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104 I N N O VAT I O N S I N T R AV E L D E M A N D M O D E L I N G , V O L U M E 2 they traverse their routes. The propagation depends on In most DTA models, the equilibrium solution is the posted speed for the links, saturation flow rates, and determined by first identifying a feasible or reasonable jam density for links. The values specified for these link path set, then allocating flow between those paths to properties define a speedflowdensity relationship for cause the path times to be equal as per the definition the link to which simulated vehicles adhere. The result- given above. In Vista, a reasonable path set is determined ing effect is that at a link level, vehicles exhibit proper by solving the dynamic user-optimal equilibrium prob- traffic flow theory properties in that they form queues, lem with the method of successive averages (MSA) pro- and queues may spill back to other links. Link travel cedure. This involves iteratively solving the CTM and times therefore consist of time required to traverse a link TDSP, then averaging the time-dependent flow solution at posted speed plus time spent delayed in queues and with the solutions from previous iterations. The weight time delayed by traffic control devices. of the most recently calculated flows is 1/N, where N is Once the vehicle simulation has finished, it is possible the iteration number, and the weight of the previous to compute link travel times at any aggregation interval averaged flows is (1 1/N). The MSA solution converges desired. A typical Vista application would include a sim- very slowly toward an equilibrium solution, but each ulation time step of 6 s, meaning that the position of iteration provides the opportunity for new routes to be vehicles in cells is updated every 6 s. The travel times on determined subject to traffic conditions established as links could then be computed for every link at every 6-s the combined effect of the other iterations. interval, or they could be computed at longer intervals Once the reasonable path set is determined, an alloca- where an average of the 6-s times would be calculated tion mechanism can be used to achieve a more exact and stored. equilibrium solution over the fixed set of reasonable Given time-dependent (by link aggregation interval) paths. Vista uses a methodology called simplicial decom- link travel times, a TDSP algorithm can be used to calcu- position, which, for any origindestinationtime interval late routes through the network. The TDSP algorithm set, causes flows from higher travel-time routes to be works much like the conventional shortest-path algo- apportioned to lower travel-time routes, and conversely, rithms used in static traffic assignment methods, except lower travel-time flows shifted to higher travel-time that the link times have a time index. At each step in the routes. The solution procedure results in a set of time- algorithm, as a link is being considered for inclusion in the dependent link flows and route times corresponding to shortest path, the criteria used includes the travel time on the dynamic user-optimal conditions. (The solution is the link at the current accumulated time along the path. In not a pure equilibrium solution. Routes with shorter other words, for a specific assignment interval, if a link is travel times may exist that were not identified in the 60 s of accumulated link time from the origin along the MSA procedure. However, the solution is probably shortest path, the link is evaluated for inclusion in the nearly an equilibrium, and it is thought to be adequately shortest path based on the link time associated with the close.) aggregate time interval that corresponds with 60 s past the The implementation described here involved itera- beginning of the assignment interval. The result of the tively building a reasonable route set and solving the TDSP is a set of routes between every origin and destina- dynamic user-equilibrium for that route set. Following tion zone starting in every assignment time interval. each dynamic user-equilibrium solution, routes that had With the capability to simulate traffic and compute previously received vehicles but no longer did were time-dependent link travel times and TDSP, a DTA pruned from the route set and new reasonable routes model can compute a dynamic network equilibrium were determined with the approximate MSA solution solution. Typically, for planning studies, a dynamic user procedure. This was followed by solving for the more equilibrium is the desired outcome. For intelligent trans- precise dynamic user-equilibrium solution for the route portation system applications, one might be more inter- set. When one of these iterations produced only a small ested in a dynamic system-equilibrium solution. The number of new routes, the model was said to have con- dynamic equilibrium is usually defined by extension of verged sufficiently. the static user-equilibrium principle that states that no used route between an origin and a destination may have a higher travel time than any unused route. By extending ANALYSIS OF VISTA RESULTS FOR ATLANTA this principle to the time-dependent case, one arrives at a similar condition: at no time along a route from an ori- A DTA model of the Atlanta region is a large problem to gin to a destination can a traveler change to a different solve. Most DTA results published to date are for much route and lower his or her travel time. In other words, smaller problems. An attempt was made to reduce the the travel times for all used routes between an origin and problem size by limiting the demand loaded on the net- a destination starting during the same time interval are work. At first trips were simulated beginning between all equal at equilibrium. 6:00 and 7:00 a.m. with the intention of increasing