Diffusion Approximation for G/G/R Machine Repair Problems with Balking and Reneging

Abstract

This paper analyzes the G/G/R machine repair problems with balking and reneging via diffusion approximation. Failed machines balk (do not enter) with a constant probability and renege (leave the queue after entering) according to a general distribution. Failure and repair times of the machines are also generally distributed. We obtain steady-state diffusion equations from the Fokker-Planck equations. In heavy traffic conditions, the approximate expressions for the diffusion parameters of the diffusion equations are obtained by the renewal theory. The analysis assumes heavy traffic conditions, that is, the number of failed machines in the repair state is nonempty in most cases all the time. We develop the expressions for the approximate probability density functions of the number of failed machines in the system. An accuracy comparison is performed between the diffusion approximation results and exact results of the M/M/R machine repair model with balking and exponential reneging times. Finally, numerical examples are given for illustration.