Simple and explicit bounds for multi-server queues with 1/(1−ρ)1/(1 - \rho) (and sometimes better) scaling

We consider the FCFS $GI/GI/n$ queue, and prove the first simple and explicit bounds that scale as $\frac{1}{1-\rho}$ (and sometimes better). Here $\rho$ denotes the corresponding traffic intensity. Conceptually, our results can be viewed as a multi-server analogue of Kingman's bound. Our main results are bounds for the tail of the steady-state queue length and the steady-state probability of delay. The strength of our bounds (e.g. in the form of tail decay rate) is a function of how many moments of the inter-arrival and service distributions are assumed finite. More formally, suppose that the inter-arrival and service times (distributed as random variables $A$ and $S$ respectively) have finite $r$th moment for some $r > 2.$ Let $\mu_A$ (respectively $\mu_S$) denote $\frac{1}{\mathbb{E}[A]}$ (respectively $\frac{1}{\mathbb{E}[S]}$). Then our bounds (also for higher moments) are simple and explicit functions of $\mathbb{E}\big[(A \mu_A)^r\big], \mathbb{E}\big[(S \mu_S)^r\big], r$, and $\frac{1}{1-\rho}$ only. Our bounds scale gracefully even when the number of servers grows large and the traffic intensity converges to unity simultaneously, as in the Halfin-Whitt scaling regime. Some of our bounds scale better than $\frac{1}{1-\rho}$ in certain asymptotic regimes. More precisely, they scale as $\frac{1}{1-\rho}$ multiplied by an inverse polynomial in $n(1 - \rho)^2.$ These results formalize the intuition that bounds should be tighter in light traffic as well as certain heavy-traffic regimes (e.g. with $\rho$ fixed and $n$ large). In these same asymptotic regimes we also prove bounds for the tail of the steady-state number in service. Our main proofs proceed by explicitly analyzing the bounding process which arises in the stochastic comparison bounds of amarnik and Goldberg for multi-server queues. Along the way we derive several novel results for suprema of random walks and pooled renewal processes which may be of independent interest. We also prove several additional bounds using drift arguments (which have much smaller pre-factors), and make several conjectures which would imply further related bounds and generalizations

As simple as possible, but no simpler: A gentle introduction to simulation modeling

Proceedings of the 2006 Winter Simulation Conference, 1-10.SEED Center PaperWe start with basic terminology and concepts of modeling, and decompose the art of modeling as a process. This overview of the process helps clarify when we should or should not use simulation models. We discuss some common missteps made by many inexperienced modelers, and propose a concrete approach for avoiding those mistakes. After a quick review of event graphs, which are a very straightforward notation for discrete event systems, we illustrate how an event graph can be translated quite directly to a computer program with the aid of a surprisingly simple library. The resulting programs are easy to implement and computationally are extremely efficient. The first half of the paper focuses principles of modeling, should be of general interest. The second half will be of interest to students, teachers, and readers who wish to know how simulations models work and how to implement them from the ground up

La utilización de la investigación de operaciones como soporte a la toma de decisiones en el sector salud: Un estado del arte

The contributions of Operations Research (OR) in the healthcare field have been extensively studied in the scientific literature since the 1960s, covering decision support tools with operational, tactical, and strategic approaches. The aim of this article is to analyze the historical development of the application of OR models in healthcare. The application trends for optimization, planning, and decision- making models are studied through a descriptive literature review and a bibliometric analysis of scientific papers published between 1952 and 2016. An upward trend in the usage of operational models is observed with the predominance of resource optimization approaches and strategic decision-making for public health.Los aportes de la Investigación de Operaciones (IO) en el campo de la salud han sido ampliamente estudiados en la literatura científica desde la década de 1960, abarcando herramientas para el soporte a la decisión en enfoques operacionales, tácticos y estratégicos. El objetivo de este artículo es analizar el avance y el desarrollo histórico del uso de modelos operativos en el campo de la salud. A través de una revisión bibliográfica descriptiva y un análisis bibliométrico de artículos científicos publicados durante el periodo 1952-2016, se estudia el comportamiento de las tendencias en la aplicación de modelos operativos para la optimización, la planificación y la toma de decisiones en el sector salud. Se evidencia una tendencia creciente en el uso de modelos de IO durante el periodo estudiado, predominando las aplicaciones orientadas a la optimización de recursos y decisiones estratégicas de salud pública

A tool for model-checking Markov chains

Markov chains are widely used in the context of the performance and reliability modeling of various systems. Model checking of such chains with respect to a given (branching) temporal logic formula has been proposed for both discrete [34, 10] and continuous time settings [7, 12]. In this paper, we describe a prototype model checker for discrete and continuous-time Markov chains, the Erlangen-Twente Markov Chain Checker EÎMC2, where properties are expressed in appropriate extensions of CTL. We illustrate the general benefits of this approach and discuss the structure of the tool. Furthermore, we report on successful applications of the tool to some examples, highlighting lessons learned during the development and application of EÎMC2

MINA - a tool for MSC-based performance analysis and simulation of distributed systems

Performance analysis can help to address quantitative system analysis from the early stages of the system development life cycle, e.g., to compare design alternatives or to identify system bottlenecks. This thesis addresses the problem of performance evaluation of distributed systems by employing a viewpoint where analytical and simulative evaluation techniques are unified in the MINA tool to make use of both techniques. We suggest a modelling tool chain to evaluate the performance of distributed systems like computer and communication systems based on an MSC description of the system. MSC-based performance evaluation of distributed systems is an approach that uses performance models, which are based on an MSC description of a system to evaluate system performance measures. To determine the system performance, these descriptions can be extended by notions for time consumption and resource usage and afterwards be included in a system performance model. Based on this unique model specification, analytical as well as simulative techniques can be applied to achieve either quick mean value results by queueing networks analysis or confidence intervals or transient measures by simulation. The applicability to real world systems and the advantages of the tool has been demonstrated by a large application example in the field of mobile communication systems, and its effectiveness has been evaluated by comparing it with other approaches. The experimental results show that the tool is scalable, the way it can model simple as well as complex systems. Moreover, it is straightforward and has the ability to find reasonable solutions in an efficient manner