arXiv subjects

Johan S. H. van Leeuwaarden

Publications and source records attributed to Johan S. H. van Leeuwaarden.

At least 19 recordsLinked to original sources

arXiv2026

Robust Optimality of Bundling Goods Beyond Finite Variance

When selling many goods with independent valuations, we develop a distributionally robust framework, consisting of a two-player game between seller and nature. The seller has only limited knowledge about the value distribution. The seller selects a revenue-maximizing mechanism, after which nature chooses a revenue-minimizing distribution from all distributions that comply with the limited knowledge. When the seller knows the mean and variance of valuations, bundling is known to be an asymptotically optimal deterministic mechanism, achieving a normalized revenue close to the mean. Moving beyond this variance assumption, we assume knowledge of the mean absolute deviation (MAD), accommodating more dispersion and heavy-tailed valuations with infinite variance. We show for a large range of MAD values that bundling remains optimal, but the seller can only guarantee a revenue strictly smaller than the mean. Another noteworthy finding is indifference to the order of play, as both the max-min and min-max versions of the problem yield identical values. This contrasts with deterministic mechanisms and the separate sale of goods, where the order of play significantly impacts outcomes. We further underscore the universality of the optimal bundling price by demonstrating its efficacy in optimizing not only absolute revenue but also the absolute regret and ratio objective among all bundling prices

cs.GT↗

arXiv2026

Distributionally robust monopoly pricing: Switching from low to high prices in volatile markets

Problem definition: Traditional monopoly pricing assumes sellers have full information about consumer valuations. We consider monopoly pricing under limited information, where a seller only knows the mean, variance and support of the valuation distribution. The objective is to maximize expected revenue by selecting the optimal fixed price. Methodology/results: We adopt a distributionally robust framework, where the seller considers all valuation distributions that comply with the limited information. We formulate a maximin problem which seeks to maximize expected revenue for the worst-case valuation distribution. The minimization problem that identifies the worst-case valuation distribution is solved using primal-dual methods, and in turn leads to an explicitly solvable maximization problem. This yields a closed-form optimal pricing policy and a new fundamental principle prescribing when to use low and high robust prices. Managerial implications: We show that the optimal policy switches from low to high prices when variance becomes sufficiently large, yielding significant performance gains compared with existing robust prices that generally decay with market uncertainty. This presents guidelines for when the seller should switch from targeting mass markets to niche markets. Similar guidelines are obtained for delay-prone services with rational utility-maximizing customers, underlining the universality and wide applicability of the novel pricing policy.

math.OC↗

arXiv2025

Fluid limits for interacting queues in sparse dynamic graphs

Consider a network of $n$ single-server queues where tasks arrive independently at each server at rate $λ_n$. The servers are connected by a graph that is resampled at rate $μ_n$ in a way that is symmetric with respect to the servers, and each task is dispatched to the shortest queue in the graph neighborhood where it appears. We aim to gain insight in the impact of the dynamic network structure on the load balancing dynamics in terms of the occupancy process which describes the empirical distribution of the number of tasks across the servers. This process evolves on the underlying dynamic graph, and its dynamics depend on the number of tasks at each individual server and the neighborhood structure of the graph. We establish that this dependency disappears in the limit as $n \to \infty$ when $λ_n / n \to λ$ and $μ_n \to \infty$, and prove that the limit of the occupancy process is given by a system of differential equations that depends solely on $λ$ and the limiting degree distribution of the graph. We further show that the stationary distribution of the occupancy process converges to an equilibrium of the differential equations, and derive properties of this equilibrium that reflect the impact of the degree distribution. Our focus is on truly sparse graphs where the maximum degree is uniformly bounded across $n$, which is natural in load balancing systems.

math.PR↗

arXiv2025

Robust Competitive Ratio for Deterministic Monopoly Pricing

We study deterministic monopoly pricing under partial knowledge of the market, where the seller has access only to summary statistics of the valuation distribution, such as the mean, dispersion, and maximum value. Using tools from distributionally robust optimization and max-min analysis, we evaluate pricing strategies based on their competitive ratio (CR). We characterize the worst-case market scenario consistent with the available information and provide a complete solution for minimizing the CR. Our analysis also covers optimal pricing under various measures of dispersion, including variance and fractional moments. Interestingly, we find that the worst-case market for CR coincides with that for expected revenue. Using proof techniques tailored to the CR framework, we further examine how dispersion and maximum valuation influence optimal deterministic pricing. These results offer practical guidance for setting robust prices when market information is limited.

math.OC↗

arXiv2024

Distribution-free expectation operators for robust pricing and stocking with heavy-tailed demand

We obtain distribution-free bounds for various fundamental quantities used in probability theory by solving optimization problems that search for extreme distributions among all distributions with the same mean and dispersion. These sharpest possible bounds depend only on the mean and dispersion of the driving random variable. We solve the optimization problems by a novel yet elementary technique that reduces the set of all candidate solutions to two-point distributions. We consider a general dispersion measure, with variance, mean absolute deviation and power moments as special cases. We apply the bounds to robust newsvendor stocking and monopoly pricing, generalizing foundational mean-variance works. This shows how pricing and order decisions respond to increased demand uncertainty, including scenarios where dispersion information allows for heavy-tailed demand distributions.

math.OC↗

arXiv2024

Multi-dimensional state space collapse in non-complete resource pooling scenarios

The present paper establishes an explicit multi-dimensional state space collapse (SSC) for parallel-processing systems with arbitrary compatibility constraints between servers and job types. This breaks major new ground beyond the SSC results and queue length asymptotics in the literature which are largely restricted to complete resource pooling (CRP) scenarios where the steady-state queue length vector concentrates around a line in heavy traffic. The multi-dimensional SSC that we establish reveals heavy-traffic behavior which is also far more tractable than the pre-limit queue length distribution, yet exhibits a fundamentally more intricate structure than in the one-dimensional case, providing useful insight into the system dynamics. In particular, we prove that the limiting queue length vector lives in a $K$-dimensional cone of which the set of spanning vectors is random in general, capturing the delicate interplay between the various job types and servers. For a broad class of systems we provide a further simplification which shows that the collection of random cones constitutes a fixed $K$-dimensional cone, resulting in a $K$-dimensional SSC. The dimension $K$ represents the number of critically loaded subsystems, or equivalently, capacity bottlenecks in heavy-traffic, with $K=1$ corresponding to conventional CRP scenarios. Our approach leverages probability generating function (PGF) expressions for Markovian systems operating under redundancy policies.

math.PR↗

arXiv2024

Server saturation in skewed networks

We consider a model inspired by compatibility constraints that arise between tasks and servers in data centers, cloud computing systems and content delivery networks. The constraints are represented by a bipartite graph or network that interconnects dispatchers with compatible servers. Each dispatcher receives tasks over time and sends every task to a compatible server with the least number of tasks, or to a server with the least number of tasks among $d$ compatible servers selected uniformly at random. We focus on networks where the neighborhood of at least one server is skewed in a limiting regime. This means that a diverging number of dispatchers are in the neighborhood which are each compatible with a uniformly bounded number of servers; thus, the degree of the central server approaches infinity while the degrees of many neighboring dispatchers remain bounded. We prove that each server with a skewed neighborhood saturates, in the sense that the mean number of tasks queueing in front of it in steady state approaches infinity. Paradoxically, this pathological behavior can even arise in random networks where nearly all the servers have at most one task in the limit.

math.PR↗

arXiv2024

Learning and balancing unknown loads in large-scale systems

Consider a system of identical server pools where tasks with exponentially distributed service times arrive as a time-inhomogenenous Poisson process. An admission threshold is used in an inner control loop to assign incoming tasks to server pools while, in an outer control loop, a learning scheme adjusts this threshold over time to keep it aligned with the unknown offered load of the system. In a many-server regime, we prove that the learning scheme reaches an equilibrium along intervals of time where the normalized offered load per server pool is suitably bounded, and that this results in a balanced distribution of the load. Furthermore, we establish a similar result when tasks with Coxian distributed service times arrive at a constant rate and the threshold is adjusted using only the total number of tasks in the system. The novel proof technique developed in this paper, which differs from a traditional fluid limit analysis, allows to handle rapid variations of the first learning scheme, triggered by excursions of the occupancy process that have vanishing size. Moreover, our approach allows to characterize the asymptotic behavior of the system with Coxian distributed service times without relying on a fluid limit of a detailed state descriptor.

math.PR↗

arXiv2024

Utility maximizing load balancing policies

Consider a service system where incoming tasks are instantaneously dispatched to one out of many heterogeneous server pools. Associated with each server pool is a concave utility function which depends on the class of the server pool and its current occupancy. We derive an upper bound for the mean normalized aggregate utility in stationarity and introduce two load balancing policies that achieve this upper bound in a large-scale regime. Furthermore, the transient and stationary behavior of these asymptotically optimal load balancing policies is characterized on the scale of the number of server pools, in the same large-scale regime.

math.PR↗

arXiv2023

Second-order bounds for the M/M/$s$ queue with random arrival rate

Consider an M/M/$s$ queue with the additional feature that the arrival rate is a random variable of which only the mean, variance, and range are known. Using semi-infinite linear programming and duality theory for moment problems, we establish for this setting tight bounds for the expected waiting time. These bounds correspond to an arrival rate that takes only two values. The proofs crucially depend on the fact that the expected waiting time, as function of the arrival rate, has a convex derivative. We apply the novel tight bounds to a rational queueing model, where arriving individuals decide to join or balk based on expected utility and only have partial knowledge about the market size.

math.PR↗

arXiv2023

Self-Learning Threshold-Based Load Balancing

We consider a large-scale service system where incoming tasks have to be instantaneously dispatched to one out of many parallel server pools. The user-perceived performance degrades with the number of concurrent tasks and the dispatcher aims at maximizing the overall quality-of-service by balancing the load through a simple threshold policy. We demonstrate that such a policy is optimal on the fluid and diffusion scales, while only involving a small communication overhead, which is crucial for large-scale deployments. In order to set the threshold optimally, it is important, however, to learn the load of the system, which may be unknown. For that purpose, we design a control rule for tuning the threshold in an online manner. We derive conditions which guarantee that this adaptive threshold settles at the optimal value, along with estimates for the time until this happens. In addition, we provide numerical experiments which support the theoretical results and further indicate that our policy copes effectively with time-varying demand patterns.

cs.PF↗

arXiv2023

Robust knapsack ordering for a partially-informed newsvendor with budget constraint

This paper studies the multi-item newsvendor problem with a constrained budget and information about demand limited to its range, mean and mean absolute deviation. We consider a minimax model that determines order quantities by minimizing the expected overage and underage costs for the worst-case demand distributions. The resulting optimization problem turns out to be solvable by a method reminiscent of the greedy algorithm that solves the continuous knapsack problem, purchasing items in order of marginal value. This method has lower computational complexity compared to directly solving the model and leads to a simple policy that (i) sorts items based on their marginal effect on the total cost and (ii) determines order quantities according to this ranking until the budget is spent.

math.OC↗

arXiv2022

Sharpest possible clustering bounds using robust random graph analysis

Complex network theory crucially depends on the assumptions made about the degree distribution, while fitting degree distributions to network data is challenging, in particular for scale-free networks with power-law degrees. We present a robust assessment of complex networks that does not depend on the entire degree distribution, but only on its mean, range and dispersion: summary statistics that are easy to obtain for most real-world networks. By solving several semi-infinite linear programs, we obtain tight (the sharpest possible) bounds for correlation and clustering measures, for all networks with degree distributions that share the same summary statistics. We identify various extremal random graphs that attain these tight bounds as the graphs with specific three-point degree distributions. We leverage the tight bounds to obtain robust laws that explain how degree-degree correlations and local clustering evolve as function of node degrees and network size. These robust laws indicate that power-law networks with diverging variance are among the most extreme networks in terms of correlation and clustering, building further theoretical foundation for widely reported scale-free network phenomena such as correlation and clustering decay.

physics.soc-ph↗

arXiv2022

Heavy-Traffic Universality of Redundancy Systems with Assignment Constraints

Service systems often face task-server assignment-constraints due to skill-based routing or geographical conditions. Redundancy scheduling responds to this limited flexibility by replicating tasks to specific servers in agreement with these assignment constraints. We gain insight from product-form stationary distributions and weak local stability conditions to establish a state space collapse in heavy traffic. In this limiting regime, the parallel-server system with redundancy scheduling operates as a multi-class single-server system, achieving full resource pooling and exhibiting strong insensitivity to the underlying assignment constraints. In particular, the performance of a fully flexible (unconstrained) system can be matched even with rather strict assignment constraints.

math.PR↗

arXiv2022

Power-of-two sampling in redundancy systems: the impact of assignment constraints

A classical sampling strategy for load balancing policies is power-of-two, where any server pair is sampled with equal probability. This does not cover practical settings with assignment constraints which force non-uniform sampling. While intuition suggests that non-uniform sampling adversely impacts performance, this was only supported through simulations, and rigorous statements have remained elusive. Building on product-form distributions for redundancy systems, we prove the stochastic dominance of uniform sampling for a four-server system as well as arbitrary-size systems in light traffic.

math.PR↗

arXiv2021

Scalable load balancing in networked systems: A survey of recent advances

The basic load balancing scenario involves a single dispatcher where tasks arrive that must immediately be forwarded to one of $N$ single-server queues. We discuss recent advances on scalable load balancing schemes which provide favorable delay performance when $N$ grows large, and yet only require minimal implementation overhead. Join-the-Shortest-Queue (JSQ) yields vanishing delays as $N$ grows large, as in a centralized queueing arrangement, but involves a prohibitive communication burden. In contrast, power-of-$d$ or JSQ($d$) schemes that assign an incoming task to a server with the shortest queue among $d$ servers selected uniformly at random require little communication, but lead to constant delays. In order to examine this fundamental trade-off between delay performance and implementation overhead, we consider JSQ($d(N)$) schemes where the diversity parameter $d(N)$ depends on $N$ and investigate what growth rate of $d(N)$ is required to asymptotically match the optimal JSQ performance on fluid and diffusion scale. Stochastic coupling techniques and stochastic-process limits play an instrumental role in establishing the asymptotic optimality. We demonstrate how this methodology carries over to infinite-server settings, finite buffers, multiple dispatchers, servers arranged on graph topologies, and token-based load balancing including the popular Join-the-Idle-Queue (JIQ) scheme. In this way we provide a broad overview of the many recent advances in the field. This survey extends the short review presented at ICM 2018 (arXiv:1712.08555).

math.PR↗

arXiv2021

Robust subgraph counting with distribution-free random graph analysis

Subgraphs such as cliques, loops and stars form crucial connections in the topologies of real-world networks. Random graph models provide estimates for how often certain subgraphs appear, which in turn can be tested against real-world networks. These subgraph counts, however, crucially depend on the assumed degree distribution. Fitting a degree distribution to network data is challenging, in particular for scale-free networks with power-law degrees. In this paper we develop robust subgraph counts that do not depend on the entire degree distribution, but only on the mean and mean absolute deviation (MAD), summary statistics that are easy to obtain for most real-world networks. By solving an optimization problem, we provide tight (the sharpest possible) bounds for the subgraph counts, for all possible subgraphs, and for all networks with degree distributions that share the same mean and MAD. We identify the extremal random graph that attains the tight bounds as the graph with a specific three-point degree distribution. We leverage the bounds to obtain robust scaling laws for how the numbers of subgraphs grow as function of the network size. The scaling laws indicate that sparse power-law networks are not the most extreme networks in terms of subgraph counts, but dense power-law networks are. The robust bounds are also shown to hold for several real-world data sets.

cs.SI↗

arXiv2020

MAD dispersion measure makes extremal queue analysis simple

A notorious problem in queueing theory is to compute the worst possible performance of the GI/G/1 queue under mean-dispersion constraints for the interarrival and service time distributions. We address this extremal queue problem by measuring dispersion in terms of Mean Absolute Deviation (MAD) instead of variance, making available recently developed techniques from Distributionally Robust Optimization (DRO). Combined with classical random walk theory, we obtain explicit expressions for the extremal interarrival time and service time distributions, and hence the best possible upper bounds, for all moments of the waiting time. {We also apply the DRO techniques to obtain tight lower bounds that together with the upper bounds provide robust performance intervals. We show that all bounds are computationally tractable and remain sharp, also when the mean and MAD are not known precisely, but estimated based on available data instead.

math.PR↗