Journal Articles (Peer-Reviewed)
Abhishek, Boon, Marko A. A., Onno J. Boxma and Rudesindo Núñez-Queija (2017): A single-server queue with batch arrivals and semi-Markov services, Queueing Systems, 86 (3-4): 217-240.
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.
Abhishek, Michel Mandjes, Boon, Marko A. A. and Rudesindo Núñez-Queija (2016): Congestion analysis of unsignalized intersections: Proceedings of 8th International Conference on Communication Systems and Networks (COMSNETS): Bangalore, India.
Abstract: This paper considers an unsignalized intersection used by two traffic streams. A stream of cars is using a primary road, and has priority over the other, low-priority, stream. Cars belonging to the latter stream cross the primary road if the gaps between two subsequent cars on the primary road is larger than their critical headways. Questions that naturally arise are: given the arrival pattern of the cars on the primary road, what is the maximum arrival rate of low-priority cars such that the number of such cars remains stable? In the second place, what can be said about the delay experienced by a typical car at the secondary road? This paper addresses such issues 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 above stability and delay issues. The contribution of this paper is twofold. First, we obtain new results for the aforementioned model with driver impatience. Secondly, we reveal some surprising aspects that have remained unobserved in the existing literature so far, many of which are caused by the fact that the capacity of the minor road cannot be expressed in terms of the mean gap size; instead more detailed characteristics of the critical headway distribution play a role.