The last departure time from an Mt/G/∞ queue with a terminating arrival process

This paper studies the last departure time from a queue with a terminating arrival process. This problem is motivated by a model of two-stage inspection in which finitely many items come to a first stage for screening. Items failing first-stage inspection go to a second stage to be examined further. Assuming that arrivals at the second stage can be regarded as an independent thinning of the departures from the first stage, the arrival process at the second stage is approximately a terminating Poisson process. If the failure probabilities are not constant, then this Poisson process will be nonhomogeneous. The last departure time from an Mt/G/∞ queue with a terminating arrival process serves as a remarkably tractable approximation, which is appropriate when there are ample inspection resources at the second stage. For this model, the last departure time is a Poisson random maximum, so that is possible to give exact expressions and develop useful approximations based on extreme-value theory

Modelling dynamic stochastic user equilibrium for urban road networks

In this study a dynamic assignment model is developed which estimates travellers' route and departure time choices and the resulting time varying traffic patterns during the morning peak. The distinctive feature of the model is that it does not restrict the geometry of the network to specific forms. The proposed framework of analysis consists of a travel time model, a demand model and a demand adjustment mechanism. Two travel time models are proposed. The first is based on elementary relationships from traffic flow theory and provides the framework for a macroscopic simulation model which calculates the time varying flow patterns and link travel times given the time dependent departure rate distributions; the second is based on queueing theory and models roads as bottlenecks through which traffic flow is either uncongested or fixed at a capacity independent of traffic density. The demand model is based on the utility maximisation decision rule and defines the time dependent departure rates associated with each reasonable route connecting, the O-D pairs of the network, given the total utility associated with each combination of departure time and route. Travellers' choices are assumed to result from the trade-off between travel time and schedule delay and each individual is assumed to first choose a departure time t, and then select a reasonable route, conditional on the choice of t. The demand model has therefore the form of a nested logit. The demand adjustment mechanism is derived from a Markovian model, and describes the day-to-day evolution of the departure rate distributions. Travellers are assumed to modify their trip choice decisions based on the information they acquire from recent trips. The demand adjustment mechanism is used in order to find the equilibrium state of the system, defined as the state at which travellers believe that they cannot increase their utility of travel by unilaterally changing route or departure time. The model outputs exhibit the characteristics of real world traffic patterns observed during the peak, i. e., time varying flow patterns and travel times which result from time varying departure rates from the origins. It is shown that increasing the work start time flexibility results in a spread of the departure rate distributions over a longer period and therefore reduces the level of congestion in the network. Furthermore, it was shown that increasing the total demand using the road network results in higher levels of congestion and that travellers tend to depart earlier in an attempt to compensate for the increase in travel times. Moreover, experiments using the queueing theory based travel time model have shown that increasing the capacity of a bottleneck may cause congestion to develop downstream, which in turn may result in an increase of the average travel time for certain O-D pairs. The dynamic assignment model is also applied to estimate the effects that different road pricing policies may have on trip choices and the level of congestion; the model is used to demonstrate the development of the shifting peak phenomenon. Furthermore, the effect of information availability on the traffic patterns is investigated through a number of experiments using the developed dynamic assignment model and assuming that guided drivers form a class of users characterised by lower variability of preferences with respect to route choice

Analysis Of Queue Characteristics At Signalized Intersections Near Highway-Railroad Grade Crossing

Analysis of traffic queues at signalized intersections which are in close proximity to highway- railroad grade crossings is of primary importance for determining if the normal signal operation needs to be preempted for railroad operations by providing a special signal mode for safe clearance of the queued vehicles from the tracks before the train arrival, and prohibiting any conflicting traffic movements towards the crossing. Such queuing analysis becomes even more critical where direct observations of traffic queues are not possible or where the assessment is needed for a future location. Inadequate estimation of queues from signalized intersections to the nearby railroad grade crossing can lead to severe safety issues. Underestimation of queue lengths may lead to an unsafe design while significantly overestimated queues may cause unnecessary traffic delays consequently leading to violations of the active traffic control devices at the crossing. In order to determine an adequate approach for reasonable estimation of queue lengths at signalized intersections near highway-railroad grade crossings, this dissertation first evaluated and compared different currently used microscopic simulation-based methods (i.e. Sim-Traffic and VISSIM) for their adequacy in estimating the queue lengths. After that several comparisons are made between the queue estimation from the simulation-based and other deterministic analytical methods including Highway Capacity Software, Synchro, and Railroad Assessment Tool. The comparisons drawn between each method helped identifying the differences and specific limitations of each method in including the impact of various important factors on the resulting queue estimation. The recommendations are provided on the basis of model capability to adequately count the impact of various significant traffic factors on queue estimation and considering minimizing the risk of underestimated queues. Based on the analysis findings, a microscopic simulation based procedure is developed using Sim-Traffic for estimating the 95th percentile queue lengths on various existing signalized intersection configurations near highway-rail grade crossings to help evaluate the need for signal preemption. In addition, recommendations are developed, if preemption is necessary, for determining queue clearance distance and minimum track clearance time. The recommended procedure is developed considering minimizing the risk of underestimated queues or unsafe design at such locations, and simplify the design and decision-making process

Continuous-time dynamics shortest path algorithms

Thesis (S.B. and M.Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1999.Includes bibliographical references (p. 116-117).by Brian C. Dean.S.B.and M.Eng

Stochastic approximation of symmetric Nash equilibria in queueing games

We suggest a novel stochastic-approximation algorithm to compute a symmetric Nash-equilibrium strategy in a general queueing game with a finite action space. The algorithm involves a single simulation of the queueing process with dynamic updating of the strategy at regeneration times. Under mild assumptions on the utility function and on the regenerative structure of the queueing process, the algorithm converges to a symmetric equilibrium strategy almost surely. This yields a powerful tool that can be used to approximate equilibrium strategies in a broad range of strategic queueing models in which direct analysis is impracticable

Performance and reliability modelling of computing systems using spectral expansion

PhD ThesisThis thesis is concerned with the analytical modelling of computing and other discrete event systems, for steady state performance and dependability. That is carried out using a novel solution technique, known as the spectral expansion method. The type of problems considered, and the systems analysed, are represented by certain two-dimensional Markov-processes on finite or semi-infinite lattice strips. A sub set of these Markov processes are the Quasi-Birth-and-Death processes. These models are important because they have wide ranging applications in the design and analysis of modern communications, advanced computing systems, flexible manufacturing systems and in dependability modelling. Though the matrixgeometric method is the presently most popular method, in this area, it suffers from certain drawbacks, as illustrated in one of the chapters. Spectral expansion clearly rises above those limitations. This also, is shown with the aid of examples. The contributions of this thesis can be divided into two categories. They are, • The theoretical foundation of the spectral expansion method is laid. Stability analysis of these Markov processes is carried out. Efficient numerical solution algorithms are developed. A comparative study is performed to show that the spectral expansion algorithm has an edge over the matrix-geometric method, in computational efficiency, accuracy and ease of use. • The method is applied to several non-trivial and complicated modelling problems, occuring in computer and communication systems. Performance measures are evaluated and optimisation issues are addressed

Border Crossing Modeling and Analysis: A Non-Stationary Dynamic Reallocation Methodology For Terminating Queueing Systems

The United States international land boundary is a volatile, security intense area. In 2010, the combined trade was $918 billion within North American nations, with 80% transported by commercial trucks. Over 50 million commercial vehicles cross the Texas/Mexico border every year, not including private vehicles and pedestrian traffic, between Brownsville and El Paso, Texas, through one of over 25 major border crossings called "ports of entry" (POE). Recently, securing our southwest border from terrorist interventions, undocumented immigrants, and the illegal flow of drugs and guns has dominated the need to efficiently and effectively process people, goods and traffic. Increasing security and inspection requirements are seriously affecting transit times. Each POE is configured as a multi-commodity, prioritized queueing network which rarely, if ever, operates in steady-state. Therefore, the problem is about finding a balance between a reduction of wait time and its variance, POE operation costs, and the sustainment of a security level. The contribution of the dissertation is three-fold. The first uses queueing theory on the border crossing process to develop a methodology that decreases border wait times without increasing costs or affecting security procedures. The outcome is the development of the Dynamic Reallocation Methodology (DRM). Currently at the POE, inspection stations are fixed and can only inspect one truck type, FAST or Non-FAST program participant. The methodology proposes moveable servers that once a threshold is met, can be switched to service the other type of truck. Particular emphasis is given to inspection (service) times under time-varying arrivals (demands). The second contribution is an analytical model of the POE, to analyze the effects of the DRM. First assuming a Markovian service time, DRM benefits are evaluated. However, field data and other research suggest a general distribution for service time. Therefore, a Coxian k-phased approximation is implemented. The DRM is analyzed under this new baseline using expected number in the system, and cycle times. A variance reduction procedure is also proposed and evaluated under DRM. Results show that queue length and wait time is reduced 10 to 33% depending on load, while increasing FAST wait time by less than three minutes

Performance Analysis of Block Codes over Finite-state Channels in Delay-sensitive Communications

As the mobile application landscape expands, wireless networks are tasked with supporting different connection profiles, including real-time traffic and delay-sensitive communications. Among many ensuing engineering challenges is the need to better understand the fundamental limits of forward error correction in non-asymptotic regimes. This dissertation seeks to characterize the performance of block codes over finite-state channels with memory and also evaluate their queueing performance under different encoding/decoding schemes. In particular, a fading formulation is considered where a discrete channel with correlation over time introduces errors. For carefully selected channel models and arrival processes, a tractable Markov structure composed of queue length and channel state is identified. This facilitates the analysis of the stationary behavior of the system, leading to evaluation criteria such as bounds on the probability of the queue exceeding a threshold. Specifically, this dissertation focuses on system models with scalable arrival profiles based on Poisson processes, and finite-state memory channels. These assumptions permit the rigorous comparison of system performance for codes with arbitrary block lengths and code rates. Based on this characterization, it is possible to optimize code parameters for delay-sensitive applications over various channels. Random codes and BCH codes are then employed as means to study the relationship between code-rate selection and the queueing performance of point-to-point data links. The introduced methodology offers a new perspective on the joint queueing-coding analysis for finite-state channels, and is supported by numerical simulations. Furthermore, classical results from information theory are revisited in the context of channels with rare transitions, and bounds on the probabilities of decoding failure are derived for random codes. An analysis framework is presented where channel dependencies within and across code words are preserved. The results are subsequently integrated into a queueing formulation. It is shown that for current formulation, the performance analysis based on upper bounds provides a good estimate of both the system performance and the optimum code parameters. Overall, this study offers new insights about the impact of channel correlation on the performance of delay-aware communications and provides novel guidelines to select optimum code rates and block lengths