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A C O M PA R I S O N O F S TAT I C A N D D Y N A M I C T R A F F I C A S S I G N M E N T U N D E R T O L L S 115 This function represents a time rate of demand, so of the flowdensity curve when flow is increasing or integrating over an interval gives the number of vehicles decreasing with volume, corresponding to the cases assigned in that interval. Because the demand function is when density is either less than or greater than the criti- piecewise linear, these integrals can be calculated using cal density, respectively. basic geometric formulas to compute the area under the curve. The curve itself is generated using these formulas: starting from a seed value, successive linear segments are COMPARING STATIC AND DYNAMIC determined to ensure that the correct number of vehicles TRAFFIC ASSIGNMENT is assigned for the major periods given as input to the algorithm, and such that the curve is everywhere non- It is difficult to compare STA with DTA because typical negative. Several curves are generated using different measures of comparison, such as volumes on individual seed values, and these are averaged to minimize the links or total system travel time (TSTT), cannot be read- impact of artifacts unique to particular seeds. ily applied due to fundamental differences between the modeling approaches. Moreover, the behavioral assump- tions are so different that parameter assumptions are not THE VISTA MODEL AND TRANSCAD'S particularly comparable, either. DTA APPROXIMATOR Clearance intervals in the DTA approximator show shorter travel times than static assignment. Clearance This work compares the results of an STA model to two intervals account for vehicles departing near the end of DTA implementations: VISTA and an add-in to Trans- the model period, and thus arriving at their destinations CAD software. beyond the model period: during these intervals, no addi- VISTA is a network-enabled software that integrates tional vehicles are assigned, but vehicles remaining on temporal network data and models for a wide range of the network are allowed to complete their trips. This transport applications. In particular, VISTA can perform results in some links experiencing flows for a longer time using a cell transmission model (CTM), a traffic flow than in STA, and an effective increase in link capacities. model developed by Daganzo (1994) as a discrete ver- This does not occur in STA because static methods are sion of the hydrodynamic traffic flow model. The CTM unable to determine when vehicles depart, and thus divides links into smaller cells, which can then be mod- assume steady-state conditions. eled individually at a fine resolution, on the order of 5 to Thus, to enable comparison, link capacities were 10 s. increased commensurate with the additional clearance A unique feature of the CTM is that flows cannot time needed for DTA. In essence, this extends the period exceed capacity; queues form to maintain flow. As vol- of analysis in STA to correspond to the added time pro- ume increases, travel time in a cell is constant until criti- vided for queue clearance in the approximator, eliminat- cal density is exceeded, after which point travel times ing the bias that exists in a direct comparison of the two. increase rapidly, corresponding to free-flow and con- Comparing STA and VISTA results is even more diffi- gested conditions, respectively. cult because the CTM used by VISTA is distinct. There- The DTA approximator is an add-in to the TransCAD fore, global measures of comparison were used. software package, and is based on an iterative algorithm Individual link flows are not comparable because of the developed by Janson and Robles (1995). Much like STA vast differences between the assignment procedures, and and unlike VISTA, it uses link performance functions to measures such as volume/capacity (v/c) ratios have dif- calculate vehicle delay. Although such functions are less ferent meanings: in VISTA, v/c is the ratio of actual flow computationally intensive, and the approximator runs to capacity and cannot exceed 1; in static assignment, v/c more quickly than VISTA, they cannot model traffic flow is the ratio of link demand to capacity (which can exceed as closely as the CTM: for instance, interaction between 1). This distinction makes v/c comparisons meaningless. links (such as queues that spill into upstream links) is not Thus, the total travel time for each of five functional modeled in the approximator, and flow on a link always classes of roadways (freeways, arterials, and so forth) increases with volume, even beyond the nominal was compared, as was the total system travel time for the capacity. entire network. Parameters used by the two models are quite differ- ent, however. The link performance functions used by STA and the DTA approximator require capacity and RESULTS free-flow time to be specified for each link, along with two calibration parameters. The CTM, on the other The DFW network contains 919 zones, 15,987 nodes, and hand, requires specification of jam density and length for 56,574 links (92 of which are tolled in this application). A each cell, as well as two parameters indicating the slopes 3-h peak period (6:00 to 9:00 a.m.) was chosen for analy-