Comparisons between observable and unobservable M/M/1 queues with respect to optimal customer behavior

We consider an M/M/1 queueing system in which the queue length may or may not be observable by a customer upon entering the system. The “observable” and “unobservable” models are compared with respect to system properties and performance measures under two different types of optimal customer behavior, which we refer to as “selfishly optimal” and “socially optimal”. We consider average customer throughput rates and show that, under both types of optimal customer behavior, the equality of effective queue-joining rates between the observable and unobservable systems results in differences with respect to other performance measures such as mean busy periods and waiting times. We also show that the equality of selfishly optimal queue-joining rates between the two types of system precludes the equality of socially optimal joining rates, and vice versa.     

Arithmetic partial differential equations, II

We continue the study of arithmetic partial differential equations initiated in [7] by classifying “arithmetic convection equations” on modular curves, and by describing their space of solutions. Certain of these solutions involve the Fourier expansions of the Eisenstein modular forms of weight 4 and 6, while others involve the Serre-Tate expansions (Mori, 1995 [13], Buium, 2003 [4]) of the same modular forms; in this sense, our arithmetic convection equations can be seen as “unifying” the two types of expansions. The theory can be generalized to one of “arithmetic heat equations” on modular curves, but we prove that they do not carry “arithmetic wave equations.” Finally, we prove an instability result for families of arithmetic heat equations converging to an arithmetic convection equation.     

On asymptotic behavior of solutions to Korteweg-de Vries type equations related to vortex filament with axial flow

We study the global existence and asymptotic behavior in time of solutions to the Korteweg-de Vries type equation called as “Hirota” equation. This equation is a mixture of cubic nonlinear Schrödinger equation and modified Korteweg-de Vries equation. We show the unique existence of the solution for this equation which tends to the given “modified” free profile by using the two asymptotic formulae for some oscillatory integrals.     

Heteroclinic and Periodic Cycles in a Perturbed Convection Model

Vivancos and Minzoni (New Choatic behaviour in a singularly perturbed model, preprint) proposed a singularly perturbed rotating convection system to model the Earth's dynamo process. Numerical simulation shows that the perturbed system is rich in chaotic and periodic solutions. In this paper, we show that if the perturbation is sufficiently small, the system can only have simple heteroclinic solutions and two types of periodic solutions near the simple heteroclinic solutions. One looks like a figure “Delta” and the other looks like a figure “Eight”. Due to the fast - slow characteristic of the system, the reduced slow system has a relay nonlinearity (“Asymptotic Method in Singularly Perturbed Systems,” Consultants Bureau, New York and London, 1994) - solutions to the slow system are continuous but their derivative changes abruptly at certain junction surfaces. We develop new types of Melnikov integral and Lyapunov-Schmidt reduction methods which are suitable to study heteroclinic and periodic solutions for systems with relay nonlinearity.     

Conservative allocation models for outbound containers in container terminals

This paper examines location assignment for outbound containers in container terminals. It is an extension to the previous modeling work of Kim et al. (2000) and Zhang et al. (2010). The previous model was an “optimistic” handling way and gave a moderate punishment for placing a lighter container onto the top of a stack already loaded with heavier containers. Considering that the original model neglected the stack height and the state-changing magnitude information when interpreting the punishment parameter and hid too much information about the specific configurations for a given stack representation, we propose two new “conservative” allocation models in this paper. One considers the stack height and the state-changing magnitude information by reinterpreting the punishment parameter and the other further considers the specific configurations for a given stack representation. Solution qualities for the “optimistic” and the two “conservative” allocation models are compared on two performance indicators. The numerical experiments indicate that both the first and second “conservative” allocation models outperform the original model in terms of the two performance indicators. In addition, to overcome computational difficulties encountered by the dynamic programming algorithm for large-scale problems, an approximate dynamic programming algorithm is presented as well.     

Exchange lemmas 1: Deng's lemma

Deng's lemma gives estimates on the behavior of solutions of ordinary differential equations in the neighborhood of a partially hyperbolic equilibrium. We prove a generalization in which “partially hyperbolic equilibrium” is replaced by “normally hyperbolic invariant manifold.”     

Existence of solutions to a new model of biological pattern formation

In this paper we study the existence of classical solutions to a new model of skeletal development in the vertebrate limb. The model incorporates a general term describing adhesion interaction between cells and fibronectin, an extracellular matrix molecule secreted by the cells, as well as two secreted, diffusible regulators of fibronectin production, the positively-acting differentiation factor (“activator”) TGF-β, and a negatively-acting factor (“inhibitor”). Together, these terms constitute a pattern forming system of equations. We analyze the conditions guaranteeing that smooth solutions exist globally in time. We prove that these conditions can be significantly relaxed if we add a diffusion term to the equation describing the evolution of fibronectin.     

Nonmonotone systems decomposable into monotone systems with negative feedback

Motivated by the work of Angeli and Sontag [Monotone control systems, IEEE Trans. Automat. Control 48 (2003) 1684-1698] and Enciso and Sontag [On the global attractivity of abstract dynamical systems satisfying a small gain hypothesis, with applications to biological delay systems, Discrete Continuous Dynamical Systems, to appear] in control theory, we show that certain finite and infinite dimensional semi-dynamical systems with “negative feedback” can be decomposed into a monotone “open-loop” system with “inputs” and a decreasing “output” function. The original system is reconstituted by “plugging the output into the input”. Employing a technique of Gouzé [A criterion of global convergence to equilibrium for differential systems with an application to Lotka-Volterra systems, Rapport de Recherche 894, INRIA] and Cosner [Comparison principles for systems that embed in cooperative systems, with applications to diffusive Lotka-Volterra models, Dynam. Cont., Discrete Impulsive Systems 3 (1997) 283-303] of imbedding the system into a larger symmetric monotone system, we are able to obtain information on the asymptotic behavior of solutions, including existence of positively invariant sets and global convergence.     

Developing real option game models

By mixing concepts from both game theoretic analysis and real options theory, an investment decision in a competitive market can be seen as a “game” between firms, as firms implicitly take into account other firms’ reactions to their own investment actions. We review two decades of real option game models, suggesting which critical problems have been “solved” by considering game theory, and which significant problems have not been yet adequately addressed. We provide some insights on the plausible empirical applications, or shortfalls in applications to date, and suggest some promising avenues for future research.     

Strategic and operational decisions in restaurant revenue management

The paper addresses restaurant revenue management from both a strategic and an operational point of view. Strategic decisions in restaurants are mainly related to defining the most profitable combination of tables that will constitute the restaurant. We propose new formulations of the so-called “Tables Mix Problem” by taking into account several features of the real setting. We compare the proposed models in a computational study showing that restaurants, with the capacity of managing tables as renewable resources and of combining different-sized tables, can improve expected revenue performances. Operational decisions are mainly concerned with the more profitable assignment of tables to customers. Indeed, the “Parties Mix Problem” consists of deciding on accepting or denying a booking request from different groups of customers, with the aim of maximizing the total expected revenue. A dynamic formulation of the “Parties Mix Problem” is presented together with a linear programming approximation, whose solutions can be used to define capacity control policies based on booking limits and bid prices. Computational results compare the proposed policies and show that they lead to higher revenues than the traditional strategies used to support decision makers.     

Forecasting accuracy of behavioural models for participation in the arts

This paper assesses the forecasting performance of count data models applied to arts attendance. We estimate participation models for two artistic activities that differ in their degree of popularity – museums and jazz concerts – with data derived from the 2002 release of the Survey of Public Participation in the Arts for the United States. We estimate a finite mixture model – a zero-inflated negative binomial model – that allows us to distinguish between “true” non-attendants and “goers” and their respective behaviour regarding participation in the arts. We evaluate the predictive (in-sample) and forecasting (out-of-sample) accuracy of the estimated model using bootstrapping techniques to compute the Brier score. Overall, the results indicate the model performs well in terms of forecasting. Finally, we draw certain policy implications from the model’s forecasting capacity, thereby allowing the identification of target populations.     

Local and global solutions for some parabolic nonlocal problems

We study local and global existence of solutions for some semilinear parabolic initial boundary value problems with autonomous nonlinearities having a “Newtonian” nonlocal term.     

Long time behavior for the weakly damped driven long-wave-short-wave resonance equations

In this paper, we consider the Cauchy problem of the long-wave-short-wave resonance equations. By making use of a Strichartz-type inequality for the solutions, decomposing suitably the solution semigroup into a decay parts and a more regular parts, and ruling out the “vanishing” and “dichotomy” of the solutions, we prove the existence of the global attractor and the asymptotic smoothing effect of the solutions.     

The distance to strong stability

The notion of “strong stability” has been introduced in a recent paper [12]. This notion is relevant for state-space models described by physical variables and prohibits overshooting trajectories in the state-space transient response for arbitrary initial conditions. Thus, “strong stability” is a stronger notion compared to alternative definitions (e.g. stability in the sense of Lyapunov or asymptotic stability). This paper defines two distance measures to strong stability under absolute (additive) and relative (multiplicative) matrix perturbations, formulated in terms of the spectral and the Frobenius norm. Both symmetric and non-symmetric perturbations are considered. Closed-form or algorithmic solutions to these distance problems are derived and interesting connections are established with various areas in matrix theory, such as the field of values of a matrix, the cone of positive semi-definite matrices and the Lyapunov cone of Hurwitz matrices. The results of the paper are illustrated by numerous computational examples.     

Representation of Quantum Brownian Motion in the Collective Coordinate Method

We consider two explicitly solvable models of quantum random processes described by the Langevin equation, namely, those for a free quantum Brownian particle and for a quantum Brownian harmonic oscillator. The Hamiltonian (string) realization of the models reveals a soliton-like structure of classical solutions. Accordingly, the zero-mode collective coordinate method turns out to be an adequate means for describing the quantum dynamics of the models.     

Macroscopic traffic models: Shifting from densities to “Celerities”

The advent of techniques to measure velocities of “GPS equipped vehicles” using satellite technology, replacing the density of the road traffic sensors, motivates and justifies the revision of conceptual, mathematical algorithms and software based models. This paper summarizes studies on the traffic evolutions achieving the minimum of a congestion function controlled “macroscopic traffic velocities” called “celerities” instead of founding traffic regulation on the measures of traffic densities. The flux valued function is the Fenchel transform of the fundamental diagram and is a convex decreasing function. We use the properties of capture basins investigated in viability theory, specifically a Lax-Hopf formula characterizing them and the “Max-Plus” morphism of capture basins for deriving the statements proved in this paper. Traffic conditions involve as well boundary conditions or conditions on trajectories inside the domain.     

Periodic solutions of the Lyness max equation

We investigate the periodic nature of solutions of a “max-type” difference equation sometimes referred to as the “Lyness max” equation. The equation we consider is x_n+1=max{x_n,A}/x_n−1, n=0,1,…, where A is a positive real parameter and the initial conditions are arbitrary positive numbers. We also present related results for a similar equation sometimes referred to as the “period 7 max” equation.     

Lindelöf spaces which are Menger, Hurewicz, Alster, productive, or D

We discuss relationships in Lindelöf spaces among the properties “Menger”, “Hurewicz”, “Alster”, “productive”, and “D”.     

Low Mach number limit for the non-isentropic Navier-Stokes equations

We study the low Mach number limit of the local in time solutions to the compressible Navier-Stokes equations with zero heat conductivity coefficient as the Mach number tends to zero. A uniform existence result for the one-dimensional initial-boundary value problem is proved provided that the initial data are “well-prepared” in the sense that the temporal derivatives up to order two are bounded initially.     

Comparison results for nonlinear degenerate Dirichlet and Neumann problems with general growth in the gradient

This paper deals with both Dirichlet and Neumann problems for a class of nonlinear degenerate elliptic equations with general growth in the gradient. First, we give an existence result of a spherically symmetric solution to the “symmetrized” problems with data depending only on the radials. Second, we prove that the solutions of the original problems can be compared, under a rearrangement, with the solutions of the “symmetrized” problems.