An invariance principle for semimartingale reflecting Brownian motions in an orthant

Semimartingale reflecting Brownian motions in an orthant (SRBMs) are of interest in applied probability because of their role as heavy traffic approximations for open queueing networks. It is shown in this paper that a process which satisfies the definition of an SRBM, except that small random perturbations in the defining conditions are allowed, is close in distribution to an SRBM. This perturbation result is called an invariance principle by analogy with the invariance principle of Stroock and Varadhan for diffusions with boundary conditions. A crucial ingredient in the proof of this result is an oscillation inequality for solutions of a perturbed Skorokhod problem. In a subsequent paper, the invariance principle is used to give general conditions under which a heavy traffic limit theorem holds for open multiclass queueing networks. This revised version was published online in June 2006 with corrections to the Cover Date.     

An empirical extension of the M/G/1 heavy traffic approximation

Numerical evaluation of waiting time distributions for M/G/1 systems is somewhat difficult. This paper examines a simple variation of the heavy traffic formula which may be useful at modest levels of traffic intensity. One can justify the heavy traffic approximation by expressing the Laplace transform of the service time distribution as a Maclaurin series and then truncating to three terms. The spectrum factorization and inversion leads in a straightforward fashion to the heavy traffic approximation. If one carries two additional terms from the Maclaurin series, the characteristic equation is a cubic with exactly one real negative root. This root provides an easy way to extend the heavy traffic formula to cases where the traffic is not so heavy. This paper studies the quality of this approximation and includes some numerical evaluation based on data actually encountered.     

Diffusion approximations for open multiclass queueing networks: sufficient conditions involving state space collapse

Certain diffusion processes known as semimartingale reflecting Brownian motions (SRBMs) have been shown to approximate many single class and some multiclass open queueing networks under conditions of heavy traffic. While it is known that not all multiclass networks with feedback can be approximated in heavy traffic by SRBMs, one of the outstanding challenges in contemporary research on queueing networks is to identify broad categories of networks that can be so approximated and to prove a heavy traffic limit theorem justifying the approximation. In this paper, general sufficient conditions are given under which a heavy traffic limit theorem holds for open multiclass queueing networks with head-of-the-line (HL) service disciplines, which, in particular, require that service within each class is on a first-in-first-out (FIFO) basis. The two main conditions that need to be verified are that (a) the reflection matrix for the SRBM is well defined and completely- S, and (b) a form of state space collapse holds. A result of Dai and Harrison shows that condition (a) holds for FIFO networks of Kelly type and their proof is extended here to cover networks with the HLPPS (head-of-the-line proportional processor sharing) service discipline. In a companion work, Bramson shows that a multiplicative form of state space collapse holds for these two families of networks. These results, when combined with the main theorem of this paper, yield new heavy traffic limit theorems for FIFO networks of Kelly type and networks with the HLPPS service discipline. This revised version was published online in June 2006 with corrections to the Cover Date.     

带时空相关性分析的行车时间估计模型

基于流体动力学方程的行车时间估计模型不能很好地反映真实的行车时间,需要对其进行一定的改进.在对交通流进行流体动力学建模的基础之上,引入对高速公路路网中不同路段之间的行车时间相关性和同一路段不同季节、不同时段的行车时间相关性分析,建立了带时空相关性分析的时间估计模型,使用统计学的方法消除动力学模型的误差.     

Traffic queue estimation

This paper deals with the urban traffic queue modelling and estimation. A general formulation of delays on a road section is used to link measured speeds at its entrance to the current queue state. This measurement model is applied to correct a posteriori the queue state distribution whose a priori evolution is ruled by a time varying Markov chain. The resulting predictor-corrector is tested with a microscopic simulation tool. Results in fixed plan control show that for long road sections (200 meters) the estimation error is unacceptable whereas for short road sections (50–100 meters) the average quadratic error is about 1 vehicle². In real time control conditions, in spite of a significant increase of the estimation quadratic error, results show a delay benefit of 15% with respect to the fixed plan control.     

Modeling demand management strategies for evacuations

Evacuations are massive operations that create heavy travel demand on road networks some of which are experiencing major congestions even with regular traffic demand. Congestion in traffic networks during evacuations, can be eased either by supply or demand management actions. This study focuses on modeling demand management strategies of optimal departure time, optimal destination choice and optimal zone evacuation scheduling (also known as staggered evacuation) under a given fixed evacuation time assumption. The analytical models are developed for a system optimal dynamic traffic assignment problem, so that their characteristics can be studied to produce insights to be used for large-scale solution algorithms. While the first two strategies were represented in a linear programming (LP) model, evacuation zone scheduling problem inevitable included integers and resulted in a mixed integer LP (MILP) one. The dual of the LP produced an optimal assignment principle, and the nature of the MILP formulations revealed clues about more efficient heuristics. The discussed properties of the models are also supported via numerical results from a hypothetical network example.     

简单闭环路网交通流定常解

基于在分岔路口满足用户均衡原理的假定,研究了由三条路段和两个交叉路口组成的简单闭环路网的交通流定常解问题,发现定常解参数及其性态依赖于路网上的车流总数:当车流总数不大于第一个临界值,或不小于第二个临界值时,定常解在每一条路段上均为密度取常数的平凡解;否则,在瓶颈路口(上游最大流量大于下游最大流量的路口)的上游路段将产生...     

主路多车道无信号交叉口混合交通流的通行能力

Highway capacity is defined as maximum volume of traffic flow through the particular highway section under given traffic conditions,road conditions and so on.Highway construction and management is judged by capacity standard.The reasonable scale and time of highway construction,rational network structure and optimal management mode of highway network can be determined by analyzing the fitness between capacity and traffic volume.All over the world,highway capacity is studied to different extent in different country. Based on the gap acceptance theory,the mixed traffic flow composed of two representative vehicle types heavy and light vehicles is analyzed with probability theory.Capacity model of the minor mixed traffic flows crossing m major lanes,on which the traffic flows fix in with M3 distributed headway,on the unsignalized intersection is set up,and it is an extension of minor lane capacity theory for one vehicle-type and one major-lane traffic flow.     

A stochastic and dynamic routing policy using branching processes with state dependent immigration

A stochastic and dynamic vehicle routing problem called the Dynamic Traveling Repairman Problem (DTRP) was introduced by Bertsimas and van Ryzin. Several routing policies were analyzed in light traffic and in heavy traffic conditions. But, the good light traffic policies become very quickly unstable with increasing traffic intensity, and the good heavy traffic policies are inefficient in light traffic conditions. In this paper, a new routing policy is defined and analyzed, using results from branching processes with state dependent immigration. This policy not only performs optimally in light traffic, but also performs very well in heavy traffic. This is important to the designer of a service system because the traffic conditions may be variable and/or be unpredictable, and having to switch routing policies could prove to be costly and difficult to implement.     

An analysis of the traffic on highways with changing surface conditions

In this paper we present an application of the theory of traffic flow formulated by Lighthill and Whitham to problems where a section of a road imposes different conditions on the flow of cars. To represent these situations mathematically, we use a different flow-density relationship which contains implicitly the changes in the properties of this special section of road.With this approach we analyze first some simplified problems of this type, the solutions of which are of use in more complicated problems. Also, we obtain the corresponding steady-state solutions and discuss their stability. Finally, to illustrate these ideas on more realistic situations, we obtain the solutions of the flow produced by heavy rain on a section of a highway and by the slowing down of a convoy of trucks moving on one lane of a three-lane highway.     

Decomposition results for general polling systems and their applications

In this paper we derive decomposition results for the number of customers in polling systems under arbitrary (dynamic) polling order and service policies. Furthermore, we obtain sharper decomposition results for both the number of customers in the system and the waiting times under static polling policies. Our analysis, which is based on distributional laws, relaxes the Poisson assumption that characterizes the polling systems literature. In particular, we obtain exact decomposition results for systems with either Mixed Generalized Erlang (MGE) arrival processes, or asymptotically exact decomposition results for systems with general renewal arrival processes under heavy traffic conditions. The derived decomposition results can be used to obtain the performance analysis of specific systems. As an example, we evaluate the performance of gated Markovian polling systems operating under heavy traffic conditions. We also provide numerical evidence that our heavy traffic analysis is very accurate even for moderate traffic. This revised version was published online in June 2006 with corrections to the Cover Date.     

Scientific Models—Marketing

After briefly introducing the gravity and interactance models, the following article goes on to discuss the status of these models and asks: Are the hypotheses reasonable; are the models logically consistent; and do the models fit the facts? The first two sections are of a theoretical nature although a number of practical examples are given to illustrate certain of the points made. The third section consists of a series of case studies covering U.K. port traffic, intra-European air traffic, West German domestic air traffic, inter-urban traffic in Africa, and urban road traffic in Oxford. Graphs are plotted to show the relationship between the various traffic model parameters.The general conclusion from the analysis is that gravity and interactance models do not provide a valid means of producing traffic forecasts in a regional environment. The concluding section is devoted to discussing other methods of traffic forecasting and suggesting the most promising areas for future research.     

Compound Transient Phenomena in Multiphase Queuing Systems. III

The author continues the work on functional limit theorems in multiphase queuing systems (QS) under heavy traffic. In this paper there are proved theorems for the waiting time of a job when at phases of a system various conditions of heavy traffic are satisfied (compound transient phenomena).     

Location-allocation models for traffic police patrol vehicles on an interurban network

This research investigates the traffic police routine patrol vehicle (RPV) assignment problem on an interurban road network through a series of integer linear programs. The traffic police RPV’s main task, like other emergency services, is to handle calls-for-service. Emergency services allocation models are generally based on the shortest path algorithm however, the traffic police RPV also handles other roles, namely patrolling to create a presence that acts as a deterrence, and issuing tickets to offenders. The RPVs need to be located dynamically on both hazardous sections and on roads with heavy traffic in order to increase their presence and conspicuousness, in an attempt to prevent or reduce traffic offences, road accidents and traffic congestion. Due to the importance of the traffic patrol vehicle’s location with regard to their additional roles, allocation of the RPVs adheres to an exogenous, legal, time-to-arrival constraint. We develop location-allocation models and apply them to a case study of the road network in northern Israel. The results of the four models are compared to each other and in relation to the current chosen locations. The multiple formulations provide alternatives that jointly account for road safety and policing objectives which aid decision-makers in the selection of their preferred RPV assignments. The results of the models present a location-allocation configuration per RPV per shift with full call-for-service coverage whilst maximizing police presence and conspicuousness as a proxy for road safety.     

Motorway traffic parameter estimation from mobile phone counts

In this paper a new method for real time estimation of vehicular flows and densities on motorways is proposed. This method is based on fusing traffic counts with mobile phone counts. The procedure used for the estimation of traffic flow parameters is based on the hypothesis that “instrumented” vehicles can be counted on specific motorway sections and traffic flow can be measured on entrance and exit ramps. The motorway is subdivided into cells, assuming that mobile phones entering and exiting every cell can be counted during the observation period. An estimate of “instrumented” vehicle concentration is obtained and propagated on the network in time and space. This allows one to estimate traffic flow parameters by sampling “instrumented” traffic flow parameters using a “concentration” (the ratio of the densities of instrumented vehicles to the density of overall traffic) propagation mechanism.     

Optimal control of assignment of jobs to processors under heavy traffic

The paper is concerned with the optimal control of the assignment of jobs from several arriving random streams to one of a bank of processors. Owing to the difficulty of the general problem, a heavy traffic approach is used. The required work depends on the processor to which it is assigned. The information that the assignment can be based on is quite flexible, and several information structures (data on which the control is based) are considered. The assignment can be made on arrival or when the job is to be processed. There can be bursty arrivals (the bursts depending on randomly varying environmental factors), rather general nonlinear cost functions and other complications. It is shown, under reasonably general conditions, that the optimal costs for the physical systems converge to the optimal cost for the heavy traffic limit problem, as the heavy traffic parameter goes to its limit. Numerical data is presented to illustrate some of the potential uses of the limit process for obtaining optimal contros, or controls satisfying optimal tradeoffs among competing criteria. The methods of proof are quite powerful tools for such optimal control problems     

A Law of the Iterated Logarithm for Extreme Queue Length in Multiphase Queues

The object of this research in queueing theory is the Law of the Iterated Logarithm (LIL) under the conditions of heavy traffic in Multiphase Queueing Systems (MQS). In this paper, the LIL is proved for extreme values of important probabilistic characteristics of the MQS investigated as well as maxima and minima of the summary queue length of customers and maxima and minima of the queue length of customers. Also, the paper presents a survey on the works for extreme values in queues and the queues in heavy traffic.      

An evaluation of freeway lane control signing using computer simulation

Lane control has been proposed as a traffic congestion alleviation method. For this method to work, a certain minimum percentage of the drivers must comply with the lane signing. The research described here has been performed to analyze the percentage of drivers that must comply with lane control. A simulation model was developed and tested. The simulation results for heavy traffic flow (1550 vehicles per hour per lane—vphpl), medium traffic flow (900 vphpl), and light traffic flow (300 vphpl) conditions indicate that lane control has little influence on congestion, regardless of the percentage of drivers that comply with the lane control signing. For heavy flow, the congestion level remains high even when all drivers comply. For medium and light flow, the congestion level remains low even when no drivers comply. The region between heavy and medium traffic flow is, however, sensitive to lane control. Four flow rates between medium and heavy flow were tested. The impact of lane control under these conditions is described.     

Heavy traffic analysis of controlled multiplexing systems

The paper develops the mathematics of the heavy traffic approach to the control and optimal control problem for multiplexing systems, where there are many mutually independent sources which feed into a single channel via a multiplexer (or of networks composed of such subsystems). Due to the widely varying bit rates over all sources, control over admission, bandwidth, etc., is needed to assure good performance. Optimal control and heavy traffic analysis has been shown to yield systems with greatly improved performance. Indeed, the heavy traffic approach covers many cases of great current interest, and provides a useful and practical approach to problems of analysis and control arising in modern high speed telecommunications. Past works on the heavy traffic approach to the multiplexing problem concentrated on the uncontrolled system or on the use of the heavy traffic limit control problem for applications, and did not provide details of the proofs. This is done in the current paper. The basic control problem for the physical system is hard, and the heavy traffic approach provides much simplification. Owing to the presence of the control, as well as to the fact that the cost function of main interest is “ergodic”, the problem cannot be fully treated with “classical” methods of heavy traffic analysis for queueing networks. A basic result is that the optimal average costs per unit time for the physical problem converge to the optimal cost per unit time for the limit stationary process as the number of sources and the time interval goes to infinity. This convergence is both in the mean and pathwise senses. Furthermore, a “nice” nearly optimal control for the limit system provides nearly optimal values for the physical system, under heavy traffic, in both a mean and pathwise sense. This revised version was published online in June 2006 with corrections to the Cover Date.     

The FCFS service discipline: Stable network topologies, bounds on traffic burstiness and delay, and control by regulators

We consider the well-known First Come First Serve (FCFS) scheduling policy under bursty arrivals. This policy is commonly used in scheduling communication networks and manufacturing systems. Recently it has been shown that the FCFS policy can be unstable for some nonacyclic network topologies.We identify some network topologies under which FCFS is stable for all arrival and service rate vectors within the system's capacity. This is done by determining a sharp bound on the burstiness of traffic exiting from a tandem section of the system, in terms of the burstiness of the incoming traffic. This burstiness bound further allows us to provide a bound on the maximum number of parts in the system, and the maximum delay. It also enables us to analyze the performance of some systems controlled by the use of traffic smoothing regulators. The maximum delay can remain bounded even in the heavy traffic limit, when all stations are 100% utilized.