An Hyper-exponential Decomposition Method for the Analysis of Production Lines with Unreliable Machines and Finite Buffers

We consider production lines consisting of a series of machines separated by finite buffers. The processing time of each machine is deterministic and all the machines have the same processing time. All machines are subject to failures. As usually the case for production systems we assume that the failures are operation dependent [3,7]. Moreover, we assume that the times to failure and the times to repair are exponentially distributed. To analyze such systems, a decomposition method was proposed by Gershwin [11]. The computational efficiency of this method was later significantly improved by the introduction of the so-called DDX algorithm [5,6]. In general, this method provides fairly accurate results. There are however cases for which the accuracy of this decomposition method may not be so good. This is the case when the reliability parameters (average failure time and average repair time) of the different machines have different orders of magnitude. Such a situation may be encountered in real production lines. In [8] an improvement of Gershwin's original decomposition method has been proposed that in general provides more accurate results in the above mentioned situation. This other method is referred to as the GE-method. The basic difference between the GE-method with that of Gershwin is that it uses a two-moment approximation instead of a one-moment approximation of the repair time distributions of the equivalent machines. There are however still cases for which the accuracy of the GE-method is not as good as expected. This is the case for example when the buffer sizes are too small in comparison with the average repair time. We present in this paper a new decomposition method that is based on a better approximation of the repair time distributions. This method uses a three-moment approximation of the repair time distributions of the equivalent machines. Numerical results show that the new method is very robust in the sense that it seems to provide accurate results in all situations.