# Performance analysis of manufacturing systems : queueing approximations and algorithms

M. Vuuren, van

Research output: ThesisPhd Thesis 1 (Research TU/e / Graduation TU/e)

## Abstract

Performance Analysis of Manufacturing Systems Queueing Approximations and Algorithms This thesis is concerned with the performance analysis of manufacturing systems. Manufacturing is the application of tools and a processing medium to the transformation of raw materials into finished goods for sale. This effort includes all intermediate processes required for the production and integration of a product’s components. For the design or improvement of manufacturing systems it is important to be able to predict their performance. For this purpose, models are being developed and analyzed. Many types of models can be distinguished. The types of models which are most common are simulation models and analytical models. Simulation models represent the events that could occur as a system operates by a sequence of steps in a computer program. The probabilistic nature of many events, such as machine failure and processing times, can be represented by sampling from a distribution representing the pattern of the occurrence of the event. Thus, to represent the typical behavior of the system, it is necessary to run the simulation model for a sufficiently long time, so that all events can occur a sufficiently large number of times. Analytical models describe the system using mathematical or symbolic relationships. These relationships are then used to derive a formula or to define an algorithm by which the performance measures of the system can be evaluated. Often it is not possible, within a reasonable amount of computer time or data storage space, to obtain the performance measure from the relationships describing the system. Further assumptions that modify these relationships then have to be made. The resulting model is then approximate rather than exact, and to validate this approximation, a simulation model may be required. This thesis was carried out in the STW-project EPT, which is a combined effort of groups from the Mechanical Engineering department and the Mathematics and Computer Science department of Eindhoven University of Technology. It aims at developing methods and techniques to analyze manufacturing systems by using the concept of the ’effective process time’ (EPT). The effective process time is the total time a job experiences at a work station; so besides the clean process time it includes all kinds of disturbances. So, it is not needed to explicitly model these disturbances in either a simulation or an analytical model. This makes it easier to collect data for simulation or analytical models and immediately makes the development of more realistic analytical models possible. The Systems Engineering group from the Mechanical Engineering department focuses on the development of simulation models, whereas the Stochastic Operations Research group from the Mathematics and Computer Science department focuses on the development of analytical models. The work in this thesis is an attempt to develop analytical methods to predict the performance of manufacturing systems. The aim is that these methods can be applied to realistic systems, and can be fed by EPT-data, yielding accurate performance predictions. As a result, these methods may serve as an alternative for, or an addition to, discrete-event simulation. To develop these methods we will make use of queueing theory and, in particular, of matrix-analytic methods. These methods, combined with decomposition techniques and aggregation methods, give us the possibility to evaluate relatively large queueing systems very efficiently and accurately. The ultimate goal is to be able to analyze complex networks with different types of nodes within a reasonable time and with a reasonable accuracy. There is still a long way to go before that goal will be reached. In this thesis some steps towards that goal are taken. In Chapter 1 we give an introduction to manufacturing systems and modeling techniques. Also, we give an overview of the types of models that are treated in this thesis, including a literature survey. In Chapter 2 an overview is given of the basics of queueing theory which is used in this thesis. As well as Markovian Arrival Processes, we introduce Quasi Birthand- Death processes and the methods to solve them using matrix-analytic methods. Also, we introduce some algorithms for determining the first two moments of the inter-arrival times of a superposed arrival process and of the maximum of a number of independent random variables. Chapter 3 is motivated by a production system that manufactures the feet of lamps. The model is a queueing system consisting of a number of multi-server stations with generally distributed service times in tandem with finite buffers in between. We develop an approximation using decomposition. Also, tests are performed to check the quality of the approximation. This chapter is concluded with a case study. Production systems with single machine workstations and very small buffers in between the workstations are common in the automotive industry. The results in the previous chapter in this specific case were not good enough. Therefore we handle a single-server tandem queue with small finite buffers in Chapter 4. By modeling the arrivals and departures in more detail compared to the approach of Chapter 3, we improve its results in the case of single server tandem queues. In Chapter 5 we deal with an assembly system. Assembly systems are frequently encountered in production lines in the automotive industry. A number of different parts arrive at queues in front of an assembly server. This assembly server assembles the parts into one product. We decompose the system into a number of subsystems for each part. A wait-to-assembly time is introduced as the time the assembly server has to wait for all parts to be available. The subsystems are solved by using a matrix-analytic method and the characteristics of the wait-to-assembly times are determined by means of an iterative algorithm. Finally, the throughput and mean sojourn times are compared with results from discrete-event simulation to test the quality of the algorithm. Chapter 6 deals with a multi-server queueing system with multiple arrival streams, this is a so-called ??GI/G/c-queue. We analyze it by aggregating the arrival process and the service process in a suitable way. Then, we describe it as a state-dependent Markov process, and solve it by using a matrix-analytic method. In this way we are able to determine an approximation for the complete queue-length distribution. The quality is tested by comparing the mean sojourn time and delay probability against the results of a discrete-event simulation. Chapter 7 is concerned with a priority system. In order-based production environments, like in semi-conductor industry, some types of products are often prioritized over others. Different types of customers arrive at a queueing system. These types each have their own priority. The priority strategy we consider is preemptive resume. Decomposing the queueing system gives a queueing system with vacations for each type of customer. By exploiting the relationship between the subsystems, we develop a method to determine the complete queue-length distribution. Again, the results of the approximation are compared with a discrete-event simulation. In Chapter 8 we present an outlook to the analysis of complete networks consisting of queues we studied in earlier chapters. We focus on two types of networks. First we discuss finitely buffered networks. Next, we consider networks with infinite buffers. Further, we also give some other directions for future research.
Original language English Doctor of Philosophy Mathematics and Computer Science Boxma, Onno J., PromotorRooda, Koos, PromotorAdan, Ivo J.B.F., Copromotor 10 May 2007 Eindhoven Technische Universiteit Eindhoven 978-90-386-0944-7 https://doi.org/10.6100/IR625074 Published - 2007

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