Resumen de Mathematical modelling of a tollbooth system with two parallel skill-based servers and two vehicle types

Baoxian Chang, Qingqing Ye, Jun Lv, Tao Jiang

• This paper considers a tollbooth system with two parallel heterogeneous servers and two vehicle types (say, cars and trucks), where one server collects tolls from vehicles of both types and the other only serves one type of them. With such characteristic, the system is referred to as “skill-based servers” for brevity in this paper. Meanwhile, vehicles are accommodated in a single common lane and get served on the “global First-Come-First-Served” basis. In fact, such a system and its variants are commonly encountered and their performance measures are of great significance to managers. We first develop a Quasi-Birth-Death process to explicitly model this tollbooth system. Then by applying spectral expansion technique, we derive the stationary probabilities for the computation of the system’s performance measures, such as mean queue size and the idle probability of each server. The Laplace–Stieltjes transforms of an arbitrary vehicle’s sojourn time is derived as well. Finally, numerical results are presented to show the impact of parameters’ selection upon performance measures, and they have intriguing managerial implications. The results also reveal that the tollbooth system with skill-based servers is much more efficient compared to the system with dedicated servers, which has been studied by Mélange et al. (Comput Oper Res 71:23–33, 2016), especially in rush hours.