arXiv subjects

Rami Atar

Publications and source records attributed to Rami Atar.

At least 37 records · Page 2Linked to original sources

arXiv2020

Customer-server population dynamics in heavy traffic

We study a many-server queueing model with server vacations, where the population size dynamics of servers and customers are coupled: a server may leave for vacation only when no customers await, and the capacity available to customers is directly affected by the number of servers on vacation. We focus on scaling regimes in which server dynamics and queue dynamics fluctuate at matching time scales, so that their limiting dynamics are coupled. Specifically, we argue that interesting coupled dynamics occur in (a) the Halfin-Whitt regime, (b) the nondegenerate slowdown regime, and (c) the intermediate, near Halfin-Whitt regime; whereas the dynamics asymptotically decouple in the other heavy traffic regimes. We characterize the limiting dynamics, which are different for each scaling regime. We consider relevant respective performance measures for regimes (a) and (b) --- namely, the probability of wait and the slowdown. While closed form formulas for these performance measures have been derived for models that do not accommodate server vacations, it is difficult to obtain closed form formulas for these performance measures in the setting with server vacations. Instead, we propose formulas that approximate these performance measures, and depend on the steady-state mean number of available servers and previously derived formulas for models without server vacations. We test the accuracy of these formulas numerically.

math.PR↗

arXiv2018

Serve the shortest queue and Walsh Brownian motion

We study a single-server Markovian queueing model with $N$ customer classes in which priority is given to the shortest queue. Under a critical load condition, we establish the diffusion limit of the workload and queue length processes in the form of a Walsh Brownian motion (WBM) living in the union of the $N$ nonnegative coordinate axes in $\mathbb{R}^N$ and a linear transformation thereof. This reveals the following asymptotic behavior. Each time that queues begin to build starting from an empty system, one of them becomes dominant in the sense that it contains nearly all the workload in the system, and it remains so until the system becomes (nearly) empty again. The radial part of the WBM, given as a reflected Brownian motion (RBM) on the half-line, captures the total workload asymptotics, whereas its angular distribution expresses how likely it is for each class to become dominant on excursions. As a heavy traffic result it is nonstandard in three ways: (i) In the terminology of Harrison (1995) it is unconventional, in that the limit is not an RBM. (ii) It does not constitute an invariance principle, in that the limit law (specifically, the angular distribution) is not determined solely by the first two moments of the data, and is sensitive even to tie breaking rules. (iii) The proof method does not fully characterize the limit law (specifically, it gives no information on the angular distribution).

math.PR↗

arXiv2017

Asymptotically optimal appointment schedules with customer no-shows

We consider the problem of scheduling appointments for a finite customer population to a service facility with customer no-shows, to minimize the sum of customer waiting time and server overtime costs. Since appointments need to be scheduled ahead of time we refer to this problem as an optimization problem rather than a dynamic control one. We study this optimization problem in fluid and diffusion scales and identify asymptotically optimal schedules in both scales. In fluid scale, we show that it is optimal to schedule appointments so that the system is in critical load; thus heavy-traffic conditions are obtained as a result of optimization rather than as an assumption. In diffusion scale, we solve this optimization problem in the large horizon limit. Our explicit stationary solution of the corresponding Brownian Optimization Problem translates the customer-delay versus server-overtime tradeoff to a tradeoff between the state of a reflected Brownian motion in the half-line and its local time at zero. Motivated by work on competitive ratios, we also consider a reference model in which an oracle provides the decision maker with the complete randomness information. The difference between the values of the scheduling problem for the two models, to which we refer as the stochasticity gap (SG), quantifies the degree to which it is harder to design a schedule under uncertainty than when the stochastic primitives (i.e., the no-shows and service times) are known in advance. In the fluid scale, the SG converges to zero, but in the diffusion scale it converges to a positive constant that we compute.

math.OC↗

arXiv2017

Law of large numbers for the many-server earliest-deadline-first queue

A many-server queue operating under the earliest deadline first discipline, where the distributions of service time and deadline are generic, is studied at the law of large numbers scale. Fluid model equations, formulated in terms of the many-server transport equation and the recently introduced measure-valued Skorohod map, are proposed as a means of characterizing the limit. The main results are the uniqueness of solutions to these equations, and the law of large numbers scale convergence to the solutions.

math.PR↗

arXiv2017

Optimality of the generalized cμ- rule in the moderate deviation regime

This paper studies a multiclass queueing system with an associated risk- sensitive cost observed in heavy traffic at the moderate deviation scale, accounting for convex queue length penalties. The main result is the asymptotic optimality of a dynamic index policy known from the diffusion scale heavy traffic literature as the generalized cμ- rule.

math.OC↗

arXiv2016

Asymptotically optimal control for a multiclass queueing model in the moderate deviation heavy traffic regime

A multi-class single-server queueing model with finite buffers, in which scheduling and admission of customers are subject to control, is studied in the moderate deviation heavy traffic regime. A risk-sensitive cost set over a finite time horizon $[0,T]$ is considered. The main result is the asymptotic optimality of a control policy derived via an underlying differential game. The result is the first to address a queueing control problem at the moderate deviation regime that goes beyond models having the so called pathwise minimality property. Moreover, despite the well known fact that an optimal control over a finite time interval is generically of a nonstationary feedback type, the proposed policy forms a stationary feedback, provided $T$ is sufficiently large.

math.PR↗

arXiv2016

A Skorokhod Map on Measure-Valued Paths with Applications to Priority Queues

The Skorokhod map on the half-line has proved to be a useful tool for studying processes with non-negativity constraints. In this work we introduce a measure-valued analog of this map that transforms each element $ζ$ of a certain class of càdlàg paths that take values in the space of signed measures on the half-line to a càdlàg path that takes values in the space of non-negative measures on $[0,\infty)$ in such a way that for each $x > 0$, the path $t \mapsto ζ_t[0,x]$ is transformed via a Skorokhod map on the half-line, and the regulating functions for different $x > 0$ are coupled. We establish regularity properties of this map and show that the map provides a convenient tool for studying queueing systems in which tasks are prioritized according to a continuous parameter. Three such well known models are the earliest-deadline-first, the shortest-job-first and the shortest-remaining-processing-time scheduling policies. For these applications, we show how the map provides a unified framework within which to form fluid model equations, prove uniqueness of solutions to these equations and establish convergence of scaled state processes to the fluid model. In particular, for these models, we obtain new convergence results in time-inhomogeneous settings, which appear to fall outside the purview of existing approaches.

math.PR↗

arXiv2016

A note on non-existence of diffusion limits for serve-the-longest-queue when the buffers are equal in size

We consider the serve-the-longest-queue discipline for a multiclass queue with buffers of equal size, operating under (i) the conventional and (ii) the Halfin-Whitt heavy traffic regimes, and show that while the queue length process' scaling limits are fully determined by the first and second order data in case (i), they depend on finer properties in case (ii). The proof of the latter relies on the construction of a {\it deterministic} arrival pattern.

math.PR↗

arXiv2015

An $ε$-Nash equilibrium with high probability for strategic customers in heavy traffic

A multiclass queue with many servers is considered, where customers make a join-or-leave decision upon arrival based on queue length information, without knowing the scheduling policy or the state of other queues. A game theoretic formulation is proposed and analyzed, that takes advantage of a phenomenon unique to heavy traffic regimes, namely Reiman's snaphshot principle, by which waiting times are predicted with high precision by the information available upon arrival. The payoff considered is given as a random variable, which depends on the customer's decision, accounting for waiting time in the queue and penalty for leaving. The notion of an equilibrium is only meaningful in an asymptotic framework, which is taken here to be the Halfin-Whitt heavy traffic regime. The main result is the identification of an $ε$-Nash equilibrium with probability approaching 1. On way to proving this result, new diffusion limit results for systems with finite buffers are obtained.

math.OC↗

arXiv2014

Information-theoretic applications of the logarithmic probability comparison bound

A well-known technique in estimating probabilities of rare events in general and in information theory in particular (used, e.g., in the sphere-packing bound), is that of finding a reference probability measure under which the event of interest has probability of order one and estimating the probability in question by means of the Kullback-Leibler divergence. A method has recently been proposed in [2], that can be viewed as an extension of this idea in which the probability under the reference measure may itself be decaying exponentially, and the Renyi divergence is used instead. The purpose of this paper is to demonstrate the usefulness of this approach in various information-theoretic settings. For the problem of channel coding, we provide a general methodology for obtaining matched, mismatched and robust error exponent bounds, as well as new results in a variety of particular channel models. Other applications we address include rate-distortion coding and the problem of guessing.

cs.IT↗

arXiv2014

A differential game for a multiclass queueing model in the moderate-deviation heavy-traffic regime

We study a differential game that governs the moderate-deviation heavy-traffic asymptotics of a multiclass single-server queueing control problem with a risk-sensitive cost. We consider a cost set on a finite but sufficiently large time horizon, and show that this formulation leads to stationary feedback policies for the game. Several aspects of the game are explored, including its characterization via a (one-dimensional) free boundary problem, the semi-explicit solution of an optimal strategy, and the specification of a saddle point. We emphasize the analogy to the well-known Harrison-Taksar free boundary problem which plays a similar role in the diffusion-scale heavy-traffic literature.

math.PR↗

arXiv2014

An asymptotic optimality result for the multiclass queue with finite buffers in heavy traffic

For a multiclass G/G/1 queue with finite buffers, admission and scheduling control, and holding and rejection costs, we construct a policy that is asymptotically optimal in the heavy traffic limit. The policy is specified in terms of a single parameter which constitutes the free boundary point from the Harrison-Taksar free boundary problem, but otherwise depends "explicitly" on the problem data. The c mu priority rule is also used by the policy, but in a way that is novel, and, in particular, different than that used in problems with infinite buffers. We also address an analogous problem where buffer constraints are replaced by throughput time constraints.

math.PR↗

arXiv2014

Fluid limits of G/G/1+G queues under the non-preemptive earliest-deadline-first discipline

A single-server queuing model is considered with customers that have deadlines. If a customer's deadline elapses before service is offered, the customer abandons the system (customers do not abandon while being served). When the server becomes available, it offers service to the customer having earliest deadline among those that are in the queue. We obtain a fluid limit of the queue length and abandonment processes and for the occupation measure of deadlines, in the form of measure-valued processes. We characterize the limit by means of a Skorohod problem in a time-varying domain, which has an explicit solution. The fluid limits also describe a certain process called the frontier, that is well known to play a key role in systems operating under this scheduling policy.

math.PR↗

arXiv2014

Control of the multiclass $G/G/1$ queue in the moderate deviation regime

A multi-class single-server system with general service time distributions is studied in a moderate deviation heavy traffic regime. In the scaling limit, an optimal control problem associated with the model is shown to be governed by a differential game that can be explicitly solved. While the characterization of the limit by a differential game is akin to results at the large deviation scale, the analysis of the problem is closely related to the much studied area of control in heavy traffic at the diffusion scale.

math.OC↗

arXiv2014

Scheduling parallel servers in the nondegenerate slowdown diffusion regime: Asymptotic optimality results

We consider the problem of minimizing queue-length costs in a system with heterogenous parallel servers, operating in a many-server heavy-traffic regime with nondegenerate slowdown. This regime is distinct from the well-studied heavy traffic diffusion regimes, namely the (single server) conventional regime and the (many-server) Halfin-Whitt regime. It has the distinguishing property that waiting times and service times are of comparable magnitudes. We establish an asymptotic lower bound on the cost and devise a sequence of policies that asymptotically attain this bound. As in the conventional regime, the asymptotics can be described by means of a Brownian control problem, the solution of which exhibits a state space collapse.

math.PR↗

arXiv2014

On the multi-dimensional skew Brownian motion

We provide a new, concise proof of weak existence and uniqueness of solutions to the stochastic differential equation for the multidimensional skew Brownian motion. We also present an application to Brownian particles with skew-elastic collisions.

math.PR↗

arXiv2013

Robust bounds on risk-sensitive functionals via Renyi divergence

We extend the duality between exponential integrals and relative entropy to a variational formula for exponential integrals involving the Renyi divergence. This formula characterizes the dependence of risk-sensitive functionals and related quantities determined by tail behavior to perturbations in the underlying distributions, in terms of the Renyi divergence. The characterization gives rise to upper and lower bounds that are meaningful for all values of a large deviation scaling parameter, allowing one to quantify in explicit terms the robustness of risk-sensitive costs. As applications we consider problems of uncertainty quantification when aspects of the model are not fully known, as well their use in bounding tail properties of an intractable model in terms of a tractable one.

math.PR↗

arXiv2010

Mutual Information, Relative Entropy, and Estimation in the Poisson Channel

Let $X$ be a non-negative random variable and let the conditional distribution of a random variable $Y$, given $X$, be ${Poisson}(γ\cdot X)$, for a parameter $γ\geq 0$. We identify a natural loss function such that: 1) The derivative of the mutual information between $X$ and $Y$ with respect to $γ$ is equal to the \emph{minimum} mean loss in estimating $X$ based on $Y$, regardless of the distribution of $X$. 2) When $X \sim P$ is estimated based on $Y$ by a mismatched estimator that would have minimized the expected loss had $X \sim Q$, the integral over all values of $γ$ of the excess mean loss is equal to the relative entropy between $P$ and $Q$. For a continuous time setting where $X^T = \{X_t, 0 \leq t \leq T \}$ is a non-negative stochastic process and the conditional law of $Y^T=\{Y_t, 0\le t\le T\}$, given $X^T$, is that of a non-homogeneous Poisson process with intensity function $γ\cdot X^T$, under the same loss function: 1) The minimum mean loss in \emph{causal} filtering when $γ= γ_0$ is equal to the expected value of the minimum mean loss in \emph{non-causal} filtering (smoothing) achieved with a channel whose parameter $γ$ is uniformly distributed between 0 and $γ_0$. Bridging the two quantities is the mutual information between $X^T$ and $Y^T$. 2) This relationship between the mean losses in causal and non-causal filtering holds also in the case where the filters employed are mismatched, i.e., optimized assuming a law on $X^T$ which is not the true one. Bridging the two quantities in this case is the sum of the mutual information and the relative entropy between the true and the mismatched distribution of $Y^T$. Thus, relative entropy quantifies the excess estimation loss due to mismatch in this setting. These results parallel those recently found for the Gaussian channel.

cs.IT↗