A Greener Heterogeneous Scheduling Algorithm via Blending Pattern of Particle Swarm Computing Intelligence and Geometric Brownian Motion

This paper focuses on the algorithms design of heterogeneous green scheduling for energy conservation and emission reduction in cloud computing. In essence, the real time, dynamic and complexity of heterogeneous scheduling require higher algorithm performance; however, the swarm intelligent algorithms although with some improvements, still exist big imbalances between local exploration and macro development or between route (solution) diversity and faster convergence. In this paper, a greener heterogeneous scheduling algorithm via blending pattern of particle swarm computing intelligence and geometric Brownian motion, is proposed, based on our earlier theoretical breakthroughs on G-Brownian motion and through a series of mathematical derivations or proofs; furthermore, in order for suitable for the hybrid processor architecture of the scheduling management server, the algorithm is designed in parallel with deep fusion of coarse-grained and master-slave models. A large number of experimental results are given. Compared with most newly published scheduling algorithms, there are significant advantages of the proposed algorithm on the dynamic optimization performance for consistent or semiconsistent and large inconsistent scheduling instances, although with lower improvement factors for small inconsistent instances.


I. INTRODUCTION
At present, the epidemic is still raging around the world, and record breaking extreme weather is also frequent. As a matter of fact, energy conservation and emission reduction is a new pressing demand of cloud heterogeneous computing. Since 2016, the annual power consumption of data centers in China (about 120 billion KWH in 2016) has exceeded the annual power generation of the Three Gorges Hydropower Station (about 100 billion KWH in 2016); and there is a huge waste of energy in China's data centers, whose PUE (Power Usage Effectiveness) is generally greater than 2.2 while that of USA is also about 1.9 in the same period. Then, around the many theoretical or technical hot spots on green scheduling coordination, a large number of studies or discussions have been widely carried out [1] .
In essence, the candidate solutions of the scheduling algorithm correspond to the candidate schemes one to one, which means that the real time, dynamic and complexity of heterogeneous scheduling optimization problems require higher optimization performance, such as solution diversity or convergence speed [2,3] .
Concretely, the improvement of swarm intelligence algorithm represented by particle swarm optimization (PSO) algorithm has been systematically carried out from different dimensions, such as parameter selection or optimization [2] , and swarm topology restructuring [3] ; among them, the fusion of different ideas is the most representative direction currently.
(1) Inspired by thermodynamic molecular motion theory, some studies introduced concepts such as group centroid, acceleration and molecular force into PSO algorithm to transform the particle velocity and displacement [4][5][6] .
(2) Other studies refer to human adaptive learning and other mechanisms to realize various information sharing, so as to improve the convergence speed or show stronger ability [7][8][9] .
(3) Moreover, the representative achievements are the effective integration of PSO algorithm and Ito process driven by the standard Brownian motion [10] .
Substantially, the PSO algorithms aforementioned in (1) or (2), are still approximately linear optimization-dynamicpatterns; the drive definition of the improved PSO algorithms in (3), is reduced to a special Markov stochastic process with constant expected drift rate and variance rate, also without the generality in the dynamic swarm intelligence simulation. All of them mean that the swarm intelligent algorithms although with some improvements, still exist big imbalances between local exploration and macro development or between route (solution) diversity and faster convergence for high dimensional multiobjective scheduling problems.
In this paper, based on our earlier theoretical breakthroughs on G-Brownian (Geometric Brownian) motion and through a series of mathematical derivations or proofs, a greener heterogeneous scheduling algorithm via blending pattern of particle swarm computing intelligence and geometric Brownian motion, i.e., PSO/RdBM G , is proposed; furthermore, in order for suitable for the hybrid processor architecture of the scheduling management server, the algorithm is designed in parallel with deep fusion of coarse-grained and master-slave models.

II. RELATED WORK
The combing of related work mainly takes two threads: ① swarm intelligence and PSO algorithms, and ② the standard Brownian motion vs. G-Brownian motion.

A. SWARM INTELLIGENCE AND PSO ALGORITHMS
The PSO algorithm is one of the most popular swarm intelligence algorithms of the computational intelligence theory in recent years. It was first proposed by Kennedy and Eberhart in 1995, and basically adopts the concepts of "group" and "evolution" to search for the optimal solution in complex space through cooperation and competition among particles.
At the same time, as the extension of traditional artificial intelligence, PSO algorithms have been widely applied because of its simple principle, profound background of traditional evolutionary computing and unique highdimensional objective optimization performance [11][12][13][14][15][16][17] .
With the deepening of application and practice, some PSO researches focus on preserving the diversity of individuals in swarm intelligence algorithms. Inspired by thermodynamic molecular motion theory, the researches [4][5][6] introduced concepts such as group centroid, acceleration and molecular force into PSO algorithm to transform formulas such as particle velocity and displacement. In other words, according to the distance between the particle and the center of mass, the switching between the inductive force and the repulsive force can be realized to control the flight direction of the particle, and the diversity of the population can be maintained to a certain extent.
Other studies refer to human adaptive learning and other mechanisms to realize various information sharing in swarm intelligence algorithms, and then to improve the convergence speed or show a stronger ability of later evolution compared with the original swarm intelligence algorithms [7][8][9] .
In recent years, the representative achievement of swarm intelligence algorithm improvement is the effective fusion of standard Brownian motion or Ito Process and PSO algorithms [10] . Some experiments show that the interdisciplinarity can improve the convergence speed or maintain the swarm diversity effectively, but at the same time, it also shows the shortcomings of the algorithm, such as the lack of stability.

B. STANDARD BROWNIAN MOTION VS. G-BROWNIAN MOTION
Standard Brownian motion was first proposed by British biologist R.Brown according to the random movement of pollen on the liquid surface (1827). Later, Wiener further studied the standard Brownian motion trajectory, and theoretically gave its spatial measure definition and other accurate descriptions (1918). Then, Kiyoshi Ito established the stochastic differential equation with the interference term of the standard Brownian motion, which was widely used in the fields of economy, management and social science; for this process, local stochastic disturbance and macroscopic drift are two obvious characteristics [18][19][20][21][22] .
On the basis of preserving the core idea of PSO algorithm, this study intends to derive the energetic particle swarm co-evolution drive equation with nonlinear expectation space and G-Brownian motion characteristics.
The team of Academician Peng, Shige from Shandong University, the cooperative unit of this paper, has made world-renowned basic theoretical researches on G-Brownian motion [23] with their unremitting efforts and systematic theoretical advancement over the past 30 years, which are powerful and instrumental in the field of nonlinear stochastic analysis.
At the same time, these preliminary works are also the valuable basis of strict theoretical derivation or proof of this topic.  In Equation (1), the first part of the formula is called the memory item, where  represents the inertial motion, that is, the influence of the past position on the present; the second part is called self-cognition, where 1 1 c r indicates that the direction of motion of particles comes from their experience; and the third part is called group cognition, where 2 2 c r reflects the cooperation and information sharing between particles (see Fig. 1).

A. THE ENERGIZED OPTIMIZATION-DYNAMIC-EQUATIONS DRIVEN BY G-BROWNIAN MOTION
In this paper, based on the core idea of original PSO algorithm (such as memory, self-cognition and social cognition) defined as Equation (1) and Equation (2), the particle swarm evolution equation is expanded to the geometric Brownian motion model with nonlinear Gexpectation, as is more generalized than the standard Brownian motion with the invariable expected drift rate or the variance rate.
Then, shown as Fig. 2, a series of strict theoretical derivation or proof is key, including ① the spatial representation of the continuous path of the particle swarm, denoted as In other words, once  are obtained, then the particles are accompanied by G-Brownian motion with nonlinear G-expectation for the more intelligent optimization swarm.
The related variables and their representative meanings are shown in Table Ⅰ.   TABLE I  THE  The Borel σ-algebra on the space The Hurst exponent of the time series

S(d)
The space of all dd  symmetric matrices Each given monotonic and nonlinear function, referred to as the G-heat function

H
The Gamma function defined in this paper as the Gheat function for a more generalized co-evolution of the particle swarms The equation definition of geometric Brownian stochastic motion with nonlinear G-expectation for a more generalized co-evolution of the particle swarms where

1) SPATIAL REPRESENTATION OF DISPLACEMENT UPDATE OF PARTICLE SWARM
By Equation (1), the PSO algorithm can be regarded as an Ito process driven by Brownian motion and drifting toward "two" attractors, which are the best position ever for Particle i until the given moment t+1( i pbest ) and the historical optimum location of the particle swarm until the given moment t+1( Then the spatial representation of the continuous path (displacement update) of Particle i, which is denoted by the following Equation (3) and Equation (4), can be obtained.
The research team of Academician Peng has made a batch of basic theoretical research results which are internationally renowned and have been shown to be powerful tools in the field of financial mathematics by virtue of nearly 30 years of unremitting efforts and systematic and solid theoretical advancement.
And these preliminary works, represented by the unique proof of solutions of backward stochastic differential equations (BSDES), the nonlinear Feynman-Kac formula for the correlation between BSDES and second-order quasilinear PDES solutions, and the nonlinear expectation theory with time consistency, are also the precious strict theoretical derivation or proof basis of this topic.

2) THE RELEVANT BOUNDED LIPSCHITZ FUNCTION DEFINITION BASED ON THE SPACE
Following that, the related bounded Lipschitz functions based on the space are defined as Equation (5), Equation (6) and Equation (7), 3 In this paper, the nonlinear function G, which is often referred to as G-heat equation, is well defined as the gamma function (see Equation (8) Then, we can get the dynamic nonlinear G-expectation space of the bounded and Lipschitz function( () X  ) as Equation (9),

4) THE ENERGIZED OPTIMIZATION DYNAMICS DRIVEN BY THE G-BROWNIAN MOTION OF PARTICLE SWARM
To summarize briefly, the optimization-dynamic-equation of particle swarm driven by the geometric Brownian stochastic motion with nonlinear G-expectation, denoted as Equation (7), Equation (8) and Equation (9), respectively, in order for a more generalized co-evolution of the particle swarms.

B. ALGORITHM DESCRIPTION
In this paper, in order for suitable for the hybrid processor architecture of the scheduling management server, the algorithm is designed in parallel with deep fusion of coarsegrained and master-slave models (see Fig. 3).

. The algorithm's parallel design with deep fusion of coarse-grained and master-slave models
Specifically, several particles subgroups can be assigned to different nodes according to coarse-grained model. A large number of particle velocity or displacement updates on each node adopt CPU-GPU collaborative master-slave parallel design. Here, the CPU is regarded as the primary server, and several threads on the GPU are the clients.
Here, C-CUDA language, MICH -VMI communication protocol and ParadisEO software component can be used.
Following that, the algorithm can be described as follows.
The PSO/RdBM G algorithm Step 1: Initialize the iteration (ι) and the subgroups, each subgroups of k particles; Step 2: Randomly initialize the velocity and the position of the particle i; Step 3: While (ι<ιmax) and (other termination criteria are not satisfied) Step 4: Do in parallel for each island /*Obtain coarsegrained model, one of parallel and distributed models */ Step 5: ι=ι+1; Step 6: Step 7: Step 8: Step 9: Step 10: Step 11: Step 12: Step 13: where 1 [ , )   ii t tt and 1, ,  ik to well define the nonlinear function G, which is often referred to as G-heat equation; Apply the Equation (9) to get the dynamic nonlinear G-expectation space of the bounded and Lipschitz function( () X  ); Generate the new subgroups; Sort the particles in the subgroups in a decreasing order of fitness values, and save the fittest particle in the external memory; Step 14: Perform local search strategies; Step 26: Output the best particle.

IV. EXPERIMENT RESULTS AND DISCUSSION
In this section, all the experiments have been carried out at National Supercomputing Center in Jinan, China.

A. SIMULATOR AND SIMULATION PARAMETERS
To ensure that the comparison between the algorithms is fair, there are not any special requirements for parameter setting between different methods; in other words, the general parameter values of the PSO algorithm are set normally.
Famous unimodal and multimodal test problems are shown in Table Ⅱ. They are roughly divided into unimodal functions with changing variables, and multimodal functions with fixed or changing variables; and they are used to verify the static optimization performance of the different methods, where unimodal functions can assist in the global convergence validation and multimodal functions can test the ability of the local search or averting premature convergence.

B. EXPERIMENTAL RESULTS AND ANALYSIS
In this subsection, the multi-objective optimization performance comparison between the different methods including the static and dynamic optimization performance results analysis, is given.

Shown as
Further, from Fig. 4, it can be seen that for the unimodal or multimodal functions, the PSO/RDBM G algorithm can quickly seek out the globally optimal solutions at the early co-evolutionary stage, while the other four methods (BLPSO [4] ,GEPSO [8] ,SPSO [9] , and OJPSO [10] ) fail to make it. Here, to compare the dynamic optimization performance of different algorithms, twelve classic instances of heterogeneous computing and cloud scheduling proposed by [26] are used.  5 summarizes the averaged efficiency improvements of the solution of PSO/RdBM G over that of BLPSO [4] , GEPSO [8] , SPSO [9] , and OJPSO [10] , for each dimension and heterogeneity model. Shown as Fig. 6, for semiconsistent instances or for consistent instances, the averaged efficiency preponderance of the PSO/RdBM G approach over the other four methods (BLPSO [4] ,GEPSO [8] ,SPSO [9] , and OJPSO [10] ) is obvious. Lower improvement factors are obtained for small inconsistent instances, but for large inconsistent instances, the advantage significantly increase.

V. CONCLUSION
As the extension of traditional artificial intelligence, PSO algorithms representing the swarm intelligence algorithms of the computational intelligence theory, have been widely used in heterogeneous multimodal optimization.
Although with some improvements, the PSO algorithms are mostly linear optimization-dynamic-models; and the PSO algorithm integrated with Ito process driven by the standard Brownian motion, is reduced to a special Markov stochastic process with constant expected drift rate and variance rate, also without the generality in the dynamic swarm intelligence simulation.
In this paper, based on the core idea of original PSO algorithm (such as memory, self-cognition and social cognition) and the strict theories such as nonlinear stochastic analysis, a greener heterogeneous scheduling algorithm via blending pattern of particle swarm computing intelligence and geometric Brownian motion, is proposed, to achieve the balance of "local exploration and macro development" or "route diversity and faster convergence".
In other words, a key series of strict theoretical derivation or proof is obtained, including ①the spatial representation of the continuous path of the particle swarm, denoted as Then, the evaluation indicators can be divided into two categories: static and dynamic optimization performance. A large number of experimental results are given. Compared with most newly published scheduling algorithms (BLPSO [4] ,GEPSO [8] ,SPSO [9] , and OJPSO [10] ), there are significant advantages of the proposed algorithm on the dynamic optimization performance for consistent or semiconsistent and large inconsistent scheduling instances, although with lower improvement factors for small inconsistent instances.