The computation of the variance of throughput rate of production systems from the central server model
Abstract
An analytical approach that models the central server model (CSM) and the modified central server model (MCSM) for the calculation of the variance of the throughput rate of production systems is proposed. In the CSM, the load and unload (L/UL) station is included in the transportation station, while in the MCSM, the L/UL station is separated out as a workstation. Both the mean and the variance of the throughput rate of the system can be obtained from this L/UL station. The benefit of using the MCSM is that the number of operations needed to finish a job, denoted as $V\sb{\rm m}$, can be a random variable rather than a constant as in the CSM. Because it is difficult to calculate the variance of the throughput rate directly, the variance of the workload for a workstation is calculated instead. The variance of the workload is calculated by integrating the autocovariances of the workload. The autocovariances of the workload are determined using spectral resolution for computing the state transition probability matrix $\rm{\bf P}({\bf h})$ at time lag h of the stationary continuous Markov process n(u) which is the number of parts residing at each workstation at time u. When the system under analysis becomes large, the number of states of the Markov process can make computation intractable. To overcome this computational intractability, two methodologies are proposed. One is to find an equivalent network with less workstations using Norton's theorem in queueing networks, and the other is to assume a finite buffer size at each workstation. Once the equivalent network is constructed with much less workstations and number of parts waiting at each workstation, the computation of the variance of the throughput rate becomes tractable. Simulation is used to obtain the variance of the throughput rate of the original system when the number of system states is too large to be computed, and the result is used as a reference when approximate analytical techniques are tested. Parametric analysis for the variance of the throughput rate is conducted with respect to the number of parts in the system, N. The results shows that the ratio between the variance and the mean of the throughput rate is an increasing function of N.
Degree
Ph.D.
Advisors
Talavege, Purdue University.
Subject Area
Industrial engineering
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