arXiv subjects

Debankur Mukherjee

Publications and source records attributed to Debankur Mukherjee.

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arXiv2019

Join-the-Shortest Queue Diffusion Limit in Halfin-Whitt Regime: Sensitivity on the Heavy-traffic Parameter

Consider a system of $N$ parallel single-server queues with unit-exponential service time distribution and a single dispatcher where tasks arrive as a Poisson process of rate $λ(N)$. When a task arrives, the dispatcher assigns it to one of the servers according to the Join-the-Shortest Queue (JSQ) policy. Eschenfeldt and Gamarnik (Math. Oper. Res., 43(3):867-886, 2018) identified a novel limiting diffusion process that arises as the weak-limit of the appropriately scaled occupancy measure of the system under the JSQ policy in the Halfin-Whitt regime, where $(N - λ(N)) / \sqrt{N} \to β> 0$ as $N \to \infty$. The analysis of this diffusion goes beyond the state of the art techniques, and even proving its ergodicity is non-trivial, and was left as an open question. Recently, exploiting a generator expansion framework via the Stein's method, Braverman (arXiv:1801.05121, 2018) established its exponential ergodicity, and adapting a regenerative approach, Banerjee and Mukherjee (Ann. Appl. Probab., 29(2):1262-1309, 2018) analyzed the tail properties of the stationary distribution and path fluctuations of the diffusion. However, the analysis of the bulk behavior of the stationary distribution, viz., the moments, remained intractable until this work. In this paper, we perform a thorough analysis of the bulk behavior of the stationary distribution of the diffusion process, and discover that it exhibits different qualitative behavior, depending on the value of the heavy-traffic parameter $β$. Moreover, we obtain precise asymptotic laws of the centered and scaled steady state distribution, as $β$ tends to 0 and $\infty$. Of particular interest, we also establish a certain intermittency phenomena in the $β\to \infty$ regime and a surprising distributional convergence result in the $β\to 0$ regime.

math.PR↗

arXiv2019

Asymptotically Optimal Load Balancing Topologies

We consider a system of $N$ servers inter-connected by some underlying graph topology $G_N$. Tasks arrive at the various servers as independent Poisson processes of rate $λ$. Each incoming task is irrevocably assigned to whichever server has the smallest number of tasks among the one where it appears and its neighbors in $G_N$. Tasks have unit-mean exponential service times and leave the system upon service completion. The above model has been extensively investigated in the case $G_N$ is a clique. Since the servers are exchangeable in that case, the queue length process is quite tractable, and it has been proved that for any $λ< 1$, the fraction of servers with two or more tasks vanishes in the limit as $N \to \infty$. For an arbitrary graph $G_N$, the lack of exchangeability severely complicates the analysis, and the queue length process tends to be worse than for a clique. Accordingly, a graph $G_N$ is said to be $N$-optimal or $\sqrt{N}$-optimal when the occupancy process on $G_N$ is equivalent to that on a clique on an $N$-scale or $\sqrt{N}$-scale, respectively. We prove that if $G_N$ is an Erdős-Rényi random graph with average degree $d(N)$, then it is with high probability $N$-optimal and $\sqrt{N}$-optimal if $d(N) \to \infty$ and $d(N) / (\sqrt{N} \log(N)) \to \infty$ as $N \to \infty$, respectively. This demonstrates that optimality can be maintained at $N$-scale and $\sqrt{N}$-scale while reducing the number of connections by nearly a factor $N$ and $\sqrt{N} / \log(N)$ compared to a clique, provided the topology is suitably random. It is further shown that if $G_N$ contains $Θ(N)$ bounded-degree nodes, then it cannot be $N$-optimal. In addition, we establish that an arbitrary graph $G_N$ is $N$-optimal when its minimum degree is $N - o(N)$, and may not be $N$-optimal even when its minimum degree is $c N + o(N)$ for any $0 < c < 1/2$.

math.PR↗

arXiv2018

Universality of Power-of-$d$ Load Balancing in Many-Server Systems

We consider a system of $N$ parallel single-server queues with unit exponential service rates and a single dispatcher where tasks arrive as a Poisson process of rate $λ(N)$. When a task arrives, the dispatcher assigns it to a server with the shortest queue among $d(N)$ randomly selected servers ($1 \leq d(N) \leq N$). This load balancing strategy is referred to as a JSQ($d(N)$) scheme, marking that it subsumes the celebrated Join-the-Shortest Queue (JSQ) policy as a crucial special case for $d(N) = N$. We construct a stochastic coupling to bound the difference in the queue length 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 the regime where $λ(N) / N \to λ< 1$ as $N \to \infty$ with $d(N) \to\infty$, along with the associated fixed point. The fluid limit turns out not to depend on the exact growth rate of $d(N)$, and in particular coincides with that for the JSQ policy. We further leverage the coupling to establish that the diffusion limit in the critical regime where $(N - λ(N)) / \sqrt{N} \to β> 0$ as $N \to \infty$ with $d(N)/(\sqrt{N} \log (N))\to\infty$ corresponds to that for the JSQ policy. These results indicate that the optimality of the JSQ policy 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↗

arXiv2018

Join-the-Shortest Queue Diffusion Limit in Halfin-Whitt Regime: Tail Asymptotics and Scaling of Extrema

Consider a system of $N$ parallel single-server queues with unit-exponential service time distribution and a single dispatcher where tasks arrive as a Poisson process of rate $λ(N)$. When a task arrives, the dispatcher assigns it to one of the servers according to the Join-the-Shortest Queue (JSQ) policy. Eschenfeldt and Gamarnik (2015) established that in the Halfin-Whitt regime where $(N-λ(N))/\sqrt{N}\toβ>0$ as $N\to\infty$, appropriately scaled occupancy measure of the system under the JSQ policy converges weakly on any finite time interval to a certain diffusion process as $N\to\infty$. Recently, it was further established by Braverman (2018) that the stationary occupancy measure of the system converges weakly to the steady state of the diffusion process as $N\to\infty$. In this paper we perform a detailed analysis of the steady state of the above diffusion process. Specifically, we establish precise tail-asymptotics of the stationary distribution and scaling of extrema of the process on large time-interval. Our results imply that the asymptotic steady-state scaled number of servers with queue length two or larger exhibits an Exponential tail, whereas that for the number of idle servers turns out to be Gaussian. From the methodological point of view, the diffusion process under consideration goes beyond the state-of-the-art techniques in the study of the steady-state of diffusion processes. Lack of any closed form expression for the steady state and intricate interdependency of the process dynamics on its local times make the analysis significantly challenging. We develop a technique involving the theory of regenerative processes that provides a tractable form for the stationary measure, and in conjunction with several sharp hitting time estimates, acts as a key vehicle in establishing the results.

math.PR↗

arXiv2018

Supermarket Model on Graphs

We consider a variation of the supermarket model in which the servers can communicate with their neighbors and where the neighborhood relationships are described in terms of a suitable graph. Tasks with unit-exponential service time distributions arrive at each vertex as independent Poisson processes with rate $λ$, and each task is irrevocably assigned to the shortest queue among the one it first appears and its $d-1$ randomly selected neighbors. This model has been extensively studied when the underlying graph is a clique in which case it reduces to the well known power-of-$d$ scheme. In particular, results of Mitzenmacher (1996) and Vvedenskaya et al. (1996) show that as the size of the clique gets large, the occupancy process associated with the queue-lengths at the various servers converges to a deterministic limit described by an infinite system of ordinary differential equations (ODE). In this work, we consider settings where the underlying graph need not be a clique and is allowed to be suitably sparse. We show that if the minimum degree approaches infinity (however slowly) as the number of servers $N$ approaches infinity, and the ratio between the maximum degree and the minimum degree in each connected component approaches 1 uniformly, the occupancy process converges to the same system of ODE as the classical supermarket model. In particular, the asymptotic behavior of the occupancy process is insensitive to the precise network topology. We also study the case where the graph sequence is random, with the $N$-th graph given as an Erdős-Rényi random graph on $N$ vertices with average degree $c(N)$. Annealed convergence of the occupancy process to the same deterministic limit is established under the condition $c(N)\to\infty$, and under a stronger condition $c(N)/\ln N\to\infty$, convergence (in probability) is shown for almost every realization of the random graph.

math.PR↗

arXiv2018

Scalable Load Balancing Algorithms in Networked Systems

A fundamental challenge in large-scale networked systems viz., data centers and cloud networks is to distribute tasks to a pool of servers, using minimal instantaneous state information, while providing excellent delay performance. In this thesis we design and analyze load balancing algorithms that aim to achieve a highly efficient distribution of tasks, optimize server utilization, and minimize communication overhead.

math.PR↗

arXiv2018

Join-Idle-Queue with Service Elasticity: Large-Scale Asymptotics of a Non-monotone System

We consider the model of a token-based joint auto-scaling and load balancing strategy, proposed in a recent paper by Mukherjee, Dhara, Borst, and van Leeuwaarden (SIGMETRICS '17, arXiv:1703.08373), which offers an efficient scalable implementation and yet achieves asymptotically optimal steady-state delay performance and energy consumption as the number of servers $N\to\infty$. In the above work, the asymptotic results are obtained under the assumption that the queues have fixed-size finite buffers, and therefore the fundamental question of stability of the proposed scheme with infinite buffers was left open. In this paper, we address this fundamental stability question. The system stability under the usual subcritical load assumption is not automatic. Moreover, the stability may not even hold for all $N$. The key challenge stems from the fact that the process lacks monotonicity, which has been the powerful primary tool for establishing stability in load balancing models. We develop a novel method to prove that the subcritically loaded system is stable for large enough $N$, and establish convergence of steady-state distributions to the optimal one, as $N \to \infty$. The method goes beyond the state of the art techniques -- it uses an induction-based idea and a "weak monotonicity" property of the model; this technique is of independent interest and may have broader applicability.

math.PR↗

arXiv2018

Large deviations analysis for the $M/H_2/n + M$ queue in the Halfin-Whitt regime

We consider the FCFS $M/H_2/n + M$ queue in the Halfin-Whitt heavy traffic regime. It is known that the normalized sequence of steady-state queue length distributions is tight and converges weakly to a limiting random variable W. However, those works only describe W implicitly as the invariant measure of a complicated diffusion. Although it was proven by Gamarnik and Stolyar that the tail of W is sub-Gaussian, the actual value of $\lim_{x \rightarrow \infty}x^{-2}\log(P(W >x))$ was left open. In subsequent work, Dai and He conjectured an explicit form for this exponent, which was insensitive to the higher moments of the service distribution. We explicitly compute the true large deviations exponent for W when the abandonment rate is less than the minimum service rate, the first such result for non-Markovian queues with abandonments. Interestingly, our results resolve the conjecture of Dai and He in the negative. Our main approach is to extend the stochastic comparison framework of Gamarnik and Goldberg to the setting of abandonments, requiring several novel and non-trivial contributions. Our approach sheds light on several novel ways to think about multi-server queues with abandonments in the Halfin-Whitt regime, which should hold in considerable generality and provide new tools for analyzing these systems.

math.PR↗

arXiv2017

Scalable Load Balancing in Networked Systems: Universality Properties and Stochastic Coupling Methods

We present an overview of scalable load balancing algorithms which provide favorable delay performance in large-scale systems, and yet only require minimal implementation overhead. Aimed at a broad audience, the paper starts with an introduction to the basic load balancing scenario, consisting of a single dispatcher where tasks arrive that must immediately be forwarded to one of $N$ single-server queues. A popular class of load balancing algorithms are so-called power-of-$d$ or JSQ($d$) policies, where an incoming task is assigned to a server with the shortest queue among $d$ servers selected uniformly at random. This class includes the Join-the-Shortest-Queue (JSQ) policy as a special case ($d = N$), which has strong stochastic optimality properties and yields a mean waiting time that vanishes as $N$ grows large for any fixed subcritical load. However, a nominal implementation of the JSQ policy involves a prohibitive communication burden in large-scale deployments. In contrast, a random assignment policy ($d = 1$) does not entail any communication overhead, but the mean waiting time remains constant as $N$ grows large for any fixed positive load. In order to examine the fundamental trade-off between performance and implementation overhead, we consider an asymptotic regime where $d(N)$ depends on $N$. We investigate what growth rate of $d(N)$ is required to match the performance of the JSQ policy on fluid and diffusion scale. The results demonstrate that the asymptotics for the JSQ($d(N)$) policy are insensitive to the exact growth rate of $d(N)$, as long as the latter is sufficiently fast, implying that the optimality of the JSQ policy can asymptotically be preserved while dramatically reducing the communication overhead. We additionally show how the communication overhead can be reduced yet further by the so-called Join-the-Idle-Queue scheme, leveraging memory at the dispatcher.

cs.PF↗

arXiv2017

Phase transitions of extremal cuts for the configuration model

The $k$-section width and the Max-Cut for the configuration model are shown to exhibit phase transitions according to the values of certain parameters of the asymptotic degree distribution. These transitions mirror those observed on Erdős-Rényi random graphs, established by Luczak and McDiarmid (2001), and Coppersmith et al. (2004), respectively.

math.CO↗

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

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↗

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

Independent Set Reconfiguration Thresholds of Hereditary Graph Classes

Traditionally, reconfiguration problems ask the question whether a given solution of an optimization problem can be transformed to a target solution in a sequence of small steps that preserve feasibility of the intermediate solutions. In this paper, rather than asking this question from an algorithmic perspective, we analyze the combinatorial structure behind it. We consider the problem of reconfiguring one independent set into another, using two different processes: (1) exchanging exactly $k$ vertices in each step, or (2) removing or adding one vertex in each step while ensuring the intermediate sets contain at most $k$ fewer vertices than the initial solution. We are interested in determining the minimum value of $k$ for which this reconfiguration is possible, and bound these threshold values in terms of several structural graph parameters. For hereditary graph classes we identify structures that cause the reconfiguration threshold to be large.

cs.DM↗

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↗