Mean value analysis of re-entrant line with batch machines and multi-class jobs

Abstract

We propose an approximate approach for estimating the performance measures of the re-entrant line with single-job machines and batch machines based on the mean value analysis (MVA) technique. Multi-class jobs are assumed to be processed in predetermined routings, in which some processes may utilize the same machines in the re-entrant fashion. The performance measures of interest are the steady-state averages of the cycle time of each job class, the queue length of each buffer, and the throughput of the system. The system may not be modeled by a product form queueing network due to the inclusion of the batch machines and the multi-class jobs with different processing times. Thus, we present a methodology for approximately analyzing such a re-entrant line using the iterative procedures based upon the MVA and some heuristic adjustments. Numerical experiments show that the relative errors of the proposed method are within 5% as compared against the simulation results.

Scope and purpose

We consider a re-entrant shop with multi-class jobs, in which jobs may visit some machines more than once at different stages of processing, as observed in the wafer fabrication process of semiconductor manufacturing. The re-entrant line also consists of both the single-job machine and the batch machine. The former refers to the ordinary machine processing one job at a time, and the latter means the machine processing several jobs together as a batch at a time. In this paper, we propose an approximation method based on the mean value analysis for estimating the mean cycle time of each class of jobs, the mean queue length of each buffer, and the throughput of the system.

Introduction

In this paper, we propose an approximate approach for analyzing the performance of the re-entrant line with batch machines and multi-class jobs using the mean value analysis (MVA) technique. In the re-entrant line, jobs visit same machines more than once at different stages of processing. The system also consists of both the single-job machine and the batch machine. The former refers to the ordinary machine processing one job at a time, and the latter means the machine processing several jobs together as a batch at a time. A well-known example of the re-entrant manufacturing systems is the semiconductor manufacturing process, especially wafer fabrication line (fab). The fab process is also characterized by the usage of the batch machines. For example, the deposition process and the furnaces accumulate jobs in the queue and then process jobs together as a batch of predetermined size. Mixture of these batch machines and single-job machines complicates the exact analysis of the system, while the flow of the multi-class jobs also requires elaborative model for analysis.

Unfortunately, such a system does not lend itself to be modeled as a product-form queueing network, and thus it is often analyzed using simulation in practice. However, the simulation of a large-scale system, such as a fab line producing the commodity semiconductor devices, often requires excessive time for modeling and running it. Managers and engineers who are interested in finding a quick answer for the system performance under certain changes in product mix or system configuration often need an alternative method. In lieu of the simulation, we present an approximate MVA approach to analyze the system based on the works of Narahari and Khan [1], [2] and Park et al. [3]. Narahari and Khan [1] first proposed the application of MVA to the re-entrant line with a single-class job and single-job machines. Park et al. [3] extended the approach and solved the case of a single-class job and mixture of both single-job and batch machines. We extend their work to consider the case of the multi-class jobs with both single-job and batch machines. The detailed description of the problem is to be given in the following section.

Re-entrant lines are explained in Kumar [4] in detail, and the research on scheduling problems and stability of re-entrant lines with single class of jobs are presented in Lu and Kumar [5], Kumar [6], Lu et al. [7], Kumar [8], Kumar and Meyn [9] and Dai and Weiss [10]. Connors et al. [11] proposed an approximation queueing model for semiconductor manufacturing systems with re-entrant characteristics and analyzed using decomposition-based approximation approach. Dai et al. [12] proposed the QNET method to find the performance measures of re-entrant lines with single class. Chaudhry and Templeton [13] summarized the general theory concerning batch arrivals and batch service queues, and Neuts [14], Powell [15] studied several rules and control policies of batch processing. Even for the case of single-class jobs, re-entrant flow with batch machines makes the system impossible to be modeled as a product-form queueing network. Extending the case to include the multi-class jobs further complicates the model, and thus we propose an approximate analysis technique in this paper. We assume the system is a closed network, in which the total work-in-process is kept constant.

For a quick review of the cases for multi-class jobs, the following cases were modeled as the closed queueing networks. Conway et al. [16] proposed the mean value analysis by chain (MVAC) for closed multi-chain product form queueing networks, which is a modification of the recursion by chain (RECAL) developed by Conway and Georganas [17]. Zhuang and Hindi [18] analyzed the FMS with limited buffers and multi-class jobs using MVA for closed queueing network models. Their system has the material handling system (MHS) and workstations with identical machines. Kim et al. [19] proposed the MVA approach for the flexible manufacturing system (FMS) with repeated visit. Pentakalos et al. [20] presented a queueing network model that can be used to carry out capacity planning studies of hierarchical mass storage systems and developed the performance model based on approximations to multiclass MVA of queueing networks. Majumdar and Woodside [21] provided an exact analysis for bounds for multiclass systems with widely differing service times.

In the following section, the problem and the assumptions are explained with an example. The proposed approach is described in Section 2. In Section 2.1, we first explain the method of obtaining the average waiting time of jobs at each buffer of the workstations. Section 2.2 presents the procedure for computing the mean cycle time and the other measures of the system. To enhance the performance of the proposed MVA method, we add some heuristic adjustment methods in Section 2.3. Section 3 presents the numerical test results obtained by comparing the results from the proposed approach and those of the simulation experiments for some sample cases. Finally, we give the summary of the paper and discussions of the further research issues.