DOI: 10.1016/j.ejor.2018.09.049 

Abstract: This paper considers an unsignalized intersection used by two traffic streams. The first stream of cars is using a primary road, and has priority over the other stream. Cars belonging to the latter stream cross the primary road if the gaps between two subsequent cars on the primary road are larger than their critical headways. A question that naturally arises relates to the capacity of the secondary road: given the arrival pattern of cars on the primary road, what is the maximum arrival rate of low-priority cars that can be sustained? This paper addresses this issue by considering a compact model that sheds light on the dynamics of the considered unsignalized intersection. The model, which is of a queueing-theoretic nature, reveals interesting insights into the impact of the user behavior on the capacity. The contributions of this paper are threefold. First, we introduce a new way to analyze the capacity of the minor road. By representing the unsignalized intersection by an appropriately chosen Markovian model, the capacity can be expressed in terms of the solution of an elementary system of linear equations. The setup chosen is so flexible that it allows us to include a new form of bunching on the main road that allows for dependence between successive gaps, which we refer to as Markov platooning; this is the second contribution. The tractability of this model facilitates studying the impact that driver impatience and various platoon formations on the main road have on the capacity of the minor road. Finally, in numerical experiments we observe various surprising features of the aforementioned model. (published online first)

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DOI: 10.1007/s11134-017-9531-4 

Abstract: We investigate the transient and stationary queue length distributions of a class of service systems with correlated service times. The classical MX/G/1 queue with semi-Markov service times is the most prominent example in this class and serves as a vehicle to display our results. The sequence of service times is governed by a modulating process J(t). The state of J(⋅) at a service initiation time determines the joint distribution of the subsequent service duration and the state of J(⋅) at the next service initiation. Several earlier works have imposed technical conditions, on the zeros of a matrix determinant arising in the analysis, that are required in the computation of the stationary queue length probabilities. The imposed conditions in several of these articles are difficult or impossible to verify. Without such assumptions, we determine both the transient and the steady-state joint distribution of the number of customers immediately after a departure and the state of the process J(t) at the start of the next service. We numerically investigate how the mean queue length is affected by variability in the number of customers that arrive during a single service time. Our main observations here are that increasing variability may reduce the mean queue length, and that the Markovian dependence of service times can lead to large queue lengths, even if the system is not in heavy traffic.

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