Heterogeneous multi-core task scheduling method based on whale optimization algorithm

Abstract

The invention discloses a heterogeneous multi-core task scheduling method based on a whale optimization algorithm. The heterogeneous multi-core task scheduling method comprises the following steps: step 1, task information acquisition and model establishment; firstly, to-be-scheduled task information needs to be obtained, and the task information comprises task types, calculation amount, memory requirements and communication overhead; then, according to the architecture characteristics of the heterogeneous multi-core processor, establishing a task scheduling system model; 2, the number of white whale individuals is initialized and corresponds to a task scheduling scheme; 3, solving an individual with the highest fitness based on a white whale optimization algorithm, and taking the individual as an optimal solution; and 4, determining a final task scheduling scheme according to the optimal solution, wherein the final task scheduling scheme comprises a processor to which each task is allocated and an execution sequence. 5, the scheduling scheme is applied to the heterogeneous multi-core system, and task scheduling is executed. According to the method, the task scheduling efficiency is remarkably improved, and the overall task completion time is shortened.

Description

Technical Field

[0001] The present invention belongs to the technical field of computer control and relates to a heterogeneous multi-core task scheduling method based on the White Whale optimization algorithm. Background Art

[0002] With the rapid development of computer technology, multi-core technology has been widely used in the computer field. Heterogeneous multi-core processors have become one of the key technologies to improve computer performance because they can provide higher computer efficiency and better energy efficiency. In heterogeneous multi-core systems, the performance differences between different processors, power consumption limitations, and dependencies between tasks make task scheduling a complex and challenging problem. How to reasonably assign tasks to different processors is a question that needs to be considered in task scheduling.

[0003] Traditional task scheduling, such as static scheduling and dynamic scheduling, performs well in some scenarios. However, when dealing with complex and changing task requirements of heterogeneous multi-core systems, it is often difficult to meet the requirements of real-time and high efficiency, and faces the problem of difficulty in finding local optimal solutions. In recent years, swarm intelligence optimization algorithms have been increasingly used in the field of task scheduling. These algorithms optimize the scheduling of tasks by simulating the behavior of biological groups in nature, such as bird migration and fish foraging. Among them, the Beluga Optimization Algorithm (BWO), as an emerging swarm intelligence optimization algorithm, is inspired by the group behavior and living habits of beluga whales, simulating the swimming, hunting and whale falling behaviors of beluga whales, and has the advantages of strong global search capabilities and fast convergence speed, providing new ideas for heterogeneous multi-core task scheduling.

[0004] Based on the above problems and ideas, this paper proposes a heterogeneous multi-core task scheduling method based on the White Whale Optimization Algorithm. The BWO algorithm is applied to heterogeneous multi-core task scheduling, and improves the efficiency and adaptability of the scheduling algorithm by simulating the social behavior of white whales, solving the shortcomings of existing algorithms in real-time, large-scale task scheduling, and energy efficiency. Summary of the invention

[0005] (I) Purpose of the invention

[0006] The purpose of the present invention is to provide a heterogeneous multi-core task scheduling method based on the White Whale optimization algorithm, aiming at the characteristics of heterogeneous multi-core processors, and realizing efficient task scheduling by simulating the group behavior of White Whales.

[0007] (II) Technical solution

[0008] In order to solve the above technical problems, the present invention provides a heterogeneous multi-core task scheduling method based on the White Whale optimization algorithm, which comprises the following steps:

[0009] Step 1: Task information acquisition and model building;

[0010] First, it is necessary to obtain the task information to be scheduled, including the task type, computational workload, memory requirements, and communication overhead. Then, a task scheduling system model is established based on the architectural characteristics of heterogeneous multi-core processors.

[0011] Step 2: Initialize the number of beluga whales and correspond to the task scheduling plan;

[0012] Step 3: Find the individual with the highest fitness based on the White Whale optimization algorithm and take it as the optimal solution;

[0013] Step 4: Determine the final task scheduling plan based on the optimal solution, including the processor assigned to each task and the execution order.

[0014] Step 5: Apply the scheduling scheme to the heterogeneous multi-core system and perform task scheduling.

[0015] Furthermore, in step 1, the task scheduling system model uses a directed acyclic graph to describe the dependencies between tasks. The model is expressed as F = {P, G, W, C}, where P = {P1, P2, ... P s}, indicating that there are s processor cores in the system, P j represents the jth processor core, 1≤j≤s; G={G1,G2,…G t} means there are t tasks, G i represents the i-th task, 1≤j≤n; Where W represents the execution time of tasks on each processor. Represents task G i In the processor core P j Execution time; C = {…C i,j …}, C represents the communication data set between tasks, C i,j represents the communication overhead between task i and task j. If task i and task j are assigned to the same processor, then C i,j =0.

[0016] Furthermore, in step 1, the requirements between tasks and processors are: all processors work normally and no failures occur during task scheduling; at the same time, a task is only executed on one processor; each task is always executed after all its predecessor tasks are completed; the priority between tasks is determined in advance.

[0017] Furthermore, in step 2, the t tasks are numbered from 0 to t-1, the s processor cores are numbered from 0 to s-1, and the Beluga optimization algorithm is initialized to generate a population containing n Beluga individuals, each Beluga represents a candidate solution, and each Beluga individual in task scheduling represents a potential task scheduling solution; the Beluga individual is represented by X k, K∈[1,n], X k =[x k,1 ,x k,2 ,...,x k,t ] is a t-dimensional array; the initial beluga population is modeled as:

[0018]

[0019] The corresponding fitness values ​​are stored as follows:

[0020]

[0021] Among them, f(x k ) is the fitness value of the i-th beluga whale;

[0022] For heterogeneous multi-core task scheduling, the shortest scheduling length SL is calculated, that is, the latest completion time of the last task is the smallest. The formula of the fitness value objective function is:

[0023] f(x)=max{eft(T i )},i=1,2,...,t

[0024] Among them, eft(T i ) is the earliest completion time of the i-th task, and the completion time of the last task on the heterogeneous multi-core is the completion time of all tasks on the entire heterogeneous system.

[0025] Furthermore, in step 2, a coding scheme based on task priority is adopted to assign different random integers between 1 and t to the t tasks, and the larger the value, the higher the priority; k =[x k,1 ,x k,2 ,...,x k,t ] is converted into a task priority sequence v = (p1, p2, ..., p t ), modeling the entire matrix as a task scheduling sequence table;

[0026] The decoding process is to allocate the solution obtained by the White Whale algorithm to the problem space through the decoding regression task, and the optimal solution is selected by calculating the fitness function.

[0027] Furthermore, in step 3, the steps of the White Whale optimization algorithm are:

[0028] S31 sets the population size N and the maximum number of iterations T;

[0029] S32 initializes the positions of individuals in the population and calculates the fitness values;

[0030] S33 determines the best individual and its fitness;

[0031] S34 calculates the balance factor B according to equations (1) and (2) respectively f and whale fall probability W f

[0032]

[0033] Where t is the current iteration number, T is the total iteration number, and parameter B0 is a random value in the range of (0, 1);

[0034]

[0035] Among them, t is the current iteration number, T is the total iteration number, and W f is a linear function;

[0036] S35 judgment B f If it is greater than 0.5, the individual position is updated according to the exploration and development phases of the White Whale algorithm;

[0037] S36 Calculation B f Is it less than W? f , if it is less than, then update the individual position according to (8), (9), and (2), that is, it is in the whale fall stage of the White Whale algorithm; if it is not less than, recalculate the fitness value and determine the best individual;

[0038]

[0039] Among them, r5, r6, and r7 are all random numbers between (0, 1), Lb and Ub are the lower and upper bounds of the optimization problem, respectively. is the position of an individual r randomly selected from the current population, It is the step length of the whale fall; C2 is the step factor, which is related to the whale fall probability and population size, C2 = 2W f ×N;

[0040] S37 determines whether the number of iterations is less than the set total number of iterations. If so, repeat steps S34-S36. If so, output the optimal solution and optimal value at this time.

[0041] Furthermore, in step 31, the population size is set according to the scale and complexity of the problem, and each individual in the population represents a possible task scheduling solution.

[0042] Furthermore, in step 32, the position of each individual in the initial population is randomly generated in the search space, the position of the individual corresponds to a specific task scheduling scheme, and the fitness value of each individual is calculated according to the fitness function.

[0043] Furthermore, in step 35, it is determined that Bf Is it greater than 0.5? If it is greater than 0.5, the individual position is updated according to formula (3), which means it is in the exploration stage of the White Whale algorithm.

[0044] During the exploration phase, beluga whales perform social behaviors in different postures, and the position of the beluga whales is updated as follows:

[0045]

[0046] in, Indicates the position of the i-th individual in the j-th dimension at the next iteration, P is a random number, the problem dimension is T, and p is a random integer in the range [1, T]; represents the position of the i-th individual on the random dimension p in the current iteration, and r also represents a random integer in the range of [1, N]. is the position of the random individual r on the random dimension p in the current iteration, r1 and r2 are both random numbers between (0, 1).

[0047] Furthermore, in step 35, it is determined that B f Is it greater than 0.5? If it is less than or equal to 0.5, the individual position is updated according to equations (4) to (7), which means it is in the development stage of the White Whale algorithm.

[0048] During the development phase, the Levy flight strategy was introduced to enhance convergence. The mathematical model is expressed as:

[0049]

[0050] in, Indicates the random jump degree, which is used to measure the strength of Levy flight. Both r3 and r4 are random numbers between (0, 1). F is a random number that conforms to the Levy distribution and is calculated using the following formula:

[0051]

[0052] Γ(x)=(x-1)! (7)

[0053] Among them, μ and ν are random numbers that obey the normal distribution, and β is a constant set to 1.5.

[0054] (III) Beneficial effects

[0055] The heterogeneous multi-core task scheduling method based on the White Whale optimization algorithm provided by the above technical solution has the following beneficial effects:

[0056] (1) The method proposed in the present invention can efficiently search for the optimal solution in a global scope by simulating the swimming, foraging and "whale fall" behaviors of beluga whales. In heterogeneous multi-core task scheduling, this algorithm can quickly find the optimal task allocation solution so that each task can be executed in parallel to the maximum extent, thereby significantly improving the efficiency of task scheduling.

[0057] (2) The White Whale optimization algorithm used in the present invention has a strong global search capability and fast convergence, and it can find the optimal solution in a short time. In heterogeneous multi-core task scheduling, this means that the execution order and allocation scheme of tasks can be determined more quickly, thereby significantly shortening the overall completion time of the task. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 It is a flow chart of a heterogeneous multi-core task scheduling method based on the White Whale optimization algorithm of the present invention.

[0059] Figure 2 It is a flow chart of the White Whale optimization algorithm adopted by the present invention. DETAILED DESCRIPTION

[0060] In order to make the purpose, content and advantages of the present invention more clear, the specific implementation methods of the present invention are further described in detail below in conjunction with the drawings and examples.

[0061] Reference Figure 1 and Figure 2 As shown, the heterogeneous multi-core task scheduling method based on the White Whale optimization algorithm in this embodiment includes the following steps:

[0062] Step 1: Obtaining task information and building a model.

[0063] First, it is necessary to obtain the task information to be scheduled, including the task type, computational workload, memory requirements, communication overhead, etc. Then, based on the architectural characteristics of heterogeneous multi-core processors, a task scheduling system model is established.

[0064] Step 2: Initialize the number of beluga individuals and correspond to the task scheduling plan.

[0065] Step 3: Based on the White Whale optimization algorithm, find the individual with the highest fitness and take it as the optimal solution.

[0066] Step 4: Determine the final task scheduling plan based on the optimal solution, including the processor assigned to each task and the execution order.

[0067] Step 5: Apply the scheduling scheme to the heterogeneous multi-core system and perform task scheduling.

[0068] In step 1, the task scheduling system model uses a directed acyclic graph (DAG) to describe the dependencies between tasks. The model is expressed as F = {P, G, W, C}. Where P = {P1, P2, ... P s}, indicating that there are s processor cores in the system, P j (1≤j≤s) represents the jth processor core. G={G1,G2,…G t} means there are t tasks, G i (1≤i≤n) represents the i-th task. Where W represents the execution time of tasks on each processor. Represents task G i In the processor core P j C={…C i,j …}, C represents the communication data set between tasks, C i,j represents the communication overhead between task i and task j. If task i and task j are assigned to the same processor, then C i,j =0.

[0069] In step 1, the requirements between tasks and processors are: all processors work normally and no failures occur during the task scheduling process; a task can only be executed on one processor at a time and is not allowed to be executed by multiple processors; each task is always executed after all its predecessor tasks are completed; the priority between tasks is determined in advance, and there is no situation where the priority changes suddenly during the scheduling process.

[0070] In step 2, the t tasks are numbered from 0 to t-1, and the s processor cores are numbered from 0 to s-1. The Beluga optimization algorithm is initialized to generate a population containing n Beluga individuals, each of which represents a candidate solution. In task scheduling, each Beluga individual represents a potential task scheduling solution. The Beluga individual is represented by X k , K∈[1,n], X k =[x k,1 ,x k,2 ,...,x k,t ] is a t-dimensional array. The initial beluga population is modeled as:

[0071]

[0072] The corresponding fitness values ​​are stored as follows:

[0073]

[0074] Among them, f(x k) is the fitness value of the i-th white whale. The purpose of heterogeneous multi-core task scheduling is to find the shortest scheduling length SL, that is, the latest completion time of the last task is the smallest. The formula of the fitness value objective function is:

[0075] f(x)=max{eft(T i )},i=1,2,...,t

[0076] Among them, eft(T i ) is the earliest completion time of the i-th task, and the completion time of the last task on the heterogeneous multi-core is the completion time of all tasks on the entire heterogeneous system.

[0077] In step 2, the White Whale optimization algorithm is used to solve the multi-core heterogeneous scheduling problem, which requires the design of a reasonable coding scheme. The present invention adopts a coding scheme based on task priority, assigning different random integers between 1 and t to t tasks. The larger the value, the higher the priority. k =[x k,1 ,x k,2 ,...,x k,t ] is converted into a task priority sequence v = (p1, p2, ..., p t ). Thus the entire matrix is ​​modeled as a task scheduling sequence table.

[0078] Table 1 shows a priority allocation method for 7 tasks:

[0079] Task <![CDATA[G1]]> <![CDATA[G2]]> <![CDATA[G3]]> <![CDATA[G4]]> <![CDATA[G5]]> <![CDATA[G6]]> <![CDATA[G7]]> Priority 6 3 1 4 5 2 7

[0080] The decoding scheme is the inverse process of the encoding scheme. The solution obtained by the White Whale algorithm is decoded and regressed into the task allocation problem space. The optimal solution is obtained by calculating the fitness function. The task priority value and the number of processors are modulo each other to obtain the task allocation of the processor. Assuming there are 4 processors, a task allocation decoding scheme is shown in Table 2.

[0081] Task <![CDATA[G1]]> <![CDATA[G2]]> <![CDATA[G3]]> <![CDATA[G4]]> <![CDATA[G5]]> <![CDATA[G6]]> <![CDATA[G7]]> processor 2 3 1 4 1 2 3

[0082] In step 3, the specific steps of the White Whale optimization algorithm are:

[0083] S31 sets the population size N and the maximum number of iterations T. According to the scale and complexity of the problem, set an appropriate population size, and each individual in the population represents a possible task scheduling solution.

[0084] S32 initializes the position of the individuals in the population and calculates the fitness value. The position of each individual in the initial population is randomly generated in the search space. The position of the individual corresponds to a specific task scheduling scheme, and the fitness value of each individual is calculated according to the fitness function.

[0085] S33 determines the best individual and its fitness.

[0086] S34 calculates the balance factor B according to equations (1) and (2) respectively f and whale fall probability W f

[0087]

[0088] Among them, t is the current iteration number, T is the total iteration number, and parameter B0 is a random value in the range of (0, 1).

[0089]

[0090] Among them, t is the current iteration number, T is the total iteration number, and W f is a linear function. S35 determines B f Is it greater than 0.5? If it is greater than 0.5, the individual position is updated according to formula (3), which means it is in the exploration stage of the White Whale algorithm. If it is less than or equal to 0.5, the individual position is updated according to formulas (4) to (7), which means it is in the development stage of the White Whale algorithm.

[0091] During the exploration phase, belugas can perform social behaviors in different postures. The position of the beluga is updated as follows:

[0092]

[0093] in, Indicates the position of the i-th individual in the j-th dimension at the next iteration. P is a random number, the problem dimension is T, and p is a random integer in the range [1, T]. Therefore, represents the position of the ith individual on the random dimension p in the current iteration, and r also represents a random integer in the range [1, N]. Therefore, is the position of the random individual r on the random dimension p in the current iteration, r1 and r2 are both random numbers between (0, 1).

[0094] In the development stage, considering the influence of the best individual and other individuals on the position update, the Levy flight strategy is introduced in the development stage to enhance the convergence. The mathematical model is expressed as:

[0095]

[0096] in, Indicates the random jump degree, which is used to measure the strength of Levy flight. Both r3 and r4 are random numbers between (0, 1). F is a random number that conforms to the Levy distribution and is calculated using the following formula:

[0097]

[0098] Γ(x)=(x-1)! (7)

[0099] Among them, μ and ν are random numbers that obey the normal distribution, and β is a constant set to 1.5.

[0100] S36 Calculation B f Is it less than W? f If it is less than , the individual position is updated according to (8), (9), and (2), that is, it is in the whale fall stage of the White Whale algorithm. If it is not less than , the fitness value is recalculated to determine the best individual.

[0101]

[0102] Among them, r5, r6, r7 are all random numbers between (0, 1), and Lb and Ub are the lower bound and upper bound of the optimization problem respectively. is the position of an individual r randomly selected from the current population, It is the step length of whale fall. C2 is the step factor, which is related to the probability of whale fall and the size of the population. C2 = 2W f ×N.

[0103] S37 determines whether the number of iterations is less than the set total number of iterations. If so, repeat steps S34-S36. If so, output the optimal solution and optimal value at this time.

[0104] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the technical principles of the present invention. These improvements and modifications should also be regarded as the scope of protection of the present invention.

Claims

1. A heterogeneous multi-core task scheduling method based on the White Whale optimization algorithm, characterized in that: The following steps are involved: Step 1: Task information acquisition and model building; First, it is necessary to obtain the task information to be scheduled, including the task type, computational workload, memory requirements, and communication overhead. Then, a task scheduling system model is established based on the architectural characteristics of heterogeneous multi-core processors. Step 2: Initialize the number of beluga individuals and correspond to the task scheduling plan; Step 3: Find the individual with the highest fitness based on the White Whale optimization algorithm and take it as the optimal solution; Step 4: Determine the final task scheduling plan based on the optimal solution, including the processor assigned to each task and the execution order. Step 5: Apply the scheduling scheme to the heterogeneous multi-core system and perform task scheduling.

2. The heterogeneous multi-core task scheduling method based on the White Whale optimization algorithm according to claim 1, characterized in that: In step 1, the task scheduling system model uses a directed acyclic graph to describe the dependencies between tasks. The model is expressed as F = {P, G, W, C}, where P = {P1, P2, ... P s }, indicating that there are s processor cores in the system, P j represents the jth processor core, 1≤j≤s; G={G1,G2,…G t } means there are t tasks, G i represents the i-th task, 1≤j≤n; Where W represents the execution time of tasks on each processor. Represents task G i In the processor core P j Execution time; C = {…C i,j …}, C represents the communication data set between tasks, C i,j represents the communication overhead between task i and task j. If task i and task j are assigned to the same processor, then C i,j =0.

3. The heterogeneous multi-core task scheduling method based on the White Whale optimization algorithm according to claim 2, characterized in that: In step 1, the requirements between tasks and processors are: all processors work normally and no failure occurs during task scheduling; a task is executed on only one processor at a time; each task is always executed after all its predecessor tasks are completed; and the priorities between tasks are determined in advance.

4. The heterogeneous multi-core task scheduling method based on the White Whale optimization algorithm according to claim 3 is characterized in that: In step 2, the t tasks are numbered from 0 to t-1, and the s processor cores are numbered from 0 to s-1. The Beluga optimization algorithm is initialized to generate a population containing n Beluga individuals, each Beluga represents a candidate solution, and each Beluga individual in task scheduling represents a potential task scheduling solution; the Beluga individual is represented by X k , K∈[1,n], X k =[x k,1 ,x k,2 ,...,x k,t ] is a t-dimensional array; the initial beluga population is modeled as: The corresponding fitness values ​​are stored as follows: Among them, f(x k ) is the fitness value of the i-th beluga whale; For heterogeneous multi-core task scheduling, the shortest scheduling length SL is calculated, that is, the latest completion time of the last task is the smallest. The formula of the fitness value objective function is: f(x)=max{eft(T i )},i=1,2,...,t Among them, eft(T i ) is the earliest completion time of the i-th task, and the completion time of the last task on the heterogeneous multi-core is the completion time of all tasks on the entire heterogeneous system.

5. The heterogeneous multi-core task scheduling method based on the White Whale optimization algorithm according to claim 4, characterized in that: In step 2, a coding scheme based on task priority is adopted to assign different random integers between 1 and t to t tasks. The larger the value, the higher the priority. k =[x k,1 ,x k,2 ,...,x k,t ] is converted into a task priority sequence v = (p1, p2, ..., p t ), modeling the entire matrix as a task scheduling sequence table; The decoding process is to allocate the solution obtained by the White Whale algorithm to the problem space through the decoding regression task, and the optimal solution is selected by calculating the fitness function.

6. The heterogeneous multi-core task scheduling method based on the White Whale optimization algorithm according to claim 5, characterized in that: In step 3, the steps of the White Whale optimization algorithm are: S31 sets the population size N and the maximum number of iterations T; S32 initializes the positions of individuals in the population and calculates the fitness values; S33 determines the best individual and its fitness; S34 calculates the balance factor B according to equations (1) and (2) respectively f and whale fall probability W f Where t is the current iteration number, T is the total iteration number, and parameter B0 is a random value in the range of (0, 1); Among them, t is the current iteration number, T is the total iteration number, and W f is a linear function; S35 judgment B f If it is greater than 0.5, the individual position is updated according to the exploration and development phases of the White Whale algorithm; S36 Calculation B f Is it less than W? f , if it is less than, then update the individual position according to (8), (9), and (2), that is, it is in the whale fall stage of the White Whale algorithm; if it is not less than, recalculate the fitness value and determine the best individual; Among them, r5, r6, and r7 are all random numbers between (0, 1), Lb and Ub are the lower and upper bounds of the optimization problem, respectively. is the position of an individual r randomly selected from the current population, It is the step length of the whale fall; C2 is the step factor, which is related to the whale fall probability and population size, C2 = 2W f ×N; S37 determines whether the number of iterations is less than the set total number of iterations. If so, repeat steps S34-S36. If so, output the optimal solution and optimal value at this time.

7. The heterogeneous multi-core task scheduling method based on the White Whale optimization algorithm according to claim 6, characterized in that: In step 31, the population size is set according to the scale and complexity of the problem, and each individual in the population represents a possible task scheduling solution.

8. The heterogeneous multi-core task scheduling method based on the White Whale optimization algorithm according to claim 7, characterized in that: In step 32, the position of each individual in the initial population is randomly generated in the search space, the position of the individual corresponds to a specific task scheduling scheme, and the fitness value of each individual is calculated according to the fitness function.

9. The heterogeneous multi-core task scheduling method based on the White Whale optimization algorithm according to claim 8, characterized in that: In step 35, it is determined that B f Is it greater than 0.5? If it is greater than 0.5, the individual position is updated according to formula (3), which means it is in the exploration stage of the White Whale algorithm. During the exploration phase, the beluga whales performed social behaviors in different postures, and the beluga whales’ positions were updated as follows: in, Indicates the position of the i-th individual in the j-th dimension at the next iteration, P is a random number, the problem dimension is T, and p is a random integer in the range [1, T]; represents the position of the i-th individual on the random dimension p in the current iteration, and r also represents a random integer in the range of [1, N]. is the position of the random individual r on the random dimension p in the current iteration, r1 and r2 are both random numbers between (0, 1).

10. The heterogeneous multi-core task scheduling method based on the White Whale optimization algorithm according to claim 9, characterized in that: In step 35, it is determined that B f Is it greater than 0.5? If it is less than or equal to 0.5, the individual position is updated according to equations (4) to (7), which means it is in the development stage of the White Whale algorithm. During the development phase, the Levy flight strategy was introduced to enhance convergence. The mathematical model is expressed as: in, Indicates the random jump degree, which is used to measure the strength of Levy flight. Both r3 and r4 are random numbers between (0, 1). F is a random number that conforms to the Levy distribution and is calculated using the following formula: Γ(x)=(x-1)! (7) Among them, μ and ν are random numbers that obey the normal distribution, and β is a constant set to 1.5.

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