Fleet-sizing and service availability for a vehicle rental system via closed queueing networks

In this paper, we address the problem of determining the optimal fleet size for a vehicle rental company and derive analytical results for its relationship to vehicle availability at each rental station in the company’s network of locations. This work is motivated by the recent surge in interest for bicycle and electric car sharing systems, one example being the French program Vélib (2010). We first formulate a closed queueing network model of the system, obtained by viewing the system from the vehicle’s perspective. Using this framework, we are able to derive the asymptotic behavior of vehicle availability at an arbitrary rental station with respect to fleet size. These results allow us to analyze imbalances in the system and propose some basic principles for the design of system balancing methods. We then develop a profit-maximizing optimization problem for determining optimal fleet size. The large-scale nature of real-world systems results in computational difficulties in obtaining this exact solution, and so we provide an approximate formulation that is easier to solve and which becomes exact as the fleet size becomes large. To illustrate our findings and validate our solution methods, we provide numerical results on some sample networks.     

基于矩阵分析方法的具有双阶段休假的排队系统驱动的流模型性能分析

基于矩阵分析方法研究了具有单重工作休假和多重休假策略M/M/1排队系统驱动的流模型.首先建立了控制该流模型的微分方程组,利用矩阵分析方法,得出了系统平稳库存量的laplace变换(LT)的矩阵阶乘表达式.进而利用LaplaceStieltjes变换(LST)得出了平稳库存量的期望.最后,通过数值例子展示了系统性能指标与参数的关系.     

Finite element approximation of a free boundary plasma problem

In this article, we study a finite element approximation for a model free boundary plasma problem. Using a mixed approach (which resembles an optimal control problem with control constraints), we formulate a weak formulation and study the existence and uniqueness of a solution to the continuous model problem. Using the same setting, we formulate and analyze the discrete problem. We derive optimal order energy norm a priori error estimates proving the convergence of the method. Further, we derive a reliable and efficient a posteriori error estimator for the adaptive mesh refinement algorithm. Finally, we illustrate the theoretical results by some numerical examples.     

Error Estimate of a New Conservative Finite Difference Scheme for the Klein-Gordon-Dirac System

In this paper, we derive and analyze a conservative Crank-Nicolson-type finite difference scheme for the Klein-Gordon-Dirac (KGD) system. Differing from the derivation of the existing numerical methods given in literature where the numerical schemes are proposed by directly discretizing the KGD system, we translate the KGD equations into an equivalent system by introducing an auxiliary function, then derive a nonlinear Crank-Nicolson-type finite difference scheme for solving the equivalent system. The scheme perfectly inherits the mass and energy conservative properties possessed by the KGD, while the energy preserved by the existing conservative numerical schemes expressed by two-level's solution at each time step. By using energy method together with the 'cut-off' function technique, we establish the optimal error estimate of the numerical solution, and the convergence rate is $\mathcal{O}(τ^2 + h^2)$ in $l^∞$-norm with time step $τ$ and mesh size $h.$ Numerical experiments are carried out to support our theoretical conclusions.     

Analysis and approximation of a reliable model

In this paper we consider the problem of approximating the dynamical system that models reliability of a system consisting of two machines separated by a finite storage buffer. The system is described as a distributed parameter system defined by a coupled partial and ordinary differential equations and formulated as an abstract Cauchy problem. To derive the dynamical solution and some instantaneous indexes of the model, we present a simple finite difference scheme and establish the convergence of this scheme by employing Trotter–Kato Theorem. Numerical results are given to illustrate the effectiveness of the scheme.     

Optimal decay estimates of a regularity-loss type system with constraint condition

In the paper [14], the authors formulated a new structural condition which includes the Kawashima–Shizuta condition, and analyzed the weak dissipative structure called the regularity-loss type for general systems which contain the Timoshenko system and the Euler–Maxwell system. However, this new structural condition can not cover all of dissipative systems. Indeed we introduce a dissipative system which does not satisfy the new condition and analyze the weaker dissipative structure in this paper. Precisely we first derive the $L^2$ decay estimate of solutions and discuss the type of the corresponding regularity-loss structure. Moreover, in order to show the optimality of the decay estimate, we analyze the expansion for the corresponding eigenvalue of our problem and derive that the solution approaches the diffusion wave as time tends to infinity.     

Quantization for an evolution equation with critical exponential growth on a closed Riemann surface

In this paper, we analyze the concentration behavior of a positive solution to an evolution equation with critical exponential growth on a closed Riemann surface, and particularly derive an energy identity for such a solution. This extends a result of Lamm-Robert-Struwe and complements that of Yang.     

Optimal hierarchical system of a grid road network

This paper develops a simple analytical model for determining the hierarchical system of road networks. The model is based on a grid road network where roads are classified into three types according to road widths and travel speeds. We derive the optimal ratios of road areas that minimize the average and maximum travel time. Minimizing the average travel time provides an efficient solution, whereas minimizing the maximum travel time provides an equitable solution. Both of the solutions are expressed in terms of road widths and travel speeds. As an application of the grid network model, we evaluate the hierarchical system of the road network of Tokyo.     

Heavy Traffic Analysis of Two Coupled Processors

We consider two parallel queues. When both are non-empty, they behave as two independent M/M/1 queues. If one queue is empty the server in the other works at a different rate. We consider the heavy traffic limit, where the system is close to instability. We derive and analyze the heavy traffic diffusion approximation for this model. In particular, we obtain simple integral representations for the joint steady state density of the (scaled) queue lengths. Asymptotic and numerical properties of the solution are studied.     

Analysis of a single server queue with semi-Markovian service interruption

In this paper we analyze a discrete-time single server queue where the service time equals one slot. The numbers of arrivals in each slot are assumed to be independent and identically distributed random variables. The service process is interrupted by a semi-Markov process, namely in certain states the server is available for service while the server is not available in other states. We analyze both the transient and steady-state models. We study the generating function of the joint probability of queue length, the state and the residual sojourn time of the semi-Markov process. We derive a system of Hilbert boundary value problems for the generating functions. The system of Hilbert boundary value problems is converted to a system of Fredholm integral equations. We show that the system of Fredholm integral equations has a unique solution. This revised version was published online in June 2006 with corrections to the Cover Date.     

Optimal strategies to diminish a pest population via bilinear controls

In this work we model and analyze two control strategies to diminish a pest population using traps. The action of any trap depends on its age. Both problems contain bilinear control rates. The large-time behavior of the model with time-periodic inflow is investigated. The first control strategy deals with a finite horizon problem while the second one is related to a time-periodic problem. We obtain Pontryagin’s principle for both control problems. A special attention is given to the periodic problem. Pontryagin’s principle is used to derive a conceptual gradient-type algorithm to approximate the optimal solution. Numerical tests are given.     

Connection between a Cauchy system representation of Kalaba and Fourier transforms

Recently R.E. Kalaba reduced the solution of a system of integral equations to the solution of a Cauchy system. In this paper we derive the same Cauchy system via the Wiener-Hopt procedure.     

Thermo-chemical modelling of the formation of a composite material

We consider a composite material composed of carbon or glass fibres included in a resin which becomes solid when it is heated up (the reaction of reticulation).

A mathematical model of the cure process is given by a kinetic equation describing the evolution of the reaction of reticulation coupled with the heat equation. The geometry of the composite material is periodic, with a small period ? >0, thus we get a coupled system of nonlinear partial differential equations.

First we prove the existence and uniqueness of a solution by using a fixed point theorem and we obtain a priori estimates. Then we derive the homogenized problem which describes the macroscopic behaviour of the material. We prove the convergence of the solution of the problem to the solution of the homogenized problem when ? tends to zero as well as the estimates for the difference of the exact and the approximate solutions.     

An Exact Riemann Solver for a Fluidized Bed Model

We study a 2 x 2 hyperbolic system of conservation laws withsource term arising in a fluidized bed model. The system issolved numerically and results are presented to demonstratethe occurrence of ‘slugging’ in the full model equations.The numerical procedure is based on operator splitting and Godunov'smethod, for which we derive the exact solution of the Riemannproblem. A second-order improvement due to Davis (1988) mayproduce small oscillations near shocks and these can be reducedif the underlying flux limiter of the Davis method is replacedby the minmod limiter.     

A numerical verification method for a periodic solution of a delay differential equation

We describe a numerical method with guaranteed accuracy to enclose a periodic solution for a system of delay differential equations. Using a certain system of equations corresponding to the original system, we derive sufficient conditions for the existence of the solution, the satisfaction of which can be verified computationally. We describe the verification procedure in detail and give a numerical example.     

Global analysis of a periodic epidemic model on cholera in presence of bacteriophage

We propose and analyze a recurrent epidemic model of cholera in the presence of bacteriophage. The model is extended by general periodic incidence functions for low‐infectious bacterium and high‐infectious bacterium, respectively. A general periodic shedding function for two infected class (phage‐positive and phage‐negative) and a generalized contact and intrinsic growth function for susceptible class are also considered. Under certain biological assumptions, we derive the basic reproduction number (R₀) in a periodic environment for the proposed model. We also observe the global stability of the disease‐free equilibrium, existence, permanence, and global stability of the positive endemic periodic solution of our proposed model. Finally, we verify our results with specific functional form. Copyright © 2016 John Wiley & Sons, Ltd.     

Coarsening dynamics of polymer droplets for a lubrication model with large slippage

We analyze the final stages of the dewetting process of nanoscopic thin polymer films on hydrophobized substrates using a lubrication model that captures the large slippage at the liquid-substrate interface. The final stages of this process are characterized by the slow-time coarsening dynamics of the remaining droplets. For this situation we derive a reduced system of equations from the lubrication model, using singular perturbation analysis. Such reduced models allow for an efficient numerical simulation of the coarsening process. The reduced model extends results of [2] for no-slip lubrication model. Apart from collapse and collision, we identify here some new coarsening dynamics. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The convergence of finite element Galerkin solution for the Roseneau equation

In this paper, we analyze the convergence of the semidiscrete solution of the Roseneau equation. We introduce the auxiliary projection of the solution, and derive the optimal convergence of the semidiscrete solution as well as the auxiliary projection inL ² normed space.     

Synchronized abandonments in a single server unreliable queue

We consider a single server unreliable queue represented by a 2-dimensional continuous-time Markov chain. At failure times, the present customers leave the system. Moreover, customers become impatient and perform synchronized abandonments, as long as the server is down. We analyze this model and derive the main performance measures using results from the basic q-hypergeometric series.     

Growth on multiple interactive-essential resources in a self-cycling fermentor: An impulsive differential equations approach

We introduce a model of the growth of a single microorganism in a self-cycling fermentor in which an arbitrary number of resources are limiting, and impulses are triggered when the concentration of one specific substrate reaches a predetermined level. The model is in the form of a system of impulsive differential equations. We consider the operation of the reactor to be successful if it cycles indefinitely without human intervention and derive conditions for this to occur. In this case, the system of impulsive differential equations has a periodic solution. We show that success is equivalent to the convergence of solutions to this periodic solution. We provide conditions that ensure that a periodic solution exists. When it exists, it is unique and attracting. However, we also show that whether a solution converges to this periodic solution, and hence whether the model predicts that the reactor operates successfully, is initial condition dependent. The analysis is illustrated with numerical examples.