The lagged cell-transmission model

Cell-transmission models of highway traffic are discrete versions of the si mple continuum (kinematic wave) model of traffic flow that are convenient f or computer implementation. They are in the Godunov family of finite differ ence approximation methods for partial differential equations. In a cell-tr ansmission scheme one partitions a highway into small sections (cells) and keeps track of the cell contents (number of vehicles) as time passes. The r ecord is updated at closely spaced instants (clock ticks) by calculating th e number of vehicles that cross the boundary separating each pair of adjoin ing cells during the corresponding clock interval. This average flow is the result of a comparison between the maximum number of vehicles that can be "sent" by the cell directly upstream of the boundary and those that can be "received" by the downstream cell. The sending (receiving) flow is a simple function of the current traffic de nsity in the upstream (downstream) cell. The particular form of the sending and receiving functions depends on the shape of the highway's flow-density relation, the proximity of junctions and on whether the highway has specia l (e.g., turning) lanes for certain (e.g., exiting) vehicles. Although the discrete and continuum models are equivalent in the limit of vanishingly sm all cells and clock ricks, the need for practically sized cells and clock i ntervals generates numerical errors in actual applications. This paper shows that the accuracy of the cell-transmission approach is enh anced if the downstream density that is used to calculate the receiving flo w(s) is read l clock intervals earlier than the current time, where l is a non-negative integer that should be chosen by means of a simple formula. Th e rationale for the introduction of this lag is explained in the paper. The lagged cell-transmission model is related (but not equivalent) to both God unov's first order method for general flow-density relations and Newell's e xact method for concave flow density relations. It is easier to apply and m ore general than the latter, and more accurate than the former. In fact, if the flow-density relation is triangular and the lag is chosen optimally, t hen the lagged cell-tansmission model is a conservative, second order, fini te difference scheme. As a result, very accurate results can be obtained wi th relatively large cells. Accuracy formulae and sample illustrations are p resented for both the triangular and the general case.