This dissertation analyzes two unsolved problems to fulfill the gap in the literature: (1). What is the vehicle delay and intersection capacity considering left-turn traffic at a pre-timed signal? (2). What are the mean and variance of delay to vehicles at a vehicle-actuated signal? The first part of this research evaluates the intersection performance in terms of capacity and delay at an isolated pre-timed signal intersection. Despite of a large body of literature on pre-timed signals, few work has examined the interactions be- tween left-turn and through vehicles. Usually a protected left-turn signal phase, before (leading) or after (lagging) through signal, is applied to a signalized inter- section when the traffic demand is relatively high. A common problem for leading left-turn operation is the blockage to left-turn vehicles by through traffic, particularly at an intersection with a short left-turn bay. During the peak hour, some vehicles on the through lane might not be able to depart at the end of a cycle, resulting in an increased probability of left-turn blockage. In turn, the blocked left-turn vehicles may also delay the through traffic to enter the intersection during the following cycle. Those problems may not exist for a lagging left-turn operation, since left-turn vehicles intend to spill out of the bay under heavy traffic. In this case, the through capacity is reduced, leading to an increase of total delay. All of these factors contribute to the difficulties of estimating the delay and capacity for an isolated intersection. In order to examine this missing part of study on the signalized intersection, two probabilistic models are proposed to deal with the left-turn bay blockage and queue spillback in a heuristic manner. Numerical case studies are also provided to test the proposed models. The second part of this research studies an isolated intersection with vehicle-actuated signal. Typically an advanced detector is located at a distance prior to the intersection such that an arriving vehicle triggers a green time extension in or- der to pass through without any stop. This extended time period actuated by the vehicle is called unit extension in this study. If no vehicle actuation occurs during a unit extension, the green phase would terminate in order to clear queues in other approaches. In this way, the actuated system dynamically allocates the green time among multiple approaches according to vehicle arrivals. And the unit extension is the only control parameter in this case. We develop a model to study the vehicle delay under a general arrival distribution with a given unit extension. Our model allows optimizing the intersection performance over the unit extension. The third part of this research applies graphical methods and diffusion approximations to the traffic signal problems. We reinterpret a graphical method which is originally proposed by Newell in order to directly measure the variance of the time for the queue clearance at a signalized intersection, which remains yet to be carefully examined in practice and would be rather challenging if only using the conventional queuing techniques. Our results demonstrate that graphical method explicitly presents both the deterministic and stochastic delay. We also illustrate that the theoretical background for the graphical methods in this particular application is inherently the diffusion approximation. Furthermore, we investigate the problems of disruptions occurred during a pre-timed traffic signal cycle. By diffusion approximation, we provide quantitative estimation on the duration that the effects of disruptions would dissipate.