Recent work has proposed using aggregate relationships between urban traffic variables--- i.e., Macroscopic Fundamental Diagrams (MFDs)---to describe aggregate traffic dynamics in urban networks. This approach is particularly useful to unveil and explore the effects of various networkwide control strategies. The majority of modeling work using MFDs hinges upon the existence of well-defined MFDs without consideration of uncertain behaviors. However, both empirical data and theoretical analysis have demonstrated that MFDs are expected to be uncertain due to inherent instabilities that exist in traffic networks. Fortunately, sufficient amounts of adaptive drivers who re-route to avoid congestion have been proven to help eliminate the instability of MFDs. Unfortunately, drivers cannot re-route themselves adaptively all the time as routing choices are controlled by multiple factors, and the presence of adaptive drivers is not something that traffic engineers can control. Since MFDs have shown promise in the design and control of urban networks, it is important to seek another strategy to mitigate or eliminate the instability of MFDs. Furthermore, it is necessary to develop a framework to account for the uncertain phenomena that emerges on the macroscopic, network-wide level to address these unavoidable stochastic behaviors.;This first half of this work investigates another strategy to eliminate inherent network instabilities and produce more reliable MFDs that is reliable and controllable from an engineering perspective---the use of adaptive traffic signals. A family of adaptive signal control strategies is examined on two abstractions of an idealized grid network using an interactive simulation and analytical model. The results suggest that adaptive traffic signals should provide a stabilizing influence that provides more well-defined MFDs. Adaptive signal control also both increases average flows and decreases the likelihood of gridlock when the network is moderately congested. The benefits achieved at these moderately congested states increase with the level of signal adaptivity. However, when the network is extremely congested, vehicle movements become more constrained by downstream congestion and queue spillbacks than by traffic signals, and adaptive traffic signals appear to have little to no effect on the network or MFD. When a network is extremely congested, other strategies should be used to mitigate the instability, like adaptively routing drivers. Therefore, without sufficient amounts of adaptive drivers, the instability of MFDs could be somewhat controlled, but it cannot be eliminated completely. This is results in more reliable MFDs until the network enters heavily congested states.;The second half of this work uses stochastic differential equations (SDEs) to depict the evolutionary dynamics of urban network while accounting for unavoidable uncertain phenomena. General analytical solutions of SDEs only exist for linear functions. Unfortunately, most MFDs observed from simulation and empirical data follow non-linear functions. Even the most simplified theoretical model is piecewise linear with breakpoints that cannot be readily accommodated by the linear SDE approach. To overcome this limitation, the SDE well-known solutions are used to develop an approximate solution method that relies on the discretization of the continuous state space. This process is memoryless and results in the development of a computationally efficient Markov Chain (MC) framework. The MC model is also supported by a well-developed theory which facilitates the estimation of future states or steady state equilibrium conditions in a network that explicitly accounts for MFD uncertainty. Due to the fact that current formalization of Markov Chains is restricted with a countable state space, some assumptions which redefine the traffic state and stochastic dynamic process need to be set for the MC model application in dynamic traffic analysis. These assumptions could be sabotaged by inappropriate parameter selections, producing excessive errors in analytical solutions. Therefore, a parametric study is performed here to illustrate how to select two key parameters, i.e. bin size and time interval to optimize the MC models and minimize errors.;The major advantage of MC models is its wide flexibility, which has been demonstrated by showing how this method could well handle a wide variety of variables. A family of numerical tests are designed to include instability of MFD model, stochastic traffic demand, different city layouts and different forms of MFDs in the scenarios under static metering strategies. The results suggest that analytical solutions derived from MC models could accurately predict the future traffic state at any moment. Furthermore, the theoretical analysis also illustrates that Markov chains could easily model dynamic traffic control based on traffic state and pre-determined time-varying strategies by adjusting the transition matrix. Overall, the developed MC models are promising in the dynamic analysis of complicated urban network control under uncertainty for which simpler algebraic solutions do not exist.
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