A Model for the Dynamics of large Queuing Networks and Supply Chains [abstract] [pdf]
Kinetic and fluid model hierarchies for supply chains SIAM Multiscale Model. Simul. 2(1), pp 43-61 2004 [abstract] [ps]
Thermalized kinetic and fluid models for re-entrant supply chains to appear in SIAM Multiscale Model. 2005 [abstract] [pdf]
A continuum model for a re-entrant factory [abstract] [pdf]
A Mesoscopic Approach to the Simulation of Semiconductor Supply Chains [ abstract] [ps] [pdf]
Control and Synchronization in Switched Arrival Systems [abstract] [pdf]
Bucket Brigades with Worker Learning submitted to European Journal of Operations Research, 2004 [ abstract] [pdf]
Bucket Brigades when Worker Speeds do not Dominate Each Other Uniformly to appear in European Journal of Operations Research, 2005 [ abstract] [ps] [pdf]
On the stability of re-entrant manufacturing systems [abstract] [html]
Periodic orbits in a class of re-entrant manufacturing systems [abstract] [ps]
Localized Solutions in Parametrically Driven Pattern Formation [abstract] [ps]
Parametrically forced pattern formation [abstract] [ps] [pdf]
Dynamics of polar reversals in spherical dynamos [ abstract] [ps] [pdf]
Does synchronization of networks of chaotic maps lead to control? to appear in Chaos, 2005 [abstract] [pdf]
Perturbed on-off intermittency [ abstract] [ps] [pdf]
Noisy Heteroclinic networks [ abstract] [pdf]
Noise and O(1) Amplitude Effects on Heteroclinic Cycles [abstract]
Analyzing the dynamics of cellular flames [abstract]
Symmetry and the Karhunen Loéve analysis [abstract]
Keywords: Supply chains, conservation laws, asymptotics.
Standard stochastic models for supply chains predict the throughput time (TPT) of a part from a statistical distribution, which is dependent on the Work in Progress (WIP) at the time the part enters the system. So, they try to predict a transient response from data which are sampled in a quasi steady state situation. For re- entrant supply chains this prediction is based on insufficient information, since subsequent arrivals can dramatically change the TPT. This paper extends these standard models, by introducing the concept of a stochastic phase velocity which dynamically updates the TPT estimate. This leads to the concepts of temperature and diffusion in the corresponding kinetic and fluid models for supply chains.
Keywords: Re-entrant supply chains, Traffic flow models, Boltzmann equation, Chapman-Enskog, Fluid limits.
We consider networks of chaotic maps with different network topologies. In each case, they are coupled in such a way as to generate synchronized chaotic solutions. By using the methods of control of chaos we are controlling a single map into a predetermined trajectory. We analyze the reaction of the network to such a control. Specifically we show that a line of 1-d logistic maps that are unidirectionally coupled can be controlled from the first oscillator whereas a ring of diffusively coupled maps cannot be controlled for more than 5 maps. We show that rings with more elements can be controlled if every third map is controlled. The dependence of unidirectionally coupled maps on noise is studied. The noise level leads to a finite synchronization lengths for which maps can be controlled by a single location. A 2-d lattice is also studied.
The dynamics of structurally stable heteroclinic cycles connecting fixed points with one dimensional unstable manifolds under the influence of noise is analyzed. Fokker Planck equations for the evolution of the probability distribution of trajectories near heteroclinc cycles are solved. The influence of the magnitude of the stable and unstable eigenvalues at the fixed points and of the amplitude of the added noise on the location and shape of the probability distribution is determined. As a consequence, the jumping of solution trajectories in and out of invariant subspaces of the deterministic system can be explained.
Keywords: dynamical systems, heteroclinic cycles, noise-induced dynamics, intermittency.
The influence of small noise on the dynamics of heteroclinic networks is studied, with a particular focus on noise-induced switching between cycles in the network. Three different types of switching are found, depending on the details of the underlying deterministic dynamics: random switching between the heteroclinic cycles determined by the linear dynamics near one of the saddle points, noise induced stability of a cycle, and intermittent switching between cycles. All three responses are explained by examining the size of the stable and unstable eigenvalues at the equilibria.
Keywords: dynamical systems, heteroclinic cycles, noise-induced dynamics, intermittency.
Video data from experiments on the dynamics of two dimensional flames are analyzed. The Karhunen Loéve (KL-) analysis is used to identify the dominant spatial structures and their temporal evolution for several dynamical regimes of the flames. A data analysis procedure to extract and process the boundaries of flame cells is described. It is shown how certain spatial structures are associated with certain temporal events. The existence of small scale, high frequency, turbulent background motion in almost all regimes is revealed.
The Karhunen Loéve (K-L) analysis is widely used to generate low dimensional dynamical systems, which have the same low dimensional attractors as some large scale simulations of PDEs. If the PDE is symmetric with respect to a symmetry group G, the dynamical system has to be equivariant under G to capture the full phase space. It is shown that symmetrizing the K-L eigenmodes instead of symmetrizing the data leads to considerable computational savings, if the K-L analysis is done in the snapshot method. The feasability of the approach is demonstrated with an analysis of Kolmogorov flow.
A chaotic model of a production flow called the switched arrival system is extended to include switching times and maintenance. The probability distribution of the chaotic return times is calculated. Scheduling maintenance, loss of production due to switching and control of the chaotic dynamics is discussed. A coupling parameter to couple switched arrival systems serially, based on lost production, is identified. Simulations of three parallel and three serial levels were performed. Global synchronization of the switching frequencies between serial levels is achieved. An analytic model allows to predict the self balancing properties of the serial system.
A two worker bucket brigade is studied where one worker has a constant speed over the whole production line and the other is slower over the first portion and faster over the second portion of the line. We analyze the dynamics and throughput of the bucket brigade under two different assumptions: (i) workers can pass each other, and (ii) workers are blocked when an upstream worker runs into a downstream worker. We show that a slight modification of the bucket brigade will always lead to a self organizing production line. The bucket brigade either balances to a fixed point or settles into a period-two orbit. Insights for the management of the bucket brigades for the various scenarios are discussed using results on the throughput performance and self-organization. Extensions to multiple skill levels and more workers are outlined.
The dynamics and throughput of a bucket brigade production system is studied when workers' speeds increase due to learning. It is shown that, if the rules of the bucket brigade system allow a re-ordering of its workers then the bucket brigade production system is very robust and will typically rebalance to a self-organizing optimal production arrangement. As workers learn only those parts of the production line that they work on, the stationary velocity distribution for the workers of a stable bucket brigade is non-uniform over the production line. Hence, depending on the initial placement of the workers, there are many different stationary velocity distributions. It is shown that all the stationary distributions lead to the same throughput.
Production flows through factories are modeled through conservation laws leading to nonlinear hyperbolic PDEs. For a linear production line, models based on conservation laws can be derived from first principles, using me thods from gas dynamics. For re-entrant manufacturing a heuristic model is presented merging a nonlocal state equation relating TPT to WIP through Little's law to produce a nonlinear nonlocal hyperbolic PDE. These two models can serve as the building blocks of fast simulations of the dynamics of capacity limited supply chains. We present simulations for a chain consisting of a re-entrant module, followed and preceded by a linear module. The response of the system to various production scenarios is discussed.
Pattern formation in a nonlinear damped Mathieu-type partial differential equation defined on one space variable is analyzed. A bifurcation analysis of an averaged equation is performed and compared to full numerical simulations. Parametric resonance leads to periodically varying patterns whose spatial structure is determined by amplitude and detuning of the periodic forcing. At onset, patterns appear subcritically and attractor crowding is observed for large detuning. The evolution of patterns under the increase of the forcing amplitude is studied. It is found that spatially homogeneous and temporally periodic solutions occur for all detuning at a certain amplitude of the forcing. Although the system is dissipative, spatial solitons are found representing domain walls creating a phase jump of the solutions. Qualitative comparisons with experiments in vertically vibrating granular media are made.
Structurally stable heteroclinic cycles (SSHC) are proposed as the mathematical structures that are responsible for the reversals of dipolar magnetic fields in spherical dynamos. The existence of SSHCs involving dipolar magnetic fields generated by convection in a spherical shell for a nonrotating sphere is rigorously proven. The possibility of SSHCs in a rotating shell is proposed and their existence in a low dimensional model of the magnetohydrodynamic equations is numerically confirmed. The resulting magnetic time series shows a dipolar magnetic field, aligned with the rotation axis that intermittently becomes unstable, changes the polar axis, starts rotating, disappears completely and eventually reestablishes itself in its original or opposite direction, chosen randomly.
A basic requirement for on-off intermittency to occur is that the system possesses an invariant subspace. We address how on-off intermittency manifests itself when a perturbation destroys the invariant subspace. In particular, we distinguish between situations where the threshold for measuring the on-off intermittency in numerical or physical experiments is much larger than or is comparable to the size of the perturbation. Our principal result is that as the perturbation parameter increases from zero, a metamorphosis in on-off intermittency occurs in the sense that scaling laws associated with physically measureable quantities change abruptly. A geometric analysis, a random-walk model, and numerical trials support the result.
This paper considers a discrete event model of reentrant manufacturing systems, forgoing the usual Poisson process/stochastic queuing theory set-up in favor of deterministic dynamical systems theory. On the more theoretical side, results obtained include a restrictive characterization of the class of dynamical systems arising in this context, and a detailed description of the variety of orbit behaviors which can arise in the case of two-machine reentrant systems. Contrary to a conjecture of Beaumarriage and Kempf, chaotic behavior does not occur
Queue changes associated with each step of a manufacturing system are modeled by constant vector fields (fluid model of a queueing network). Observing these vector fields at fixed events reduces them to a set of piecewise linear maps. It is proved that these maps show only periodic or eventually periodic orbits. An algorithm to determine the period of the orbits is presented. The dependence of the period on the processing rates is shown for a 3(4)-step, 2-machine problem.
The Mathieu partial differential equation is analyzed as a prototypical model for pattern formation due to parametric resonance. After averaging and scaling it is shown to be a perturbed Nonlinear Schr\"{o}dinger Equation. Adiabatic perturbation theory for solitons is applied to determine which solitons of the NLS survive the perturbation due to damping and parametric forcing. Numerical simulations compare the perturbation results to the dynamics of the Mathieu PDE. Stable and weakly unstable soliton solutions are identified. They are shown to be closely related to oscillons found in parametrically driven sand experiments.
High volume, multi stage continuous production flow through a re-entrant factory is modeled through a conservation law for a continous density variable on a continuous production line augmented by a state equation for the speed of the production along the production line. The resulting nonlinear nonlocal hyperbolic conservation law allows fast and accurate simulations. Little's law is built into the model. It is argued that the state equation for a re-entrant factory should be nonlinear. It is shown that for any nonlinear state equation the model has two steady states of production below a critical start rate: A high volume - high cycle time state and a low volume - low cycle time state. The stability of the low volume state is proved. Output is controlled by adjusting the start rate to a changed demand rate. Two linear factories and a re-entrant factory each one modeled by a hyperbolic conservation law are linked to provide proof of concept for efficient supply chain simulations. Instantaneous density and flux through the supply chain as well as Work In Progress (WIP) and output as a function of time are presented. Extensions to include multiple product flows and preference rules for products and dispatch rules for re-entrant choices are discussed.
We present a model hierarchy for queuing networks and supply chains, analogous to the hierarchy leading from the many body problem to the equations of gas dynamics. Various possible mean field models for the interaction of individual parts in the chain are presented. For the case of linearly ordered queues the mean filed models and fluid approximations are verified numerically.