Variance of the throughput of an \(N\)-station production line with no intermediate buffers and time dependent failures.

*(English)*Zbl 0916.90119Summary: The variance of the throughput of an \(N\)-station production line with no intermediate buffers and time dependent failures is analytically determined. Time to failure and time to repair distributions are assumed to be exponential. The analytical method yields a closed-form expression for the variance of the throughput. The method is based on determining the limiting variance of the total residence (sojourn) time in a specific state of an irreducible recurrent Markov process from the probability of visiting that state at time \(t\) given an initial state. This probability function is the instantaneous availability of a production system in the reliability context. A production line with no inter-station buffers and time-dependent failures is basically a series system with hot standby. The same procedure can be applied to determine the variance of the throughputs of various arrangements of workstations including series, parallel, series-parallel systems provided that the instantaneous availabilities of these systems can be written explicitly. Numerical experiments show that, although the expected throughput decreases monotonically, the variance of the throughput may increase and then decrease as the number of stations in the line increases depending on the system parameters. Numerical experiments that investigate this phenomenon and also the dependence of the coefficient of variation on the number of stations are also presented in this study.

##### MSC:

90B25 | Reliability, availability, maintenance, inspection in operations research |

##### Keywords:

availability analysis; reliability; tandem queues; time to failure; variance of the throughput; \(N\)-station production line; time to repair; irreducible recurrent Markov process
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