Cycle Time Approximations for the G/G/m Queue Subject to Server Failures and Cycle Time Offsets with Applications

Abstract

Approximate queueing formulae are often employed for the practical evaluation of manufacturing system performance. I. INTRODUCTIONQueueing, or waiting in line for an available server, is a common phenomenon in many industries including computing, manufacturing and customer service and has been studied for perhaps 100 years (Erlang published his first paper on queueing theory in 1909, [1]). Simple closed form approximation formulae for the mean cycle time behavior of the G/G/m queue (potentially subject to server failures) have been developed and employed over the course of decades to predict, understand and plan for manufacturing system behavior, see as a start [2,3,4] and the references contained therein. Rough cut approximations have been employed, as in [5], in the performance evaluation of networks of queues by the use of approximate coupling formulae intended to capture the interaction between the arrival and departure processes within the network. Approximations are often used in lieu of exact numerical procedures or performance bounds. The intuitive value of simple expressions for system performance coupled with their ease of use has resulted in their ascendance as the primary tool for understanding cycle time behavior in IBM's 200mm semiconductor wafer fabricator. Much work has been devoted to successfully bridging the gap between measures of actual system performance and the predictions obtained from the M/D/1 (or the more generic G/G/m) queue [6,7,8,9,10,11].

Performance evaluation of single and multi-class production systems using an approximating queuing network

“…An accurate prediction of the cycle time distribution as a function of the throughput is then essential. A prediction model has to incorporate the workstation behavior such as integrated processing (i.e., processing multiple products at the same time in the various process clusters/chambers), predicted/unpredicted breakdowns and dispatching rules [1], [2]. It is also desired that the proposed prediction model requires little development and maintenance effort and the model evaluations are computationally cheap [2], [3].…”

“…For predicting the cycle time in high-tech manufacturing, there are two common categories of models including 1) "Simple" analytical models (i.e., models that require only a few easily-used and easily-estimated parameters) and 2) (discrete-event) Simulation models. Commonly-used simple analytical models are closed form G/G/m queueing models [2]- [12]. Despite the usefulness and easiness of queueing models, they suffer from the lack of accuracy for complex workstations/equipment.…”

“…Despite the usefulness and easiness of queueing models, they suffer from the lack of accuracy for complex workstations/equipment. However, some modifications have been proposed by Morrison and Martin [2], [7], [8] for predicting the cycle time of cluster tools in semiconductor manufacturing. Simulation models, on the other hand, are alternatively used to accurately represent the processing of workstations [13]- [15].…”

“…Another approach to providing an abstraction of a detailed simulation model is aggregation [2]. Aggregation can be done by simplifying the complex components/assumptions of the system as [20], and [21] replaced non-bottleneck workstations by a constant delay, but they did not use the simplified model for cycle time distribution prediction.…”

“…The flow line models of [29] have been only proposed for a single-class workstation/equipment while considering multiple products needs to reconstruct the models and take into account the inter-product correlations. In another work, Morrison and Martin [2] considered the processing workstations as general G/G/m queues and proposed a closed deterministic formulation for approximating the cycle time of the workstations subject to server failures and cycle time offsets (e.g., events such as travel and hold upstream of a workstation). The authors then developed flow line models for clustered photolithography tools and conducted simulations to assess the quality of the models.…”