Last Buffer First Served Research Articles

Characteristic Curves and Cycle Time Control of Re-Entrant Lines

The characteristic curve (CC) of a production system is a function describing the behavior of cycle time ( CT ) versus throughput ( TP ). In systems with unlimited buffers, this function typically has a knee-type shape. Operating the system below the knee is not efficient, since TP can be increased without an appreciable increase in CT . Operating above the knee is also counterproductive— CT becomes extremely large without a significant increase of TP . Thus, the desirable operating point is at the knee. In order to operate the system at or close to the knee, some knowledge of its CC is necessary. In this paper, an analytical/empirical method for calculating CCs of single-product re-entrant lines is developed using the so-called bottleneck workcenter model of systems at hand. Based on this method, open- and closed-loop raw material release policies, which ensure operating at the desired point of CC, are provided. The development is carried out in terms of the First Buffer First Served (FBFS) and Last Buffer First Served (LBFS) dispatch policies, although CCs for other dispatch policies can be investigated in a similar manner. In addition, it is shown by simulations that the method developed can be used for evaluating CC of a model of a scaled-down wafer fab, which includes multiple workcenters and batching, as long as the severity of the bottleneck workcenter is sufficiently high.

Non-product form equilibrium probabilities in a class of two-station closed reentrant queueing networks

While many single station queues possess explicit forms for their equilibrium probabilities, queueing networks are more problematic. Outside of the class of product form networks (e.g., Jackson, Kelly, and BCMP networks), one must resort to bounds, simulation, asymptotic studies or approximations. By focusing on a class of two-station closed reentrant queueing networks under the last buffer first served (LBFS) policy, we show that non-product form equilibrium probabilities can be obtained. When the number of customer classes in the network is five or fewer, explicit solutions can be obtained. Otherwise, we require the roots of a characteristic polynomial and a matrix inversion that depend only on the network topology. The approach relies on two key points. First, under LBFS, the state space can be reduced to four dimensions independent of the number of buffers in the system. Second, there is a sense of spatial causality in the global balance equations that can then be exploited. To our knowledge, these two-station closed reentrant queueing networks under LBFS represent the first class of queueing networks for which explicit non-product form equilibrium probabilities can be constructed (for five customer classes or less), the generic form of the equilibrium probabilities can be deduced and matrix analytic approaches can be applied. As discussed via example, there may be other networks for which related observations can be exploited.

Equilibria, Stability, and Transients in Re-Entrant Lines Under FBFS and LBFS Dispatch and Constant Release

A model of a re-entrant line, consisting of the bottleneck workcenter and time delays representing other workcenters, is considered. Its mathematical description is provided and performance metrics are introduced. The steady states of this model and their stability properties are investigated under two dispatch policies-first buffer first served (FBFS) and last buffer first served (LBFS)-and under constant release rate. The transients due to machine downtime are also analyzed. It is shown that, although LBFS may be viewed as having superior steady-state characteristics, it induces longer and more volatile transients than FBFS and, in some cases, periodic and chaotic regimes.

Statics and Dynamics of Re-entrant Production Lines

AbstractA model of a re-entrant line, consisting of the bottleneck workcenter and time delays representing other workcenters, is considered. Its mathematical description is provided and performance metrics are introduced. The steady states of this model and their stability properties are investigated under two dispatch policies – First Buffer First Served (FBFS) and Last Buffer First Served (LBFS) – and under constant release rate. The transients due to machine downtime are also analyzed. It is shown that, although LBFS may be viewed as having superior steady state characteristics, it induces longer and more volatile transients than FBFS and, in some cases, periodic and chaotic regimes.

STABILITY OF THE MULTI-TYPE REENTRANT LINES WITH TYPE PRIORITY SERVICE DISCIPLINE

This paper studies the fluid model of a special kind of multi-type queueing network with type priority. The fluid of the same type in the same station is operated under last buffer first served (LBFS) policy. The stability of this kind of fluid model is proved via the Lyapunov function.

Dynamic load balancing with flexible workers

We study the problem of dynamically allocating flexible workers to stations in tandem or serial manufacturing systems. Workers are trained to do a subset of consecutive tasks. We show that the optimal policy is often LBFS (last buffer first-served) or FBFS (first buffer first-served). These results generalize earlier results on the optimality of the pick-and-run, expedite, and bucket brigade-type policies. We also show that, for exponential processing times and general manufacturing networks, the optimal policy will tend to have several workers assigned to the same station.

Dynamic load balancing with flexible workers

We study the problem of dynamically allocating flexible workers to stations in tandem or serial manufacturing systems. Workers are trained to do a subset of consecutive tasks. We show that the optimal policy is often LBFS (last buffer first-served) or FBFS (first buffer first-served). These results generalize earlier results on the optimality of the pick-and-run, expedite, and bucket brigade-type policies. We also show that, for exponential processing times and general manufacturing networks, the optimal policy will tend to have several workers assigned to the same station.

Stability analysis of buffer priority scheduling policies using Petri nets

A Petri net approach to determining the conditions for stability of a re-entrant system with buffer priority scheduling policy is described in this paper. The concept of buffer boundedness based on the dynamic behavior of the markings in the system model is emphasized. The method is used to demonstrate the stability of the first buffer first served (FBFS) and the last buffer first served (LBFS) scheduling policies. Finally a sufficient condition for instability of systems with a positive feedback loop (PFL) is established, and an example is given.

Optimal control of tandem reentrant queues

We consider optimal policies for reentrant queues in which customers may be served several times at the same station. We show that for tandem reentrant queues the last-buffer first-served (LBFS) policy stochastically maximizes the departure process.

The QNET Method for Re-Entrant Queueing Networks with Priority Disciplines

This paper is concerned with the estimation of performance measures of two priority disciplines in a d-station re-entrant queueing network. Such networks arise from complex manufacturing systems such as wafer fabrication facilities. The priority disciplines considered are First-Buffer-First-Served (FBFS) and Last-Buffer-First-Served (LBFS). An analytical method is developed to estimate the long-run average workload at each station and the mean sojourn time in the network. When the first-buffer-first-served discipline is used, a refined estimate of the mean sojourn time is also developed. The workload estimation has two steps. In the first step, following Harrison and Williams (Harrison, J. M., R. J. Williams. 1992. Brownian models of feedforward queueing networks: Quasireversibility and product form solutions. Anns. Appl. Prob. 2 263–293.), we use a d-dimensional reflecting Brownian motion (RBM) to model the workload process. We prove that the RBM exists and is unique in distribution and that it has a unique stationary distribution. We then use an algorithm of Dai and Harrison (Dai, J. G., J. M. Harrison. 1992. Reflected Brownian motion in an orthant: Numerical methods for steady-state analysis. Anns. Appl. Prob. 2 65–86.) to compute the stationary distribution of the RBM. Our method uses both the first and second moment information, and it is rooted in heavy traffic theory. It is closely related to the QNET method of Harrison and Nguyen (Harrison, J. M., V. Nguyen. 1993. Brownian models of multiclass queueing networks: Current status and open problems. Queueing Systems: Theory and Appl. 13 5–40.) for two-moment analysis of First-In-First-Out (FIFO) discipline. Our performance estimates of several example problems are compared to the simulation estimates to illustrate the effectiveness of our method.

Stability and Instability of Fluid Models for Reentrant Lines

Reentrant lines can be used to model complex manufacturing systems such as wafer fabrication facilities. As the first step to the optimal or near-optimal scheduling of such lines, we investigate their stability. In light of a recent theorem of Dai (Dai, J. G. 1995. On positive Harris recurrence of multiclass queueing networks: A unified approach via fluid models. Ann. Appl. Probab. 5 49–77.) which states that a scheduling policy is stable if the corresponding fluid model is stable, we study the stability and instability of fluid models. To do this we utilize piecewise linear Lyapunov functions. We establish stability of First-Buffer-First-Served (FBFS) and Last-Buffer-First-Served (LBFS) disciplines in all reentrant lines, and of all work-conserving disciplines in any three buffer reentrant lines. For the four buffer network of Lu and Kumar we characterize the stability region of the Lu and Kumar policy, and show that it is also the global stability region for this network. We also study stability and instability of Kelly-type networks. In particular, we show that not all work-conserving policies are stable for such networks; however, all work-conserving policies are stable in a ring network.