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

机译:G / G / R剥皮和剥皮的机器维修问题的扩散近似

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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.
机译:本文通过扩散近似分析了G / G / R机器维修中出现的回和退回问题。出现故障的机器将以恒定的概率拒绝(不进入)并根据一般分布放弃(进入后离开队列)。机器的故障和维修时间通常也被分配。我们从Fokker-Planck方程获得稳态扩散方程。在交通繁忙的情况下,通过更新理论获得了扩散方程的扩散参数的近似表达式。该分析假设交通状况繁重,即在大多数情况下,始终处于修复状态的故障机器数是非空的。我们开发系统中故障机器数量的近似概率密度函数的表达式。在扩散近似结果与M / M / R机器维修模型的精确结果之间,进行了精确的比较,并得出了停滞时间和指数修正时间。最后,给出了数字示例进行说明。

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