The Efficiency of Two-stage Series Lines

In thi; paper, we consider the procedure to ,~stimate the various effects of the design factors m the twostage series lines in which the first stage has an arbitrary service time. First, the system states are represented by the imbedded Markov .;hain and the steady state probabilities are solved by the Laplace transform with respect to the arbitrary service time distribution. Next, the efficiency, the idling time distribution, the blocking time distribution and the number of in-process works distribution are considered from the system states. Moreover the relations between the dual models are discussed. 1 . In troduct i on Many papers have been published concerning the two-stage series lines. Hunt and others estimate the efficiencies and the mean number of in-process works for the lines with exponential or erlang service times by the Markov model [6], [7], [8], [ll), [12], [13]. However, most of the papers discuss on1y the efficiencies for the lines with the arbitrary service times by the approximation methods [1], [3], [10]. On the other hand, the queueing system M/G/l with a finite waiting room which is related to the dual model considered in this paper is discussed by Hashida and others, and the various results are presented [4], [5]. In this paper we consider the procedure to estimate the effects of the design factors in the two-stage series lines in which the first stage has an arbitrary service time and the second stage has an exponential service tj~me. First, we show that the system states can be represented by the irnbedded Markov chain like GI/M/l or M/G/l queueing model and that the state probabilities in the steady state condition can be solved by the Laplace transform with respect to the arbitrary service time distribution. And from the state probabilities the efficiency, the idling time distribution, the blocking time distribu:ion and 321 © 1979 The Operations Research Society of Japan