Eginhard J. Muth

The production rate of a series of work stations with variable service times

SUMMARY A mathematical model for the interdemand time at the first station of a production line with stochastic service times is presented. Production rate is expressed as the reciprocal of mean interdemand time. The analysis is not restricted to the assumption of exponentially distributed service times. General solutions for the distribution of interdemand time are derived for two and three work stations. For any number of work stations a lower bound on production rate is presented graphically. It is shown that production rates obtained with exponentially distributed service times are unrealistically low

A General Model of a Production Line with Intermediate Buffer and Station Breakdown

Abstract A general model for a class of production lines with two unreliable stations, a finite capacity interstation buffer, discrete items, constant cycle time, and synchronous transfer is presented. The model is applicable to a broad range of production lines with various item service mechanisms, item transfer mechanisms, station breakdown mechanisms, and station repair mechanisms. The model provides insights into the general properties of production lines and gives rise to a general recursive solution for the steady-state probabilities. Sufficient conditions are given for the two special cases, the matrix geometric case and the scalar geometric case, which permit the efficient computation of the production rate.

The bowl phenomenon revisited

A novel method of analysing serial production lines has been developed. This method lets one compute the throughput rate of lines composed of dissimilar stations, as well as for a large class of distributions of station service times. Several distribution-free models of 3-station lines are presented. These models are used to compute the throughput rate of unbalanced lines in which the sum of the mean service times is constant. Results are shown as contour plots of constant throughput rate. The bowl phenomenon is reviewed in the light of this capacity to model with a greater degree of freedom.

Conveyor Theory: A Survey

Abstract This paper surveys the research which has been published in the area of conveyor theory. Deterministic and probabilistic approaches, as well as descriptive and normative approaches to modeling conveyor systems are discussed. Areas for further study are suggested.

Stochastic processes and their network representations associated with a production line queuing model

Abstract The passage of items through the stations of a production line has associated with it different types of random points in time that induce different stochastic point processes. The process of holding times of items at each of k stations is defined; we call it a generalized vector renewal process. The steady-state properties of this process are represented by an activity network. Relationships among random variables of interest are simplified through the use of equivalent networks. Special results are derived for k = 2, k = 3, and k = 4.

Analysis of Closed-Loop Conveyor Systems

Abstract A closed-loop conveyor system having a single loading station, a single unloading station, and operating with time-varying input and output flow rates, is analyzed. The balance of flow on the conveyor is represented by a difference equation. Solutions of that difference equation appear naturally in terms of a Fourier series expansion. An important description of the system is its frequency response. Singularities in the frequency response represent cases of incompatibility. Incompatibility is shown to depend on the ratio T/P of conveyor period to work-cycle period, and on the presence of harmonics in input and output flow rates. Solutions for several specific cases are presented graphically.

Analysis of Closed-Loop Conveyor Systems, the Discrete Flow Case

Abstract A closed-loop conveyor system having a single loading station, a single unloading station, and discrete, time-varying input and output flows, is analyzed. The sequence of material flow on the return leg of the conveyor is represented by a difference equation. That equation is reduced to a matrix formulation. Incompatibility is related to the eigenvectors and eigenvalues of the matrix involved, and a general solution procedure is given. The ratio r/p, which is the remainder of the ratio conveyor period to work-cycle period, is shown to be an important criterion for compatibility and optimization. An example is worked out and several solutions are presented graphically.

Modelling and system analysis of multistation closed-loop conveyors†

SUMMARY A discrete-flow, closed-loop conveyor system with multiple loading and unloading stations is considered. An inherent capability of such a system is that certain cyclic variations in the input and output flow rates can be accommodated through the storage feature of the closed loop. The system is reduced to an equivalent one-station system for which the balance of flow is described by a difference equation. The solution of this equation yields feasibility conditions as well as guidelines for conveyor design optimization and for conveyor operation. A numerical example solution for a three-station conveyor is included.

A Model of a Closed-Loop Conveyor with Random Material Flow

Abstract A closed-loop conveyor with one loading station and one unloading station is considered. The material arriving at the unloading station is modeled as a stationary stochastic process. An analytical solution for the probability distribution of the material flow leaving the unloading station is presented. It is shown that the storage feature of the closed loop can be utilized to smooth the random fluctuations present in the input flow. This smoothing is quantified by the variance reduction factor, which is the ratio of the variances of output flow and input flow. The variance reduction factor is related to the conveyor capacity and a linear decision rule for unloading.

The Throughput Rate of Multistation Production Lines with Stochastic Servers

A serial production line with an arbitrary number of stations is analyzed using the holding-time model. In contrast to Markovian state models, the holding-time model does not require the service times to be exponentially distributed. The distribution of the interdemand time at the first station is obtained as a function of the distribution of station service times which are of a general nature. The throughput rate of the line is the reciprocal of the mean interdemand time. For service times of phase type, two numerical solution methods, one exact and the other approximate, are presented. Numerical results are given for up to ten stations.