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136 ply (e.g., due to incidents, road work, weather conditions, and road geometry) on freeways; on urban arterials, travel time is significantly influenced by-besides fluctuations in traffic demand and supplytraffic control at, stochastic arrivals to, and departures from the intersection. Many of these factors are stochastic, which results in variable travel times. However, a complete and valid model to predict travel times that takes into account all these stochastic influencing factors seems unfeasible so far. In addition, most research done toward examining the variability in travel times has concentrated on long-term variability (e.g., peak hours, off-peak hours, and daily and seasonal variability). Traffic control has a special effect on the predictability of travel times. Two consecutive vehicles that enter the network at nearly the same time can have completely different travel times when the first vehicle just passes an intersection at the end of the green phase and the following vehicle has to stop. This may have an impact on delays at the next intersections, so that the first vehicle may have a much shorter travel time than the second one. This phenomenon, where small differences in the initial state can have large differences in the final condition, is called "bifurcation." Systems with bifurcations are ill-predictable.He et al. (7) examined the short-term temporal-spatial variations and correlations of travel times among urban links by using vehicletracking data. The results indicated strong evidence of significant correlation between link travel time and arrival time. The variability of travel time as a function of traffic demand was proposed by Liu et al. (8). Individual travel times were used to construct travel time distributions at different traffic demand levels. The fuzzy k-means method was applied to classify traffic patterns based on traffic flows. One limitation of this method is that it is data driven, and there is no analytical analysis to support the underlining traffic process on the urban road network. As Viti and van Zuylen (9) discussed, the stochastic delays at the signalized intersection constitute a large part of travel times on urban links. Understanding of the vehicle delay evolution or delay variability at signalized intersections can lead to more insights into the variability of urban link travel times and give more possibilities for travel time estimation and prediction.Delays at signalized intersections have been widely investigated, and researchers have taken great effort to approximate the analytical expression for the mean delay [Webster and Cobbe (10), Miller (11), Newell (12), and Akcelik (13)]. Much less work has been done to quantify delay variability at signalized intersections. Investigation into the delay variability can be done in two main ways: (a) by modeling the variance of delay and (b) by obtaining the delay distribution. To quantify the delay variability in an analytical way, Fu developed a model to predict the variance of overall delay (14). The methodology...

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