed scaling factor in a stable operation mode is often the best discipline (self-loop in WTG). However, alternating modes, where the DVFS modes change over a predefined pattern, sometimes outperform the stable ones.

A preliminary version of this paper appeared in July 2015 at the International Symposium on Low Power Electronics and Design (ISLPED), under the title of “Modeling and Power Optimization of Cyber-Physical Systems with Energy-Workload Tradeoff” [

The authors declare that they have no competing interests.

This work was supported by ICT R&D program of MSIP/IITP (B0101-15-0661, the research and development of the self-adaptive software framework for various IoT devices), Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2013R1A2A2A01067907), and the new faculty research fund of Ajou University.