Evolutionary game for task mapping in resource constrained heterogeneous environments

Highlights

• Task mapping is formulated as multi-objective problem based on evolutionary game theory on networks.

• Ecosystem’s heterogeneity is accounted by the model parameters.

• Power, workload imbalance, resource affinity and data offloading costs are optimized.

• Our model outperforms first-fit (up to 34.6%) and best-fit methods (up to 35.7%).

Abstract

Power-aware computing is becoming popular using heterogeneous ecosystem. For recent execution units, the heterogeneity is exhibited via the hybrid cores, as well as via virtual and physical asymmetric cores. The inherent performance disparity between different types of execution units at their different clock frequencies offers a great resources scheduling challenge. Multiple metrics (such as throughput, latency, energy cost) are used to decide whether the scheduling is an optimal solution or not. However, in heterogeneous ecosystem, tasks distribution aiming at optimizing costs is not trivial. During the task mapping, one of the primary challenges is to dynamically identify and map the inherent advantages/features of the heterogeneous or hybrid architectures for each individual task. In this work we deal with the task mapping problem using a multi-objective formulation based on evolutionary game theory to optimize a suitable payoff function. This payoff accounts for the power, workload imbalance, task resource affinity and data offloading costs (from host to accelerator). Here, we report that in a very restrictive resource usage scenario (supporting both over and under subscription), the proposed formulation based on Evolutionary Games on Network equation (EGN) can outperform the traditional resource allocation heuristics (such as best-fit, first-fit). Using an extensive set of simulations, we show that our proposed model can outperform first-fit algorithm from more than 5% up to 34.6% and best-fit algorithm from 4% up to 35.7%.

Introduction

The recent evolutionary trend in silicon industry has made it possible to develop heterogeneous system architecture (HSA). Presently, there has been a shift of focus from absolute performances to performances per unit energy. The traditional multitasking approach minimizes the loss of execution cycles by running several tasks simultaneously, while HSA-based platform aims at accomplishing several tasks with multiple resource requirements to further improve the overall power cost and performance (such as latency and throughput). In [1], it has been shown that even heterogeneous instruction set architecture (ISA) can outperform both the homogeneous and the single-ISA heterogeneous-based architecture. In general, the heterogeneous systems can offer more performance improvements [2] and also improve energy consumption [3]. Thus heterogeneous computing could be the answer to Exascale challenge (such as power, concurrency and hardware technology) [4]. However, the performance of execution units shows differences based on the size or type of the applications [5]. On the other hand, the proper tuning of applications can dramatically reduce the performance gap between the heterogeneous execution units [6].

In general, multiple users run different types of tasks using the computational resources of a distributed heterogeneous environment (such as Cloud, HPC platforms). The users individually want to optimize their cost, while the commercial service providers want to improve their profit margins. It has already been shown that traditional load balancing algorithms are not enough to arrive at an optimal energy-aware resource allocation configurations [7]. Mapping of tasks in the distributed environment is not trivial, while the optimal mapping (defined as matching and scheduling) of independent tasks onto heterogeneous systems is a NP-complete problem [8]. However, achieving fair resource allocation in the heterogeneous environment is not trivial as well because (i) the allocation must be both equitable and Pareto efficient [9] and (ii) also during the run time, task migration involves a costly transfer of task-related information.

Application of game theory in distributed computing domain is not new because it offers an efficient framework for tackling the task mapping problems. Specifically, game theory is already been used for: (i) efficient resource management (or load balancing) [10], [11], [12], [13], [14], [15], [16], [17], (ii) energy optimization [18], [19], [20]; (iii) price optimization [21], [22], [23].

Interestingly, evolutionary game theory (EGT) provides a dynamical point-of-view on the standard game theory. Specifically, EGT deals with the dynamical evolution of strategies in a population, under the influence of mechanisms based on the natural selection [24]. In this context, a sub-optimal choice (known as the Nash equilibrium (NE)¹  [25]) can be seen as an emerging phenomenon achievable after a sufficiently long time, thanks to the repeated game interactions among the members of the population. Indeed, according to the outcomes obtained after each game interaction, players are able to change their strategies with more outperforming ones. This mechanism is accounted for by the replicator equation (RE), which is a nonlinear set of an ordinary differential equation (ODE) [24], [25], [26]. The state variable of RE is represented by the distribution of strategies over the population and it is largely applied for studying biological systems [27], social networks [28], [29], [30], as well as control theory [31], telecommunications [32], and neuroscience [33]. Remarkably, finding a NE of a static game is not a trivial process [34]. However, it turns out that an attractive steady state of the RE corresponds to a NE of the underlying static game [25], [26]. Therefore, the fascinating idea of a dynamics which naturally converges toward an attractive steady state can be profitably used for finding a NE of a given game (see for example [35], [36], where this approach has been successfully used for solving a graph matching problem for artificial intelligence purposes).

Recently, an extension of RE accounting for networked populations of players has been introduced in [28], [29], [37], namely the Evolutionary Games on Network equation (EGN). In this paper, we exploit this relation between EGN and NE providing a formulation for a task mapper (or mapper) to place the incoming tasks (submitted during the job submission window, see Fig. 1) on the available resources of a distributed system in order to minimize the overall cost. To this aim, we formulate the allocation problem as an instance of a non-cooperative game [36], [38], where the different tasks (players) are competing with each other to satisfy their own requirements, trying at the same time to minimize the related costs. The total payoff consists of power cost, task’s data offload profile (data movement costs from host to the accelerators such as GPUs, PHIs), task’s resource affinity (to the specific hardware type) and also the workload distribution among the (heterogeneous) processing units. To properly account all the different cost sources, we develop a multi-objective payoff function for the proposed game, based on suitable payoff matrices, which depend on the resource features (such as power consumption unitary cost, offload overhead unitary cost, and so on). Moreover, additional nonlinear artificial costs have been added to the multi-objective payoff function to take into account the resource capacity constraints. Finally, the developed EGN is exploited to find out NE (i.e., a sub-optimal assignment of resources to tasks) of the evolutionary game in an efficient way. To summarize, the main contributions of the paper are:

• We propose a formulation of the task mapping problem using EGN. In particular, this game is based on a payoff function composed of four conflicting objective functions.

• The EGN equation is used for finding NE of the game, and hence sub-optimal solutions for the task mapping problem.

• The effectiveness of the proposed approach is also demonstrated after comparing the EGN equation with well-known heuristics such as the best-fit algorithm (BFA), and the first-fit algorithm (FFA). We have shown that our EGN-based model can outperform FFA $5.1\text{\%}$ (when the resource usage limit is set to $50\text{\%}$), $24.1\text{\%}$ (resource usage limit is set to $80\text{\%}$), $\approx 34.6\text{\%}$ (limit is $100\text{\%}$) and $34.1\text{\%}$ (over-subscription, limit set to $120\text{\%}$). Similarly, our model performs $4.1\text{\%}$ (usage limit is set to $50\text{\%}$), $31.7\text{\%}$ (limit increased to $50\text{\%}$), $35.7\text{\%}$ (limit is $100\text{\%}$), and $31.3\text{\%}$ (limit set to $120\text{\%}$) better compared to the BFA.

Rest of the paper is organized as follows: Section 2 presents the relevant related research works. The general formulation of the task mapping problem is described in Section 3, while Section 4 contains the fundamental elements of evolutionary game theory used in this work. Later, the proposed model is introduced in Section 5 and Section 6 reports the numerical results obtained by simulating the mathematical model together with elaborated discussion. Section 7 presents results on the robustness of the proposed model. Concluding remarks and plausible future developments are mentioned in Section 8.