arXiv subjects

Marko Boon

Publications and source records attributed to Marko Boon.

At least 19 recordsLinked to original sources

arXiv2022

Estimating Probability Distributions of Travel Times by Fitting a Markovian Velocity Model

To improve the routing decisions of individual drivers and the management policies designed by traffic operators, one needs reliable estimates of travel time distributions. Since congestion caused by both recurrent patterns (e.g., rush hours) and non-recurrent events (e.g., traffic incidents) leads to potentially substantial delays in highway travel times, we focus on a framework capable of incorporating both effects. To this end, we propose to work with the Markovian Velocity model (MVM), based on an environmental background process that tracks both random and (semi-)predictable events affecting the vehicle speeds in a highway network. We show how to operationalize this flexible data-driven model in order to obtain the travel time distribution for a vehicle departing at a known day and time to traverse a given path. Specifically, we detail how to structure the background process and set the speed levels corresponding to the different states of this process. First, for the inclusion of non-recurrent events, we study incident data to describe the random durations of the incident and inter-incident times. Both of them depend on the time of day, but we identify periods in which they can be considered time-independent. Second, for an estimation of the speed patterns in both incident and inter-incident regime, loop detector data for each of the identified periods is studied. In numerical examples that use road network detector data of the Dutch highway network, we obtain the travel time distribution estimates that arise under different traffic regimes, and illustrate the advantages compared to traditional travel-time prediction methods.

physics.soc-ph↗

arXiv2021

A Framework for Efficient Dynamic Routing under Stochastically Varying Conditions

Despite measures to reduce congestion, occurrences of both recurrent and non-recurrent congestion cause large delays in road networks with important economic implications. Educated use of Intelligent Transportation Systems (ITS) can significantly reduce travel times. We focus on a dynamic stochastic shortest path problem: our objective is to minimize the expected travel time of a vehicle, assuming the vehicle may adapt the chosen route while driving. We introduce a new stochastic process that incorporates ITS information to model the uncertainties affecting congestion in road networks. A Markov-modulated background process tracks traffic events that affect the speed of travelers. The resulting continuous-time routing model allows for correlation between velocities on the arcs and incorporates both recurrent and non-recurrent congestion. Obtaining the optimal routing policy in the resulting semi-Markov decision process using dynamic programming is computationally intractable for realistic network sizes. To overcome this, we present the EDSGER* algorithm, a Dijkstra-like shortest path algorithm that can be used dynamically with real-time response. We develop additional speed-up techniques that reduce the size of the network model. We quantify the performance of the algorithms by providing numerical examples that use road network detector data for The Netherlands.

math.OC↗

arXiv2021

Optimal capacity allocation for heavy-traffic fixed-cycle traffic-light queues and intersections

Setting traffic light signals is a classical topic in traffic engineering, and important in heavy-traffic conditions when green times become scarce and longer queues are inevitably formed. For the fixed-cycle traffic-light queue, an elementary queueing model for one traffic light with cyclic signaling, we obtain heavy-traffic limits that capture the long-term queue behavior. We leverage the limit theorems to obtain sharp performance approximations for one queue in heavy traffic. We also consider optimization problems that aim for optimal division of green times among multiple conflicting traffic streams. We show that inserting heavy-traffic approximations leads to tractable optimization problems and close-to-optimal signal prescriptions. The same type of limiting result can be established for several vehicle-actuated strategies which adds to the general applicability of the framework presented in this paper.

math.OC↗

arXiv2021

Dynamic Appointment Scheduling

This paper considers appointment scheduling in a setting in which at every client arrival the schedule of all future clients can be adapted. Starting our analysis with an explicit treatment of the case of exponentially distributed service times, we then develop a phase-type-based approach to also cover cases in which the service times' squared coefficient of variation differs from 1. The approach relies on dynamic programming, with the state information being the number of clients waiting, the elapsed service time of the client in service, and the number of clients still to be scheduled. The use of dynamic schedules is illustrated through a set of numerical experiments, showing (i) the effect of wrongly assuming exponentially distributed service times, and (ii) the gains (over static schedules, that is) achieved by rescheduling.

math.PR↗

arXiv2020

Containment strategies after the first wave of COVID-19 using mobility data

In their response to the COVID-19 outbreak, governments face the dilemma to balance public health and economy. Mobility plays a central role in this dilemma because the movement of people enables both economic activity and virus spread. We use mobility data in the form of counts of travelers between regions, to extend the often-used SEIR models to include mobility between regions. We quantify the trade-off between mobility and infection spread in terms of a single parameter, to be chosen by policy makers, and propose strategies for restricting mobility so that the restrictions are minimal while the infection spread is effectively limited. We consider restrictions where the country is divided into regions, and study scenarios where mobility is allowed within these regions, and disallowed between them. We propose heuristic methods to approximate optimal choices for these regions. We evaluate the obtained restrictions based on our trade-off. The results show that our methods are especially effective when the infections are highly concentrated, e.g., around a few municipalities, as resulting from superspreading events that play an important role in the spread of COVID-19. We demonstrate our method in the example of the Netherlands. The results apply more broadly when mobility data is available.

physics.soc-ph↗

arXiv2019

A Novel Data-driven Algorithm for the Automated Detection of Unexpectedly High Traffic Flow in Uncongested Traffic States

We present an algorithm to identify days that exhibit the seemingly paradoxical behaviour of high traffic flow and, simultaneously, a striking absence of traffic jams. We introduce the notion of high-performance days to refer to these days. The developed algorithm consists of three steps: step 1, based on the fundamental diagram (i.e. an empirical relation between the traffic flow and traffic density), we estimate the critical speed by using robust regression as a tool for labelling congested and uncongested data points; step 2, based on this labelling of the data, the breakdown probability can be estimated (i.e. the probability that the average speed drops below the critical speed); step 3, we identify unperturbed moments (i.e. moments when a breakdown is expected, but does not occur) and subsequently identify the high-performance days based on the number of unperturbed moments. Identifying high-performance days could be a building block in the quest for traffic jam reduction; using more detailed data one might be able to identify specific characteristics of high-performance days. The algorithm is applied to a case study featuring the highly congested A15 motorway in the Netherlands.

physics.soc-ph↗

arXiv2018

Heavy-traffic analysis of the $M^X/\text{semi-Markov}/1$ queue

In this paper we analyze a single server queue with batch arrivals and semi-Markovian service times. We also include the feature that the first service of each busy period might have a different distribution than subsequent service times. Our generating function based approach allows us to determine the heavy traffic limit of the scaled queue-length distribution. It turns out that this distribution converges to an exponential distribution. Nonsurprisingly, the exceptional first service does not influence this limiting distribution. We identify a sufficient and necessary condition under which the dependence between successive service times disappears in the limit, which we illustrate in a numerical example.

math.PR↗

arXiv2018

Applications of the $M^X/\text{semi-Markov}/1$ queue to road traffic

The single server queue with multiple customer types and semi-Markovian service times, sometimes referred to as the $M/SM/1$ queue, has been well-studied since its introduction by Neuts in 1966. In this paper, we apply an extension of this model, with batch arrivals and exceptional first service, to road traffic situations involving multiple streams of conflicting traffic. In particular, we use it in the context of gap acceptance models where low-priority traffic needs to cross (or, depending on the application, merge with) another traffic flow of higher priority. Traditionally, gap acceptance models are based on the $M/G/1$ queue with exceptional first service, in this application area commonly referred to as the $M/G2/1$ queue. In an earlier study, we showed how the $M^X/SM/1$ queue with exceptional first service can be applied in this context to extend the model with driver impatience and more realistic merging behaviour. In this paper, we show how this same queueing system can be used to model a Markov modulated Poisson arrival process of the high-priority traffic stream. Due to its flexibility, this arrival process is very relevant in this application, particularly because it allows the modelling of platoon forming of vehicles. The correlated inter-arrival times of these high priority vehicles cause the merging times of two subsequent low priority vehicles to become dependent as well (as they correspond with the service times in the underlying queueing model). We derive the waiting time and sojourn time distributions of an arbitrary customer, showing that these depend on the position of the customer inside the batch, as well as on the type of the first customer in the batch.

math.PR↗

arXiv2018

Generalized gap acceptance models for unsignalized intersections

This paper contributes to the modeling and analysis of unsignalized intersections. In classical gap acceptance models vehicles on the minor road accept any gap greater than the CRITICAL gap, and reject gaps below this threshold, where the gap is the time between two subsequent vehicles on the major road. The main contribution of this paper is to develop a series of generalizations of existing models, thus increasing the model's practical applicability significantly. First, we incorporate {driver impatience behavior} while allowing for a realistic merging behavior; we do so by distinguishing between the critical gap and the merging time, thus allowing MULTIPLE vehicles to use a sufficiently large gap. Incorporating this feature is particularly challenging in models with driver impatience. Secondly, we allow for multiple classes of gap acceptance behavior, enabling us to distinguish between different driver types and/or different vehicle types. Thirdly, we use the novel M$^X$/SM2/1 queueing model, which has batch arrivals, dependent service times, and a different service-time distribution for vehicles arriving in an empty queue on the minor road (where `service time' refers to the time required to find a sufficiently large gap). This setup facilitates the analysis of the service-time distribution of an arbitrary vehicle on the minor road and of the queue length on the minor road. In particular, we can compute the MEAN service time, thus enabling the evaluation of the capacity for the minor road vehicles.

math.PR↗

arXiv2018

Congestion analysis of unsignalized intersections: The impact of impatience and Markov platooning

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 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 stability. The contributions of this paper are threefold. First, we obtain new results for the aforementioned model that includes 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 \emph{mean} gap size; instead more detailed characteristics of the critical headway distribution play a crucial role. The third contribution is the introduction of a new form of bunching on the main road, called Markov platooning. The tractability of this model allows us to study the impact of various platoon formations on the main road on the capacity of the minor road.

math.OC↗

arXiv2018

Congestion analysis of unsignalized intersections

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 \emph{mean} gap size; instead more detailed characteristics of the critical headway distribution play a role.

math.OC↗

arXiv2018

A single server queue with batch arrivals and semi-Markov services

We investigate the transient and stationary queue-length distributions of a class of service systems with correlated service times. The classical $M^X/G/1$ 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 $J(\cdot)$ at a service initiation time determines the joint distribution of the subsequent service duration and the state of $J(\cdot)$ 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 {\em 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.

math.PR↗

arXiv2017

Queue-length balance equations in multiclass multiserver queues and their generalizations

A classical result for the steady-state queue-length distribution of single-class queueing systems is the following: the distribution of the queue length just before an arrival epoch equals the distribution of the queue length just after a departure epoch. The constraint for this result to be valid is that arrivals, and also service completions, with probability one occur individually, i.e., not in batches. We show that it is easy to write down somewhat similar balance equations for {\em multidimensional} queue-length processes for a quite general network of multiclass multiserver queues. We formally derive those balance equations under a general framework. They are called distributional relationships, and are obtained for any external arrival process and state dependent routing as long as certain stationarity conditions are satisfied and external arrivals and service completions do not simultaneously occur. We demonstrate the use of these balance equations, in combination with PASTA, by (i) providing very simple derivations of some known results for polling systems, and (ii) obtaining new results for some queueing systems with priorities. We also extend the distributional relationships for a non-stationary framework.

math.PR↗

arXiv2017

Pollaczek contour integrals for the fixed-cycle traffic-light queue

The fixed-cycle traffic-light (FCTL) queue is the standard model for intersections with static signaling, where vehicles arrive, form a queue and depart during cycles controlled by a traffic light. Classical analysis of the FCTL queue based on transform methods requires a computationally challenging step of finding the complex-valued roots of some characteristic equation. Building on the recent work of Oblakova et al. (Exact expected delay and distribution for the fixed-cycle traffic-light model and similar systems in explicit form, 2016), we obtain a contour-integral expression, reminiscent of Pollaczek integrals for bulk-service queues, for the probability generating function of the steady-state FCTL queue. We also show that similar contour integrals arise for generalizations of the FCTL queue introduced in Oblakova et al. (2016) that relax some of the classical assumptions. Our results allow to compute the queue-length distribution and all its moments using algorithms that rely on contour integrals and avoid root-finding procedures.

math.PR↗

arXiv2016

Networks of fixed-cycle intersections

We present an algorithmic method for analyzing networks of intersections with static signaling, with as primary example a line network that allows traffic flow over several intersections in one main direction. The method decomposes the network into separate intersections and treats each intersection in isolation using an extension of the fixed-cycle traffic-light (FCTL) queue. The network effects are modeled by matching the output process of one intersection with the input process of the next (downstream) intersection. This network analysis provides insight into wave phenomena due to vehicles experiencing progressive cascades of green lights and sheds light on platoon forming in case of imperfections. Our algorithm is shown to match results from extensive discrete-event simulations and can also be applied to more complex network structures.

math.PR↗

arXiv2016

Heavy traffic analysis of roving server networks

This paper studies the heavy-traffic (HT) behaviour of queueing networks with a single roving server. External customers arrive at the queues according to independent renewal processes and after completing service, a customer either leaves the system or is routed to another queue. This type of customer routing in queueing networks arises very naturally in many application areas (in production systems, computer- and communication networks, maintenance, etc.). In these networks, the single most important characteristic of the system performance is oftentimes the path time, i.e. the total time spent in the system by an arbitrary customer traversing a specific path. The current paper presents the first HT asymptotic for the path-time distribution in queueing networks with a roving server under general renewal arrivals. In particular, we provide a strong conjecture for the system's behaviour under HT extending the conjecture of Coffman et al. [E.G. Coffman Jr., A.A. Puhalskii, M.I. Reiman 1995 and 1998] to the roving server setting of the current paper. By combining this result with novel light-traffic asymptotics we derive an approximation of the mean path-time for arbitrary values of the load and renewal arrivals. This approximation is not only highly accurate for a wide range of parameter settings, but is also exact in various limiting cases.

math.PR↗

arXiv2016

Design Heuristic for Parallel Many Server Systems under FCFS-ALIS

We study a parallel queueing system with multiple types of servers and customers. A bipartite graph describes which pairs of customer-server types are compatible. We consider the service policy that always assigns servers to the first, longest waiting compatible customer, and that always assigns customers to the longest idle compatible server if on arrival, multiple compatible servers are available. For a general renewal stream of arriving customers and general service time distributions, the behavior of such systems is very complicated. In particular, the calculation of matching rates, the fraction of services of customer-server type, is intractable. We suggest through a heuristic argument that if the number of servers becomes large, the matching rates are well approximated by matching rates calculated from the tractable bipartite infinite matching model. We present simulation evidence to support this heuristic argument, and show how this can be used to design systems with desired performance requirements.

math.PR↗

arXiv2015

Critically loaded k-limited polling systems

We consider a two-queue polling model with switch-over times and $k$-limited service (serve at most $k_i$ customers during one visit period to queue $i$) in each queue. The major benefit of the $k$-limited service discipline is that it - besides bounding the cycle time - effectuates prioritization by assigning different service limits to different queues. System performance is studied in the heavy-traffic regime, in which one of the queues becomes critically loaded with the other queue remaining stable. By using a singular-perturbation technique, we rigorously prove heavy-traffic limits for the joint queue-length distribution. Moreover, it is observed that an interchange exists among the first two moments in service and switch-over times such that the HT limits remain unchanged. Not only do the rigorously proven results readily carry over to $N$($\geq2$) queue polling systems, but one can also easily relax the distributional assumptions. The results and insights of this note prove their worth in the performance analysis of Wireless Personal Area Networks (WPAN) and mobile networks, where different users compete for access to the shared scarce resources.

math.PR↗