A Model of a Traffic Jam Behind a Bottleneck

This paper discusses a model for the stochastic motion of the slow-down points of cars behind a bottleneck. The model considered assumes cars to have nonzero length, and employs a simple “car-following-slowdown rale.” The case where “gaps” between cars form a stationary reversible sequence of random variables is discussed, and the main result consists of an explicit expression for the steady-state distribution of the physical length of the stream of cars behind the bottleneck when “gaps” are independent and identically distributed (iid) with exponential tail and atom at the origin. When this atom has zero mass (i.e., the gaps are iid exponential) it turns out that the result yields the steady-state distribution of waiting time in an M / D /1 queue and agrees with the known result. When this atom is nonzero, the main result in the queuing context yields the steady-state distribution of waiting time in an M ′/ D /1 queue, where M ′ denotes an arrival stream with interarrival times being iid with exponential tail and atom at the origin.