International Journal of Operations Research International Journal of Operations Research Vol. 7, No. 2, 1-18 (2010)
Matrix Geometric Analysis of Throughput Processes of Queues with Applications to A Pull Serial Line Hsing Luh Department of Mathematical Sciences, National Chengchi University Taipei 106, Taiwan, R.O.C.
Received April 2010; Revised May 2010; Accepted May 2010
Abstract In this paper, we focus on the behavior of a queue in a pull serial line at a throughput process under correlated demands. In order to compute the performance measures of the throughput process, we propose a numeric model and an algorithm which is an extension of the matrix geometric analysis method. By constructing a recursive procedure for calculating the joint distribution of an arbitrary number of consecutive interdeparture times in a PH/G/1/K queue, we obtain explicitly the covariance of nonadjacent interdeparture times, and display the correlation coefficients that reveal the long-range dependence. It confirms some structure properties and produces numerical examples for the lag-n autocorrelation of interdeparture times for several different demand distributions, exhibiting both positive and negative autocorrelation.
Keywords PH/G/1 Queue, matrix geometric solutions, covariance, throughput processes. 1.
INTRODUCTION
In recent years, there has been considerable interest in the study and analysis of pull systems. Analytical solutions exist almost exclusively for the pull serial lines with deterministic or exponentially distributed times. Bardinelli (1992) studied continuous review policies and Gershwin (1987) derived a method for decomposing a tandem queue. Otherwise, it may be studied with simulation approaches. For instances, Aytug and Dogan (1998) used a simulation generator to evaluate more complex systems. In order to study a pull serial line analytically, modelling and characterizing the output process of a line is an essential step. This is because that an output process will become an input process of the next station in the line. In addition, characterization of the output process allows an evaluation of the smoothing effect of varied demands and operations. The main problem when using the output process of the previous station as an input to the next station is that after a few operations at stations the resulting process becomes very complicated and hence untractable. In this paper, researches are focused on both performance analysis and structure properties under the ergodicity. Among them, a very important property is the autocorrelation of the output process of production, i.e., the correlation between interdeparture times since most of performance measures are functionally related to it. To imitate a pull serial line of production, consider a generic model of kanban systems in manufacturing where a pull production line that consists of a serial of stations. Each station is composed of an inbound node, a bulletin board and an outbound node. Assume external demand arrives only at the last station in a single unit and there is a single type of items produced in each station. Source of materials is unlimited and demands that cannot be filled immediately are backlogged. Jobs flow through the stations in sequence and one operation is performed at each station in which there is one machine or a server. Both the lot size and the batch size is one. The service time in each station and the interarrival time of external demand are assumed to be independent identically distributed. There is no transit time for the movement of items between stations, namely, no scrap or defection, and no down time. In a station, both an inbound node and a bulletin board can be described in a single server queueing model. Thus, each station in the pull serial line consists of three queues, i.e., Q1 , Q 2 and Q 3 as shown in Figure 1. A specified station owns a finite number of cards (kanbans) which are collected in Q 3 . If there is at least one card in Q 3 and at least one job in the Q1 , then one of these jobs is moved to Q 2 . Here, that job is tied with one card from Q 3 . The pair
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2 Luh: Matrix Geometric Analysis of Throughput Processes of Queues with Applications to A Pull Serial Line IJOR Vol. 7, No. 2, 1−18 (2010)
enters the buffer at Q 2 to wait for operation. Namely, if the server is busy, the pair has to wait; otherwise a process starts. After processing the pair is separated and the card is returned to Q 2 and the job is moved to next station. From the description above, it is easily formulated at each station as a single server queue with general service and interarrival times. The detail of modelling is given in Section 2. For an M/G/1 queue, Conolly (1975) gave the joint distribution for two consecutive interdeparture times D0 and D1 , from which the covariance Cov( D0 , D1 ) is derived. Daley (1968) derived the generating function of the sum of covariance for the sequence {Cov( D0 , Dn ) ; n 1, 2, . . .} , where Dn is the nth lag of interdeparture time after D0. Jenkins (1966) analyzed the correlation of consecutive interdeparture times for an M/Em/1 queue, where Em denotes an Erlang distribution with m phases, indicating the evaluation for n > 2 by his method rapidly becomes unwieldy. Daley and Shanbhag (1975) and Ishikawa (1991) have studied the correlation structure of the M/G/1/K queue. King (1975) also had an investigation on the covariance structure of the output process. But their results were only limited on the case where the arrival process is Poisson. Zhang et al. (2005) derived the departure process of a BMAP/MAP/1 queue. Lim et al. (2006) studied a departure process of a single server queueing system with Markov renewal input. Most recently, Kempaa (2010) addressed results for departure process in the MX/G/1 queueing system with a single vacation and exhaustive service. There is little discussion on the covariance structure of the departure. In the paper, we will construct a procedure for calculating the joint distribution for the arbitrary number of consecutive interdeparture times D0 , D1 , , Dn -1 for a general arrival process. Our paper provides a different approach by taking advantage of a recursive structure in the set of interdeparture times, implying the correlation coefficient of the departure. The rest of the paper is organized as follows. In Section 2, we introduce notation and give as a preliminary a set of equations for calculating the queue size distribution at departure times. In Section 3, we present a procedure for calculating the joint distribution of n consecutive interdeparture times from that of n−1 consecutive interdeparture times. In Section 4, we show the explicit expressions for the covariance of nonadjacent interdeparture times in PH/G/1 queue. We also display some numerical results for covariance, and give a number of few remarks. Section 5 concludes the paper. 2.
Queue-length at departure instants
Notations Consider a PH/G/1/K queue with the First-Come-First-Served discipline. The arrival process follows a stochastic stream having the average arrival rate
