Traffic flows along an infinitely long straight two-lane road without intersections or visual obstructions. Vehicles travel along the road in both directions, each vehicle travelling at its desired speed until it catches up with a slower vehicle. Upon catching up it will immediately assume the speed of the slower vehicle until an opportunity to overtake arises. Thus because vehicles are conserved the only way that a queue can change length is to amalgamate with another queue, or for one of the trapped vehicles to overtake the head of the queue and draw away. The rate at which queues break up in one lane is dependent on the frequency and duration of the gaps between queues in the second lane. The p.d.f. for queue length in one lane is therefore dependent on the occurrence of gaps in the second lane. A transition matrix procedure is developed using this dependence to enable the p.d.f. for queue length to be evaluated conditional on these gaps. When this procedure is then applied to the two lanes alternately two sequences of p.d.f.'s are obtained. These sequences are shown to converge to limiting distributions which describe the conditions in the two lanes at equilibrium. A simple model is introduced to illustrate the technique. The parameters in the model are assigned fixed numerical values except for the parameters relating to the volume of traffic. Once the volume of traffic reaches a certain level two stable equilibrium states are found. A tentative explanation for this behaviour is given and a graphical method for representing the stability of the system is suggested. (a) for the covering abstract of the symposium please see irrd abstract no. 224453.
Samenvatting