Stochastic Processes and Optimization Problems in Assemblage Systems

Assemblage systems e.g. arise from production systems, where k different pieces are delivered by k parallel production lines. At some place this pieces are assembled in order to form the desired good, taking exactly one piece from every of the k single lines. According to random fluctuations of the production process parallel queues are formed by single pieces waiting to be processed at the assemblage station. If one neglects the time needed to transform a complete group of k different parts into the final assembled good, a pure assemblage system emerges. It is always possible, to separate an assemblage system into two stages: firstly a pure assemblage system, followed by a queueing system with a single waiting line of complete sets of k pieces. In this paper pure assemblage systems are considered. If such systems have unlimited waiting-room for each line, an equilibrium distribution of queue lenghts never exists. Therefore control measures such as limitation of the waiting-room or partial reduction of production speed are taken and give raise to various optimization problems. With few exceptions, such problems seem to be rather difficult for k > 2 production lines.In this article the specific reasons of the difficulties of multiline systems should be pointed out and the following results should be given: Firstly, an algorithm in matrix form for the calculation of the equilibrium distribution in the case of three production lines; secondly, approximations for the distribution of the number of single parts waiting in front of the assemblage system; this approximations are needed for explicit handling of optimization problems.