We consider a Markovian queueing model for computing the traffic density and travel times in a city at a macroscopic scale during rush hour. Accounting for the speed/density relation of the macroscopic fundamental diagram of traffic flow, we assume that the service rates of the queueing model at hand are state-dependent. We focus on the fluid limit and obtain a set of differential equations that describe the evolution of the traffic density at the level of neighbourhoods. We also calculate the time-dependent travel times for specific flows in the city and consider the rational time-dependent choice between public and private transport, assuming that there is a congestion-free public alternative to private transportation. Numerical examples reveal that a small reduction in peak traffic can significantly reduce the average travel times.

我们考虑使用马尔可夫排队模型来计算高峰时段宏观尺度的城市交通密度和旅行时间。考虑到交通流宏观基本图的速度/密度关系，我们假设手头的排队模型的服务率是状态相关的。我们专注于流体限制并获得一组微分方程，这些方程描述了社区级别的交通密度演变。我们还计算了城市中特定流量的时间相关旅行时间，并考虑了公共交通和私人交通之间的合理时间相关选择，假设有一个无拥堵的公共交通替代私人交通。数值例子表明，高峰流量的小幅减少可以显着减少平均旅行时间。