Abstract

Generally, how to satisfy the deadline constraint is the major issue in solving real-time scheduling. Recently, neural network using competitive learning rule provides a highly effective method and deriving a sound solution for scheduling problem with less network complexity. However, due to the availability of resources, the machines may not reach full utilization. To facilitate the problem the extra neuron is introduced to the competitive neural network (CHNN). This study tries to impose slack neuron on CHNN with respect to process time and deadline constraints. Simulation results reveal that the competitive neural network imposed on the proposed energy function with slack neurons integrated ensures an appropriate approach of solving this class of scheduling problems of single or multiple identical machines.

Keywords: Scheduling; Slack neuron; Competitive learning; Hopfield neural network

Nomenclature

N

total number of jobs/processes to be scheduled

M

total number of machines/processors to be operated

T

deadline of the jobs

i, j, k

denotes the “job”, “machine”, and “time” variables, respectively

x, y, z

denotes the “job”, “machine”, and “time” variables, respectively

V_ijk, V_xyz

represents the binary states of neurons (i, j, k) and (x, y, z) on Hopfield neural network

P_i

denotes the total execution time required by process i

di

deadline of the process i

G_ijk

defined to examine that whether the time of process x finished in processor i later than the time limit

H(G_ijk)

unit step function. Defined to check if the timing constraint satisfied. A non-zero value indicates the assigned schedule violating the timing constraint. On the other hand, a zero value is yield as meet the timing requirement

W_xyzijk

synaptic weight between neuron (x, y, z) and neuron (i, j, k)

θ_ijk

input bias from outside of neuron (i, j, k)

Net_ijk

net value of neuron (i, j, k), a neuron (i, j, k) receives a community of neuron with interconnection strength W_xyzijk and an input bias, θ_ijk, from outside

δ(a, b)

Kronecker delta function. The value is 1 if a equals b. Otherwise, the value is zero

E

energy function

C₁, C₂, C₃, C₄, C₅

weighting factors of energy terms

Article Outline

Nomenclature

1. Introduction

2. Energy function of the scheduling problem

3. Competitive algorithm

4. Experimental simulations

5. Conclusions

Appendix A. Convergence of the energy function

References