Triggering Scheduler

In this paper we present the performance analysis of two nite queues coupled by a triggering scheduler Customers waiting for service have rst to pass the scheduler and then experience service by their dedicated server The scheduler is permitted to gather a custo mer from a particular queue only if the corresponding server is idle An application of this model can be found in job shop like manufacturing systems where parts have to be xed before being processed We derive the steady state probabilities of the corresponding continuous time Mar kov chain and present numerical results for major system characteristics In order to stress the behaviour of the system various load situations are considered Introduction In the recent years a great change in the design and the analysis of manufacturing systems has appeared Since modern mass production systems are quite complex simulation models yield a limited insight into the behaviour of such systems For that reason and considering the large range of system parameters simulation does not appear as an adequate method for designing and controlling modern manufacturing systems In a number of studies queueing theory has been applied to investigate manu facturing systems Queueing models are often superior to simulations since they provide an insight to the interaction of the multiple factors determining the per formance of manufacturing systems A comprehensive survey can be found in the special issue Queueing Models in Manufacturing Systems of the journal Queueing Systems presenting queueing theory applied to manufacturing systems There attention has been devoted to various types of manufacturing systems including manufacturing ow lines and transfer lines job shop like ma nufacturing systems and exible machining systems Traditional manufacturing systems are organized as job shops or ow lines see In job shops a variety of products is manufactured on machines of di erent types whereby machines performing similar operations are grouped together into machine centers Thereby some operations on products of a par ticular type can be performed by the same machine In contrast to ow lines This paper was published in Walke B and Spaniol O Messung Modellierung und Bewertung von Rechen und Kommunikationssytemen ITG GI Fachtagung Aachen September Verlag der Augustinus Buchhandlung Aachen ISBN which the jobs are processed by the machines has not to be the same for all types of products The existing queueing models applied to job shops are based on the assump tion that a job or a customer requires only one processing unit server at a time A survey of general stochastic models applied to job shops can be found in the book of Buzacott and Shanthikumar and a review of open queueing network models in manufacturing systems can be found in the publication of Bitran and Dasu In this paper we consider a machine center of a job shop consisting of di erent machines and an operator Every machine is processing parts of a single type and a bu er of nite capacity is maintained for every type of parts Before parts can be processed by the machines they have to be loaded by the operator rst The operator is permitted to gather a part of a particu lar type only if the corresponding machine is idle Thus a part waiting for for service needs two processing resources the operator and its dedicated machine Notice that this model can also be applied to investigate the performance of a central server computer system where the operator corresponds to the central server and the machines represent peripherical devices We evaluate the performance of such a system in order to stress the impact of the arrival and service times on system characteristics such as waiting times queue lengths and machine and operator utilizations The paper is organized as follows In Section the basic model is described in detail whereas in Section we present the performance analysis and de ne the system characteristics such as mean waiting times Numerical results are discussed in Section Model description The structure of the basic queueing model with two di erent types of customers is depicted in Fig Customers of type arrive at queue Q and those of type at queue Q The arrival processes are assumed to be Poisson with intensity i i Both customer types wait for being served by the triggering serverO After service completion by the scheduler a customer of type i is instantaneously passed to server Mi where it will be served before leaving the system In the context of a job shop like manufacturing system one can think of the triggering server O as an operator and of the servers Mi as machines respectively The customers are parts xed by the operator onto machines for processing Both the operator and the machines can process only one part at a time A machine can only be scheduled in the case of being idle and the operator nds at least one customer in the corresponding queue The service times of the operator and service times of the machines are assumed to be exponentially distributed with mean and i respectively The arrival processes and both service processes are supposed to be independent of each other Further the queues are of nite capacity where the capacity of queue Qi is denoted by Si Let us now describe how the operator proceeds starting with an empty system the queues are empty and the machines are idle When for example a customer