Markovian bulk-arrival and bulk-service queues with general state-dependent control

We study a modified Markovian bulk-arrival and bulk-service queue incorporating general state-dependent control. The stopped bulk-arrival and bulk-service queue is first investigated, and the relationship between this stopped queue and the full queueing model is examined and exploited. Using this relationship, the equilibrium behaviour for the full queueing process is studied and the probability generating function of the equilibrium distribution is obtained. Queue length behaviour is also examined, and the Laplace transform of the queue length distribution is presented. The important questions regarding hitting times and busy period distributions are answered in detail, and the Laplace transforms of these distributions are presented. Further properties regarding the busy period distributions including expectation and conditional expectation of busy periods are also explored

Markov denumerable process and queue theory

In this thesis, we study a modified Markovian batch-arrival and bulk- service queue including finite states for dependent control. We first consider the stopped batch-arrival and bulk-service queue process Q∗, which is the process with the restriction of the state-dependent control. After we obtain the expression of the Q∗-resolvent, the extinction probability and the mean extinction time are explored. Then, we apply a decomposition theorem to resume the stopped queue process back to our initial queueing model, that is to find the expression of Q-resolvent. After that, the criteria for the recurrence and ergodicity are also explored, and then, the generating function of equilibrium distribution is obtained. Additionally, the Laplace transform of the mean queue length is presented. The hitting time behaviors including the hitting probability and the hitting time distribution are also established. Furthermore, the busy period distribution is also obtained by the expression of Laplace transform. To conclude the discussion of the queue properties, a special case that m = 3 for our queueing model is discussed. Furthermore, we consider the decay parameter and decay properties of our initial queue process. First of all, similarly we consider the case of the stopped queue process Q∗. Based on this q-matrix, the exact value of the decay parameter λC is obtained theoretically. Then, we apply this result back to our initial queue model and find the decay parameter of our initial queueing model. More specifically, we prove that the decay parameter can be expressed accurately. After that, under the assumption of transient Q, the criteria for λC -recurrence are established. For λC -positive recurrent examples, the generating function of the λC-invariant measure and vector are explored. Finally, a simple example is provided to end this thesis

Quasi-stationary distributions

This paper contains a survey of results related to quasi-stationary distributions, which arise in the setting of stochastic dynamical systems that eventually evanesce, and which may be useful in describing the long-term behaviour of such systems before evanescence. We are concerned mainly with continuous-time Markov chains over a finite or countably infinite state space, since these processes most often arise in applications, but will make reference to results for other processes where appropriate. Next to giving an historical account of the subject, we review the most important results on the existence and identification of quasi-stationary distributions for general Markov chains, and give special attention to birth-death processes and related models. Results on the question of whether a quasi-stationary distribution, given its existence, is indeed a good descriptor of the long-term behaviour of a system before evanescence, are reviewed as well. The paper is concluded with a summary of recent developments in numerical and approximation methods

Queue methods for variability in congested traffic

Time-dependent queue methods are extended to calculate variances of stochastic queues along with their means, and thereby provide a tool for evaluation and better understanding of travel time variability and reliability in congested traffic networks and other systems, including through probability distributions estimated from moments. Objectives include developing computationally efficient analytical methods, and achieving robustness by reflecting the underlying structure of queuing systems rather than relying on statistical fitting, New deterministic and equilibrium formulae for queue variance are developed, acting also as constraints on estimating time-dependent queues generated by a range of processes, enabling improved accuracy and reliability estimates. New methods for approximating equilibrium and dynamic probability distributions use respectively doubly-nested geometric distributions and exponentially-weighted combinations of exponential and Normal functions, avoiding the need to rely on empirical functions, costly simulation, or equilibrium distributions inappropriate in dynamic cases. For growing queues, corrections are made to the popular sheared approximation, that combines deterministic and Pollaczek-Khinchin equilibrium mean formulae in one time-dependent function. For decaying queues, a new exponential approximation is found to give better results, possibly through avoiding implicit quasi-static assumption in shearing. Predictions for M/M/1 (yield) and M/D/1 (signal) processes applied to 34 oversaturated peaks show good agreement when tested against Markov simulations based on recurrence relations. Looking to widen the range of queues amenable to time-dependent methods, dependence of stochastic signal queues on green period capacity is confirmed by an extended M/D/1 process, for which new formulae for equilibrium moments are obtained and compared with earlier approximations. A simple formulation of queuing on multiple lanes with shared service is developed, two-lane examples with turning movements showing fair match to simulation. The main new methods are implemented in a spreadsheet demonstrator program, incorporating a database of time-sliced peak cases together with a procedure for estimating dynamic probability distributions from moments

Vehicle-based modelling of traffic . Theory and application to environmental impact modelling

This dissertation addresses vehicle-based approaches to traffic flow modelling. Having regard to the inherent dynamic nature of traffic, the investigations are mainly focused on the question, how this is captured by different model classes. In the first part, the dynamics of a microscopic car-following model (SKM), presented in, is studied by means of computer simulations and analytical calculations. A classification of the model's behaviour is given with respect to the stability of high-flow states and the outflow from jam. The effects of anticipatory driving on the model's dynamics is explored, yielding results valid in general for this model class. In the second part, a new approach is introduced based on queueing theory. It can be regarded as a microscopic implementation of a state-dependent queueing model, using coupled queues where the service rates additionally depend on the conditions downstream. The concept is shown to reproduce the dynamics of free flow and wide-moving jams. This is demonstrated by comparison with the SKM and real world measurements. An analytical treatment is given as well. The phenomena of boundary induced phase transitions is further addressed, giving the complete phase diagrams of both models. Finally, the application of the queueing approach within simulation-based traffic assignment is demonstrated in regard to environmental impact modelling

Decay properties and quasi-stationary distributions for stopped Markovian bulk-arrival and bulk-service queues

We consider decay properties including the decay parameter, invariant measures and quasi-stationary distributions for a Markovian bulk-arrival and bulk-service queue which stops when the waiting line is empty. Investigating such a model is crucial for understanding the busy period and other related properties of the Markovian bulk-arrival and bulk-service queuing processes. The exact value of the decay parameter λ C is first obtained. We show that the decay parameter can be easily expressed explicitly. The invariant measures and quasi-distributions are then revealed. We show that there exists a family of invariant measures indexed by λ ∈ [0,λ C]. We then show that under some mild conditions there exists a family of quasi-stationary distributions also indexed by λ ∈ [0,λ C]. The generating functions of these invariant measures and quasi-stationary distributions are presented. We further show that this stopped Markovian bulk-arrival and bulk-service queueing model is always λ C-transient. Some deep properties regarding λ C-transience are examined and revealed. The clear geometric interpretation of the decay parameter is explained. A few examples are then provided to illustrate the results obtained in this paper. © 2010 Springer Science+Business Media, LLC.link_to_subscribed_fulltex