随机交通均衡配流模型及其等价的变分不等式问题

本文讨论了交通网络系统的随机用户均衡原理的数学表述问题.在路段出行成本是流量的单调函数的较弱条件下,对具有固定需求和弹性需求的模式,首次证明了随机均衡配流模型可表示为一个变分不等式问题,同时也说明了该变分不等式问题与相应的互补问题以及一个凸规划问题之间的等价关系.     

A Multi-Class Dynamic User Equilibrium Model for Queuing Networks with Advanced Traveler Information Systems

This paper presents a formulation and solution algorithm for a composite dynamic user-equilibrium assignment problem with multi-user classes, in order to assess the impacts of Advanced Traveler Information Systems (ATIS) in general networks with queues. Suppose that users equipped with ATIS will receive complete information and hence be able to choose the best departure times and routes in a deterministic manner, while users not equipped with ATIS will have incomplete information and hence may make decisions on departure times and routes in a stochastic manner. This paper proposes a discrete-time, finite-dimensional variational inequality formulation that involves two criteria regarding the route and departure time choice behaviors, i.e., the deterministic dynamic user equilibrium and the nested logit-based stochastic dynamic user equilibrium. The formulation is then converted to an equivalent “zero-extreme value” minimization problem. A heuristic algorithm based on route/time-swapping process is proposed, which iteratively adjusts the route and departure time choices to reach closely to an extreme point of the minimization problem. A numerical example is used to demonstrate the effectiveness of the proposed approach for assessing the ATIS impacts such as changes in individual travel costs, departure times, route inflows, queuing peaks and total network travel cost. This revised version was published online in July 2006 with corrections to the Cover Date.     

Solving the logit-based stochastic user equilibrium problem with elastic demand based on the extended traffic network model

This paper proposes a novel extended traffic network model to solve the logit-based stochastic user equilibrium (SUE) problem with elastic demand. In this model, an extended traffic network is established by properly adding dummy nodes and links to the original traffic network. Based on the extended traffic network, the logit-based SUE problem with elastic demand is transformed to the SUE problem with fixed demand. Such problem is then further converted to a linearly constrained convex programming and addressed by a predictor–corrector interior point algorithm with polynomial complexity. A numerical example is provided to compare the proposed model with the method of successive averages (MSA). The numerical results indicate that the proposed model is more efficient and has a better convergence than the MSA.     

Estimation of Transit Origin–Destination Matrices from Passenger Counts Using a Frequency-Based Approach

This paper investigates the transit passenger origin–destination (O–D) estimation problem in congested transit networks where updated passenger counts and outdated O–D matrices are available. The bi-level programming approach is used for the transit passenger O–D estimation problem. The upper level minimizes the sum of error measurements in passenger counts and O–D matrices, and the lower level is a new frequency-based stochastic user equilibrium (SUE) assignment model that can determine simultaneously the passenger overload delays and passenger route choices in congested transit network together with the resultant transit line frequencies. The lower-level problem can be formulated as either a logit-type or probit-type SUE transit assignment problem. A heuristic solution algorithm is developed for solving the proposed bi-level programming model which is applicable to congested transit networks. Finally, a case study on a simplified transit network connecting Kowloon urban area and the Hong Kong International Airport is provided to illustrate the applications of the proposed bi-level programming model and solution algorithm. This revised version was published online in July 2006 with corrections to the Cover Date.     

ATIS和道路收费下混合SUE相对于SSO的效率损失

在ATIS和道路收费共同作用的异质性交通网络中,基于用户在信息接受程度与时间价值上的异质性,对用户进行合理分类,所有用户均按照随机方式进行择路.构建了多用户混合随机均衡等价的变分不等式模型,以及多用户随机社会最优模型.以用户感知的总出行成本作为系统性能评价的指标,当收费作为系统总成本的一部分时,分别在时间准则与费用准则下研究了多用户混合随机均衡相对于随机社会最优的绝对效率损失问题.研究结果表明,时间准则下的绝对效率损失上界与路段出行时间函数和混合随机均衡时系统的实际总出行时间有关,费用准则下的绝对效率损失上界还与出行者的社会经济特性和随机社会最优时系统的实际总出行时间有关.     

Stochastic persistence and stability analysis of a modified Holling–Tanner model

The article aims to study the basic dynamical features of a modified Holling–Tanner prey–predator model with ratio‐dependent functional response. We have proved the global existence of the solution for the deterministic model. The parametric restriction for persistence of both species is also obtained along with the proof of local asymptotic stability of the interior equilibrium point(s). Conditions for local bifurcations of interior equilibrium points are provided. The global dynamic behavior is examined thoroughly with supportive numerical simulation results. Next, we have formulated the stochastic model by perturbing the intrinsic growth rates of prey and predator populations with white noise terms. The existence uniqueness of solutions for stochastic model is established. Further, we have derived the parametric restrictions required for the persistence of the stochastic model. Finally, we have discussed the stochastic stability results in terms of the first and second order moments. Numerical simulation results are provided to support the analytical findings. Copyright © 2012 John Wiley & Sons, Ltd.     

The influence of forestry resources on rainfall: A deterministic and stochastic model

In this paper, we propose and analyze a deterministic model along with its stochastic version to address the problem of scanty rainfall by means of forestry resources. For deterministic model, boundedness of the system, feasibility of equilibria and their stability behavior are discussed. For stochastic model, boundedness, existence, uniqueness of global positive solution and sufficient conditions for the existence of unique stationary distribution are obtained. Model analysis reveals that the stability of the forest cover equilibrium state depends only on the model parameters in the deterministic case, however it also depends on the magnitude of the intensities of white noise terms in the stochastic case. To validate analytically obtained results and see the effect of key parameters, we have simulated proposed models using Indian annual rainfall data. The proposed model suggests that for the parameter values given in Table 2, the plantation of trees with slight higher intrinsic growth rate is beneficial to increase the rainfall.     

User equilibrium traffic network assignment with stochastic travel times and late arrival penalty

The classical Wardrop User Equilibrium (UE) assignment model assumes traveller choices are based on fixed, known travel times, yet these times are known to be rather variable between trips, both within and between days; typically, then, only mean travel times are represented. Classical Stochastic User Equilibrium (SUE) methods allow the mean travel times to be differentially perceived across the population, yet in a conventional application neither the UE or SUE approach recognises the travel times to be inherently variable. That is to say, there is no recognition that drivers risk arriving late at their destinations, and that this risk may vary across different paths of the network and according to the arrival time flexibility of the traveller. Recent work on incorporating risky elements into the choice process is seen either to neglect the link to the arrival constraints of the traveller, or to apply only to restricted problems with parallel alternatives and inflexible travel time distributions. In the paper, an alternative approach is described based on the ‘schedule delay’ paradigm, penalising late arrival under fixed departure times. The approach allows flexible travel time densities, which can be fitted to actual surveillance data, to be incorporated. A generalised formulation of UE is proposed, termed a Late Arrival Penalised UE (LAPUE). Conditions for the existence and uniqueness of LAPUE solutions are considered, as well as methods for their computation. Two specific travel time models are then considered, one based on multivariate Normal arc travel times, and an extended model to represent arc incidents, based on mixture distributions of multivariate Normals. Several illustrative examples are used to examine the sensitivity of LAPUE solutions to various input parameters, and in particular its comparison with UE predictions. Finally, paths for further research are discussed, including the extension of the model to include elements such as distributed arrival time constraints and penalties.     

On a stochastic disease model with vaccination

We propose a stochastic disease model where vaccination is included and such that the immunity isn’t permanent. The existence, uniqueness and positivity of the solution and the stability of disease free equilibrium is studied. The numerical simulation is done.     

A turning restriction design problem in urban road networks

Turning restriction is one of the commonest traffic management techniques and an effective low cost traffic improvement strategy in urban road networks. However, the literature has not paid much attention to the turning restriction design problem (TRDP), which aims to determine a set of intersections where turning restrictions should be implemented. In this paper, a bi-level programming model is proposed to formulate the TRDP. The upper level problem is to minimize the total travel cost from the viewpoint of traffic managers, and the lower level problem is to depict travelers’ route choice behavior based on stochastic user equilibrium (SUE) theory. We propose a branch and bound method (BBM), based on the sensitivity analysis algorithm (SAA), to find the optimal turning restriction strategy. A branch strategy and a bound strategy are applied to accelerate the solution process of the TRDP. The computational experiments give promising results, showing that the optimal turning restriction strategy can obviously reduce system congestion and are robust to the variations of both the dispersion parameter of the SUE problem and the level of demand.     

同时具有logistic出生和Markov切换的随机SIRS传染病模型的动力学

研究了一类同时具有logistic出生和Markov切换的随机SIRS传染病模型.首先通过构造合适的V函数，利用It?公式分析了随机传染病模型全局正解的存在唯一性，继而讨论出了该模型的解存在一个遍历平稳分布的结果，以及疾病灭绝的充分条件，最后给出了数值例子来说明本文得出的结论.     

无限时滞中立型随机泛函微分方程解的存在唯一性

有限时滞随机泛函微分方程的存在唯一性已经得到较多的研究,但对于无限时滞随机泛函微分方程的性质极少.本文在不需要线性增长条件,在一致Lipschitz条件下证明了无限时滞中立型随机泛函微分方程的存在唯一性,给出了精确解和近似解的误差估计,最后给出了解的矩估计.     

A stochastic interactive model for the diffusion of information

Two important generalizations of information diffusion models are the presence of stochastic effects and the possibility of arbitrary patterns of influence among individuals. A Markov random fields model includes both of these features. Under very weak assumptions, there is a unique equilibrium distribution of information patterns for given stochastic (local) interactions among a finite population. This has implications for policies to influence the transmission of information. The dynamic behavior of a special and simple case of the model tends to approximate the standard (logistic) diffusion curve. For an infinite population, uniqueness of equilibrium distributions may fail; some sufficient conditions to ensure uniqueness are given.     

The concert queueing game: strategic arrivals with waiting and tardiness costs

We consider the noncooperative choice of arrival times by individual users, who seek service at a first-come first-served queueing system that opens up at a given time. Each user wishes to obtain service as early as possible, while minimizing the expected wait in the queue. This problem was recently studied within a simplified fluid-scale model. Here, we address the unscaled stochastic system, assuming a finite (possibly random) number of homogeneous users, exponential service times, and linear cost functions. In this setting, we establish that there exists a unique Nash equilibrium, which is symmetric across users, and characterize the equilibrium arrival-time distribution of each user in terms of a corresponding set of differential equations. We further establish convergence of the Nash equilibrium solution to that of the associated fluid model as the number of users is increased. We finally consider the price of anarchy in our system and show that it exceeds 2, but converges to this value for a large population size.     

A demand model with departure time choice for within-day dynamic traffic assignment

A within-day dynamic demand model is formulated, embodying, in addition to the classic generation, distribution and modal split stages, an actual demand model taking into account departure time choice. The work focuses on this last stage, represented through an extension of the discrete choice framework to a continuous choice set. The dynamic multimodal supply and equilibrium model based on implicit path enumeration, which have been developed in previous work are outlined here, to define within-day dynamic elastic demand stochastic multimodal equilibrium as a fixed point problem on users flows and transit line frequencies. A MSA algorithm capable, in the case of Logit route choice models, of supplying equilibrium flows and frequencies on real dimension networks, is presented, as well as the specific procedures implementing the departure time choice and actual demand models. Finally, the results obtained on a test network are presented and conclusions are drawn.     

On the theorem of T. Yamada and S. Watanabe

The well-known theorem of T. Yamada and S. Watanabe asserts that (weak) existence and pathwise uniqueness of the solution of a stochastic equation implies the existence of a strong solution. This is the most powerful tool for proving that a stochastic equation possesses a strong solution. However, pathwise uniqueness is far from being a necessary condition for this. Even if the solution is not unique in law it is also of interest to ask for strong solutions. In the present note, we will discuss in more detail the connection between pathwise uniqueness and the existence of a strong solution. We will state a condition which is not only sufficient but also necessary for the existence of a strong solution.     

奇异Hamilton-Jacobi-Bellman方程粘性解的存在唯一性

研究系数在边界点有奇性的一类Hamilt on- Jacobi- Bellman (HJB)方程的粘性解的存在唯一性问题及解的渐近估计,这类问题包括波动系数振荡或爆破的情况.奇异HJB方程在随机最优控制和金融数学等许多领域都有重要的应用,包括金融数学中的随机利率模型.应用粘性上下解理论建立了一类奇异HJB方程的比较原理,给出了粘性解存在唯一性的条件.     

Stationary distribution of a stochastic Alzheimer's disease model

In this paper, based on the pathogenesis of Alzheimer's disease, we investigate a stochastic mathematical model, focusing on the dynamics of β-amyloid (Aβ) plaques, Aβ oligomers, PrP^C proteins, and the Aβ-x-PrP^C complex. Within the framework of the Lyapunov method, we first show existence and uniqueness of global positive solution of the model and then establish the sufficient conditions for existence of an ergodic stationary distribution of the positive solution. Ultimately, numerical examples are presented to illustrate the effectiveness of theoretical results.     

Stationary distribution and extinction of a stochastic nutrient-phytoplankton-zooplankton model with cell size

In this paper, we study a stochastic nutrient-phytoplankton-zooplankton model with cell size that represents the interaction between internal mechanism of species and external environment. We first investigate the existence and uniqueness of the global positive solution with positive initial values. Then we construct sufficient conditions for the existence of an ergodic stationary distribution of positive solution. Once more, we find that large noise intensities cause the extinctions of phytoplankton and zooplankton. Finally, numerical simulations are given to verify the correctness of theoretical results.     

Optimal control problems constrained by the stochastic Navier–Stokes equations with multiplicative Lévy noise

We consider the controlled stochastic Navier–Stokes equations in a bounded multidimensional domain, where the noise term allows jumps. In order to prove existence and uniqueness of an optimal control w.r.t. a given control problem, we first need to show the existence and uniqueness of a local mild solution of the considered controlled stochastic Navier–Stokes equations. We then discuss the control problem, where the related cost functional includes stopping times dependent on controls. Based on the continuity of the cost functional, we can apply existence and uniqueness results provided in [4], which enables us to show that a unique optimal control exists.