A Numerical Analysis of the M D b N Queueing System

In this paper we provide a numerical analysis of the M D N queueing system This system consists of an in nite bu er and N identical batch servers with batch capacity b It serves as a model for various types of workstations in the manufacturing context Using an approach presented by Crommelin we derive the distribution of the number of lots waiting in queue at arbitrary time instants and the distribution of the waiting time of lots The results are of an exact nature and the computation is of low complexity concerning run time and memory usage Introduction Batch servers or bulk servers can be found in many areas of manufacturing For example in the manufacturing of semiconductors lots of wafers are usually grouped together into batches at workstations like oxidation ovens or chemical deposition Furthermore identical servers are commonly grouped together as workstations All servers at a workstation are fed by a single queue In this way the idle times of the servers are minimized Analytical approaches for single batch servers are well known Neuts presented an analytical study of batch service systems that operate according to a minimum batch size rule Chaudhry and Templeton provide an extensive discussion of bulk service systems and their analysis In and batch service systems in push and pull manufacturing environments are analyzed using embedded Markov chain techniques The method of analysis proposed in this paper follows an approach that Crommelin used to derive the distribution of the waiting time in an M D N system His method has been adapted to the discrete time equivalent Geo D N system by Vicari and Tran Gia This paper is structured as follows Section provides a description of the queueing systemM D N In Section we propose a method of analysis for this system and derive performance measures of interest We present some numerical results in Section and summarize the paper in Section The M D N queueing model We consider the queueing system depicted in Figure It consists of a queue of in nite length and N identical batch servers Each of the servers has a batch capacity of b