Analytical Solution for Time-Dependent Queue-Size Behavior in the Manufacturing Line with Finite Buffer Capacity and Machine Setup and Closedown Times

Abstract

A single reliable-machine manufacturing line with finite buffer capacity is considered, in which each idle period is preceded by a random closedown time, and each busy period is preceded by a random setup time. The stream of arriving jobs is described by a simple Poisson process, while processing (service), closedown and setup times are generally distributed random variables. A system of Volterra-type integral equations for the transient queue-size distribution conditioned by the initial level of buffer saturation is found by using the Markov property of successive service completion epochs and the continuous version of total probability law. A parallel system written for Laplace transforms is obtained and written in a specific form. The solution of the latter system is derived in a compact-form applying the linear algebraic approach. The final representations are found in terms of “input” system parameters (transforms of processing, closedown and setup times’ distributions) and certain sequence defined recursively.