Large deviations analysis for the M/H2/n+MM/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

Breaking the dimensionality curse in multi-server queues

International audiencePh/Ph/c and and Ph/Ph/c/N queues can be viewed as a common model of multi-server facilities. We propose a simple approximate solution for the equilibrium probabilities in such queues based on a reduced state description in order to circumvent the well-known and dreaded combinatorial growth of the number of states inherent in the classical state description. The number of equations to solve in our approach increases linearly with the number of servers and phases in the service time distribution. A simple fixed-point iteration is used to solve these equations. Our approach applies both to open models with unrestricted buffer size and to queues with finite-size buffers. The results of a large number of empirical studies indicate that the overall accuracy of the proposed approximation appears very good. For instance, the median relative error for the mean number in the queue over thousands of examples is below 0.1% and the relative error exceeds 5% in less than 1.5% of cases explored. The accuracy of the proposed approximation becomes particularly good for systems with more than 8 servers, and tends to become excellent as the number of servers increases

Solving the ME/ME/1 queue with state-space methods and the matrix sign function

Cataloged from PDF version of article.Matrix exponential (ME) distributions not only include the well-known class of phase-type distributions but also can be used to approximate more general distributions (e.g., deterministic, heavy-tailed, etc.). In this paper, a novel mathematical framework and a numerical algorithm are proposed to calculate the matrix exponential representation for the steady-state waiting time in an ME/ME/1 queue. Using state–space algebra, the waiting time calculation problem is shown to reduce to finding the solution of an ordinary differential equation in state–space form with order being the sum of the dimensionalities of the inter-arrival and service time distribution representations. A numerically efficient algorithm with quadratic convergence rates based on the matrix sign function iterations is proposed to find the boundary conditions of the differential equation. The overall algorithm does not involve any transform domain calculations such as root finding or polynomial factorization, which are known to have potential numerical stability problems. Numerical examples are provided to demonstrate the effectiveness of the proposed approach. © 2004 Elsevier B.V. All rights reserved

A Robust Queueing Network Analyzer Based on Indices of Dispersion

In post-industrial economies, modern service systems are dramatically changing the daily lives of many people. Such systems are often complicated by uncertainty: service providers usually cannot predict when a customer will arrive and how long the service will be. Fortunately, useful guidance can often be provided by exploiting stochastic models such as queueing networks. In iterating the design of service systems, decision makers usually favor analytical analysis of the models over simulation methods, due to the prohibitive computation time required to obtain optimal solutions for service operation problems involving multidimensional stochastic networks. However, queueing networks that can be solved analytically require strong assumptions that are rarely satisfied, whereas realistic models that exhibit complicated dependence structure are prohibitively hard to analyze exactly. In this thesis, we continue the effort to develop useful analytical performance approximations for the single-class open queueing network with Markovian routing, unlimited waiting space and the first-come first-served service discipline. We focus on open queueing networks where the external arrival processes are not Poisson and the service times are not exponential. We develop a new non-parametric robust queueing algorithm for the performance approximation in single-server queues. With robust optimization techniques, the underlying stochastic processes are replaced by samples from suitably defined uncertainty sets and the worst-case scenario is analyzed. We show that this worst-case characterization of the performance measure is asymptotically exact for approximating the mean steady-state workload in G/G/1 models in both the light-traffic and heavy-traffic limits, under mild regularity conditions. In our non-parametric Robust Queueing formulation, we focus on the customer flows, defined as the continuous-time processes counting customers in or out of the network, or flowing from one queue to another. Each flow is partially characterized by a continuous function that measures the change of stochastic variability over time. This function is called the index of dispersion for counts. The Robust Queueing algorithm converts the index of dispersion for counts into approximations of the performance measures. We show the advantage of using index of dispersion for counts in queueing approximation by a renewal process characterization theorem and the ordering of the mean steady-state workload in GI/M/1 models. To develop generalized algorithm for open queueing networks, we first establish the heavy-traffic limit theorem for the stationary departure flows from a GI/GI/1 model. We show that the index of dispersion for counts function of the stationary departure flow can be approximately characterized as the convex combination of the arrival index of dispersion for counts and service index of dispersion for counts with a time-dependent weight function, revealing the non-trivial impact of the traffic intensity on the departure processes. This heavy-traffic limit theorem is further generalized into a joint heavy-traffic limit for the stationary customer flows in generalized Jackson networks, where the external arrival are characterized by independent renewal processes and the service times are independent and identically distributed random variables, independent of the external arrival processes. We show how these limiting theorems can be exploited to establish a set of linear equations, whose solution serves as approximations of the index of dispersion for counts of the flows in an open queueing network. We prove that this set of equations is asymptotically exact in approximating the index of dispersion for counts of the stationary flows. With the index of dispersion for counts available, the network is decomposed into single-server queues and the Robust Queueing algorithm can be applied to obtain performance approximation. This algorithm is referred to as the Robust Queueing Network Analyzer. We perform extensive simulation study to validate the effectiveness of our algorithm. We show that our algorithm can be applied not only to models with non-exponential distirbutions but also to models with more complex arrival processes than renewal processes, including those with Markovian arrival processes

Decomposition of discrete-time open tandem queues with Poisson arrivals and general service times

In der Grobplanungsphase vernetzter Logistik- und Produktionssysteme ist man häufig daran interessiert, mit geringem Berechnungsaufwand eine zufriedenstellende Approximation der Leistungskennzahlen des Systems zu bestimmen. Hierbei bietet die Modellierung mittels zeitdiskreter Methoden gegenüber der zeitkontinuierlichen Modellierung den Vorteil, dass die gesamte Wahrscheinlichkeitsverteilung der Leistungskenngrößen berechnet werden kann. Da Produktions- und Logistiksysteme in der Regel so konzipiert sind, dass sie die Leistung nicht im Durchschnitt, sondern mit einer bestimmten Wahrscheinlichkeit (z.B. 95%) zusichern, können zeitdiskrete Warteschlangenmodelle detailliertere Informationen über die Leistung des Systems (wie z.B. der Warte- oder Durchlaufzeit) liefern. Für die Analyse vernetzter zeitdiskreter Bediensysteme sind Dekompositionsmethoden häufig der einzig praktikable und recheneffiziente Ansatz, um stationäre Leistungsmaße in den einzelnen Bediensystemen zu berechnen. Hierbei wird das Netzwerk in die einzelnen Knoten zerlegt und diese getrennt voneinander analysiert. Der Ansatz basiert auf der Annahme, dass der Punktprozess des Abgangsstroms stromaufwärts liegender Stationen durch einen Erneuerungsprozess approximiert werden kann, und so eine unabhängige Analyse der Bediensysteme möglich ist. Die Annahme der Unabhängigkeit ermöglicht zwar eine effiziente Berechnung, führt jedoch zu teilweise starken Approximationsfehlern in den berechneten Leistungskenngrößen. Der Untersuchungsgegenstand dieser Arbeit sind offene zeitdiskrete Tandem-Netzwerke mit Poisson-verteilten Ankünften am stromaufwärts liegenden Bediensystem und generell verteilten Bedienzeiten. Das Netzwerk besteht folglich aus einem stromaufwärts liegenden M/G/1-Bediensystem und einem stromabwärts liegenden G/G/1-System. Diese Arbeit verfolgt drei Ziele, (1) die Defizite des Dekompositionsansatzes aufzuzeigen und dessen Approximationsgüte mittels statistischer Schätzmethoden zu bestimmen, (2) die Autokorrelation des Abgangsprozesses des M/G/1-Systems zu modellieren um die Ursache des Approximationsfehlers erklären zu können und (3) einen Dekompositionsansatz zu entwickeln, der die Abhängigkeit des Abgangsstroms berücksichtigt und so beliebig genaue Annäherungen der Leistungskenngrößen ermöglicht. Im ersten Teil der Arbeit wird die Approximationsgüte des Dekompositionsverfahrens am stromabwärts liegenden G/G/1-Bediensystem mit Hilfe von Linearer Regression (Punktschätzung) und Quantilsregression (Intervallschätzung) bestimmt. Beide Schätzverfahren werden jeweils auf die relativen Fehler des Erwartungswerts und des 95%-Quantils der Wartezeit im Vergleich zu den simulierten Ergebnissen berechnet. Als signifikante Einflussfaktoren auf die Approximationsgüte werden die Auslastung des Systems und die Variabilität des Ankunftsstroms identifiziert. Der zweite Teil der Arbeit fokussiert sich auf die Berechnung der Autokorrelation im Abgangsstroms des M/G/1-Bediensystems. Aufeinanderfolgende Zwischenabgangszeiten sind miteinander korreliert, da die Abgangszeit eines Kunden von dem Systemzustand abhängt, den der vorherige Kunde bei dessen Abgang zurückgelassen hat. Die Autokorrelation ist ursächlich für den Dekompositionsfehler, da die Ankunftszeiten am stromabwärts liegenden Bediensystem nicht unabhängig identisch verteilt sind. Im dritten Teil der Arbeit wird ein neuer Dekompositionsansatz vorgestellt, der die Abhängigkeit im Abgangsstroms des M/G/1-Systems mittels eines semi-Markov Prozesses modelliert. Um eine explosionsartige Zunahme des Zustandsraums zu verhindern, wird ein Verfahren eingeführt, das den Zustandsraum der eingebetteten Markov-Kette beschränkt. Numerischen Auswertungen zeigen, dass die mit stark limitierten Zustandsraum erzielten Ergebnisse eine bessere Approximation bieten als der bisherige Dekompositionsansatz. Mit zunehmender Größe des Zustandsraums konvergieren die Leistungskennzahlen beliebig genau

Multi-threshold Control of the BMAP/SM/1/K Queue with Group Services

We consider a finite capacity queue in which arrivals occur according to a batch Markovian arrival process (BMAP). The customers are served in groups of varying sizes. The services are governed by a controlled semi-Markovian process according to a multithreshold strategy. We perform the steady-state analysis of this model by computing (a) the queue length distributions at departure and arbitrary epochs, (b) the Laplace-Stieltjes transform of the sojourn time distribution of an admitted customer, and (c) some selected system performance measures. An optimization problem of interest is presented and some numerical examples are illustrated

Performance analysis of priority queueing systems in discrete time

The integration of different types of traffic in packet-based networks spawns the need for traffic differentiation. In this tutorial paper, we present some analytical techniques to tackle discrete-time queueing systems with priority scheduling. We investigate both preemptive (resume and repeat) and non-preemptive priority scheduling disciplines. Two classes of traffic are considered, high-priority and low-priority traffic, which both generate variable-length packets. A probability generating functions approach leads to performance measures such as moments of system contents and packet delays of both classes

Variability-aware request replication for latency curtailment

Processing time variability is commonplace in distributed systems, where resources display disparate performance due to, e.g., different workload levels, background processes, and contention in virtualized environments. However, it is paramount for service providers to keep variability in response time under control in order to offer responsive services. We investigate how request replication can be used to exploit processing time variability to reduce response times, considering not only mean values but also the tail of the response time distribution. We focus on the distributed setup, where replication is achieved by running copies of requests on multiple servers that otherwise evolve independently, and waiting for the first replica to complete service. We construct models that capture the evolution of a system with replicated requests using approximate methods and observe that highly variable service times offer the best opportunities for replication ¿¿¿ reducing the response time tail in particular. Further, the effect of replication is non-uniform over the response time distribution: gains in one metric, e.g., the mean, can be at the cost of another, e.g., the tail percentiles. This is demonstrated in wide range of numerical virtual experiments. It can be seen that capturing service time variability is key to the evaluation of latency tolerance strategies and in their design