A tight bound on the throughput of queueing networks with blocking

In this paper, we present a bounding methodology that allows to compute a tight lower bound on the cycle time of fork--join queueing networks with blocking and with general service time distributions. The methodology relies on two ideas. First, probability masses fitting (PMF) discretizes the service time distributions so that the evolution of the modified network can be modelled by a Markov chain. The PMF discretization is simple: the probability masses on regular intervals are computed and aggregated on a single value in the orresponding interval. Second, we take advantage of the concept of critical path, i.e. the sequence of jobs that covers a sample run. We show that the critical path can be computed with the discretized distributions and that the same sequence of jobs offers a lower bound on the original cycle time. The tightness of the bound is shown on computational experiments. Finally, we discuss the extension to split--and--merge networks and approximate estimations of the cycle time.queueing networks, blocking, throughput, bound, probability masses fitting, critical path.

Manufacturing flow line systems: a review of models and analytical results

The most important models and results of the manufacturing flow line literature are described. These include the major classes of models (asynchronous, synchronous, and continuous); the major features (blocking, processing times, failures and repairs); the major properties (conservation of flow, flow rate-idle time, reversibility, and others); and the relationships among different models. Exact and approximate methods for obtaining quantitative measures of performance are also reviewed. The exact methods are appropriate for small systems. The approximate methods, which are the only means available for large systems, are generally based on decomposition, and make use of the exact methods for small systems. Extensions are briefly discussed. Directions for future research are suggested.National Science Foundation (U.S.) (Grant DDM-8914277

Probability masses fitting in the analysis of manufacturing flow lines

A new alternative in the analysis of manufacturing systems with finite buffers is presented. We propose and study a new approach in order to build tractable phase-type distributions, which are required by state-of-the-art analytical models. Called "probability masses fitting" (PMF), the approach is quite simple: the probability masses on regular intervals are computed and aggregated on a single value in the corresponding interval, leading to a discrete distribution. PMF shows some interesting properties: it is bounding, monotonic and it conserves the shape of the distribution. After PMF, from the discrete phase-type distributions, state-of-the-art analytical models can be applied. Here, we choose the exactly model the evolution of the system by a Markov chain, and we focus on flow lines. The properties of the global modelling method can be discovered by extending the PMF properties, mainly leading to bounds on the throughput. Finally, the method is shown, by numerical experiments, to compute accurate estimations of the throughput and of various performance measures, reaching accuracy levels of a few tenths of percent.stochastic modelling, flow lines, probability masses fitting, discretization, bounds, performance measures, distributions.

Nested Fork-Join Queuing Networks and Their Application to Mobility Airfield Operations Analysis

A single-chain nested fork-join queuing network (FJQN) model of mobility airfield ground processing is proposed. In order to analyze the queuing network model, advances on two fronts are made. First, a general technique for decomposing nested FJQNs with probabilistic forks is proposed, which consists of incorporating feedback loops into the embedded Markov chain of the synchronization station, then using Marie\u27s Method to decompose the network. Numerical studies show this strategy to be effective, with less than two percent relative error in the approximate performance measures in most realistic cases. The second contribution is the identification of a quick, efficient method for solving for the stationary probabilities of the λn/Ck/r/N queue. Unpreconditioned Conjugate Gradient Squared is shown to be the method of choice in the context of decomposition using Marie\u27s Method, thus broadening the class of networks where the method is of practical use. The mobility airfield model is analyzed using the strategies described above, and accurate approximations of airfield performance measures are obtained in a fraction of the time needed for a simulation study. The proposed airfield modeling approach is especially effective for quick-look studies and sensitivity analysis

On Fork-Join Queues and Maximum Ratio Cliques

This dissertation consists of two parts. The first part delves into the problem of response time estimation in fork-join queueing networks. These systems have been seen in literature for more than thirty years. The estimation of the mean response time in these systems has been found to be notoriously hard for most forms of these queueing systems. In this work, simple expressions for the mean response time are proposed as conjectures. Extensive experiments demonstrate the remarkable accuracy of these conjectures. Algorithms for the estimation of response time using these conjectures are proposed. For many of the networks studied in this dissertation, no approximations are known in literature for estimation of their response time. Therefore, the contribution of this dissertation in this direction marks significant progress in the analysis of fork-join queues. The second part of this dissertation introduces a fractional version of the classical maximum weight clique problem, the maximum ratio clique problem, which is to find a maximal clique that has the largest ratio of benefit and cost weights associated with the cliques vertices. This problem is formulated to model networks in which the vertices have a benefit as well as a cost associated with them. The maximum ratio clique problem finds applications in a wide range of areas including social networks, stock market graphs and wind farm location. NP-completeness of the decision version of the problem is established, and three solution methods are proposed. The results of numerical experiments with standard graph instances, as well as with real-life instances arising in finance and energy systems, are reported

Efficient buffer design algorithms for production line profit maximization

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 447-465).A production line is a manufacturing system where machines are connected in series and separated by buffers. The inclusion of buffers increases the average production rate of the line by limiting the propagation of disruptions, but at the cost of additional capital investment, floor space of the line, and inventory. Production lines are also a special case of assembly/disassembly systems as well as closed-loop systems. This thesis makes contributions to production system profit maximization. The profit of a production line is the revenue associated with the production rate minus the buffer space cost and average inventory holding cost. We assume that machines have already been chosen and therefore our only decision variables are the buffer sizes and the loop population. The difficulties of the research come from evaluation and optimization. We improve evaluation of loop systems. The optimization problem is hard since both the objective function and the constraints are nonlinear. Our optimization problem, where we consider the nonlinear production rate constraint and average inventory cost, is new. We present an accurate, fast, and reliable algorithm for maximizing profits through buffer space optimization for production lines, and extend the algorithm to closed-loop systems and production lines with an additional maximum part waiting time constraint. A nonlinear programming approach is adopted to solve the optimization problem. Two necessary modifications are proposed to improve the accuracy of the existing loop evaluation method before optimization of loops is studied. An analytical formulation of the part waiting time distribution is developed for two-machine one-buffer lines. It is used in the profit maximization for production lines with both the production rate constraint and the maximum part waiting time constraint. Numerical experiments are provided to show the accuracy and efficiency of the proposed algorithms. Finally, a segmentation method and an additive property of production line optimization are studied. They enable us to optimize very long lines rapidly and accurately.by Chuan Shi.Ph.D

Equivalence, Reversibility, Symmetry and Concavity Properties in Fork/Join Queueing Networks with Blocking

In this paper we study quantitative as well as qualitative properties of Fork/Join Queueing Networks with Blocking (FJQN/B's). Specifically, we prove results regarding the equivalence of the behavior of a FJQN/B and that of its duals and a strongly connected marked graph. In addition, we obtain general conditions that must be satisfied by the service times to guarantee the existence of a long term throughput and its independence on the initial configuration. We also establish conditions under which the reverse of a FJQN/B has the same throughput as the original network. By combining the equivalence result for duals and the reversibility result, we establish a symmetry property for the throughput of a FJQN/B. Last, we establish that the throughput is a concave function of the buffer sizes and the initial marking, provided that the service times are mutually independent random variables belonging to the class of PERT distributions that includes the Erlang distributions. This last result ..

The roles of random boundary conditions in spin systems

Random boundary conditions are one of the simplest realizations of quenched disorder. They have been used as an illustration of various conceptual issues in the theory of disordered spin systems. Here we review some of these result