Linear stability of a generalized multi-anticipative car following model with time delays

In traffic flow, the multi-anticipative driving behavior describes the reaction of a vehicle to the driving behavior of many vehicles in front where as the time delay is defined as a physiological parameter reflecting the period of time between perceiving a stimulus of leading vehicles and performing a relevant action such as acceleration or deceleration. A lot of effort has been undertaken to understand the effects of either multi-anticipative driving behavior or time delays on traffic flow dynamics. This paper is a first attempt to analytically investigate the dynamics of a generalized class of car-following models with multi-anticipative driving behavior and different time delays associated with such multi-anticipations. To this end, this paper puts forwards to deriving the (long-wavelength) linear stability condition of such a car-following model and study how the combination of different choices of multi-anticipations and time delays affects the instabilities of traffic flow with respect to a small perturbation. It is found that the effect of delays and multi-anticipations are model-dependent, that is, the destabilization effect of delays is suppressed by the stabilization effect of multi-anticipations. Moreover, the weight factor reflecting the distribution of the driver's sensing to the relative gaps of leading vehicles is less sensitive to the linear stability condition of traffic flow than the weight factor for the relative speed of those leading vehicles.