Analysing queueing at toll plazas using a coupled, multiple-queue, queueing system model: application to toll plaza design

A vehicle approaching a toll plaza observes the queues at each of the available toll-lanes before choosing which to join. This choice process, the arrival process of vehicles and the service characteristics of the toll-booths, affect the queues and delay the drivers. In this paper, queueing at a toll plaza is modelled as a multiple-queue queueing system where the arrival process to a queue (toll-lane) is dependent on the state of all the queues. In the past, such systems have been modelled mathematically only for two queues and are not applicable for toll plazas with three or more toll-lanes. The proposed model determines the steady-state probability density function (pdf) for the queues at large toll plazas. This study is used to determine the number of toll-lanes or the length of the upstream queueing area required to achieve certain user-specified levels-of-service. Expected delay and maximum queue length are used as level-of-service measures. Indicative design charts are also provided.