1) The document discusses methods to obtain more accurate approximations for performance measures like the probability of an empty queue in the Halfin-Whitt regime for an M/D/s queue.
2) The main idea is to view the M/D/s queue through the prism of the Gaussian random walk and obtain asymptotic series expansions involving terms including the Riemann zeta function.
3) The series expansions quantify the relationship between the limiting system and finite queue sizes, and the first few terms have the correct behavior as the number of servers grows large.
This document investigates the relationship between queue length and workload processes in Markovian shortest remaining processing time (SRPT) queues through simulation. Prior work found that under heavy traffic conditions, the diffusion-scaled queue length converges to zero while the workload converges to a non-degenerate limit. The author explores if varying the processing rate affects the limiting variance of the queue length by simulating SRPT queues with exponential processing times and varying arrival rates. Simulation results suggest that under a logarithmic scaling, the queue length and workload processes converge to the same limit, with the limiting variance depending on the arrival rate.
This document summarizes a numerical study of single-phase and two-phase flow through sudden contractions in mini channels. Two-phase computational fluid dynamics simulations using an Eulerian-Eulerian model were performed to calculate pressure drop across contractions for water, air, and air-water mixtures. The pressure drop was determined by extrapolating pressure profiles upstream and downstream. Results were validated against experimental data and used to develop a correlation for two-phase pressure drop due to contraction.
HOW TO PREDICT HEAT AND MASS TRANSFER FROM FLUID FRICTIONbalupost
In this paper, the „Generalized Lévêque Equation (GLE)“, which allows to calculate heat or mass transfer coefficients – or the corresponding Nusselt and Sherwood numbers – from frictional pressure drop or friction forces in place of the flow rates or Reynolds numbers is used in external flow situations, such as a single sphere or a single cylinder in cross flow.
This document presents results from a lattice QCD calculation of the proton isovector scalar charge (gs) at two light quark masses. The calculation uses domain-wall fermions and Iwasaki gauge actions on a 323x64 lattice with a spacing of 0.144 fm. Ratios of three-point to two-point correlation functions are formed and fit to a plateau to extract gs. Values of gs are obtained for quark masses of 0.0042 and 0.001, and all-mode averaging is used for the lighter mass. Chiral perturbation theory will be used to extrapolate gs to the physical quark mass. Preliminary results for gs at the unphysical quark masses are reported in lattice units.
1) The document discusses finite element methods for solving the incompressible Navier-Stokes equations numerically. It covers spatial discretization using finite elements, time discretization, solving the resulting algebraic systems, theoretical analysis, error control, and extension to weakly compressible flows.
2) It provides examples of viscous flows that can be computed using these methods, such as cavity flow, vortex shedding behind a cylinder, and leapfrogging vortex rings.
3) The goal is to develop efficient and accurate numerical tools for computing laminar viscous flows.
On Approach of Estimation Time Scales of Relaxation of Concentration of Charg...Zac Darcy
In this paper we generalized recently introduced approach for estimation of time scales of mass transport.
The approach have been illustrated by estimation of time scales of relaxation of concentrations of charge
carriers in high-doped semiconductor. Diffusion coefficients and mobility of charge carriers and electric
field strength in semiconductor could be arbitrary functions of coordinate.
River stage forecasting using wavelet analysisVinit Sehgal
The document describes research on using wavelet regression (WR) models and artificial neural network (ANN) models to forecast river stages of the Kosi River in India. WR models were developed using discrete wavelet transform to decompose historical river stage data into wavelet components, which were then used as inputs to autoregressive models. The WR models achieved higher coefficients of correlation and lower root mean square errors than ANN models when evaluating predictions against verification data, indicating WR models more accurately forecasted river stages of the Kosi River.
This document analyzes multicarrier transmission in time-varying channels. It considers two cases: when the channel is completely known at the transmitter and receiver, and when only channel statistics are known at the transmitter and the channel is known at the receiver. For the first case, it develops a vector coding scheme that is shown to be asymptotically optimal. For the second case, it derives the expected mutual information for OFDM in time-varying environments to study the effect of time variation on OFDM design. It also illustrates transmission overhead requirements through a numerical example.
This document investigates the relationship between queue length and workload processes in Markovian shortest remaining processing time (SRPT) queues through simulation. Prior work found that under heavy traffic conditions, the diffusion-scaled queue length converges to zero while the workload converges to a non-degenerate limit. The author explores if varying the processing rate affects the limiting variance of the queue length by simulating SRPT queues with exponential processing times and varying arrival rates. Simulation results suggest that under a logarithmic scaling, the queue length and workload processes converge to the same limit, with the limiting variance depending on the arrival rate.
This document summarizes a numerical study of single-phase and two-phase flow through sudden contractions in mini channels. Two-phase computational fluid dynamics simulations using an Eulerian-Eulerian model were performed to calculate pressure drop across contractions for water, air, and air-water mixtures. The pressure drop was determined by extrapolating pressure profiles upstream and downstream. Results were validated against experimental data and used to develop a correlation for two-phase pressure drop due to contraction.
HOW TO PREDICT HEAT AND MASS TRANSFER FROM FLUID FRICTIONbalupost
In this paper, the „Generalized Lévêque Equation (GLE)“, which allows to calculate heat or mass transfer coefficients – or the corresponding Nusselt and Sherwood numbers – from frictional pressure drop or friction forces in place of the flow rates or Reynolds numbers is used in external flow situations, such as a single sphere or a single cylinder in cross flow.
This document presents results from a lattice QCD calculation of the proton isovector scalar charge (gs) at two light quark masses. The calculation uses domain-wall fermions and Iwasaki gauge actions on a 323x64 lattice with a spacing of 0.144 fm. Ratios of three-point to two-point correlation functions are formed and fit to a plateau to extract gs. Values of gs are obtained for quark masses of 0.0042 and 0.001, and all-mode averaging is used for the lighter mass. Chiral perturbation theory will be used to extrapolate gs to the physical quark mass. Preliminary results for gs at the unphysical quark masses are reported in lattice units.
1) The document discusses finite element methods for solving the incompressible Navier-Stokes equations numerically. It covers spatial discretization using finite elements, time discretization, solving the resulting algebraic systems, theoretical analysis, error control, and extension to weakly compressible flows.
2) It provides examples of viscous flows that can be computed using these methods, such as cavity flow, vortex shedding behind a cylinder, and leapfrogging vortex rings.
3) The goal is to develop efficient and accurate numerical tools for computing laminar viscous flows.
On Approach of Estimation Time Scales of Relaxation of Concentration of Charg...Zac Darcy
In this paper we generalized recently introduced approach for estimation of time scales of mass transport.
The approach have been illustrated by estimation of time scales of relaxation of concentrations of charge
carriers in high-doped semiconductor. Diffusion coefficients and mobility of charge carriers and electric
field strength in semiconductor could be arbitrary functions of coordinate.
River stage forecasting using wavelet analysisVinit Sehgal
The document describes research on using wavelet regression (WR) models and artificial neural network (ANN) models to forecast river stages of the Kosi River in India. WR models were developed using discrete wavelet transform to decompose historical river stage data into wavelet components, which were then used as inputs to autoregressive models. The WR models achieved higher coefficients of correlation and lower root mean square errors than ANN models when evaluating predictions against verification data, indicating WR models more accurately forecasted river stages of the Kosi River.
This document analyzes multicarrier transmission in time-varying channels. It considers two cases: when the channel is completely known at the transmitter and receiver, and when only channel statistics are known at the transmitter and the channel is known at the receiver. For the first case, it develops a vector coding scheme that is shown to be asymptotically optimal. For the second case, it derives the expected mutual information for OFDM in time-varying environments to study the effect of time variation on OFDM design. It also illustrates transmission overhead requirements through a numerical example.
This is a post-preprint of a contributed paper at the XXII International conference on High Energy Physics Leipzig, July 19-25, 1984. Some of the ideas developed in this paper may be useful in the interpretation of the standard model of modern cosmology
Closed-form Solutions of Generalized Greenshield Relations for the Social and...Michael Maroun
This document presents closed-form expressions for the economic optimization of traffic flow on a single-entry road modeled by the Lighthill-Whitham-Richards fluid model with a generalized Greenshields relation. It provides solutions for the social and user optimum flow rates for all values of gamma, the parameter in the generalized Greenshields relation. The document derives the continuity equation governing traffic density and flow over time and space on the road. It presents the method of characteristics to solve this partial differential equation and obtain the density and flow functions, subject to given initial conditions. Tables summarize the initial conditions used to solve the continuity equation for this traffic flow model.
1. The document derives a general differential equation for fluid flow problems in rectangular Cartesian coordinates using a shell momentum balance.
2. It considers flow between parallel plates where the velocity depends only on the x-coordinate and derives expressions for the shear stress distribution, velocity profile, maximum velocity, average velocity, and mass flow rate for a Newtonian fluid.
3. The shear stress is found to be linearly proportional to x, the velocity profile parabolic, the maximum velocity occurs at the center, and the average velocity is 2/3 of the maximum velocity.
Joint blind calibration and time-delay estimation for multiband rangingTarik Kazaz
In this presentation, we focus on the problem of blind joint calibration of multiband transceivers and time-delay (TD) estimation of multipath channels. We show that this problem can be formulated as a particular case of covariance matching. Although this problem is severely ill-posed, prior information about radio-frequency chain distortions and multipath channel sparsity is used for regularization. This approach leads to a biconvex optimization problem, which is formulated as a rank-constrained linear system and solved by a simple group Lasso algorithm.
% This method is general and can be also applied for calibration of sensors arrays and in direction of arrival estimation.
Numerical experiments show that the proposed algorithm provides better calibration and higher resolution for TD estimation than current state-of-the-art methods.
This project investigates fluid flow near a flat plate that is suddenly accelerated. The velocity profile is obtained using the similarity method proposed by Stokes, which reduces the partial differential equations to an ordinary differential equation. The equation is solved numerically using Simpson's approximation. The results show that the velocity profiles for varying times are similar when scaled appropriately. The velocity and shear stress at the wall are also examined for different times. The shear stress decreases with increasing time as diffusion causes the flow to develop.
This document summarizes a journal article that models a dam break over a rubble mound using the shallow water equations and the Lax-Friedrichs numerical scheme. The model is validated against experimental data. A grid refinement study is performed using three grid sizes to calculate the grid convergence index, which indicates the smallest grid size needed for a grid-independent solution. The model accurately simulates stationary water and matches experimental data for a dam break over a triangular obstacle.
Análisis de los resultados experimentales del aleteo de turbomaquinaria utili...BrianZamora7
This paper presents a new rotor-stator coupling method for frequency domain analysis of unsteady flow in turbomachinery. The method, called time and space mode decomposition and matching method, is based on coordinate transformation and Fourier transformation. It extracts relevant time and spatial modes from the flow variables using frequency, nodal diameter, and Fourier coefficients. Detailed procedures and formulas are established to identify matching modes across interfaces and calculate mode coefficients. The method was tested on a transonic compressor by comparing frequency domain and time domain solutions.
A Markov Chain Monte Carlo approach to the Steiner Tree Problem in water netw...Carlo Lancia
The document describes using a Metropolis algorithm and Markov chain Monte Carlo (MCMC) approach to solve the minimal Steiner tree problem, which models finding the shortest water distribution network for a farm district. It formulates the problem on a graph and defines the optimization problem and statistical mechanics analogy. It then describes designing a Markov chain with transition probabilities following the Metropolis rule and a Hamiltonian function incorporating edge costs and penalties for configurations violating constraints.
This document provides an introduction to kernel density estimation for non-parametric density estimation. It discusses how kernel density estimation works by placing a kernel over each data point and summing the kernels to estimate the probability density function without parametric assumptions. The key steps are: (1) using a kernel function like the Parzen window to determine how many points fall within a region of size h centered on the point x to estimate; (2) the estimate is the sum of the kernel values divided by the sample size N and volume h^D; and (3) the bandwidth h acts as a smoothing parameter, with a wider h producing a smoother estimate.
This document describes the development of a numerical tool to simulate gas flow and heat transfer in a Wankel rotary engine. The tool comprises a 2D/3D grid generator for the engine geometry, an implicit finite element method to handle pressure-velocity coupling, and robust multigrid solvers on distorted meshes. These components are implemented in a new finite element software package called Hi-Flow++, which currently contains a 2D solver for the stationary compressible Navier-Stokes equations in the low-Mach number approximation. The project aims to extend this to nonstationary flows and develop a 3D solver.
The document discusses the design of pipe networks for water distribution. It describes various methods for analyzing pressure in distribution systems, including the equivalent pipe method, Hardy Cross method, and graphical method. The equivalent pipe method involves replacing a complex pipe system with a single hydraulically equivalent pipe. The document provides detailed steps for applying the equivalent pipe method to pipes placed in series and parallel. It also describes the Hardy Cross method which balances heads by iteratively correcting assumed pipe flows until the total head loss equals zero.
Setting and Usage of OpenFOAM multiphase solver (S-CLSVOF)takuyayamamoto1800
The S-CLSVOF solver in OpenFOAM uses a coupled volume of fluid (VOF) and level set method to simulate multiphase flows. It uses a level set function to track the interface and reinitialize it, improving on the standard VOF method. The solver has been implemented in OpenFOAM versions 2.0.x and higher but boundary conditions for the level set function have not been fully developed. The document provides information on setting up and running a dam break tutorial case using the S-CLSVOF solver by modifying an existing interFoam case.
An improved dft based channel estimationsakru naik
This document proposes an improved DFT-based channel estimation method for MIMO-OFDM systems. The conventional DFT method causes energy leakage in non-sample-spaced multipath channels. The improved method extends the LS estimate using symmetry, calculates the changing rate of leakage energy, and selects useful paths based on this rate to reduce leakage energy. Simulation results show the improved method reduces leakage energy more efficiently and provides better channel estimation performance than LS and conventional DFT algorithms.
The document summarizes computational fluid dynamics (CFD) simulations performed to optimize the mesh for the Siemens 4th generation distributed burner. Three meshes were analyzed: a coarse 0.55 million cell mesh, an intermediate 4.3 million cell mesh, and a fine 35 million cell mesh. Results showed negligible differences in velocity and equivalence ratio between the 4.3 million and 35 million cell meshes inside the main burner area. An optimized 1.4 million cell mesh was also created, showing similar results to the finer meshes while greatly reducing computational cost. Overall, the study demonstrated that the mesh in the main burner region could be optimized without loss of accuracy to better balance resolution and computation time.
DSD-NL 2015, Geo Klantendag D-Series, 5 Grote deformaties bij paalinstallatiesDeltares
The document discusses using the Material Point Method (MPM) to model soil-structure interaction problems involving large deformations, including cone penetration testing (CPT), pile installation, and lateral pile load testing. It describes how MPM can model (1) partially drained CPT to better estimate soil permeability from measurements, (2) undrained CPT using an anisotropic clay model to relate tip resistance to undrained shear strength, and (3) pile installation and subsequent lateral loading to validate numerical simulations against field test results. The goal is to use MPM to improve understanding of soil behavior and parameter estimation for geotechnical analysis and design.
This document summarizes John Obuch's paper on modeling crystallization processes in one dimension. It presents four models of crystal growth: (1) the classic Avrami model where crystals grow at a constant rate until impinging on neighbors, (2) a modified model where growth stops when crystals are a distance Δ apart, (3) a model where growth stops at a fixed width, and (4) a combined model of (2) and (3). It derives an expression for the expected crystallized volume V(t) at time t using the Beta distribution of gap sizes between nucleation sites. The document then describes simulating the models by randomly placing nucleation sites and calculating the critical times when crystals impinge.
A REVIEW OF OPTIMUM SPEED MODEL An Assignment On Advanced Traffic Engineering...DAUDA SANUSI
A REVIEW OF OPTIMUM SPEED MODEL
An Assignment On Advanced Traffic Engineering (CIV8329)
by
Sanusi Dauda
SPS/16/MCE/00027
Submitted to
Prof. H. M. Alhassan
Highway and Transportation Engineering (Option)
Department of Civil Engineering
Faculty of Engineering
Bayero University, Kano
19TH May, 2017
This document presents and compares three approximation methods for thin plate spline mappings that reduce the computational complexity from O(p3) to O(m3), where m is a small subset of points p. Method 1 uses only the subset of points to estimate the mapping. Method 2 uses the subset of basis functions with all target values. Method 3 approximates the full matrix using the Nyström method. Experiments on synthetic grids show Method 3 has the lowest error, followed by Method 2, with Method 1 having the highest error. The three methods trade off accuracy, computation time, and the ability to do principal warp analysis.
we demonstrate that it can be
advantageous from a computational point of view
to use a two-stage realization instead of a single stage
realization for sample rate conversions with
prime numbers. One of the stages performs a
conversion by a factor of two using linear-phase, or
approximately linear-phase, half-band filters. The
other stage changes the sample rate by the
rational factor N/2 using a linear-phase FIR filter.
The actual filtering can be performed at the lowest
of the two sample frequencies involved. It is also
possible to exploit the coefficient symmetry of the
linear-phase FIR filter in the stage that changes
the rate by a rational factor. The overall workload
of the two-stage realization can therefore be
reduced compared with the corresponding single stage
realization.
Palm Tree Hostel Medellín has been welcoming guests since 2001 and is a top choice for travelers due to its great facilities. It is located in a central area near public transportation and offers a calm, clean, and inexpensive hostel experience with a warm, home-like atmosphere. The hostel aims to make guests' stay in Medellín an unforgettable experience by providing information about the city and opportunities to meet other travelers from around the world.
Este documento describe un estudio que evalúa la presencia de anticuerpos IgE, IgG e IgA específicos para carne de vaca en pacientes alérgicos con diferentes manifestaciones clínicas. Los resultados mostraron que la carne de vaca fue el alimento con mayor inmunogenicidad e identificaron la coexistencia de los tres tipos de anticuerpos en los pacientes alérgicos, lo que podría ayudar a comprender mejor las patologías alérgicas y mejorar el tratamiento.
Este documento presenta la colección de vestidos 2015 de Bella Moda, incluyendo 10 modelos diferentes con detalles como velos, colas y escotes. Cada vestido se describe con su precio original, color, tallas disponibles y características distintivas, además de mostrar su nuevo precio de venta rebajado.
This is a post-preprint of a contributed paper at the XXII International conference on High Energy Physics Leipzig, July 19-25, 1984. Some of the ideas developed in this paper may be useful in the interpretation of the standard model of modern cosmology
Closed-form Solutions of Generalized Greenshield Relations for the Social and...Michael Maroun
This document presents closed-form expressions for the economic optimization of traffic flow on a single-entry road modeled by the Lighthill-Whitham-Richards fluid model with a generalized Greenshields relation. It provides solutions for the social and user optimum flow rates for all values of gamma, the parameter in the generalized Greenshields relation. The document derives the continuity equation governing traffic density and flow over time and space on the road. It presents the method of characteristics to solve this partial differential equation and obtain the density and flow functions, subject to given initial conditions. Tables summarize the initial conditions used to solve the continuity equation for this traffic flow model.
1. The document derives a general differential equation for fluid flow problems in rectangular Cartesian coordinates using a shell momentum balance.
2. It considers flow between parallel plates where the velocity depends only on the x-coordinate and derives expressions for the shear stress distribution, velocity profile, maximum velocity, average velocity, and mass flow rate for a Newtonian fluid.
3. The shear stress is found to be linearly proportional to x, the velocity profile parabolic, the maximum velocity occurs at the center, and the average velocity is 2/3 of the maximum velocity.
Joint blind calibration and time-delay estimation for multiband rangingTarik Kazaz
In this presentation, we focus on the problem of blind joint calibration of multiband transceivers and time-delay (TD) estimation of multipath channels. We show that this problem can be formulated as a particular case of covariance matching. Although this problem is severely ill-posed, prior information about radio-frequency chain distortions and multipath channel sparsity is used for regularization. This approach leads to a biconvex optimization problem, which is formulated as a rank-constrained linear system and solved by a simple group Lasso algorithm.
% This method is general and can be also applied for calibration of sensors arrays and in direction of arrival estimation.
Numerical experiments show that the proposed algorithm provides better calibration and higher resolution for TD estimation than current state-of-the-art methods.
This project investigates fluid flow near a flat plate that is suddenly accelerated. The velocity profile is obtained using the similarity method proposed by Stokes, which reduces the partial differential equations to an ordinary differential equation. The equation is solved numerically using Simpson's approximation. The results show that the velocity profiles for varying times are similar when scaled appropriately. The velocity and shear stress at the wall are also examined for different times. The shear stress decreases with increasing time as diffusion causes the flow to develop.
This document summarizes a journal article that models a dam break over a rubble mound using the shallow water equations and the Lax-Friedrichs numerical scheme. The model is validated against experimental data. A grid refinement study is performed using three grid sizes to calculate the grid convergence index, which indicates the smallest grid size needed for a grid-independent solution. The model accurately simulates stationary water and matches experimental data for a dam break over a triangular obstacle.
Análisis de los resultados experimentales del aleteo de turbomaquinaria utili...BrianZamora7
This paper presents a new rotor-stator coupling method for frequency domain analysis of unsteady flow in turbomachinery. The method, called time and space mode decomposition and matching method, is based on coordinate transformation and Fourier transformation. It extracts relevant time and spatial modes from the flow variables using frequency, nodal diameter, and Fourier coefficients. Detailed procedures and formulas are established to identify matching modes across interfaces and calculate mode coefficients. The method was tested on a transonic compressor by comparing frequency domain and time domain solutions.
A Markov Chain Monte Carlo approach to the Steiner Tree Problem in water netw...Carlo Lancia
The document describes using a Metropolis algorithm and Markov chain Monte Carlo (MCMC) approach to solve the minimal Steiner tree problem, which models finding the shortest water distribution network for a farm district. It formulates the problem on a graph and defines the optimization problem and statistical mechanics analogy. It then describes designing a Markov chain with transition probabilities following the Metropolis rule and a Hamiltonian function incorporating edge costs and penalties for configurations violating constraints.
This document provides an introduction to kernel density estimation for non-parametric density estimation. It discusses how kernel density estimation works by placing a kernel over each data point and summing the kernels to estimate the probability density function without parametric assumptions. The key steps are: (1) using a kernel function like the Parzen window to determine how many points fall within a region of size h centered on the point x to estimate; (2) the estimate is the sum of the kernel values divided by the sample size N and volume h^D; and (3) the bandwidth h acts as a smoothing parameter, with a wider h producing a smoother estimate.
This document describes the development of a numerical tool to simulate gas flow and heat transfer in a Wankel rotary engine. The tool comprises a 2D/3D grid generator for the engine geometry, an implicit finite element method to handle pressure-velocity coupling, and robust multigrid solvers on distorted meshes. These components are implemented in a new finite element software package called Hi-Flow++, which currently contains a 2D solver for the stationary compressible Navier-Stokes equations in the low-Mach number approximation. The project aims to extend this to nonstationary flows and develop a 3D solver.
The document discusses the design of pipe networks for water distribution. It describes various methods for analyzing pressure in distribution systems, including the equivalent pipe method, Hardy Cross method, and graphical method. The equivalent pipe method involves replacing a complex pipe system with a single hydraulically equivalent pipe. The document provides detailed steps for applying the equivalent pipe method to pipes placed in series and parallel. It also describes the Hardy Cross method which balances heads by iteratively correcting assumed pipe flows until the total head loss equals zero.
Setting and Usage of OpenFOAM multiphase solver (S-CLSVOF)takuyayamamoto1800
The S-CLSVOF solver in OpenFOAM uses a coupled volume of fluid (VOF) and level set method to simulate multiphase flows. It uses a level set function to track the interface and reinitialize it, improving on the standard VOF method. The solver has been implemented in OpenFOAM versions 2.0.x and higher but boundary conditions for the level set function have not been fully developed. The document provides information on setting up and running a dam break tutorial case using the S-CLSVOF solver by modifying an existing interFoam case.
An improved dft based channel estimationsakru naik
This document proposes an improved DFT-based channel estimation method for MIMO-OFDM systems. The conventional DFT method causes energy leakage in non-sample-spaced multipath channels. The improved method extends the LS estimate using symmetry, calculates the changing rate of leakage energy, and selects useful paths based on this rate to reduce leakage energy. Simulation results show the improved method reduces leakage energy more efficiently and provides better channel estimation performance than LS and conventional DFT algorithms.
The document summarizes computational fluid dynamics (CFD) simulations performed to optimize the mesh for the Siemens 4th generation distributed burner. Three meshes were analyzed: a coarse 0.55 million cell mesh, an intermediate 4.3 million cell mesh, and a fine 35 million cell mesh. Results showed negligible differences in velocity and equivalence ratio between the 4.3 million and 35 million cell meshes inside the main burner area. An optimized 1.4 million cell mesh was also created, showing similar results to the finer meshes while greatly reducing computational cost. Overall, the study demonstrated that the mesh in the main burner region could be optimized without loss of accuracy to better balance resolution and computation time.
DSD-NL 2015, Geo Klantendag D-Series, 5 Grote deformaties bij paalinstallatiesDeltares
The document discusses using the Material Point Method (MPM) to model soil-structure interaction problems involving large deformations, including cone penetration testing (CPT), pile installation, and lateral pile load testing. It describes how MPM can model (1) partially drained CPT to better estimate soil permeability from measurements, (2) undrained CPT using an anisotropic clay model to relate tip resistance to undrained shear strength, and (3) pile installation and subsequent lateral loading to validate numerical simulations against field test results. The goal is to use MPM to improve understanding of soil behavior and parameter estimation for geotechnical analysis and design.
This document summarizes John Obuch's paper on modeling crystallization processes in one dimension. It presents four models of crystal growth: (1) the classic Avrami model where crystals grow at a constant rate until impinging on neighbors, (2) a modified model where growth stops when crystals are a distance Δ apart, (3) a model where growth stops at a fixed width, and (4) a combined model of (2) and (3). It derives an expression for the expected crystallized volume V(t) at time t using the Beta distribution of gap sizes between nucleation sites. The document then describes simulating the models by randomly placing nucleation sites and calculating the critical times when crystals impinge.
A REVIEW OF OPTIMUM SPEED MODEL An Assignment On Advanced Traffic Engineering...DAUDA SANUSI
A REVIEW OF OPTIMUM SPEED MODEL
An Assignment On Advanced Traffic Engineering (CIV8329)
by
Sanusi Dauda
SPS/16/MCE/00027
Submitted to
Prof. H. M. Alhassan
Highway and Transportation Engineering (Option)
Department of Civil Engineering
Faculty of Engineering
Bayero University, Kano
19TH May, 2017
This document presents and compares three approximation methods for thin plate spline mappings that reduce the computational complexity from O(p3) to O(m3), where m is a small subset of points p. Method 1 uses only the subset of points to estimate the mapping. Method 2 uses the subset of basis functions with all target values. Method 3 approximates the full matrix using the Nyström method. Experiments on synthetic grids show Method 3 has the lowest error, followed by Method 2, with Method 1 having the highest error. The three methods trade off accuracy, computation time, and the ability to do principal warp analysis.
we demonstrate that it can be
advantageous from a computational point of view
to use a two-stage realization instead of a single stage
realization for sample rate conversions with
prime numbers. One of the stages performs a
conversion by a factor of two using linear-phase, or
approximately linear-phase, half-band filters. The
other stage changes the sample rate by the
rational factor N/2 using a linear-phase FIR filter.
The actual filtering can be performed at the lowest
of the two sample frequencies involved. It is also
possible to exploit the coefficient symmetry of the
linear-phase FIR filter in the stage that changes
the rate by a rational factor. The overall workload
of the two-stage realization can therefore be
reduced compared with the corresponding single stage
realization.
Palm Tree Hostel Medellín has been welcoming guests since 2001 and is a top choice for travelers due to its great facilities. It is located in a central area near public transportation and offers a calm, clean, and inexpensive hostel experience with a warm, home-like atmosphere. The hostel aims to make guests' stay in Medellín an unforgettable experience by providing information about the city and opportunities to meet other travelers from around the world.
Este documento describe un estudio que evalúa la presencia de anticuerpos IgE, IgG e IgA específicos para carne de vaca en pacientes alérgicos con diferentes manifestaciones clínicas. Los resultados mostraron que la carne de vaca fue el alimento con mayor inmunogenicidad e identificaron la coexistencia de los tres tipos de anticuerpos en los pacientes alérgicos, lo que podría ayudar a comprender mejor las patologías alérgicas y mejorar el tratamiento.
Este documento presenta la colección de vestidos 2015 de Bella Moda, incluyendo 10 modelos diferentes con detalles como velos, colas y escotes. Cada vestido se describe con su precio original, color, tallas disponibles y características distintivas, además de mostrar su nuevo precio de venta rebajado.
Este documento contiene un listado de 16 contactos con sus nombres, apellidos, teléfono, edad, celular, correo electrónico y página web. La información incluye datos personales como nombre, apellidos, edad y de contacto como teléfono, correo y página web.
The document discusses how poor data quality can negatively impact marketing ROI and provides strategies for addressing data quality issues. It suggests enhancing CRM data through data cleaning and integration with external sources to improve customer targeting, segmentation, and campaign analysis. Regular data maintenance is important to reduce costs from data issues and maximize returns from sales and marketing databases.
El documento describe un proyecto para mejorar las habilidades de lectoescritura de los estudiantes de una escuela rural mediante el uso de las TIC. El proyecto busca determinar si la escasa producción de textos de los estudiantes refleja una baja competencia lectora y escritora. Los objetivos son motivar a los estudiantes a crear sus propios textos usando herramientas informáticas y desarrollar sus habilidades comunicativas y de lectoescritura. La metodología propuesta es activa y participativa usando el método de
Este documento lista varios lugares y edificios notables en una ciudad española, incluyendo una iglesia, ayuntamiento, estación de tren, biblioteca, convento, ermita, fuentes, fragmentos de murallas antiguas y esculturas.
Este número de la revista Abades Magazine incluye los siguientes temas:
1) Una entrevista con la artista Pasión Vega donde habla sobre el éxito de su último trabajo musical y su próxima gira.
2) Un reportaje sobre la inauguración de una nueva área de servicio de Abades en Córdoba.
3) Una agenda con eventos culturales y ferias que tendrán lugar en los próximos meses en diferentes ciudades de Andalucía.
К докладу "Разработка форматов мобильного библиотечного обслуживания для различных целевых групп" Добрыниной И.А., директора Центральной универсальной научной библиотеки им. Н.А. Некрасова (г. Москва) (семинар-практикум «Развитие системы мобильного библиотечного обслуживания населения». Москва, октябрь 2014)
Feedback processes in online learning environments: main findings from EdOnline Research Group
Espasa, A.; Guasch, T.; Martínez Melo. M. & Mayordomo, R.
1st International Workshop on Technology-Enhanced Assessment, Analytics and Feedback (TEAAF2014)
Die Bedeutung der inneren Haltung bei FührungskräftenCulture Work GmbH
Ein entscheidender Erfolgsfaktor eines Unternehmens ist eine optimale Führung.
Menschen mit gut entwickelten Führungsfähigkeiten schaffen es, einer Organisation Identität zu geben und ihre Mitarbeiter durch Inspiration und Wertschätzung zu Höchstleistung und Freude an der Arbeit zu motivieren. Entscheidend dafür ist die eigene Innere Haltung und das Vermögen, eine einladende Kommunikationskultur zu schaffen.
Neuro-Systemisch betrachtet besteht eine Organisation oder ein Unternehmen nicht aus der Hierarchie, die sich üblicherweise in einem Organigramm abgebildet wird, sondern aus den tagtäglichen Kommunikations-und Interaktionsmustern, die die beteiligten Personen – auf Basis ihrer inneren Haltungen – dort initiieren. Im Rahmen von Selbstorganisationsphänomenen entstehen dann funktionale und weniger funktionale Muster, die den unternehmerischen und individuellen Erfolg bestimmen.
Für viele Menschen sind die eigenen Lernerfahrungen von der Bewertung durch Andere geprägt, insbesondere durch Bezugspersonen. Diese erlebten Erfahrungen sind als emotional gekoppelte Netzwerke im Frontalhirn angelegt und bestimmen die eigenen Einstellungen, Bewertungen, Wahrnehmungen und Handlungsmuster. Sie bilden die innere Haltung, die „Brille“, die bestimmt welche Dinge eine Person wahrnimmt und in welchem Maße sie diese mit Bedeutung „auflädt“.
Für eine positive Beziehungsgestaltung zu Mitarbeitern und Kollegen ist es für Führungskräfte daher von besonderem Interesse die eigenen inneren Haltungen und Bewertungen kennen zu lernen und ihre Wirkung auf andere einschätzen können. Denn nur wer sich selbst verstehen kann, weiß, wie er mit anderen in Beziehung treten kann. Darüber hinaus ist es hilfreich Kompetenzen zu entwickeln, mit denen man die für das Verhalten anderer bedeutsamen Haltungen erkennt. Dann kann jede Führungskraft ihre Mitarbeiter oder andere Menschen so ansprechen, bzw. einladen, dass eine offene wertschätzende Haltung entsteht, die durch wechselseitige Rückkoppelungsprozesse auch die Stimmung in der Gesamtorganisation positiv beeinflusst.
Der Vortrag möchte Führungskräften grundlegende Neuro-systemische Modelle vorstellen und so Werkzeuge und Anregungen für eine nachhaltig positive Beziehungsgestaltung geben.
El documento describe las principales fuentes de energía según su origen, dividiéndolas en no renovables y renovables. Entre las no renovables se encuentran el carbón, el petróleo, el gas natural y la energía nuclear. El carbón se forma a partir de restos vegetales fosilizados, mientras que el petróleo y el gas natural se originan también de restos orgánicos acumulados en el subsuelo. La energía nuclear proviene de reacciones nucleares como la fisión y fusión. El documento también explica brevemente los problemas med
Este documento presenta una introducción a una serie de lecciones sobre la felicidad impartidas por la maestra Kwan Yin. El objetivo es abrir los corazones de las personas y guiarlas hacia la armonía. Se propone dar las bases para una terapia efectiva que transforme las emociones negativas en la raíz de los problemas mentales. Las lecciones buscan generar reflexión profunda para iniciar un proceso de autocuración. Se explica que la percepción de las personas está influenciada por sus experiencias pasadas, creando un círculo vicioso que las
Start-up Stage - E-Commerce - Presentation by Daniel Thung, Co-Founder of Brillen.de at the NOAH 2015 Conference in London, Old Billingsgate on the 12th of November 2015.
Personas con afasia. Comunicación en el entorno sanitario. Cuaderno de Apoyo. Ceapat de Imserso
"Cuaderno de Apoyo a la comunicación en el entorno sanitario. Personas con afasia."
Este cuaderno es parte del trabajo llevado a cabo en el proyecto «Yo te cuento, cuenta conmigo», desarrollado entre 2012 y 2013, en el que se han elaborado también los siguientes recursos de apoyo para la comunicación:
Cuaderno de apoyo para la comunicación con el paciente y
Cuaderno de apoyo a la comunicación con el paciente (discapacidad intelectual)
Este cuaderno está diseñado para apoyar la comunicación con el paciente con afasia en el entorno sanitario. Consiste en una serie de fichas en las que se recoge el vocabulario básico sobre salud, expresado con pictogramas y con la palabra escrita.
Este documento proporciona información sobre la influenza. Explica que la influenza es una enfermedad respiratoria contagiosa causada por virus de influenza A, B y C. Describe los diferentes subtipos de virus de influenza A y cómo surgió un nuevo virus H1N1 en 2009. También detalla cómo se transmite la influenza de persona a persona y los síntomas comunes. Finalmente, cubre temas como el tratamiento, las pruebas de laboratorio y la importancia de la vacunación contra la influenza.
An Open Shop Approach in Approximating Optimal Data Transmission Duration in ...cscpconf
In the past decade Optical WDM Networks (Wavelength Division Multiplexing) are being used
quite often and especially as far as broadband applications are concerned. Message packets
transmitted through such networks can be interrupted using time slots in order to maximize
network usage and minimize the time required for all messages to reach their destination.
However, preempting a packet will result in time cost. The problem of scheduling message
packets through such a network is referred to as PBS and is known to be NP-Hard. In this paper
we have reduced PBS to Open Shop Scheduling and designed variations of polynomially
solvable instances of Open Shop to approximate PBS. We have combined these variations and
called the induced algorithm HSA (Hybridic Scheduling Algorithm). We ran experiments to
establish the efficiency of HSA and found that in all datasets used it produces schedules very
close to the optimal. To further establish HSA’s efficiency we ran tests to compare it to SGA,
another algorithm which when tested in the past has yielded excellent results.
AN OPEN SHOP APPROACH IN APPROXIMATING OPTIMAL DATA TRANSMISSION DURATION IN ...csandit
This document presents a hybrid algorithm (HSA) for approximating optimal data transmission duration in WDM networks. HSA reduces the preemptive bipartite scheduling problem (PBS) to open shop scheduling problems that can be solved in polynomial time. HSA combines two such algorithms, POSA and OS01PT, to minimize makespan and number of preemptions respectively. Experimental results show HSA produces schedules very close to optimal and outperforms another efficient algorithm (SGA) for PBS, with an approximation ratio up to 8% better. Future work could aim to improve HSA's time complexity or prove a better approximation ratio.
Alexander Litvinenko's research interests include developing efficient numerical methods for solving stochastic PDEs using low-rank tensor approximations. He has made contributions in areas such as fast techniques for solving stochastic PDEs using tensor approximations, inexpensive functional approximations of Bayesian updating formulas, and modeling uncertainties in parameters, coefficients, and computational geometry using probabilistic methods. His current research focuses on uncertainty quantification, Bayesian updating techniques, and developing scalable and parallel methods using hierarchical matrices.
This document proposes a fast and robust bootstrap method for inference using the least trimmed squares (LTS) estimator in regression analysis. The classical bootstrap is computationally intensive and lacks robustness when applied to LTS. The proposed method draws bootstrap samples but approximates the LTS solution in each sample using information from the original LTS estimate, rather than recomputing LTS from scratch. This avoids the need for multiple initial subsets and is shown via simulations to perform well, providing accurate confidence intervals while being both fast and robust compared to the classical bootstrap for LTS.
This document proposes methods for evaluating the quality of unsupervised anomaly detection algorithms when labeled data is unavailable. It introduces two label-free performance criteria called Excess-Mass (EM) and Mass-Volume (MV) curves, which are based on existing concepts but adapted here. To address issues with high-dimensional data, a feature subsampling methodology is described. An experiment evaluates three anomaly detection algorithms on various datasets using the proposed EM and MV criteria, finding they accurately discriminate algorithm performance compared to labeled ROC and PR criteria.
1. Dimensional analysis and the concept of similitude allow experiments using scale models to be used to study full-scale systems. Dimensional analysis uses Buckingham pi theorem to determine the minimum number of dimensionless groups needed to describe a phenomenon in terms of the variables involved.
2. For a model to accurately simulate a prototype system, the dimensionless pi groups that describe the phenomenon must be equal between the model and prototype. This establishes the modeling laws or similarity requirements that a model must satisfy.
3. Common dimensionless groups in fluid mechanics include the Reynolds number, Froude number, Strouhal number, and Weber number. These groups arise frequently in analyzing experimental data from fluid mechanics problems.
This document summarizes a new stochastic optimization method called Complex Simultaneous Perturbation Stochastic Approximation (CSPSA) that can directly optimize real-valued functions of complex variables. CSPSA estimates the complex gradient of the target function within the field of complex numbers and generates a sequence of complex estimates that converges to the optimal solution. The method has advantages over existing approaches that optimize in the real domain, as calculations are simpler using complex variables and the complex structure can improve performance. Numerical tests on quantum state tomography demonstrate CSPSA achieves solutions orders of magnitude closer to the true minimum compared to other methods using the same resources.
This document summarizes an exercise on using computational methods to solve partial differential equations (PDEs). It first discusses using relaxation techniques to solve the Laplace equation for modeling electric fields around parallel plate capacitors. The Jacobi method was used to iteratively calculate potentials on a grid until convergence within a specified tolerance. Smaller tolerances and larger grids required more iterations to converge. Longer capacitor plates produced more uniform fields resembling the theoretical infinite case. The document demonstrates how computational methods can effectively solve important physical problems modeled by PDEs.
Quantum algorithm for solving linear systems of equationsXequeMateShannon
Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b, find a vector x such that Ax=b. We consider the case where one doesn't need to know the solution x itself, but rather an approximation of the expectation value of some operator associated with x, e.g., x'Mx for some matrix M. In this case, when A is sparse, N by N and has condition number kappa, classical algorithms can find x and estimate x'Mx in O(N sqrt(kappa)) time. Here, we exhibit a quantum algorithm for this task that runs in poly(log N, kappa) time, an exponential improvement over the best classical algorithm.
This document discusses different types of regression analysis techniques including linear regression, polynomial regression, support vector regression, decision tree regression, ridge regression, lasso regression, and logistic regression. Linear regression finds the relationship between a continuous dependent variable and one or more independent variables. Polynomial regression handles nonlinear relationships through higher-order terms. Support vector regression and decision tree regression can handle both linear and nonlinear data. Ridge and lasso regression are regularization techniques used to prevent overfitting. Logistic regression is for classification rather than regression problems.
This document summarizes a research paper that proposes using H-infinity optimization to derive a causal approximation for spline interpolation. Spline interpolation is commonly used in image processing but requires filtering past and future data, making it non-causal. The paper formulates designing a causal approximation as an H-infinity optimization problem to minimize the worst-case error over all possible input signals. For cubic splines, a closed-form optimal causal filter is derived. Numerical methods can solve for optimal filters for higher-order splines or when constraining the filter to be finite impulse response. An example is provided to demonstrate the effectiveness of the proposed causal approximation using H-infinity optimization.
Forecasting Default Probabilities in Emerging Markets and Dynamical Regula...SSA KPI
The document discusses robust conic generalized partial linear models (RCGPLMs) and their application in forecasting default probabilities in emerging markets. It provides background on regression techniques like MARS and CMARS. It then introduces generalized linear models (GLMs) and generalized partial linear models (GPLMs), describing how conic GPLMs combine linear and nonlinear regression methods. The document outlines the development of robust conic GPLMs to account for uncertainty in input data, and provides examples of applying RCGPLMs to real-world default prediction problems.
Discrete wavelet transform-based RI adaptive algorithm for system identificationIJECEIAES
In this paper, we propose a new adaptive filtering algorithm for system identifica- tion. The algorithm is based on the recursive inverse (RI) adaptive algorithm which suffers from low convergence rates in some applications; i.e., the eigenvalue spread of the autocorrelation matrix is relatively high. The proposed algorithm applies discrete-wavelet transform (DWT) to the input signal which, in turn, helps to overcome the low convergence rate of the RI algorithm with relatively small step-size(s). Different scenarios has been investigated in different noise environments in system identification setting. Experiments demonstrate the advantages of the proposed DWT recursive inverse (DWT-RI) filter in terms of convergence rate and mean-square-error (MSE) compared to the RI, discrete cosine transform LMS (DCT-LMS), discretewavelet transform LMS (DWT-LMS) and recursive-least-squares (RLS) algorithms under same conditions.
This paper proposes and analyzes the performance of a selection decode-and-forward cooperative free-space optical communication system using adaptive subcarrier quadrature amplitude modulation. The system employs selective relaying to choose the best intermediate node based on channel state information. Novel expressions are derived for outage probability, spectral efficiency, and bit error rate considering Gamma-Gamma atmospheric turbulence fading. Numerical results show that the proposed adaptive system has improved performance compared to non-adaptive systems and all-active relaying schemes.
Adjusting PageRank parameters and comparing results : REPORTSubhajit Sahu
This is my report on Adjusting PageRank parameters and comparing results (version 1).
While doing research work under Prof. Dip Banerjee, Prof. Kishore Kothapalli.
Web graphs unaltered are reducible, and thus the rate of convergence of the power-iteration method is the rate at which αk → 0, where α is the damping factor, and k is the iteration count. An estimate of the number of iterations needed to converge to a tolerance τ is logα τ. For τ = 10-6 and α = 0.85, it can take roughly 85 iterations to converge. For α = 0.95, and α = 0.75, with the same tolerance τ = 10-6, it takes roughly 269 and 48 iterations respectively. For τ = 10-9, and τ = 10-3, with the same damping factor α = 0.85, it takes roughly 128 and 43 iterations respectively. Thus, adjusting the damping factor or the tolerance parameters of the PageRank algorithm can have a significant effect on the convergence rate, both in terms of time and iterations. However, especially with the damping factor α, adjustment of the parameter value is a delicate balancing act. For smaller values of α, the convergence is fast, but the link structure of the graph used to determine ranks is less true. Slightly different values for α can produce very different rank vectors. Moreover, as α → 1, convergence slows down drastically, and sensitivity issues begin to surface [langville04].
Linear regression [Theory and Application (In physics point of view) using py...ANIRBANMAJUMDAR18
Machine-learning models are behind many recent technological advances, including high-accuracy translations of the text and self-driving cars. They are also increasingly used by researchers to help in solving physics problems, like Finding new phases of matter, Detecting interesting outliers
in data from high-energy physics experiments, Founding astronomical objects are known as gravitational lenses in maps of the night sky etc. The rudimentary algorithm that every Machine Learning enthusiast starts with is a linear regression algorithm. In statistics, linear regression is a linear approach to modelling the relationship between a scalar response (or dependent variable) and one or more explanatory variables (or independent
variables). Linear regression analysis (least squares) is used in a physics lab to prepare the computer-aided report and to fit data. In this article, the application is made to experiment: 'DETERMINATION OF DIELECTRIC CONSTANT OF NON-CONDUCTING LIQUIDS'. The entire computation is made through Python 3.6 programming language in this article.
1) The document describes methods for optimizing the widths of radial basis functions in regression analysis models.
2) It presents an efficient computational method for re-estimating the regularization parameter based on generalized cross-validation that utilizes eigendecomposition.
3) The method is also extended to optimize the basis function width by testing multiple trial values and selecting the width with the smallest cross-validation value. Testing on practical problems showed the method improved prediction performance over fixed-width approaches.
The document describes an algorithm for efficiently finding shortest paths between two points (the point-to-point or P2P problem) in a graph by allowing preprocessing. It improves on previous reach-based approaches by introducing bidirectional variants that use implicit lower bounds and adding shortcut arcs to reduce vertex reaches. The resulting algorithm, called RE, has similar performance to the best previous method (hh) but is simpler and combines better with A∗ search (REAL algorithm), yielding significantly faster query times, especially on road networks.
Similar to Corrected asymptotics for a multi-server queue in the Halfin-Whitt regime (20)
Heuristicas para problemas de ruteo de vehiculosdolimpica
Este documento presenta un resumen de diferentes heurísticas para resolver problemas de ruteo de vehículos. Describe primero las características de estos problemas, incluyendo clientes, depósitos y vehículos. Luego revisa heurísticas clásicas como el algoritmo de ahorros, heurísticas de inserción y métodos de asignar primero-rutear después. También cubre metaheurísticas como algoritmos de hormigas, tabu search y algoritmos genéticos. Finalmente, analiza extensiones de las heurísticas clásicas
Sandoya fernando métodos exactos y heurísticos para el vrp jornadasolimpica
Este documento describe métodos exactos y heurísticos para resolver el Problema del Agente Viajero (TSP) y el Problema de Ruteo de Vehículos (VRP). El TSP busca encontrar la ruta más corta para visitar todas las ciudades, mientras que el VRP busca encontrar rutas óptimas para una flota de vehículos para distribuir bienes desde depósitos a clientes. Se presentan formulaciones matemáticas de ambos problemas y se describen heurísticas como el vecino más cercano para aproximar soluciones dado que son
Estudio de técnicas de búsqueda por vecindad a muy gran escalaolimpica
Este documento describe técnicas de búsqueda por vecindad a gran escala para resolver problemas de optimización combinatoria que son intratables mediante métodos exactos. Se clasifican tres tipos de algoritmos de búsqueda por vecindad a gran escala: 1) métodos de profundidad variable que realizan búsquedas heurísticas en vecindades exponencialmente grandes, 2) algoritmos basados en flujos de redes que usan técnicas de flujo para identificar mejoras en vecindades grandes, y 3) vecindades inducidas
Este documento presenta un estudio del espacio de soluciones del problema del cajero viajante (TSP). Los autores analizaron 20 instancias del TSP tomadas de una base de datos estándar y obtuvieron muestras del espacio de soluciones usando dos métodos: un algoritmo de optimización local y un método propuesto para generar muestras uniformemente distribuidas. El análisis de los resultados fue consistente con la conjetura de que el espacio de soluciones del TSP tiene una estructura globalmente convexa, lo que significa
Cost versus distance_in_the_traveling_sa_79149olimpica
The document analyzes solutions to the Traveling Salesman Problem (TSP) on a 532-city instance using five local search heuristics. It finds that lower-cost solutions tend to be closer to the optimal tour and to other good solutions, supporting the idea that TSP solution spaces have a "globally convex" or "big valley" structure. The optimal tour is located near the center of the main cluster of good solutions.
Este documento presenta una introducción a los conceptos de heurísticas y problemas combinatorios. Explica que las heurísticas son técnicas que aumentan la eficiencia de la búsqueda de soluciones a problemas complejos al sacrificar en ocasiones la optimalidad. También define problemas combinatorios como aquellos con un número finito pero muy grande de soluciones posibles. Finalmente, distingue entre heurísticas de construcción, que encuentran una primera solución, y heurísticas de mejoramiento, que mejoran soluciones existentes.
This document summarizes a study that developed algorithms to optimize road freight transportation routes in Spain while accounting for environmental costs. The algorithms, called Algorithms with Environmental Criteria (AEC), incorporate estimates of environmental costs like noise and air pollution alongside traditional routing costs like distance and delivery expenses. The researchers applied the AEC algorithms to real delivery data from a logistics company in Navarre, Spain, finding routes that minimized total costs including environmental externalities.
Este documento presenta los resultados de un estudio sobre los efectos económicos y de seguridad de los vehículos en el transporte de mercancías de operadores logísticos. El estudio analiza los problemas de seguridad y medioambientales en el transporte por carretera y desarrolla algoritmos para la construcción de rutas que incorporan los costes de seguridad y medioambientales. El informe concluye que es adecuado incorporar estos costes en la gestión logística de las empresas para mejorar la seguridad y reducir los impactos med
Este documento describe una modificación al Algoritmo Genético Estocástico (StGA) llamada StGA2, la cual utiliza una tasa de mutación variable por bit basada en la aptitud para mejorar la capacidad de escapar de óptimos locales. El StGA2 se aplica a la estimación de la dirección de arribo de señales en comunicaciones móviles usando antenas inteligentes. Adicionalmente, se discute la convergencia del algoritmo y su precisión para funciones de 2 a 30 dimensiones.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
Your Skill Boost Masterclass: Strategies for Effective Upskilling
Corrected asymptotics for a multi-server queue in the Halfin-Whitt regime
1. Corrected asymptotics for a multi-server queue
in the Halfin-Whitt regime
A.J.E.M. Janssen∗ J.S.H. van Leeuwaarden⋄ B. Zwart♭
First version: May 12, 2007
Second version: October 8, 2007
Abstract
To obtain insight in the quality of heavy-traffic approximations for queues with many
servers, we consider the steady-state number of waiting customers in an M/D/s queue
as s → ∞. In the Halfin-Whitt regime, it is well known that this random variable con-
verges to the supremum of a Gaussian random walk. This paper develops methods that
yield more accurate results in terms of series expansions and inequalities for the proba-
bility of an empty queue, and the mean and variance of the queue length distribution.
This quantifies the relationship between the limiting system and the queue with a small
or moderate number of servers. The main idea is to view the M/D/s queue through
the prism of the Gaussian random walk: as for the standard Gaussian random walk, we
provide scalable series expansions involving terms that include the Riemann zeta function.
Keywords: M/D/s queue; Halfin-Whitt scaling; Gaussian random walk; all-time maxi-
mum; Riemann zeta function; Lerch’s transcendent; Spitzer’s identity; queues in heavy
traffic; Lambert W Function; corrected diffusion approximation.
AMS 2000 Subject Classification: 11M06, 30B40, 60G50, 60G51, 65B15.
1 Introduction
Heavy-traffic analysis is a popular tool to analyze stochastic networks, since the analysis of
a complicated network often reduces to the analysis of a much simpler (reflected) diffusion,
which may be of lower dimension than the original system. This makes the analysis of complex
systems tractable, and from a mathematical point of view, these results are appealing since
they can be made rigorous.
A downside of heavy-traffic analysis is that the results are of an asymptotic nature, and
only form an approximation for a finite-sized system. In a pioneering paper, Siegmund [31]
proposed a corrected diffusion approximation for the waiting time in a single-server queue
∗
Philips Research. Digital Signal Processing Group, WO-02, 5656 AA Eindhoven, The Netherlands. Email:
a.j.e.m.janssen@philips.com.
⋄
Eindhoven University of Technology. Math and Computer Science department, 5612 AZ Eindhoven, The
Netherlands. Email: j.s.h.v.leeuwaarden@tue.nl (corresponding author).
♭
Georgia Institute of Technology. H. Milton Stewart School of Industrial and Systems Engineering, 765 Ferst
Drive, 30332 Atlanta, USA. Email: bertzwart@gatech.edu.
We thank M. Vlasiou for pointing out a connection to the Lambert W function. JvL was supported by the
Netherlands Organisation for Scientific Research (NWO).
1
2. (actually, Siegmund formulated his result in terms of a random walk). In heavy traffic,
the workload distribution is approximated by an exponential distribution. Siegmund gave
a precise estimate of the correction term, nowadays a classic result and textbook material,
cf. Asmussen [2], p. 369. Siegmund’s first order correction has been extended recently by
Blanchet & Glynn [5], who give a full series expansion for the tail probability of the GI/GI/1
waiting time distribution in heavy traffic.
The results in [5, 31] correspond to the conventional heavy-traffic scaling. The present
paper considers corrected diffusion approximations for a heavy-traffic scaling known as the
Halfin-Whitt [12] regime. This regime considers queues where the number of servers grows
large as the system becomes critically loaded. The number of servers s is chosen according
√
to s = λ + β λ, where β is some positive constant. As the√ scaling parameter λ tends to
infinity, the traffic intensity tends to one according to 1 − O(1/ λ). The Halfin-Whitt regime
is also known as the QED (Quality and Efficiency Driven) regime, due to the fact that, in
the limit, a system can be highly utilized (efficiency) while the√ waiting times stay relatively
small (quality). Also, setting the number of servers as s = λ + β λ is often called square-root
staffing. This terminology is motivated by the emergence of large customer contact centers
which need to be staffed with agents, thus calling for accurate and scalable approximations
of multi-server queues. We refer to Gans et al. [11] and Mandelbaum [26] for overviews.
The Halfin-Whitt regime was formally introduced in [12] for a system with exponential
service times (G/M/s queue), although in [19] the same type of scaling was already applied
to the Erlang loss model (M/M/s/s queue). The extension of the results on the G/M/s
queue to non-exponential service times turned out to be challenging. The past few years
have witnessed a substantial effort to rise to this challenge, resulting in several papers on the
Halfin-Whitt regime for various types of service time distributions, cf. Puhalskii & Reiman
[29], Jelenkovi´ et al. [20], Whitt [40], Mandelbaum & Momˇilovic [25], and Reed [30].
c c
Although these results offer important qualitative insights and are useful to solve concrete
staffing problems, one would like to have a better understanding of the quality of the asymp-
totic approximations. For instance, how fast does convergence to the heavy-traffic limit take
place? It would be helpful to have asymptotic estimates or even inequalities from which we
could judge just how close the scaled queueing model is to its heavy-traffic limit. Borst et
al. [4] consider optimal staffing of an M/M/s queue in the Halfin-Whitt regime, and show
numerically that optimizing the system based on the Halfin-Whitt approximation (with s
infinite instead of finite) of the cost function is rarely off by more than a single agent from
systems with as few as 10 servers. As mentioned in the conclusions of [4], these observations
call for a theoretical foundation−a task we take up in the present paper.
1.1 Goals, results and insights
We now give a general description of the results obtained in this paper. We consider a multi-
server queue with arrival rate λ, s servers and deterministic service times (set to 1). We let
√
the arrival rate of the system grow large and set s = λ + β λ for some constant β > 0. Our
main performance measure is the probability that the queue is empty. The model at hand
has been considered before by Jelenkovi´ et al. [20] who showed that the scaled number of
c
ˆ
waiting customers Qλ converges to the maximum Mβ of a Gaussian random walk with drift
√
−β, for which the emptiness probability is known. As λ → ∞, for β < 2 π, there is the
2
3. result ∞
√ β ζ(1/2 − r) −β 2 r
ˆ
P(Qλ = 0) → P(Mβ = 0) = 2β exp √ , (1)
2π r!(2r + 1) 2
r=0
with ζ the Riemann zeta function, see Chang & Peres [7], and Janssen & Van Leeuwaarden [15,
16]. The limiting result for P(Mβ = 0) has the appealing property that the time to compute
it does not depend on the number of servers, which is the case for standard computational
procedures for the M/D/s queue, see e.g. Tijms [37] and Franx [10] and references therein.
The main aim of this paper is to obtain series expansions refining this asymptotic result.
These series expansions can be used in two ways. First of all, the series expansions quantify
the relationship between the limiting system and the queue with a small or moderate number
of servers. In addition, the first term (or first few terms) of these expansions have the correct
behavior as the number of servers grows large.
One insight we find particularly interesting is that our approximations are not based on the
parameter β, but on a modification of it, which depends on s and is given by
α(s) = (−2s(1 − ρ + ln ρ))1/2 , (2)
with ρ = λ/s. This function converges to β as s → ∞, cf. Lemma 7. Another insight we obtain
ˆ
is that the resulting approximation P(Mα(s) = 0) is, in fact, a lower bound for P(Qλ = 0).
We also obtain an upper bound, again involving the function α(s).
The model we consider may seem rather specific, but one should realize that obtaining
series expansions and bounds of this type is by no means a trivial task. The state of the
art for traditional corrected diffusion approximations does not go beyond the random walk,
and relies on the availability of the Wiener-Hopf factorization. In the Halfin-Whitt regime,
the limiting no-wait probability has been found in two cases only, namely for exponential
service times and for deterministic service times. We believe that the latter case is the most
challenging one.
We apply the methods developed in this paper to the M/M/s queue in [17], in which case the
Halfin-Whitt regime results in a non-degenerate limit for the Erlang C formula (probability
that a customer has to wait). There we obtain the same important insight: the Halfin-Whitt
approximation can be substantially improved when β is replaced with α(s); this function is
the same for both models.
We finally like to point out that the results in this paper are all formulated for the special
case of Poisson arrivals, but the methodology we develop is applicable to more general models
(see Section 6). An additional motivation for considering deterministic service times is that
the number of waiting customers in the queue is related to a discrete bulk-service queue, which
has proven its relevance in a variety of applications (see [24], Chapter 5, for an overview).
1.2 Methodology
We now turn to a discussion and motivation of the techniques we use and develop in this
ˆ
paper. The ratio of P(Qλ = 0) and P(Mβ = 0) serves as a primary measure of convergence
and should tend to one as λ grows large. This ratio can be expressed as (using Spitzer’s
identity, cf. (17))
ˆ ∞ √ √
P(Qλ = 0) 1 ˆ
= exp (P(Aλl ≤ β l) − P (β l)) , (3)
P(Mβ = 0) l
l=1
3
4. √
ˆ
where Aλl = (Alλ − lλ)/ lλ and Alλ a Poisson random variable with mean lλ, and
x
1 2 /2
P (x) = √ e−u du (4)
2π −∞
the normal distribution function. To estimate (3) one can use Berry-Esseen bounds, but these
do not lead to sharp results (cf. Lemma 1). In order to get more precise estimates one can use
classical approximations for sums of i.i.d. random variables like saddlepoint approximations
√
ˆ
or Edgeworth expansions (see [3, 21]). However, these require each quantity P(Aλl ≤ β l) −
√
P (β l) to be approximated separately and uniformly in l.
ˆ
√ get convenient asymptotic expansions, we follow a different approach: we bring P(Aλl ≤
To
β l) into quasi-Gaussian form, a method that is standard in asymptotic analysis (for an
illuminating discussion see De Bruijn [6], pp. 67-71). The resulting asymptotic expansion for
e.g. the probability of an empty queue then contains terms of the type
∞ ∞ 1 2
Gk (a) = lk+1/2 e− 2 lsx z(x)dx, a, s ∈ R+ , k ∈ Z, (5)
l=1 a
where z(x) is some function that does not depend on l. This approach seems technical at
first sight but we believe it to be elegant and even intuitively appealing, as there is a clear
interpretation in terms of a change-of-measure argument, see the end of Section 2.
A large part of this paper deals with obtaining the quasi-Gaussian form, analyzing z(x),
and reformulating and estimating Gk (a) which is done in Section 4. A key result is Theorem
3, which gives a representation of Gk (a) for a large class of functions z(x); the only condition
that is imposed on z(x) is that z : [0, ∞) → C is a continuous function satisfying z(x) =
O(exp(εx2 )) for any ε > 0 and that z(x) has a Taylor series around zero. To illustrate the
generality of our result, we note that Chang & Peres’ result (1) on the Gaussian random walk
can be viewed as a special case by taking z(x) ≡ 1.
We focus on the case in which Aλ has a Poisson distribution, which ensures a particularly
tractable form of z(x) yielding convenient computational schemes. This form is given in
Subsection 2.2 and studied in detail in Appendix A. The derivative of z(x) is related to
the Lambert W function; our treatment is self-contained, produces some important auxiliary
results, and is based on earlier results obtained by Szeg¨ [33]. We include our analysis in a
o
separate appendix, since we believe it is interesting in its own right.
Theorem 3 yields a series expansion which can be truncated at a convenient point to obtain
high precision estimates of performance measures. Using classical methods, we can even
estimate the optimal truncation point of the series expansion. We illustrate these general
ideas by specializing them to the M/D/s queue in Subsection 4.3.
1.3 Organization
This paper is organized as follows. In Section 2 we introduce our model and provide short
proofs of results which can also be found in [20]. In particular we establish convergence of
the number of waiting customers to the maximum of the Gaussian random walk, and give a
rough Berry-Esseen bound. These results form a point of departure for the rest of the paper.
We also explain in Section 2 how our asymptotic analysis will be carried out. In Section 3,
for the emptiness probability, and the mean and variance of the queue length distribution,
we rewrite the Spitzer-type expressions into quasi-Gaussian expressions. The reformulation
4
5. and estimation of Gk (a) is carried out in Section 4. Section 5 focuses on lower and upper
bounds which have the correct asymptotic behavior in the Halfin-Whitt regime. We use the
quasi-Gaussian expression for the emptiness probability obtained in Section 3 to derive these
bounds. Conclusions and possible extensions are presented in Section 6.
2 The M/D/s queue and the Halfin-Whitt regime
We consider the M/D/s queue and keep track of the number of customers waiting in the
queue (without those in service) at the end of intervals equal to the constant service time
(which we set to one). Customers arrive according to a Poisson process with rate λ and are
served by at most s servers. Let Qλ,n denote the number of customers waiting in the queue
at the end of interval n. The queueing process is then described by
Qλ,n+1 = (Qλ,n + Aλ,n − s)+ , n = 0, 1, . . . . (6)
where x+ = max{0, x}, and Aλ,n denotes the number of customers that arrived at the queue
during interval n. Obviously, the Aλ,n are i.i.d. for all n, and copies of a Poisson random
variable Aλ with mean λ. It should be noted that due to the assumption of constant service
times, the customers which are being serviced at the end of the considered interval should
start within this interval, and for the same reason, the customers whose service is completed
during this interval should start before its beginning.
Assume that EAλ,n = λ < s and let Qλ denote the random variable that follows the
stationary queue length distribution, i.e., Qλ is the weak limit of Qλ,n . Let
√
s = λ + β λ, β > 0. (7)
Let {Sn : n ≥ 0} be a random walk with S0 = 0, Sn = X1 + . . . + Xn and X, X1 , X2 , . . .
i.i.d. random variables with EX < 0, and let M := max{Sn : n ≥ 0} denote the all-time
maximum. When X is normally distributed with mean −β < 0 and variance 1 we speak
of the Gaussian random walk and denote its all-time maximum by Mβ . We often use the
following notation which is standard in asymptotic analysis:
∞
f (x) ∼ fn (x),
n=0
by which we denote that, for every fixed integer k ≥ 1,
k−1
f (x) − fn (x) = fk (x)(1 + o(1)).
n=0
d
Let → denote convergence in distribution.
2.1 Basic results
The following theorem can be proved using a similar approach as in Jelenkovi´ et al. [20].
c
√
ˆ
Theorem 1. Define Qλ = Qλ / λ. As λ → ∞,
5
6. ˆ d
(i) Qλ → Mβ ;
ˆ
(ii) P(Qλ = 0) → P(Mβ = 0);
ˆ
(iii) E[Qk ] → E[Mβ ] for any k > 0.
k
λ
Proof. Proof of (i): Note that
ˆ d ˆ ˆ
Qλ = (Qλ + Aλ − β)+ , (8)
√
ˆ ˆ
with Aλ = (Aλ − λ)/ λ. Since Aλ converges in distribution to the standard normal random
ˆ
variable as λ → ∞, (i) follows from Theorem X.6.1 in Asmussen [2], if the family (Aλ , λ ≥ 0)
ˆ
is uniformly integrable. But this follows simply from the fact that E[Aλ2 ] = 1 for all λ.
Proof of (ii): The result lim supλ→∞ P(Q ˆ λ = 0) ≤ P(Mβ = 0) follows from (i). To show the
lim inf, note that from Spitzer’s identity (see (14))
∞
ˆ 1 ˆ
ln P(Qλ = 0) = − P(Alλ > lβ). (9)
l
l=1
√
ˆ
Taking the lim inf, applying Fatou’s lemma, and using that P(Alλ > lβ) → P (−β l) yields
∞ √
ˆ 1
lim inf ln P(Qλ = 0) ≥ − P (−β l) = ln P(Mβ = 0), (10)
λ→∞ l
l=1
which proves (ii). Statement (iii) follows from (i) if we can prove the additional uniform
ˆ
integrability condition supλ>N E[Qk ] < ∞ for some constant N and any k. To prove this,
λ
ˆ
note that the Cram´r-Lundberg-Kingman inequality states that P(Qλ > x) ≤ e−sx , for any
e
ˆλ −β)
s > 0 such that E[es(A ] ≤ 1. After some straightforward computation, this inequality can
be rewritten into
√ s
λ es/ λ − 1 − √ − sβ ≤ 0. (11)
λ
Since ex − 1 − x ≤ 1 x2 ex , we see that any s is admissible that satisfies
2
s2 s/√λ
e − sβ ≤ 0. (12)
2
It is easy to see that s = β satisfies this inequality if λ ≥ N := (β/ ln 2)2 . We conclude that
ˆ
P(Qλ > x) ≤ e−βx (13)
ˆ
for any x ≥ 0 and any λ > N . The uniform integrability condition supλ>N E[Qk ] < ∞ now
λ
ˆ ∞ ˆ
follows directly using for example the formula E[Qk ] = 0 kxk−1 P(Qλ > x)dx.
λ
ˆ
As a consequence of Theorem 1 we know that P(Qλ = 0) (which equals P(Qλ = 0)) tends
to P(Mβ = 0) as λ tends to infinity. We are interested in how fast the M/D/s queue in
the Halfin-Whitt regime approaches the Gaussian random walk, and so we take the ratio of
P(Qλ = 0) and P(Mβ = 0) as our measure of convergence. From Spitzer’s identity for random
walks (see Theorem 3.1 in [32]) we have
∞
1
− ln P(M = 0) = P(Sl > 0), (14)
l
l=1
6
7. which gives for the M/D/s queue
∞ ∞ ∞
1 1 (lλ)j
− ln P(Qλ = 0) = P(Alλ > ls) = e−lλ , (15)
l l j!
l=1 l=1 j=ls+1
√
where we choose λ such that s = λ+β λ is integer-valued, i.e. λ = 1 (2s+β 2 −(4sβ 2 +β 4 )1/2 )
2
with s = 1, 2, . . .. For the Gaussian random walk we have ln P(Mβ = 0) as in (10). The
following can be proved using a Berry-Esseen bound.
Lemma 1. For ω := 4 ζ( 3 ) ≈ 2.0899 there are the bounds
5 2
−ω P(Qλ = 0) ω
exp √ ≤ ≤ exp √ . (16)
λ P(Mβ = 0) λ
Proof. Along the same lines as Theorem 2 in [20]. From (15) and (10) we get
∞ √
P(Qλ = 0) 1
= exp (P (−β l) − P(Alλ > ls))
P(Mβ = 0) l
l=1
∞ √
1
≤ exp |P (−β l) − P(Alλ > ls)| . (17)
l
l=1
Rewriting √ √ √
ˆ
|P (−β l) − P(Alλ > ls)| = |P(Aλl ≤ β l) − P (β l)| (18)
√
ˆ
with Aλl = (Aλl − λl)/ λl and using the Berry-Esseen bound for the Poisson case (see Michel
[27])
√ √ 4 30.6 1 4
ˆ
|P(Aλl ≤ β l) − P (β l)| ≤ min , √ ≤ √ . (19)
5 1 + β 3 l3/2 lλ 5 lλ
yields, upon substituting (19) into (17), the second inequality in (16). The first inequality in
(16) follows in a similar way.
We should stress that the occurrence of ζ( 3 ) in Lemma 1 is unrelated to the result (1) of
2
Chang & Peres [7].
2.2 Quasi-Gaussian form: motivation and outline
The bound in (16) does not reveal much information, except that convergence takes place at
√
rate O(1/ λ). In order to get more precise estimates one can use a saddlepoint approximation
or an Edgeworth expansion. √
√ However, these are not very convenient, as they require each
ˆ
element P(Aλl ≤ β l) − P (β l) to be approximated separately due to its dependence on
l. One example would be the Edgeworth expansion for the Poisson distribution (see [3],
Eq. (4.18) on p. 96)
√ √ 1 1 2
ˆ
P(Alλ ≤ β l) = P (β l) − √ e− 2 β l (β 2 l − 1) + O(1/λl), (20)
6 2πlλ
which leads to the approximation
∞ √ √ ∞ ∞
1 ˆ 1 β 2 − 1 β2l 1 1 2l
P(Aλl ≤ β l) − P (β l) ≈ − √ e 2 − e− 2 β . (21)
l=1
l 6 2πλ l=1
l1/2 l=1
l3/2
7
8. It may not come as a surprise that (21) is not a good approximation because we neglect
all O(1/λl) terms in (20). Although including more terms in the Edgeworth expansion is
√
ˆ
an option, we choose to get more convenient asymptotic expansions for P(Aλl ≤ β l) by
bringing it into quasi-Gaussian form.
Specifically, we prove the following theorem in Section 3.
Theorem 2.
∞ ∞ √
p(ls) 1 2 /2
− ln P(Qλ = 0) = √ √ e−x y ′ (x/ ls)dx, (22)
l 2π α l
l=1
in which
λ λ 1/2
α = − 2s 1 − + ln , (23)
s s
α → β as λ → ∞, √
p(n) = nn e−n 2πn/n!, (24)
√
and y ′ is a function analytic in |x| < 2 π (see Appendix A, (138)).
For p there is Stirling’s formula, see Abramowitz-Stegun [1], 6.1.37 on p. 257,
∞
1 1 pk
p(n) ∼ 1 − + + ... = , n → ∞, (25)
12n 288n2 nk
k=0
and for y ′ there is the power series representation
∞
2 1 √
y (x) = 1 − x + x2 + . . . =
′
bi xi , |x| < 2 π. (26)
3 12
i=0
From an aesthetic viewpoint, expression (22) conveys much understanding about the character
of the convergence, since we have complete agreement with the Gaussian random walk (10)
when we would have λ → ∞. The deviations from the quasi-Gaussian random walk are
embodied by p ≡ 1, y ′ ≡ 1 and α ≡ β. From (22) we see that there is the asymptotic
expansion
∞
1 √
− ln P(Qλ = 0) ∼ √ pk s−k+1/2 G−(k+1) (α/ s), (27)
2π k=0
where
∞ ∞ 1 2
Gk (a) = lk+1/2 e− 2 lsx y ′ (x)dx. (28)
l=1 a
Similar expressions, though somewhat more complicated than the one in (27), exist for
EQλ and VarQλ (see Subsection 3.2) and these involve Gk with k = 0, −1, −2 . . . and
k = 1, 0, −1 . . ., respectively. We shall study Gk thoroughly, leading to series expansions,
asymptotics and bounds.
We close this section by giving an interpretation of the quasi-Gaussian form (22). Using,
see Appendix A, (132),
∞
1 n 1 2
= e− 2 nx y ′ (x)dx (29)
p(n) 2π −∞
8
9. 3
2.5
2
y ′ (x)
1.5
1
0.5
0
−2 −1 0 1 2 3 4
Figure 1: The function y ′ (x) for x ∈ [−2, 4].
and ∞
2 /2 √ √ ∞
2 /2
√ e−x y ′ (x/ ls)dx = ls √
e−lsx y ′ (x)dx (30)
α l α/ s
we find from (22) that
∞ ∞ −lx2 /2 ′ √
1 α e y (x/ s)dx
− ln P(Qλ = 0) = ∞ −lx2 /2 ′ √ . (31)
l −∞ e y (x/ s)dx
l=1
As mentioned in the introduction, the resulting formula reveals that the summands of the
random walk associated with the M/D/s queue, and the summands of the Gaussian ran-
dom walk are absolutely continuous with respect to each other. The connecting measure
√
between the two densities has a density as well, and equals y ′ (·/ s). Another interpretation
ˆ
is that P(Qλ = 0) is obtained by twisting the Gaussian distribution associated with Mβ . The
√
associated Radon-Nikodym derivative can again be described in terms of y ′ (·/ s).
3 From Spitzer-type expressions to quasi-Gaussian forms
In this section we show how to obtain the expression (22). In addition, we present similar
results for the mean and variance of the queue length.
3.1 Proof of Theorem 2
For n = 0, 1, . . . we let
n
zk
sn (z) = , z ∈ C. (32)
k!
k=0
With ρ = λ/s and n = ls (so that λl = nρ), and
q(ξ) = e1−ξ ξ, ξ ∈ C, (33)
we then have from Szeg¨ [33], p. 50 (also see Abramowitz-Stegun [1], 6.5.13 on p. 262),
o
∞ ρ
(lλ)j nn+1 e−n
e−lλ = 1 − e−λl sn (λl) = q n (ξ)dξ. (34)
j! n! 0
j=n+1
9
10. Using this relation we can rewrite the Spitzer-type expression (15) as
∞ ρ
1/2 p(ls)
− ln P(Qλ = 0) = s √ q ls (ξ)dξ, (35)
l=1
2πl 0
with p(n) as defined in (25). We then consider the equation
f (y) := − ln q(1 − y) = 1 x2 ,
2 (36)
with x ∈ C from which y is to be solved. We note that
f (y) = 1 y 2 + 1 y 3 + 1 y 4 + . . . ,
2 3 4 (37)
whence there is an analytic solution y(x) around x = 0 that satisfies y(x) = x + O(x2 ) as
x → 0. Furthermore, since f increases from 0 to ∞ as y increases from 0 to 1, we have
that y(x) increases from 0 to ∞, and for any x ≥ 0 there is a unique non-negative solution
y(x) = y of (36). Furthermore, we let
γ = −2(1 − ρ + ln ρ), α = (sγ)1/2 . (38)
Then it holds that
1 1 2
q ls (ρ) = e− 2 lsγ = e− 2 α l , (39)
and ρ ∞ 1 2 1 ∞ 1 2 √
q ls (ξ)dξ = √
e− 2 lsx y ′ (x)dx = √ √ e− 2 x y ′ (x/ ls)dx. (40)
0 γ ls α l
Substituting (40) into (35) yields (22).
Lemma 2. The parameters α and β are related according to
√
√ y(α/ s)
β/ s = √ . (41)
(1 − y(α/ s))1/2
√
Proof. Follows from 1 − ρ = y(γ 1/2 ) = y(α/ s), see Table 3.2, and
1/2
s−λ s−λ λ 1 β β √
1−ρ = = √ √ = √ ρ1/2 = √ (1 − y(α/ s))1/2 . (42)
s λ s s s s
We have that y(x)(1 − y(x))−1/2 = x + 1 x2 +
6
5 3
72 x + . . . (see Appendix A), and hence
β =α+ √ α2
1
+ 5 3
6 s 72s α + .... (43)
10
11. 3.2 Mean and variance of the queue length
Our primary characteristic in this paper is the probability of an empty queue. However, the
techniques that we develop can be applied to other characteristics like the mean and variance
of the queue length. From Spitzer’s identity it follows that the mean and variance of the
maximum M are given by ∞ 1 E((Sl+ )k ) with k = 1 and k = 2, respectively. For the
l=1 l
M/D/s queue this yields
∞ ∞
1 (lλ)j
EQλ = (j − ls)e−lλ , (44)
l j!
l=1 j=ls+1
∞ ∞
1 (lλ)j
VarQλ = (j − ls)2 e−lλ . (45)
l j!
l=1 j=ls+1
This leads after considerable rewriting to
∞ ρ
1/2 p(ls)
EQλ = s √ ρq ls (ρ) − ls(1 − ρ) q ls (ξ)dξ (46)
l=1
2πl 0
and
∞ ρ
1/2 p(ls)
VarQλ = s √ − ρ(ls(1 − ρ) − 1)q ls (ρ) + ((1 − ρ)2 l2 s2 + lsρ) q ls (ξ)dξ . (47)
l=1
2πl 0
In a similar way as for P(Qλ = 0), (46) and (47) can then be brought into the forms
∞ 1 2 ∞ √
√ e− 2 α l p(ls) ∞
2 /2
EQλ = s ρp(ls) √ − αR(ρ) √ √ e−x y ′ (x/ ls)dx , (48)
l=1
2πl l=1
2π α l
∞ ∞ √
p(ls) 2 /2
VarQλ = s (α2 lR2 (ρ) + ρ) √ √ e−x y ′ (x/ ls)dx
l=1
2π α l
∞ √ √ 1 2
e− 2 α l
− α lρR(ρ) − ρ/ ls p(ls) √ , (49)
l=1
2π
where
1−ρ 1
R(ρ) = √ = 1 − (1 − ρ) + . . . . (50)
γ 3
For the Gaussian random walk we have that (see [15])
∞ 1 2
e− 2 β l 1 ∞
2 /2
EMβ = √ − β√ √ e−x dx , (51)
l=1
2πl 2π β l
∞ ∞
1 2 /2 β 1 2
VarMβ = (β 2 l + 1) √ √ e−x dx − √ l1/2 e− 2 β l . (52)
l=1
2π β l 2π
√
Ignoring the factors s and s, we again have complete agreement with the Gaussian random
walk when λ → ∞. The deviations from the Gaussian random walk are embodied by p ≡ 1,
y ′ ≡ 1, β ≡ α and the fact that R(ρ) ≡ 1 when ρ < 1. The introduced notation is summarized
in Table 3.2.
11
12. √
s = λ+β λ
ρ = λ/s
γ = −2(1 − ρ + ln ρ)
√
α = sγ
√ √
a = α/ s = γ
√
y(x) = x − 1 x2 + 36 x3 + . . . ; |x| < 2 π
1
√ 3 √ √ −1/2
β = s y(α/ s)(1 − y(α/ s))
√ √
β = α + 1 s−1/2 α2 + 72 s−1 α3 + . . . ; |α/ s| < 2 π
6 √
5
1 1
p(n) = nn e−n 2πn/n! ∼ 1 + 12 n−1 + 288 n−2 + . . .
Table 1: Interrelations between some parameters and functions.
4 Results for Gk
In this section we give a reformulation of the function Gk in terms of a principal series
expansion. The level of generality is higher than needed for the M/D/s queue, as we consider
a large class of functions z(x) of which y ′ (x) is just a special case. In Subsection 4.1 we derive
the Taylor series for the most general case. We also discuss some special cases that lead
to considerable reductions in complexity of the expressions. The principal series expansion
√
comprises terms involving s, z, a = α/ s and elementary functions, as well as a constant
√
Lk , not depending on a = α/ s, which is more complicated. For this Lk we present an
asymptotic series as s → ∞ that can be used conveniently when the radius of convergence
√
r0 of z(x) = ∞ bj z j is not small (for instance 2 π as in the pilot case z(x) = y ′ (x)). In
j=0
Subsection 4.1 we derive the principal series expansion. In Subsection 4.2 we investigate the
numerical evaluation of Lk in terms of the optimal truncation value of the series expansions.
In Subsection 4.3 we use this general result for the specific case of z(x) = y ′ (x) to derive series
expansions and asymptotics for P(Qλ = 0). There is a clear connection with the Gaussian
random walk. In fact, results for the Gaussian random walk involve the function Gk for the
special case z(x) ≡ 1.
4.1 Principal series expansion
We let z : [0, ∞) → C be a continuous function satisfying z(x) = O(exp(εx2 )) for any ε > 0,
and we assume that there is an r0 > 0 such that z(x) is represented by its Taylor series
∞ j
j=0 bj x for 0 ≤ x < r0 . We consider for s > 0 and integer k the function
∞ ∞ 1 2
Gk (a) = lk+1/2 e− 2 lsx z(x)dx, a > 0. (53)
l=1 a
In the case that z(x) = xi we have
i+1 √
Gk (a) = s− 2 Tk,i (a s), (54)
where Tk,i is defined as
∞ ∞ 1 2
Tk,i (b) = lk+1/2 e− 2 lx xi dx (55)
l=1 b
12
13. with i = 0, 1, . . . and k ∈ Z. The functions Tk,i have been thoroughly investigated in [16],
Section 5, leading to analytic expressions. We now generalize this result to Gk .
Theorem 3. For k ∈ Z and a < 2 π/s we have that
k+3/2 2k+1 a 2k+2 j
2 bj aj−2k−2 z(x) − j=0 bj x
Gk (a) = Γ(k + 3/2) − b2k+2 ln a − dx
s 2k + 2 − j 0 x2k+3
j=0
∞
(− 1 s)r a
+ Lk − ζ(−k − r − 1/2) 2
x2r z(x)dx, (56)
r! 0
r=0
where
∞ 2k+2 2k+1 j+1
∞
k+1/2 1
− 2 lsx2 j1 j+1 2 2
Lk = l e z(x) − bj x dx + bj Γ ζ(−k + j/2)
0 2 2 s
l=1 j=0 j=0
k+3/2 k
2 1 √
+ b2k+2 Γ(k + 3/2) − ln 2s . (57)
s 2j + 1
j=0
Proof. We have
∞
1 2
G′ (a) = −z(a)
k lk+1/2 e− 2 lsa . (58)
l=1
The right-hand side of (58) can be expressed in terms of Lerch’s transcendent Φ, defined as
the analytic continuation of the series
∞
Φ(z, t, v) = (v + n)−t z n , (59)
n=0
which converges for any real number v = 0, −1, −2, . . . if z and t are any complex numbers
with either |z| < 1, or |z| = 1 and Re(t) > 1. Note that ζ(t) = Φ(1, t, 1). Thus,
1 2 1 2
G′ (a) = −z(a)e− 2 sa Φ(e− 2 sa , −k − 1 , 1).
k 2 (60)
We then use the important result derived by Bateman [9], §1.11(8) (with ζ(t, v) := Φ(1, t, v)
the Hurwitz zeta function)
∞
Γ(1 − t) (ln z)r
Φ(z, t, v) = (ln 1/z)t−1 + z −v ζ(t − r, v) , (61)
zv r!
r=0
which holds for | ln z| < 2π, t = 1, 2, 3, . . ., and v = 0, −1, −2, . . ., as to obtain
k+3/2 ∞
2 (− 1 sa2 )r
G′ (a) = −z(a) Γ(k + 3/2)
k a−2k−3 + ζ(−k − r − 1/2) 2
. (62)
s r=0
r!
Therefore
k+3/2 2k+2
2
G′ (a) + Γ(k + 3/2)
k bj aj−2k−3 =
s
j=0
k+3/2 2k+2 ∞
2 (− 1 sa2 )r
−Γ(k + 3/2) z(a) − bj aj a−2k−3 − z(a) ζ(−k − r − 1/2) 2
.
s r!
j=0 r=0
(63)
13
14. √
The series on the second line of (63) converges uniformly in a with s1/2 a ∈ [0, c] and c < 2 π,
so upon integrating the identity in (63) we get for 1 sa2 ≤ c < 2π
2
k+3/2 2k+1
2 bj aj−2k−2
Gk (a) + Γ(k + 3/2) + b2k+2 ln a =
s j − 2k − 2
j=0
k+3/2 a 2k+2
2
Lk − Γ(k + 3/2) z(x) − bj xj x−2k−3 dx
s 0 j=0
∞
(− 1 s)r a
− ζ(−k − r − 1/2) 2
x2r z(x)dx, (64)
r=0
r! 0
where
k+3/2 2k+1
2 bj aj−2k−2
Lk = lim Gk (a) + Γ(k + 3/2) + b2k+2 ln a . (65)
a↓0 s j − 2k − 2
j=0
We shall determine Lk . It holds that, as a ↓ 0,
∞ ∞ 2k+2 2k+2 ∞ ∞
1 2 1 2
Gk (a) = lk+1/2 e− 2 lsx z(x) − bi xi dx + bi lk+1/2 e− 2 lsx xi dx
l=1 a i=0 i=0 l=1 a
∞ ∞ 2k+2 2k+2
1 2 i+1 √
= lk+1/2 e− 2 lsx z(x) − bi xi dx + o(1) + bi s− 2 Tk,i (a s).
l=1 0 i=0 i=0
(66)
2k+2 i
Here (54) has been used and the o(1) comes from the fact that z(x) − i=0 bi x = O(x2k+3 )
so that
a 2k+2
1 2 1
e− 2 lsx z(x) − bi xi dx = O . (67)
0 (ls)k+2
i=0
Now from Janssen & van Leeuwaarden [16], Section 5,
i+1 √ i+1 Γ(k + 3/2) √
s− 2 Tk,i (a s) = s− 2 2k+3/2 (a s)i−2k−2 + Lk,i + O(a)
2k + 2 − i
2 k+3/2 ai−2k−2 i+1
= Γ(k + 3/2) + s− 2 Lk,i + O(a) (68)
s 2k + 2 − i
for i = 0, 1, . . . , 2k + 1 and
2k+3 √ 2k+3 √
s− 2 Tk,2k+2 (a s) = s− 2 − Γ(k + 3/2)2k+3/2 ln(a s) + Lk,2k+2 + O(a)
k+3/2
2 √
= −Γ(k + 3/2) ln(a s) + s−(k+3/2) Lk,2k+2 + O(a).
s
(69)
Here
1 i + 1 i+1
Lk,i = Γ 2 2 ζ(−k + i/2), i = 0, 1, . . . , 2k + 1, (70)
2 2
14
15. k
k+3/2 1 1
Lk,2k+2 = Γ(k + 3/2)2 − ln 2 . (71)
2j + 1 2
j=0
Therefore, as a ↓ 0, we get
∞ ∞ 2k+2
k+1/2 − 1 lsx2
Gk (a) = l e 2 z(x) − bi xi dx + o(1)
l=1 0 i=0
2k+1 i+1
k+3/2
2 ai−2k−2 1 i+1 2 2
+ bi Γ(k + 3/2) + Γ ζ(−k + i/2)
s 2k + 2 − i 2 2 s
i=0
k+3/2 k
2 √ 1 √
+ b2k+2 Γ(k + 3/2) − ln a − ln s+ − ln 2 . (72)
s 2j + 1
j=0
Adding
k+3/2 2k+1
2 bj aj−2k−2
Γ(k + 3/2) + b2k+2 ln a (73)
s j − 2k − 2
j=0
at either side of (72) and letting a ↓ 0, we find that Lk has the required value (57). Then (56)
follows from (64).
Some values of the Riemann zeta function ζ are given in Table 4.1.
x ζ(x) x ζ(x)
-5.5 -0.00267145801990 0.0 -0.50000000000000
-5.0 -0.00396825396825 0.5 -1.46035450880959
-4.5 -0.00309166924722 1.5 2.61237534868549
-4.0 0 2.0 1.64493406684823
-3.5 0.00444101133548 2.5 1.34148725725092
-3.0 0.00833333333333 3.0 1.20205690315959
-2.5 0.00851692877785 3.5 1.12673386731706
-2.0 0 4.0 1.08232323371114
-1.5 -0.02548520188983 4.5 1.05470751076145
-1.0 -0.08333333333333 5.0 1.03692775514337
-0.5 -0.20788622497735 5.5 1.02520457995469
Table 2: Some values of the Riemann zeta function ζ.
We now give several special cases of Theorem 3. The next two corollaries focus on negative
values of k.
Corollary 1. For 1 sa2 < 2π and k = −2, −3, . . . we have that
2
k+3/2 a
2
Gk (a) = − Γ(k + 3/2) x−2k−3 z(x)dx
s 0
∞
(− 1 s)r a
+ Lk − ζ(−k − r − 1/2) 2
x2r z(x)dx, (74)
r! 0
r=0
∞ 1 2
where Lk = ∞ lk+1/2 0 e− 2 lsx z(x)dx (which follows from the definition of Lk in (57) in
l=1
which all series over j vanish for k = −2, −3, . . .).
15
16. Corollary 2. For 1 sa2 < 2π and k = −1 we have that
2
1/2 a
2π z(x) − b0
G−1 (a) = − ln a + dx
s 0 x
∞
(− 1 s)r a
+ L−1 − ζ(−r + 1/2) 2
x2r z(x)dx, (75)
r! 0
r=0
∞ −1/2 ∞ − 1 lsx2 2π 1/2
√
where L−1 = l=1 l 0 e
2 (z(x) − b0 )dx − s ln 2s.
Theorem 3 is meant for the case that a and the convergence radius r0 of ∞ bj xj are gen-
j=0
eral. In the case that a < r0 the expressions can be simplified considerably, as demonstrated
below. If a < r0 we have
a 2k+2 j ∞
z(x) − j=0 bj x bj aj−2k−2
dx = , (76)
0 x2k+3 j − 2k − 2
j=2k+3
a ∞
bj aj+2r+1
x2r z(x)dx = . (77)
0 j + 2r + 1
j=0
As a consequence of (76) we have that the expression on the first line of (56)
2k+1 a 2k+2 j
bj aj−2k−2 z(x) − j=0 bj x
− b2k+2 ln a − dx (78)
2k + 2 − j 0 x2k+3
j=0
simplifies to
∞
bj aj−2k−2
− b2k+2 ln a. (79)
2k + 2 − j
j=0,j=2k+2
Together with (77) this gives expressions for Gk (a) that are, apart from the Lk to which we
turn next, convenient for computation when a is small.
Lemma 3. For the first line of (57)
∞ 2k+2 2k+1 j+1
∞
k+1/2 − 1 lsx2 j 1 j+1 2 2
l e 2 z(x) − bj x dx + bj Γ ζ(−k + j/2) (80)
0 2 2 s
l=1 j=0 j=0
there is the asymptotic expression
∞ j+1
1 j+1 2 2
∼ bj Γ ζ(−k + j/2), s → ∞. (81)
2 2 s
j=0,j=2k+2
j+1
In case that bj Γ 2 = O(B j ) for some B > 0, the asymptotic series in (81) is convergent
when s > 2B 2 , with sum equal to (80).
16
17. Proof. Using
∞ (j+1)/2 ∞ (j+1)/2
− 1 lsx2 j 1 2 −u (j−1)/2 1 2 j+1
e 2 x dx = e u du = Γ , (82)
0 2 ls 0 2 ls 2
we find that
∞ ∞ 2k+2 ∞ ∞ ∞
1 2 1 2
lk+1/2 e− 2 lsx z(x) − bj xj dx ∼ bj lk+1/2 e− 2 lsx xj dx
l=1 0 j=0 j=2k+3 l=1 0
∞ (j+1)/2 ∞
1 2 j+1
= bj Γ l−j/2+k . (83)
2 s 2
j=2k+3 l=1
∞ −j/2+k
This yields (81) since ζ(j/2 − k) = l=1 l .
Remark 4. The series expansion (56) for Gk (a) comprises, as a ↓ 0, leading order terms
involving aj−2k−2 , j = 0, 1, . . . , 2k + 1, and ln a when k = −1, 0, 1, . . ., and Gk (a) stays
bounded as a ↓ 0 for k = −2, −3, −4, . . .. In most cases we are interested in, the value of a
is quite small (say ≤ 0.1). The formula in (56) can be used conveniently for computation of
√
Gk (a) for values of a from 0 to as large as π. For larger values of a, we present in Appendix
A formula (163) as an attractive alternative to compute Gk (a). This alternative shows, for
instance, quite clearly an exp(− 1 (s + 1)a2 )-behavior of Gk (a) as a gets large.
2
Remark 5. Chang & Peres [7], Theorem 1.1, proved that
∞
√ β ζ(1/2 − r) −β 2 r
P(Mβ = 0) = 2β exp √ , (84)
2π r!(2r + 1) 2
r=0
√ √
for 0 < β < 2 π. This result follows easily from Theorem 3, for the case z(x) ≡ 1, a = β/ s
and k = −1.
√
For general k, setting z(x) ≡ 1 and a = β/ s in Theorem 3 leads to the following result.
√
Lemma 4. For β < 2 π and k ∈ Z we have that
∞ ∞
√ β ζ(−k − r − 1 ) −β 2 r
lk P (−β l) = − √ 2
+ Rk (β), (85)
2π r!(2r + 1) 2
l=1 r=0
√
where R−1 (β) = − ln 2β and
1 Γ(k + 3 ) k+ 3 −2k−2 1
2
Rk (β) = √ 2 2β + ζ(−k), k = −1. (86)
2π 2k + 2 2
4.2 Optimal truncation value
Lemma 3 can be deployed in two ways. We can take only the first few terms to get a good
idea of how things behave (see Subsection 4.3), or for the numerical evaluation of Lk , we take
as many terms as needed using optimal truncation. The optimal truncation value J of (81)
17
18. is so large (see developments below) that we can replace ζ(−k + J/2) by 1. The truncation
error made by approximating (80) by
J (j+1)/2
1 j+1 2
bj Γ ζ(−k + j/2) (87)
2 2 s
j=0,j=2k+2
is of the order
(J+2)/2
1 J +2 2
bJ+1 Γ . (88)
2 2 s
We replace, furthermore, bJ+1 = (J + 2)aJ+2 by its asymptotic bound, see Appendix A,
Lemma 13,
J + 2 −1/2 1 J+2
|bJ+1 | ≤ √ . (89)
2 2 π
Thus J +2
1/2 (J+2)/2
1 J +2 2 2 1 J +2 1
bJ+1 Γ ≤ Γ . (90)
2 2 s 2(J + 2) 2 2πs
The factor (1/2(J + 2))1/2 is rather unimportant for determination of the optimal truncation
value J, and we focus on
J +2 1 (J+2)/2
DJ = Γ . (91)
2 2πs
Noting that Γ(J/2 + 3/2)/Γ(J/2 + 1) ≈ (J/2 + 1)1/2 , we see that
1/2 1/2
DJ+1 J +2 1
≈ . (92)
DJ 2 2πs
The right-hand side of (92) decreases in J until J/2 + 1 = 2πs; this J is (near to) the optimal
truncation point. At this point we estimate the right-hand side of (90) by Stirling’s formula
as
1/2 2πs 1/2 √ 2πs
1 1 1 1 e−2πs
Γ(2πs) ≈ (2πs)2πs−1/2 e−2πs 2π = √ . (93)
8πs 2πs 8πs 2πs s 8π
For instance, for s = 10 this equals 10−29 .
Remark 6. Observe how important it is that we have managed to show the good bound (89)
√
on |bJ+1 |. If, for instance, the 1/2 π in this bound were to be replaced by 1, the e−2πs on the
√
far right of (93) would have to be replaced by e−s/2 and the resulting quantity 2πe−s/2 /s
would be 0.0017 for s = 10.
4.3 Accurate approximations for the M/D/s queue
We can apply Theorem 3 to obtain accurate approximations for the emptiness probability
and the mean and variance of the queue length. By way of illustration, we do this in some
detail for P(Qλ = 0) and briefly indicate at the end of this section how one can proceed for
the other cases.
18
19. We have from (25) and (27) that
∞
1 √
− ln P(Qλ = 0) ∼ √ pk s−k+1/2 G−(k+1) (α/ s)
2π k=0
1 √ √ √
= √ s1/2 G−1 (α/ s) − 1 −1/2
12 s G−2 (α/ s) + 1 −3/2
288 s G−3 (α/ s) + . . . .
2π
(94)
√
The G−2 , G−3 , . . . are bounded functions of a = α/ s while G−1 (a) behaves like
2π 1/2 √
− ln a 2s as a ↓ 0. (95)
s
Accurate approximations to − ln P(Qλ = 0) are obtained by including 1, 2, 3, . . . terms of the
second line of (94) in which the G’s must be approximated. For the number of terms of the
asymptotic series in (94) to be included one could follow a truncation strategy (based on
π
(139), (152) and the bound G−k (a) ≤ ( 2s )1/2 ζ(k), k = 2, 3, . . .) pretty much as was done in
Subsection 4.2. We shall not pursue this point here.
We shall compute accurate approximations to G−k (a) for k = 1, 2, . . .. We have from (74)
√
and (75) for α < 2 π
√
1/2 α/ s
√ 2π √ y ′ (x) − 1
G−1 (α/ s) = − ln α/ s + dx
s 0 x
∞ √
(− 1 s)r α/ s
+ L−1 − ζ(−r + 1 ) 2
2 x2r y ′ (x)dx, (96)
r! 0
r=0
and for k = 2, 3, . . .,
√
−k+3/2 α/ s
√ 2
G−k (α/ s) = − Γ(−k + 3/2) x2k−3 y ′ (x)dx
s 0
∞ √
1 r α/ s
1 (− 2 s)
+ L−k − ζ(k − r − 2) x2r y ′ (x)dx. (97)
r! 0
r=0
Here,
∞ 1/2
∞ 1 2 2π √
L−1 = l−1/2 e− 2 lsx (y ′ (x) − 1)dx − ln 2s, (98)
0 s
l=1
and for k = 2, 3, . . .,
∞ ∞ 1 2
L−k = l−k+1/2 e− 2 lsx y ′ (x)dx. (99)
l=1 0
Below we specify the missing ingredients in (96)-(99).
• We have
√ √
α/ s α/ s ∞ ∞ j
y ′ (x) − 1 bj α
dx = bj xj−1 dx = √ , (100)
0 x 0 j s
j=1 j=1
√ √
and the computation of the series is feasible when 0 ≤ α/ s ≤ 2 π, the bj being
√ j
computable and O(1/(2 π) ).
19
20. β = 0.01 (0.0141) β = 0.1 (0.1334)
s α true (94)-1 (94)-2 (94)-3 α true (94)-1 (94)-2 (94)-3
1 0.0100 0.0268 0.0256 0.0267 0.0267 0.0983 0.2351 0.2265 0.2345 0.2343
2 0.0100 0.0225 0.0219 0.0225 0.0225 0.0988 0.2022 0.1980 0.2021 0.2021
5 0.0100 0.0190 0.0188 0.0190 0.0190 0.0993 0.1747 0.1730 0.1747 0.1746
10 0.0100 0.0174 0.0173 0.0174 0.0174 0.0995 0.1617 0.1609 0.1617 0.1617
20 0.0100 0.0164 0.0163 0.0164 0.0164 0.0996 0.1529 0.1525 0.1529 0.1529
50 0.0100 0.0155 0.0155 0.0155 0.0155 0.0998 0.1455 0.1453 0.1455 0.1455
100 0.0100 0.0151 0.0150 0.0151 0.0151 0.0998 0.1419 0.1418 0.1419 0.1419
200 0.0100 0.0148 0.0148 0.0148 0.0148 0.0999 0.1393 0.1393 0.1393 0.1393
500 0.0100 0.0145 0.0145 0.0145 0.0145 0.0999 0.1371 0.1371 0.1371 0.1371
β = 0.2 (0.2518) β = 0.5 (0.5293)
s α true (94)-1 (94)-2 (94)-3 α true (94)-1 (94)-2 (94)-3
1 0.1932 0.4105 0.3979 0.4092 0.4089 0.4573 0.7182 0.7049 0.7137 0.7134
2 0.1952 0.3613 0.3549 0.3611 0.3610 0.4699 0.6656 0.6586 0.6642 0.6641
5 0.1970 0.3185 0.3159 0.3185 0.3185 0.4811 0.6156 0.6125 0.6151 0.6151
10 0.1979 0.2979 0.2966 0.2979 0.2978 0.4867 0.5899 0.5883 0.5897 0.5897
20 0.1985 0.2838 0.2831 0.2837 0.2837 0.4906 0.5719 0.5710 0.5717 0.5717
50 0.1991 0.2716 0.2714 0.2716 0.2716 0.4941 0.5560 0.5557 0.5560 0.5560
100 0.1993 0.2657 0.2655 0.2657 0.2657 0.4958 0.5481 0.5479 0.5481 0.5481
200 0.1995 0.2615 0.2615 0.2615 0.2615 0.4970 0.5426 0.5425 0.5425 0.5425
500 0.1997 0.2579 0.2579 0.2579 0.2579 0.4981 0.5377 0.5376 0.5377 0.5377
β = 1 (0.8005) β = 2 (0.9762)
s α true (94)-1 (94)-2 (94)-3 α true (94)-1 (94)-2 (94)-3
1 0.8299 0.9055 0.8973 0.8948 0.8945 1.3670 0.9835 0.9793 0.9636 0.9633
2 0.8790 0.8787 0.8746 0.8737 0.8736 1.5296 0.9799 0.9787 0.9674 0.9672
5 0.9236 0.8511 0.8493 0.8489 0.8489 1.6948 0.9774 0.9770 0.9703 0.9703
10 0.9462 0.8364 0.8354 0.8352 0.8352 1.7835 0.9766 0.9764 0.9723 0.9723
20 0.9622 0.8259 0.8253 0.8252 0.8252 1.8473 0.9763 0.9762 0.9738 0.9738
50 0.9762 0.8165 0.8163 0.8162 0.8162 1.9040 0.9762 0.9761 0.9750 0.9750
100 0.9832 0.8118 0.8117 0.8117 0.8117 1.9324 0.9762 0.9761 0.9755 0.9755
200 0.9881 0.8085 0.8084 0.8084 0.8084 1.9524 0.9762 0.9762 0.9759 0.9759
500 0.9925 0.8056 0.8055 0.8055 0.8055 1.9700 0.9762 0.9762 0.9761 0.9761
Table 3: Series expansions for P(Qλ = 0) based on (94). The values of P(Mβ = 0) are given
between brackets.
• We have
√ √
α/ s α/ s ∞ ∞ n+j+1
n ′ bj α
x y (x)dx = bj xn+j dx = √ , (101)
0 0 n+j+1 s
j=0 j=0
√ √
and the computation of the series is feasible when 0 ≤ α/ s ≤ 2 π. Furthermore
√ ∞
α/ s j+1
bj α
(− 1 s)r
2
2r ′
x y (x)dx = (− 1 α2 )r
2
√ . (102)
0 2r + j + 1 s
j=0
Since, see [15], Sec. 6, ζ(−r + 1 )/r! = O(1/(2π)r ), the computation of the series over r
2 √
at the right-hand side of (96) is feasible when α < 2 π. A similar result holds for the
series over r at the right-hand side of (97).
20
21. • We have by Lemma 3
∞ ∞ j+1
∞
−1/2 1
− 2 lsx2 ′ 1 j+1 2 2
l e (y (x) − 1)dx ∼ bj Γ ζ(1 + j/2), (103)
0 2 2 s
l=1 j=1
∞ j+1
1 j+1 2 2
L−k ∼ bj Γ ζ(k + j/2), k = 2, 3, . . . , (104)
2 2 s
j=0
for the series expressions at the right-hand sides of (98) and (99). The left-hand sides
of (103) and (104) can be accurately approximated by using the optimal truncation
approach of Subsection 4.2. Alternatively, assume that we include all three terms on
the second line of (94) (so that the truncation error is O(s−7/2 )). We then include in
the right-hand side of (103) the terms with j = 1, 2, 3, 4, and in the right-hand side of
(104) the terms with j = 1, 2.
When we want to compute accurate approximations to EQλ and VarQλ , we can use (48)
and (49), and then it becomes necessary to approximate Gk (a) with k = 0 and k = 1 as
well. This can still be done by using Theorem 3 with its simplifications as pointed out in
√
Corollary 2 since z(x) = y ′ (x) has bj = O((2 π)−j ). Of course, there are a variety of ways to
proceed here, just like in case of − ln P(Qλ = 0) treated above. For the latter case, we have
just worked out one of the more straightforward methods.
Table 3 displays approximations to P(Qλ = 0) based on the series expansion (94). Results
are given for 1, 2, and 3 terms of the second line of (94), and the G’s are approximated as
described in this subsection. Clearly, the expansions provide sharp approximations, and in
most cases, one term suffices to get accurate results, i.e.,
√
s √
P(Qλ = 0) ≈ exp − √ G−1 (α/ s) . (105)
2π
5 Bounds and approximations for the emptiness probability
The Gaussian form (22) for P(Qλ = 0) is rather complicated due to the presence of p(ls) and
z(x) = y ′ (x), which both can be expressed as infinite series. In this section we obtain bounds
on P(Qλ = 0) by using inequalities for p(ls) and y ′ (x).
Lemma 5.
∞ ∞
1/2 1 1 2
P(Qλ = 0) ≥ exp −s √ √
e− 2 lsx y ′ (x)dx =: LB, (106)
l=1
2πl γ
∞ ∞
1 1 1 2
P(Qλ = 0) ≤ exp − s1/2 √ 1− √
e− 2 lsx y ′ (x)dx =: U B. (107)
2πl 12ls γ
l=1
Proof. Follows directly from rewriting (22) as
∞ ∞
1/2 p(ls) 1 2
P(Qλ = 0) = exp −s √ √
e− 2 lsx y ′ (x)dx (108)
l=1
2πl γ
21