Resonator stability subject to dynamic random-tilt aberration

We derive the behavior of the average exit time (i.e., the number of reflections before escape) of a ray path traveling between two perfect mirrors subject to dynamic random-tilt aberrations. Our calculation is performed in the paraxial approximation. When small random tilts are taken into account, we may consider an asymptotic regime that generically reduces the problem to the study of the exit time from an interval for a harmonic, frictionless oscillator driven by Gaussian white noise. Despite its apparent simplicity, the exact solution of this problem remains an open mathematical challenge, and we propose here a simple approximation scheme. For flat mirrors, the natural frequency of the oscillator vanishes, and, in this case, the average exit time is known exactly. It exhibits a 2/3 scaling-law behavior in terms of the variance of the random tilts. This behavior also follows from our approximation scheme, which establishes the consistency of the scaling law. Our mathematical results are confirmed with simulation experiments.     

Optimal Stopping and Gittins' Indices for Piecewise Deterministic Evolution Processes

Westudy the optimal stopping problem for a class of continuoustime random evolutions described by stochastic differential equationswith alternating renewal processes as noise sources. The exactsolution of this stopping problem provides, in explicit form,an expression for the Gittins' indices needed to derive the optimalscheduling of a class of multi-armed bandit problems in continuoustime. The underlying random processes to which the bandits' armsobey are random velocity models. Such processes are commonlyused to describe, in the fluid limit, the random production flowsdelivered by failure prone machines.     

Hedging Point for Non-Markovian Piecewise Deterministic Production Processes

We consider a single non-markovian failure prone machine which delivers a single product. The operating policy of the machine is chosen to be of the hedging point type. In the infinite horizon limit, we calculate the position of the hedging point that minimizes a convex cost function.     

Fluctuations of the production output of transfer lines

By using a fluid modeling approach, we study the fluctuations around the average throughput delivered by simple production system. A special attention is paid to the buffered production dipole for which an explicit estimation for an stationary variance of the throughput is calculated.     

Optimal threshold control for failure-prone tandem production systems

We consider the flow dynamics of a tandem production system formed by two failure-prone machines separated by a buffer stock. The production rates of the machines are regulated by a feedback mechanism which solves an associated optimal control problem with an average cost criterion. The cost structure penalizes both the entrance into and the sojourn on the buffer boundaries. The generic structure of the optimal control involves four buffer content thresholds. When the buffer content crosses these thresholds, the production rates are tuned to reduce the tendency to enter into the buffer boundaries. Using the fluid modelling framework, we obtain analytical results for the stationary buffer level distribution in the case where an operating machine can produce with, either a “nominal” or a “reduced” rate. In the stationary regime, the optimal positions of the buffer thresholds, the throughput and the average buffer content are presented.     

Optimal hysteresis for a class of deterministic deteriorating two-armed Bandit problem with switching costs

We derive the optimal policy for the dynamic scheduling of a class of deterministic, deteriorating, continuous time and continuous state two-armed Bandit problems with switching costs. Due to the presence of switching costs, the scheduling policy exhibits an hysteretic character. Using this exactly solvable class of models, we are able to explicitly observe the performance of a sub-optimal policy derived from a set of generalized priority indices (generalized Gittins’ indices) similar to those first introduced in a contribution of Asawa and Teneketzis (IEE Trans. Automat. Control 41 (1996) 328).