arXiv subjects

Johan S. H. van Leeuwaarden

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

At least 55 records · Page 3Linked to original sources

arXiv2017

Corrected mean-field model for random sequential adsorption on random geometric graphs

A notorious problem in mathematics and physics is to create a solvable model for random sequential adsorption of non-overlapping congruent spheres in the $d$-dimensional Euclidean space with $d\geq 2$. Spheres arrive sequentially at uniformly chosen locations in space and are accepted only when there is no overlap with previously deposited spheres. Due to spatial correlations, characterizing the fraction of accepted spheres remains largely intractable. We study this fraction by taking a novel approach that compares random sequential adsorption in Euclidean space to the nearest-neighbor blocking on a sequence of clustered random graphs. This random network model can be thought of as a corrected mean-field model for the interaction graph between the attempted spheres. Using functional limit theorems, we characterize the fraction of accepted spheres and its fluctuations.

math.PR↗

arXiv2017

Big jobs arrive early: From critical queues to random graphs

We consider a queue to which only a finite pool of $n$ customers can arrive, at times depending on their service requirement. A customer with stochastic service requirement $S$ arrives to the queue after an exponentially distributed time with mean $S^{-α}$ for some $α\in[0,1]$; so larger service requirements trigger customers to join earlier. This finite-pool queue interpolates between two previously studied cases: $α= 0$ gives the so-called $Δ_{(i)}/G/1$ queue and $α= 1$ is closely related to the exploration process for inhomogeneous random graphs. We consider the asymptotic regime in which the pool size $n$ grows to infinity and establish that the scaled queue-length process converges to a diffusion process with a negative quadratic drift. We leverage this asymptotic result to characterize the head start that is needed to create a long period of activity. We also describe how this first busy period of the queue gives rise to a critically connected random forest.

math.PR↗

arXiv2017

Optimal Service Elasticity in Large-Scale Distributed Systems

A fundamental challenge in large-scale cloud networks and data centers is to achieve highly efficient server utilization and limit energy consumption, while providing excellent user-perceived performance in the presence of uncertain and time-varying demand patterns. Auto-scaling provides a popular paradigm for automatically adjusting service capacity in response to demand while meeting performance targets, and queue-driven auto-scaling techniques have been widely investigated in the literature. In typical data center architectures and cloud environments however, no centralized queue is maintained, and load balancing algorithms immediately distribute incoming tasks among parallel queues. In these distributed settings with vast numbers of servers, centralized queue-driven auto-scaling techniques involve a substantial communication overhead and major implementation burden, or may not even be viable at all. Motivated by the above issues, we propose a joint auto-scaling and load balancing scheme which does not require any global queue length information or explicit knowledge of system parameters, and yet provides provably near-optimal service elasticity. We establish the fluid-level dynamics for the proposed scheme in a regime where the total traffic volume and nominal service capacity grow large in proportion. The fluid-limit results show that the proposed scheme achieves asymptotic optimality in terms of user-perceived delay performance as well as energy consumption. Specifically, we prove that both the waiting time of tasks and the relative energy portion consumed by idle servers vanish in the limit. At the same time, the proposed scheme operates in a distributed fashion and involves only constant communication overhead per task, thus ensuring scalability in massive data center operations.

math.PR↗

arXiv2017

Robust heavy-traffic approximations for service systems facing overdispersed demand

Arrival processes to service systems often display fluctuations that are larger than anticipated under the Poisson assumption, a phenomenon that is referred to as overdispersion. Motivated by this, we analyze a class of discrete stochastic models for which we derive heavy-traffic approximations that are scalable in the system size. Subsequently, we show how this leads to novel capacity sizing rules that acknowledge the presence of overdispersion. This, in turn, leads to robust approximations for performance characteristics of systems that are of moderate size and/or may not operate in heavy traffic.

math.PR↗

arXiv2017

Critical window for the configuration model: finite third moment degrees

We investigate the component sizes of the critical configuration model, as well as the related problem of critical percolation on a supercritical configuration model. We show that, at criticality, the finite third moment assumption on the asymptotic degree distribution is enough to guarantee that the sizes of the largest connected components are of the order $n^{2/3}$ and the re-scaled component sizes (ordered in a decreasing manner) converge to the ordered excursion lengths of an inhomogeneous Brownian Motion with a parabolic drift. We use percolation to study the evolution of these component sizes while passing through the critical window and show that the vector of percolation cluster-sizes, considered as a process in the critical window, converge to the multiplicative coalescent process in the sense of finite dimensional distributions. This behavior was first observed for Erdős-Rényi random graphs by Aldous (1997) and our results provide support for the empirical evidences that the nature of the phase transition for a wide array of random-graph models are universal in nature. Further, we show that the re-scaled component sizes and surplus edges converge jointly under a strong topology, at each fixed location of the scaling window.

math.PR↗

arXiv2016

The restricted Erlang-R Queue: Finite-size effects in service systems with returning customers

Motivated by health care systems with repeated services that have both personnel (nurse and physician) and space (beds) constraints, we study a restricted version of the Erlang-R model. The space restriction policies we consider are blocking or holding in a pre-entrant queue. We develop many-server approximations for the system performance measures when either policy applies, and explore the connection between them. We show that capacity allocation of both resources should be determined simultaneously, and derive the methodology to determine it explicitly. We show that the system dynamics is captured by the fraction of needy time in the network, and that returning customers should be accounted for both in steady-state and time-varying conditions. We demonstrate the application of our policies in two case studies of resource allocation in hospitals.

math.PR↗

arXiv2016

Hierarchical Configuration Model

We introduce a class of random graphs with a community structure, which we call the hierarchical configuration model. On the inter-community level, the graph is a configuration model, and on the intra-community level, every vertex in the configuration model is replaced by a community: i.e., a small graph. These communities may have any shape, as long as they are connected. For these hierarchical graphs, we find the size of the largest component, the degree distribution and the clustering coefficient. Furthermore, we determine the conditions under which a giant percolation cluster exists, and find its size.

math.PR↗

arXiv2016

Mesoscopic scales in hierarchical configuration models

To understand mesoscopic scaling in networks, we study the hierarchical configuration model (HCM), a random graph model with community structure. The connections between the communities are formed as in a configuration model. We study the component sizes of the hierarchical configuration model at criticality when the inter-community degrees have a finite third moment. We find the conditions on the community sizes such that the critical component sizes of the HCM behave similarly as in the configuration model. Furthermore, we study critical bond percolation on the HCM. We show that the ordered components of a critical HCM on $N$ vertices are of sizes $O(N^{2/3})$. More specifically, the rescaled component sizes converge to the excursions of a Brownian motion with parabolic drift, as for the scaling limit for the configuration model under a finite third moment condition.

math.PR↗

arXiv2016

Asymptotic Optimality of Power-of-$d$ Load Balancing in Large-Scale Systems

We consider a system of $N$ identical server pools and a single dispatcher where tasks arrive as a Poisson process of rate $λ(N)$. Arriving tasks cannot be queued, and must immediately be assigned to one of the server pools to start execution, or discarded. The execution times are assumed to be exponentially distributed with unit mean, and do not depend on the number of other tasks receiving service. However, the experienced performance (e.g. in terms of received throughput) does degrade with an increasing number of concurrent tasks at the same server pool. The dispatcher therefore aims to evenly distribute the tasks across the various server pools. Specifically, when a task arrives, the dispatcher assigns it to the server pool with the minimum number of tasks among $d(N)$ randomly selected server pools. This assignment strategy is called the JSQ$(d(N))$ scheme, as it resembles the power-of-$d$ version of the Join-the-Shortest-Queue (JSQ) policy, and will also be referred to as such in the special case $d(N) = N$. We construct a stochastic coupling to bound the difference in the system occupancy processes between the JSQ policy and a scheme with an arbitrary value of $d(N)$. We use the coupling to derive the fluid limit in case $d(N) \to \infty$ and $λ(N)/N \to λ$ as $N \to \infty$, along with the associated fixed point. The fluid limit turns out to be insensitive to the exact growth rate of $d(N)$, and coincides with that for the JSQ policy. We further leverage the coupling to establish that the diffusion limit corresponds to that for the JSQ policy as well, as long as $d(N)/\sqrt{N} \log(N) \to \infty$, and characterize the common limiting diffusion process. These results indicate that the JSQ optimality can be preserved at the fluid-level and diffusion-level while reducing the overhead by nearly a factor O($N$) and O($\sqrt{N}/\log(N)$), respectively.

math.PR↗

arXiv2016

Epidemic spreading on complex networks with community structures

Many real-world networks display a community structure. We study two random graph models that create a network with similar community structure as a given network. One model preserves the exact community structure of the original network, while the other model only preserves the set of communities and the vertex degrees. These models show that community structure is an important determinant of the behavior of percolation processes on networks, such as information diffusion or virus spreading: the community structure can both \textit{enforce} as well as \textit{inhibit} diffusion processes. Our models further show that it is the mesoscopic set of communities that matters. The exact internal structures of communities barely influence the behavior of percolation processes across networks. This insensitivity is likely due to the relative denseness of the communities.

physics.soc-ph↗

arXiv2016

Local clustering in scale-free networks with hidden variables

We investigate the presence of triangles in a class of correlated random graphs in which hidden variables determine the pairwise connections between vertices. The class rules out self-loops and multiple edges and allows for negative degree correlations (disassortative mixing) due to infinite-variance degrees controlled by a structural cutoff $h_s$ and natural cutoff $h_c$. We show that local clustering decreases with the hidden variable (or degree). We also determine how the average clustering coefficient $C$ scales with the network size $N$, as a function of $h_s$ and $h_c$. For scale-free networks with exponent $2<τ<3$ and the default choices $h_s\sim N^{1/2}$ and $h_c\sim N^{1/(τ-1)}$ this gives $C\sim N^{2-τ}\ln N$ for the universality class at hand. We characterize the extremely slow decay of $C$ when $τ\approx 2$ and show that for $τ=2.1$, say, clustering only starts to vanish for networks as large as $N=10^{11}$.

math.PR↗

arXiv2016

Generalized random sequential adsorption on Erdős-Rényi random graphs

We investigate Random Sequential Adsorption (RSA) on a random graph via the following greedy algorithm: Order the $n$ vertices at random, and sequentially declare each vertex either active or frozen, depending on some local rule in terms of the state of the neighboring vertices. The classical RSA rule declares a vertex active if none of its neighbors is, in which case the set of active nodes forms an independent set of the graph. We generalize this nearest-neighbor blocking rule in three ways and apply it to the Erdős-Rényi random graph. We consider these generalizations in the large-graph limit $n\to\infty$ and characterize the jamming constant, the limiting proportion of active vertices in the maximal greedy set.

math.PR↗

arXiv2016

Finite-pool queues with heavy-tailed services

We consider the $Δ_{(i)}/G/1$ queue, in which a a total of $n$ customers independently demand service after an exponential time. We focus on the case of heavy-tailed service times, and assume that the tail of the service time distribution decays like $x^{-α}$, with $α\in (1,2)$. We consider the asymptotic regime in which the population size grows to infinity and establish that the scaled queue length process converges to an α-stable process with a negative quadratic drift. We leverage this asymptotic result to characterize the headstart that is needed to create a long period of activity. This result should be contrasted with the case of light-tailed service times, which was shown to have a similar scaling limit, but then with a Brownian motion instead of an α-stable process.

math.PR↗

arXiv2016

Power-law relations in random networks with communities

Most random graph models are locally tree-like - do not contain short cycles - rendering them unfit for modeling networks with a community structure. We introduce the hierarchical configuration model (HCM), a generalization of the configuration model that includes community structures, while properties such as the size of the giant component, and the size of the giant percolating cluster under bond percolation can still be derived analytically. Viewing real-world networks as realizations of the HCM, we observe two previously unobserved power-law relations: between the number of edges inside a community and the community sizes, and between the number of edges going out of a community and the community sizes. We also relate the power-law exponent $τ$ of the degree distribution with the power-law exponent of the community size distribution $γ$. In the special case of extremely dense communities (e.g., complete graphs), this relation takes the simple form $τ=γ-1$.

physics.soc-ph↗

arXiv2016

Steady-state analysis of shortest expected delay routing

We consider a queueing system consisting of two non-identical exponential servers, where each server has its own dedicated queue and serves the customers in that queue FCFS. Customers arrive according to a Poisson process and join the queue promising the shortest expected delay, which is a natural and near-optimal policy for systems with non-identical servers. This system can be modeled as an inhomogeneous random walk in the quadrant. By stretching the boundaries of the compensation approach we prove that the equilibrium distribution of this random walk can be expressed as a series of product-forms that can be determined recursively. The resulting series expression is directly amenable for numerical calculations and it also provides insight in the asymptotic behavior of the equilibrium probabilities as one of the state coordinates tends to infinity.

math.PR↗

arXiv2015

Dominant poles and tail asymptotics in the critical Gaussian many-sources regime

The dominant pole approximation (DPA) is a classical analytic method to obtain from a generating function asymptotic estimates for its underlying coefficients. We apply DPA to a discrete queue in a critical many-sources regime, in order to obtain tail asymptotics for the stationary queue length. As it turns out, this regime leads to a clustering of the poles of the generating function, which renders the classical DPA useless, since the dominant pole is not sufficiently dominant. To resolve this, we design a new DPA method, which might also find application in other areas of mathematics, like combinatorics, particularly when Gaussian scalings related to the central limit theorem are involved.

math.PR↗

arXiv2015

Heavy-traffic analysis through uniform acceleration of queues with diminishing populations

We consider a single server queue that serves a finite population of $n$ customers that will enter the queue (require service) only once, also known as the $Δ_{(i)}/G/1$ queue. This paper presents a method for analyzing heavy-traffic behavior by using uniform acceleration, which simultaneously lets $n$ and the service rate grow large, while the initial resource utilization approaches one. A key feature of the model is that, as time progresses, more customers have joined the queue, and fewer customers can potentially join. This diminishing population gives rise to a class of reflected stochastic processes that vanish over time, and hence do not have a stationary distribution. We establish that, when the arrival times are exponentially distributed, by suitably rescaling space and time, the queue length process converges to a Brownian motion with parabolic drift, a stochastic-process limit that captures the effect of a diminishing population by a negative quadratic drift. When the arrival times are generally distributed, our techniques provide information on the typical queue length and the first busy period.

math.PR↗

arXiv2015

Giant component sizes in scale-free networks with power-law degrees and cutoffs

Scale-free networks arise from power-law degree distributions. Due to the finite size of real-world networks, the power law inevitably has a cutoff at some maximum degree $Δ$. We investigate the relative size of the giant component $S$ in the large-network limit. We show that $S$ as a function of $Δ$ increases fast when $Δ$ is just large enough for the giant component to exist, but increases ever more slowly when $Δ$ increases further. This makes that while the degree distribution converges to a pure power law when $Δ\to\infty$, $S$ approaches its limiting value at a slow pace. The convergence rate also depends on the power-law exponent $τ$ of the degree distribution. The worst rate of convergence is found to be for the case $τ\approx2$, which concerns many of the real-world networks reported in the literature.

physics.soc-ph↗