随机过程与排队论
大作业
姓名：李嘉文
学号：1150349310087 日期：2016-01-12 指导教师：石剑虹老师
The Application of Stochastic Process in
Transportation System
1. Intruduction
Economic and social factors haveprofound influences on the level and pattern of travel demand and the choices of travelerswithin a given transport infrastructure. They also impact on the ability of responsibleauthorities to fund the maintenance and improvement of infrastructure, and to conducteffective travel demand management and control policies. It is just at such stages of majorchange and uncertainty that those planning future transport policies most need support inmaking their decisions, but in general this is exactly when most of the modelling tools weadopt fail to offer support, with their assumptions based on either an unchanging world, orone in which the future follows deterministically from the present. Even in periods ofrelative economic/social stability, such assumptions are increasingly difficult to support;this is most notable in cities where continued demand growth has outpaced the expansionin capacity of the transport infrastructure, with the transport system highly sensitive todaily and seasonal fluctuations in demand and capacities.
The question then arises as to how we might develop modelling approaches to better deal with such situations. One approach to such problems is that of ‘worst-case planning’whereby the models suggest actions for a planner to take so as to minimize the impacts under a worst-case scenario.
At its simplestmost stripped down level the Stochastic Process SP) approach could besaid to comprise three main elements for representing the epoch-to-epoch changes in atransport system:
1. A learning model, to describe how travellers learn from their travel experiences in pasttime epochs.
2. A decision model, to describe how travellers make decisions, given their learntexperiences.
3. A supply model, to describe the experiences of travellers in a particular time epoch.
2. Model Establishment
The elements that described in the previous section are described by probability statements or probabilitydistributions, and when brought together they provide a single, self-consistent frameworkfor representing the mutual interactions between the uncertain components of thetransport system. Just as we demand of equilibrium transportation analysis, we can ask towhat extent this combination of elements may produce a well-defined and unique ‘output’(if the long-run is indeed what interests us), but whereas in equilibrium systems we referto a unique flow state, in the SP approach we refer to a unique probability distribution offlows. That is to say, the result of the modelling approach is to
provide the planner withprobability distributions, not with single point estimates.
2.1 Representation of time
Perhaps the most fundamental decision to be made in representing transportation systemdynamics is how to represent time. In order to understand this issue we need first todefine the application context of the class of modelling problems we shall consider, namelyto problems of transport planning over some given future planning horizon of weeks oryears (though often it may not be explicitly defined). This application domain distinguishesthem in particular from operational models that be used for optimizing short-term systemperformance over a periods of a few minutes or hours. Within this context, a centralelement of the models we shall consider will be the adaptive behaviour of travellers overtime, as they repeat the requirement to make certain travel decisions. For example, thismay be a traveller, or group of travellers, deciding on a residential or work location (adecision perhaps reviewed over periods of years). Alternatively, the traveller may be acommuter choosing a mode and route/service to work each morning, a decision whichmight be reviewed over days or weeks; even if this choice is stable, the decision tocontinue with a previous choice is then also being made, at least in our models if notconsciously by travellers. In the latter (commuting) example ’day-to-day dynamics’ hasbeen suggested as a term to capture this idea of repeated decision-making and adaptation.
2.2 Representation of state & distributions
Having restricted attention to discrete time processes, as explained in section 2.1, we moveon to the second element of Table 1, namely the definition of ",state",, and at this stage wecan begin to introduce some notation. We denote the discrete time epochs (betweenepochtime or day-to-day time) by the letter t (for t = 0, 1, 2,...), the state vectordescribing epoch t as x(t), and the state-space to which any state must belong as a set , i.e.x(t)(for t = 0, 1, 2,...). This compound state vector may contain several different kinds of‘entity’such as choices and memories of travellers, and experiences of traveltimesmeasured at some chosen level of aggregation, and describing within-epoch‘within-day time’) variations in time and space. The details are important, and we shalldelve into them in the remainder of the paper, but for the moment the key issue is that x(t)is a sufficient description in two respects:A1 if we know x(t) then we know (or can infer) everything we might want to know fromthe model for the purposes of design or evaluation; andA2 if we know x(t) then we have sufficient information to write down a probability lawthat determines the probabilities of all future states of the modelled transportationsystem for times t +1, t + 2, ....
Assumption A1 implies that model outputs are sufficient for the intended purpose, both interms of what they measure and their level of aggregation, but does not preclude thesubsequent application of sub-models to infer further outputs; for example, it may be thatx(t) itself does not contain a pollution level variable, but contains information onexplanatory variables (flows, speeds, etc.)