The Skorokhod problem in a time-dependent interval

We consider the Skorokhod problem in a time-varying interval. We prove existence and uniqueness of the solution. We also express the solution in terms of an explicit formula. Moving boundaries may generate singularities when they touch. Under the assumption that the first time τ$\tau$ when the moving boundaries touch after time zero is strictly positive, we derive two sets of conditions on the moving boundaries. We show that the variation of the local time of the associated reflected Brownian motion on [0,τ]$[0,\tau ]$ is finite under the first set of conditions and infinite under the second set of conditions. We also apply these results to study the semimartingale property of a class of two-dimensional reflected Brownian motions.     

A Multiclass Feedback Queueing Network with a Regular Skorokhod Problem

We consider a four-class two-station network with feedback, with fluid inputs and a head-of-the-line generalized processor sharing discipline at each station. We derive the Skorokhod Problem associated with the network and obtain algebraic sufficient conditions for Lipschitz continuity of the associated Skorokhod Map. This provides the first example of a multiclass network with feedback for which the associated Skorokhod Problem has been proved to be regular. As an elementary application, we show that under the conditions which guarantee Lipschitz continuity the network is stable if and only if the usual load conditions apply.     

Chain conformation of rod-like polymers in the melt: Small-angle neutron scattering of poly(benzoyl paraphenylene)

The chain conformation of a rigid rod polymer, poly(benzoyl paraphenylene), is determined in the melt using small-angle neutron scattering. The coherent scattering cross-section from blends of partially deuterated and hydrogenous poly(benzoyl paraphenylene) agree well with ideal rod scattering for q > 0.02 Å⁻¹, indicating that the polymer chains are highly extended. Comparison of the results to a single chain-scattering function for chains of arbitrary stiffness yield a persistence length of ca. 130 Å. Pure component scattering at the lowest scattering lengths indicate that the melt is not molecularly homogenous, but is comprised of domains, potentially reflecting localized groupings of chains with similar backbone orientation. Furthermore, this mesoscopic structure depends on the processing history of the polymer in the melt state. © 1998 John Wiley & Sons, Inc. J. Polym. Sci. B Polym. Phys. 36: 2449–2459, 1998