Scheduling analysis with martingales

This paper proposes a new characterization of queueing systems by bounding a suitable exponential transform with a martingale. The constructed martingale is quite versatile in the sense that it captures queueing systems with Markovian and autoregressive arrivals in a unified manner; the second class is particularly relevant due to Wold’s decomposition of stationary processes. Moreover, using the framework of stochastic network calculus, the martingales allow for a simple handling of typical queueing operations: (1) flows’ multiplexing translates into multiplying the corresponding martingales, and (2) scheduling translates into time-shifting the martingales. The emerging calculus is applied to estimate the per-flow delay for FIFO, SP, and EDF scheduling. Unlike state-of-the-art results, our bounds capture a fundamental exponential leading constant in the number of multiplexed flows, and additionally are numerically tight

Martingale proofs of many-server heavy-traffic limits for Markovian queues

This is an expository review paper illustrating the ``martingale method'' for proving many-server heavy-traffic stochastic-process limits for queueing models, supporting diffusion-process approximations. Careful treatment is given to an elementary model -- the classical infinite-server model $M/M/\infty$, but models with finitely many servers and customer abandonment are also treated. The Markovian stochastic process representing the number of customers in the system is constructed in terms of rate-1 Poisson processes in two ways: (i) through random time changes and (ii) through random thinnings. Associated martingale representations are obtained for these constructions by applying, respectively: (i) optional stopping theorems where the random time changes are the stopping times and (ii) the integration theorem associated with random thinning of a counting process. Convergence to the diffusion process limit for the appropriate sequence of scaled queueing processes is obtained by applying the continuous mapping theorem. A key FCLT and a key FWLLN in this framework are established both with and without applying martingales.Comment: Published in at http://dx.doi.org/10.1214/06-PS091 the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Estimation of Gini Index within Pre-Specied Error Bound

Gini index is a widely used measure of economic inequality. This article develops a general theory for constructing a confidence interval for Gini index with a specified confidence coefficient and a specified width. Fixed sample size methods cannot simultaneously achieve both the specified confidence coefficient and specified width. We develop a purely sequential procedure for interval estimation of Gini index with a specified confidence coefficient and a fixed margin of error. Optimality properties of the proposed method, namely first order asymptotic efficiency and asymptotic consistency are proved. All theoretical results are derived without assuming any specific distribution of the data

Analysis of Multiserver Retrial Queueing System: A Martingale Approach and an Algorithm of Solution

The paper studies a multiserver retrial queueing system with $m$ servers. Arrival process is a point process with strictly stationary and ergodic increments. A customer arriving to the system occupies one of the free servers. If upon arrival all servers are busy, then the customer goes to the secondary queue, orbit, and after some random time retries more and more to occupy a server. A service time of each customer is exponentially distributed random variable with parameter $\mu_1$. A time between retrials is exponentially distributed with parameter $\mu_2$ for each customer. Using a martingale approach the paper provides an analysis of this system. The paper establishes the stability condition and studies a behavior of the limiting queue-length distributions as $\mu_2$ increases to infinity. As $\mu_2\to\infty$, the paper also proves the convergence of appropriate queue-length distributions to those of the associated `usual' multiserver queueing system without retrials. An algorithm for numerical solution of the equations, associated with the limiting queue-length distribution of retrial systems, is provided.Comment: To appear in "Annals of Operations Research" 141 (2006) 19-52. Replacement corrects a small number of misprint