arXiv subjects

Michel Mandjes

Publications and source records attributed to Michel Mandjes.

At least 55 records · Page 3Linked to original sources

arXiv2024

Trusting: Alone and together

We study the problem of an agent continuously faced with the decision of placing or not placing trust in an institution. The agent makes use of Bayesian learning in order to estimate the institution's true trustworthiness and makes the decision to place trust based on myopic rationality. Using elements from random walk theory, we explicitly derive the probability that such an agent ceases placing trust at some point in the relationship, as well as the expected time spent placing trust conditioned on their discontinuation thereof. We then continue by modelling two truster agents, each in their own relationship to the institution. We consider two natural models of communication between them. In the first (``observable rewards'') agents disclose their experiences with the institution with one another, while in the second (``observable actions'') agents merely witness the actions of their neighbour, i.e., placing or not placing trust. Under the same assumptions as in the single agent case, we describe the evolution of the beliefs of agents under these two different communication models. Both the probability of ceasing to place trust and the expected time in the system elude explicit expressions, despite there being only two agents. We therefore conduct a simulation study in order to compare the effect of the different kinds of communication on the trust dynamics. We find that a pair of agents in both communication models has a greater chance of learning the true trustworthiness of an institution than a single agent. Communication between agents promotes the formation of long term trust with a trustworthy institution as well as the timely exit from a trust relationship with an untrustworthy institution. Contrary to what one might expect, we find that having less information (observing each other's actions instead of experiences) can sometimes be beneficial to the agents.

physics.soc-ph↗

arXiv2023

A queueing-based approach for integrated routing and appointment scheduling

This paper aims to address the integrated routing and appointment scheduling (RAS) problem for a single service provider. The RAS problem is an operational challenge faced by operators that provide services requiring home attendance, such as grocery delivery, home healthcare, or maintenance services. While considering the inherently random nature of service and travel times, the goal is to minimize a weighted sum of the operator's travel times and idle time, and the client's waiting times. To handle the complex search space of routing and appointment scheduling decisions, we propose a queueing-based approach to effectively deal with the appointment scheduling decisions. We use two well-known approximations from queueing theory: first, we use an approach based on phase-type distributions to accurately approximate the objective function, and second, we use an heavy-traffic approximation to derive an efficient procedure to obtain good appointment schedules. Combining these two approaches results in a fast and sufficiently accurate hybrid approximation, thus essentially reducing RAS to a routing problem. Moreover, we propose the use a simple yet effective large neighborhood search metaheuristic to explore the space of routing decisions. The effectiveness of our proposed methodology is tested on benchmark instances with up to 40 clients, demonstrating an efficient and accurate methodology for integrated routing and appointment scheduling.

math.OC↗

arXiv2023

Accurate and efficient approximation of large-scale appointment schedules

Setting up optimal appointment schedules requires the computation of an inherently involved objective function, typically requiring distributional knowledge of the clients' waiting times and the server's idle times (as a function of the appointment times of the individual clients). A frequently used idea is to approximate the clients' service times by their phase-type counterpart, thus leading to explicit expressions for the waiting-time and idle-time distributions. This method, however, requires the evaluation of the matrix exponential of potentially large matrices, which already becomes prohibitively slow from, say, 20 clients on. In this paper we remedy this issue by recursively approximating the distributions involved relying on a two-moments fit. More specifically, we approximate the sojourn time of each of the clients by a low-dimensional phase-type, Weibull or Lognormal random variable with the desired mean and variance. Our computational experiments show that this elementary, yet highly accurate, technique facilitates the evaluation of optimal appointment schedules even if the number of clients is large. The three ways to approximate the sojourn-time distribution turn out to be roughly equally accurate, except in certain specific regimes, where the low-dimensional phase-type fit performs well across all instances considered. As this low-dimensional phase-type fit is by far the fastest of the three alternatives, it is the approximation that we recommend.

math.PR↗

arXiv2023

Bankruptcy probabilities under non-Poisson inspection

This paper concerns an insurance firm's surplus process observed at renewal inspection times, with a focus on assessing the probability of the surplus level dropping below zero. For various types of inter-inspection time distributions, an explicit expression for the corresponding transform is given. In addition, Cramér-Lundberg type asymptotics are established. Also, an importance sampling based Monte Carlo algorithm is proposed, and is shown to be logarithmically efficient.

math.PR↗

arXiv2023

Stability of a Stochastic Ring Network

In this paper we establish a necessary and sufficient stability condition for a stochastic ring network. Such networks naturally appear in a variety of applications within communication, computer, and road traffic systems. They typically involve multiple customer types and some form of priority structure to decide which customer receives service. These two system features tend to complicate the issue of identifying a stability condition, but we demonstrate how the ring topology can be leveraged to solve the problem.

math.PR↗

arXiv2023

The Cramér-Lundberg model with a fluctuating number of clients

This paper considers the Cramér-Lundberg model, with the additional feature that the number of clients can fluctuate over time. Clients arrive according to a Poisson process, where the times they spend in the system form a sequence of independent and identically distributed non-negative random variables. While in the system, every client generates claims and pays premiums. In order to describe the model's rare-event behaviour, we establish a sample-path large-deviation principle. This describes the joint rare-event behaviour of the reserve-level process and the client-population size process. The large-deviation principle can be used to determine the decay rate of the time-dependent ruin probability as well as the most likely path to ruin. Our results allow us to determine whether the chance of ruin is greater with more or with fewer clients and, more generally, to determine to what extent a large deviation in the reserve-level process can be attributed to an unusual outcome of the client-population size process.

math.PR↗

arXiv2023

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↗

arXiv2022

Externalities in queues as stochastic processes: The case of FCFS M/G/1

Externalities are the costs that a user of a common resource imposes on others. For example, consider a FCFS M/G/1 queue and a customer with service demand of $x\geq0$ minutes who arrived into the system when the workload level was $v\geq0$ minutes. Let $E_v(x)$ be the total waiting time which could be saved if this customer gave up on his service demand. In this work, we analyse the \textit{externalities process} $E_v(\cdot)=\left\{E_v(x):x\geq0\right\}$. It is shown that this process can be represented by an integral of a (shifted in time by $v$ minutes) compound Poisson process with positive discrete jump distribution, so that $E_v(\cdot)$ is convex. Furthermore, we compute the LST of the finite-dimensional distributions of $E_v(\cdot)$ as well as its mean and auto-covariance functions. We also identify conditions under which, a sequence of normalized externalities processes admits a weak convergence on $\mathcal{D}[0,\infty)$ equipped with the uniform metric to an integral of a (shifted in time by $v$ minutes) standard Wiener process. Finally, we also consider the extended framework when $v$ is a general nonnegative random variable which is independent from the arrival process and the service demands. This leads to a generalization of an existing result from a previous work of Haviv and Ritov (1998).

math.PR↗

arXiv2022

On fluctuation-theoretic decompositions via Lindley-type recursions

Consider a Lévy process $Y(t)$ over an exponentially distributed time $T_β$ with mean $1/β$. We study the joint distribution of the running maximum $\bar{Y}(T_β)$ and the time epoch $G(T_β$) at which this maximum last occurs. Our main result is a fluctuation-theoretic distributional equality: the vector ($\bar{Y}(T_β),G(T_β)$) can be written as a sum of two independent vectors, the first one being ($\bar{Y}(T_{β+ω}),G(T_{β+ω})$) and the second one being the running maximum and corresponding time epoch under the restriction that the Lévy process is only observed at Poisson($ω$) inspection epochs (until $T_β$). We first provide an analytic proof for this remarkable decomposition, and then a more elementary proof that gives insight into the occurrence of the decomposition and into the fact that $ω$ only appears in the right hand side of the decomposition. The proof technique underlying the more elementary derivation also leads to further generalizations of the decomposition, and to some fundamental insights into a generalization of the well known Lindley recursion.

math.PR↗

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↗

arXiv2022

Road traffic estimation and distribution-based route selection

In route selection problems, the driver's personal preferences will determine whether she prefers a route with a travel time that has a relatively low mean and high variance over one that has relatively high mean and low variance. In practice, however, such risk aversion issues are often ignored, in that a route is selected based on a single-criterion Dijkstra-type algorithm. In addition, the routing decision typically does not take into account the uncertainty in the estimates of the travel time's mean and variance. This paper aims at resolving both issues by setting up a framework for travel time estimation. In our framework, the underlying road network is represented as a graph. Each edge is subdivided into multiple smaller pieces, so as to naturally model the statistical similarity between road pieces that are spatially nearby. Relying on a Bayesian approach, we construct an estimator for the joint per-edge travel time distribution, thus also providing us with an uncertainty quantification of our estimates. Our machinery relies on establishing limit theorems, making the resulting estimation procedure robust in the sense that it effectively does not assume any distributional properties. We present an extensive set of numerical experiments that demonstrate the validity of the estimation procedure and the use of the distributional estimates in the context of data-driven route selection.

math.OC↗

arXiv2022

Graphon-valued processes with vertex-level fluctuations

We consider a class of graph-valued stochastic processes in which each vertex has a type that fluctuates randomly over time. Collectively, the paths of the vertex types up to a given time determine the probabilities that the edges are active or inactive at that time. Our focus is on the evolution of the associated empirical graphon in the limit as the number of vertices tends to infinity, in the setting where fluctuations in the graph-valued process are more likely to be caused by fluctuations in the vertex types than by fluctuations in the states of the edges given these types. We derive both sample-path large deviation principles and convergence of stochastic processes. We demonstrate the flexibility of our approach by treating a class of stochastic processes where the edge probabilities depend not only on the fluctuations in the vertex types but also on the state of the graph itself.

math.PR↗

arXiv2022

Estimating customer impatience in a service system with unobserved balking

This paper studies a service system in which arriving customers are provided with information about the delay they will experience. Based on this information they decide to wait for service or to leave the system. Specifically, every customer has a patience threshold and they balk if the observed delay is above the threshold. The main objective is to estimate the parameters of the customers' patience-level distribution and the corresponding potential arrival rate, using knowledge of the actual queue-length process only. The main complication, and distinguishing feature of our setup, lies in the fact that customers who decide not to join are not observed, remarkably, we manage to devise a procedure to estimate the underlying patience and arrival rate parameters. The model is a multi-server queue with a Poisson stream of customers, enabling evaluation of the corresponding likelihood function of the state-dependent effective arrival process. We establish strong consistency of the MLE and derive the asymptotic distribution of the estimation error. Several applications and extensions of the method are discussed. The performance is further assessed through a series of numerical experiments. By fitting parameters of hyperexponential and generalized-hyperexponential distributions our method provides a robust estimation framework for any continuous patience-level distribution.

math.PR↗

arXiv2022

Optimal departure time advice in road networks with stochastic disruptions

Due to recurrent (e.g. daily or weekly) patterns and non-recurrent disruptions (e.g. caused by incidents), travel times in road networks are time-dependent and inherently random. This is challenging for travelers planning a future trip, aiming to ensure on-time arrival at the destination, while also trying to limit the total travel-time budget spent. The focus of this paper lies on determining their optimal departure time: the latest time of departure for which a chosen on-time arrival probability can be guaranteed. To model the uncertainties in the network, a Markovian background process is used, tracking events affecting the driveable vehicle speeds on the links, thus enabling us to incorporate both recurrent and non-recurrent effects. It allows the evaluation of the travel-time distribution, given the state of this process at departure, on each single link. Then, a computationally efficient algorithm is devised that uses these individual link travel-time distributions to obtain the optimal departure time for a given path or origin-destination pair. Since the conditions in the road network, and thus the state of the background process, may change between the time of request and the advised time of departure, we consider an online version of this procedure as well, in which the traveler receives departure time updates while still at the origin. Finally, numerical experiments are conducted to exemplify a selection of properties of the optimal departure time and, moreover, quantify the performance of the presented algorithms in an existing road network -- the Dutch highway network.

math.OC↗

arXiv2022

Central limit theorem for the principal eigenvalue and eigenvector of Chung-Lu random graphs

A Chung-Lu random graph is an inhomogeneous Erdős-Rényi random graph in which vertices are assigned average degrees, and pairs of vertices are connected by an edge with a probability that is proportional to the product of their average degrees, independently for different edges. We derive a central limit theorem for the principal eigenvalue and the components of the principal eigenvector of the adjacency matrix of a Chung-Lu random graph. Our derivation requires certain assumptions on the average degrees that guarantee connectivity, sparsity and bounded inhomogeneity of the graph.

math.PR↗

arXiv2022

A relative approach to opinion formation

Formal models of opinion formation commonly represent an individual's opinion by a value on a fixed opinion interval. We propose an alternative modeling method wherein interpretation is only provided to the relative positions of opinions vis-à-vis each other. This method is then considered in a similar setting as the discrete-time Altafini model (an extension of the well-known DeGroot model), but with more general influence weights. Even in a linear framework, the model can describe, in the long run, polarization, dynamics with a periodic pattern, and (modulus) consensus formation. In addition, in our alternative approach key characteristics of the opinion dynamic can be derived from real-valued square matrices of influence weights, which immediately allows one to transfer matrix theory insights to the field of opinion formation dynamics under more relaxed conditions than in the DeGroot or discrete-time Altafini models. A few specific themes are covered: (i) We demonstrate how stable patterns in relative opinion dynamics are identified which are hidden when opinions are considered in an absolute opinion framework. (ii) For the two-agent case, we provide an exhaustive closed-form description of the relative opinion model's dynamic in the long run. (iii) We explore group dynamics analytically, in particular providing a non-trivial condition under which a subgroup's asymptotic behavior carries over to the entire population.

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

A spectral signature of breaking of ensemble equivalence for constrained random graphs

For random systems subject to a constraint, the microcanonical ensemble requires the constraint to be met by every realisation ("hard constraint"), while the canonical ensemble requires the constraint to be met only on average ("soft constraint"). It is known that for random graphs subject to topological constraints breaking of ensemble equivalence may occur when the size of the graph tends to infinity, signalled by a non-vanishing specific relative entropy of the two ensembles. We investigate to what extent breaking of ensemble equivalence is manifested through the largest eigenvalue of the adjacency matrix of the graph. We consider two examples of constraints in the dense regime: (1) fix the degrees of the vertices (= the degree sequence); (2) fix the sum of the degrees of the vertices (= twice the number of edges). Example (1) imposes an extensive number of local constraints and is known to lead to breaking of ensemble equivalence. Example (2) imposes a single global constraint and is known to lead to ensemble equivalence. Our working hypothesis is that breaking of ensemble equivalence corresponds to a non-vanishing difference of the expected values of the largest eigenvalue under the two ensembles. We verify that, in the limit as the size of the graph tends to infinity, the difference between the expected values of the largest eigenvalue in the two ensembles does not vanish for (1) and vanishes for (2). A key tool in our analysis is a transfer method that uses relative entropy to determine whether probabilistic estimates can be carried over from the canonical ensemble to the microcanonical ensemble, and illustrates how breaking of ensemble equivalence may prevent this from being possible.

math.PR↗