## Kø-netværksmodeller til analyse af FMS anlæg

### Anders Laage Kragh

This work contains a study of how to modelling a FMS (Flexible Manufacturing Systems) as a queueing network. The work has been prompted by the great interest abroad trying to modelling a FMS by a queueing network. At the institute where I prepared my M.S. the interest for simulation were greater than for queueing theory. Therefore this thesis is an attempt to show how to model a FMS as a queueing network.

The first two chapters contain a discussion about flexible manufacturing system and a description of the most important distribution in queueing theory, the exponential distribution. Later it will be shown, that quasi-reversible and symmetric queues are very important in the theory of queueing network. These two queues are defined in chapter 2 and some examples of quasi-reversible and symmetric are illustrated. In chapter 3 and 4 the M/D/1 queue with a limited number of queueing places and the M/D/1 with several kinds of customers are mentioned. For the M/D/1 queue with a limited number of queueing places the state probabilities and the waiting time distribution are determined , which have not been done before. The M/D/ 1 queue with several types of customers, is the most realistic queue for modelling a machine in a FMS. This queue is not one of the known types of queues, which can be a part of a queueing network with product form solution. However, in chapter 4 it is shown, that the M/D/1 queue with several types of customers looks very much like the M/G/1 queue with processor sharing and this gives hope for being able to make some good models with simple solutions.

In chapter 5 the well known models for queueing network with product form solution are described. When one is able to make a model with product form solution, it is possible to have a simple solution to the model. The algorithms for the solution of this models are described in chapter 6. Here the MVA should be mentioned, because the development of this algorithm focus on some other properties of a queueing network. The algorithms based on convolution should be mentioned because they make it possible to model queueing network with blocking and with state dependent routing. This is shown in the thesis.

In a queueing network it is not possible to have some queues to stop servicing because of blocking. Some are claiming that this is a fundamental lack, others, on the other hand, are claiming that it is of no consequence. The truth is probably in between. In chapter 8 are mentioned approximate methods for modelling queueing network with queues which stop to operate because of blocking. In connexion to this, the WIPAC curve is defined and discussed. Based on the WIPAC curve another definition of the FMS as a production system with unbalanced machines and low level of work in process is given. If this definition of FMS is used, the FMS concept will be more usefull and many of the misunderstandings which are limiting the use of queueing network in modelling FMS will be cleared.

At the end of the thesis a realistic FMS is modelled as a queueing network and analysed by the means of the methods mentioned in chapter 6. The results of these analysis are compared to simulations of the same system. The two methods led to almost the same findings. This is due to the great variance on the result of the simulation, which stresses one of the major sources of errors, which are ignored by many when prefering the simulation models to the analytical models.

The conclusion of thesis is that the strict definition of FMS is too restrictive for Danish conditions so that the possibilities of using queueing networks to model Danish FMS are reduced. The more wider definition of FMS includes many more and greater production facilities within the FMS conception. To model these facilities the theory for queueing network would be a good supplement to the simulation of the facilities. A model based on the theory of queueing network would reduce the number of alternative solutions, which has to be investigated by a detail simulation.