Stationary distributions and optimal control of queues with batch Markovian arrival process under multiple adaptive vacations

We consider an infinite-buffer single server queue with batch Markovian arrival process (BMAP) and exhaustive service discipline under multiple adaptive vacation policy. That is, the server serves until system emptied and after that server takes a random maximum number H different vacations until either he finds at least one customer in queue or the server have exhaustively taken all the vacations. The maximum number H of vacations taken by the server is a discrete random variable. We obtain queue-length distributions at various epochs such as, service completion/vacation termination, pre-arrival, arbitrary, post-departure and pre-service. The proposed analysis is based on the use of matrix-analytic procedure to obtain queue-length distribution at a post-departure epoch. Later we use supplementary variable method and simple algebraic manipulations to obtain the queue-length distribution at other epochs using queue-length distribution at post-departure epoch. Some important performance measures, like mean queue lengths and mean waiting times have been obtained. Several other vacation queueing models can be obtained as a special case of our model, e.g., single-, multiple-vacation model and queues with exceptional first vacation time. Finally, the total expected cost function per unit time is considered to determine a locally optimal multiple adaptive vacation policy at a minimum cost.     

Analytic and numerical aspects of batch service queues with single vacation

This paper deals with an M/G/1 batch service queue where customers are served in batches of maximum size b with a minimum threshold value a. The server takes a single vacation when he finds less than a customers after the service completion. The vacation time of the server is arbitrarily distributed. Using the supplementary variable method we obtain the probability generating functions of the queue length distributions at various epochs. We also obtain relations among queue length distributions at arbitrary, service (vacation) termination epochs. Further their evaluation is also discussed. Finally, some numerical results and graphs are presented.     

Analysis of GI/M/s/c queues using uniformisation

We combine uniformisation, a powerful numerical technique for the analysis of continuous time Markov chains, with the Markov chain embedding technique to analyze GI/M/s/c queues. The main steps of the proposed approach are the computation of

• (1)the mixed-Poisson probabilities associated to the number of arrival epochs in the uniformising Poisson process between consecutive customer arrivals to the system; and
• (2)the conditional embedded uniformised transition probabilities of the number of customers in the queueing system immediately before customer arrivals to the system.

To show the performance of the approach, we analyze queues with Pareto interarrival times using a stable recursion for the associated mixed-Poisson probabilities whose computation time is linear in the number of computed coefficients. The results for queues with Pareto interarrival times are compared with those obtained for queues with other interarrival time distributions, including exponential, Erlang, uniform and deterministic interarrival times. The obtained results show that much higher loss probabilities and mean waiting times in queue may be obtained for queues with Pareto interarrival times than for queues with the other mentioned interarrival time distributions, specially for small traffic intensities.     

Performance analysis of a finite-buffer discrete-time queue with bulk arrival,bulk service and vacations

This paper considers a finite-buffer discrete-time Geo^X/G^Y/1/K+B queue with multiple vacations that has a wide range of applications including high-speed digital telecommunication systems and various related areas. The main purpose of this paper is to present a performance analysis of this system. For this purpose, we first derive a set of linear equations to compute the steady-state departure-epoch probabilities based on the embedded Markov chain technique. Next, we present numerically stable relationships for the steady-state probabilities of the queue lengths at three different epochs: departure, random, and arrival. Finally, based on these relationships, we present various useful performance measures of interest such as the moments of the number of packets in the queue at three different epochs, the mean delay in the queue of a packet, the loss probability and the probability that server is busy with computational experiences.     

Optimal management for infinite capacity N-policy M/G/1 queue with a removable service station

In this article, we consider an infinite capacity N-policy M/G/1 queueing system with a single removable server. Poisson arrivals and general distribution service times are assumed. The server is controllable that may be turned on at arrival epochs or off at service completion epochs. We apply a differential technique to study system sensitivity, which examines the effect of different system input parameters on the system. A cost model for infinite capacity queueing system under steady-state condition is developed, to determine the optimal management policy at minimum cost. Analytical results for sensitivity analysis are derived. We also provide extensive numerical computations to illustrate the analytical sensitivity properties obtained. Finally, an application example is presented to demonstrate how the model could be used in real applications to obtain the optimal management policy.     

Queue size distribution in a discrete-time D-BMAP/G/1 retrial queue

We consider a discrete-time batch Markovian arrival process (D-BMAP)/G/1 retrial queue. We find the light-tailed asymptotics for the stationary distributions of the number of customers at embedded epochs and at arbitrary time. Using these tail asymptotics we propose a method for calculating the stationary distributions of the number of customers at embedded epochs and at arbitrary time. Numerical examples are presented to illustrate our results.     

Steady state analysis of the GI/M/1/N queue with a variant of multiple working vacations

This paper studies the GI/M/1/N queue with a variant of multiple working vacations, where the server leaves for a working vacation as soon as the system becomes empty. The server takes at most H consecutive working vacations if the system remains empty after the end of a working vacation. Employing the supplementary variable and embedded Markov chain methods, we obtain the queue length distribution at different time epochs. Based on the various system length distribution, the probability of blocking, mean waiting times and mean system lengths have been derived. Finally, numerical results are discussed.     

On asymmetric generalised t stochastic volatility models

In stochastic volatility (SV) models, asset returns conditional on the latent volatility are usually assumed to have a normal, Student-t or exponential power (EP) distribution. An earlier study uses a generalised t (GT) distribution for the conditional returns and the results indicate that the GT distribution provides a better model fit to the Australian Dollar/Japanese Yen daily exchange rate than the Student-t distribution. In fact, the GT family nests a number of well-known distributions including the commonly used normal, Student-t and EP distributions. This paper extends the SV model with a GT distribution by incorporating general volatility asymmetry. We compare the empirical performance of nested distributions of the GT distribution as well as different volatility asymmetry specifications. The new asymmetric GT SV models are estimated using the Bayesian Markov chain Monte Carlo (MCMC) method to obtain parameter and log-volatility estimates. By using daily returns from the Standard and Poors (S&P) 500 index, we investigate the effects of the specification of error distributions as well as volatility asymmetry on parameter and volatility estimates. Results show that the choice of error distributions has a major influence on volatility estimation only when volatility asymmetry is not accounted for.     

A perishable inventory system with modified (S−1,S) policy and arbitrary processing times

In this article we analyze a lost sales (S−1,S) perishable system, under Poisson demands and exponential lifetimes, in which the reorders are placed at every demand epoch so as to take the inventory position back to its maximum level S. The items are replenished one at a time and the resupply time has arbitrary distribution. The various operating characteristics are obtained using Markov renewal techniques. A matrix recursive scheme is developed to determine the stationary distribution of the underlying Markov chain. The efficiency of this procedure in the determination of optimal S that minimizes the long run expected cost rate is discussed. Sensitivity analysis of various system parameters is also carried out.Scope and purposeThe analysis of perishable inventory systems, which has potential applications in various sectors of industry, is far more difficult than their infinite lifetime counterparts. In the study of perishable systems under traditional (S−1,S) policy, reorders for items are placed at demand as well as failure epochs. However, it will be more meaningful not to place orders at failure epochs, which is also practical and economical since it avoids continuous monitoring of items. This paper deals with a continuous review perishable system under a more realistic base stock policy, with variable ordering quantity and arbitrary unit resupply times. The operating characteristics of this complex model are derived using the techniques of semi-regenerative processes. A matrix recursive scheme developed to determine the stationary distribution has enabled us to code an efficient numerical program for cost optimization.     

Almost k-wise independence versus k-wise independence

We say that a distribution over {0,1}ⁿ is (ε,k)-wise independent if its restriction to every k coordinates results in a distribution that is ε-close to the uniform distribution. A natural question regarding (ε,k)-wise independent distributions is how close they are to some k-wise independent distribution. We show that there exist (ε,k)-wise independent distributions whose statistical distance is at least n^O(k)·ε from any k-wise independent distribution. In addition, we show that for any (ε,k)-wise independent distribution there exists some k-wise independent distribution, whose statistical distance is n^O(k)·ε.     

Steady state analysis and computation of the GI/M/1/L queue with multiple working vacations and partial batch rejection

This paper analyzes a finite-buffer bulk-arrival bulk-service queueing system with multiple working vacations and partial batch rejection in which the inter-arrival and service times are, respectively, arbitrarily and exponentially distributed. Using the supplementary variable and the embedded Markov chain techniques, we obtain the waiting queue-length distributions at pre-arrival and arbitrary epochs. We also present Laplace–Stiltjes transform of the actual waiting-time distribution in the queue. Finally, several performance measures and a variety of numerical results in the form of tables and graphs are discussed.     

A new approach to the queue length and waiting time of BMAP/G/1 queues

We apply the factorization principle to derive the generating function of the queue length and the vector Laplace–Stieltjes transform of the waiting time of a BMAP/G/1 queue. The mean performance measures are provided with a computational experience.Scope and purposeThe classical method of obtaining the queue length and waiting time distributions of BMAP/G/1 queues starts with the analysis of the imbedded Markov renewal process at departure epochs. This method is intricate and time consuming when the idle period process is complicated. In this paper, we demonstrate that the factorization property can be applied efficiently and effectively to derive the queue length distributions of BMAP/G/1 queueing systems by avoiding the conventional standard procedures. The approach demonstrated in this paper can be applied to the analysis of many other BMAP/G/1 queueing systems with higher behavioral complexities.     

Photofragment image analysis using the Onion-Peeling Algorithm

With the growing popularity of the velocity map imaging technique, a need for the analysis of photoion and photoelectron images arose. Here, a computer program is presented that allows for the analysis of cylindrically symmetric images. It permits the inversion of the projection of the 3D charged particle distribution using the Onion Peeling Algorithm. Further analysis includes the determination of radial and angular distributions, from which velocity distributions and spatial anisotropy parameters are obtained. Identification and quantification of the different photolysis channels is therefore straightforward. In addition, the program features geometry correction, centering, and multi-Gaussian fitting routines, as well as a user-friendly graphical interface and the possibility of generating synthetic images using either the fitted or user-defined parameters.

Program summary

Title of program: Glass OnionCatalogue identifier: ADRYProgram Summary URL:http://cpc.cs.qub.ac.uk/summaries/ADRYProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions: noneComputer: IBM PCOperating system under which the program has been tested: Windows 98, Windows 2000, Windows NTProgramming language used: Delphi 4.0Memory required to execute with typical data: 18 MwordsNo. of bits in a word: 32No. of bytes in distributed program, including test data, etc.: 9 911 434Distribution format: zip fileKeywords: Photofragment image, onion peeling, anisotropy parametersNature of physical problem: Information about velocity and angular distributions of photofragments is the basis on which the analysis of the photolysis process resides. Reconstructing the three-dimensional distribution from the photofragment image is the first step, further processing involving angular and radial integration of the inverted image to obtain velocity and angular distributions. Provisions have to be made to correct for slight distortions of the image, and to verify the accuracy of the analysis process.Method of solution: The “Onion Peeling” algorithm described by Helm [Rev. Sci. Instrum. 67 (6) (1996)] is used to perform the image reconstruction. Angular integration with a subsequent multi-Gaussian fit supplies information about the velocity distribution of the photofragments, whereas radial integration with subsequent expansion of the angular distributions over Legendre Polynomials gives the spatial anisotropy parameters. Fitting algorithms have been developed to centre the image and to correct for image distortion.Restrictions on the complexity of the problem: The maximum image size (1280×1280) and resolution (16 bit) are restricted by available memory and can be changed in the source code. Initial centre coordinates within 5 pixels may be required for the correction and the centering algorithm to converge. Peaks on the velocity profile separated by less then the peak width may not be deconvolved. In the charged particle image reconstruction, it is assumed that the kinetic energy released in the dissociation process is small compared to the energy acquired in the electric field. For the fitting parameters to be physically meaningful, cylindrical symmetry of the image has to be assumed but the actual inversion algorithm is stable to distortions of such symmetry in experimental images.Typical running time: The analysis procedure can be divided into three parts: inversion, fitting, and geometry correction. The inversion time grows approx. as R³, where R is the radius of the region of interest: for R=200 pixels it is less than a minute, for R=400 pixels less then 6 min on a 400 MHz IBM personal computer. The time for the velocity fitting procedure to converge depends strongly on the number of peaks in the velocity profile and the convergence criterion. It ranges between less then a second for simple curves and a few minutes for profiles with up to twenty peaks. The time taken for the image correction scales as R² and depends on the curve profile. It is on the order of a few minutes for images with R=500 pixels.Unusual features of the program: Our centering and image correction algorithm is based on Fourier analysis of the radial distribution to insure the sharpest velocity profile and is insensitive to an uneven intensity distribution. There exists an angular averaging option to stabilize the inversion algorithm and not to loose the resolution at the same time.     

A heavy-tailed empirical Bayes method for replicated microarray data

DNA microarray has been recognized as being an important tool for studying the expression of thousands of genes simultaneously. These experiments allow us to compare two different samples of cDNA obtained under different conditions. A novel method for the analysis of replicated microarray experiments based upon the modelling of gene expression distribution as a mixture of α-stable distributions is presented. Some features of the distribution of gene expression, such as Pareto tails and the fact that the variance of any given array increases concomitantly with an increase in the number of genes studied, suggest the possibility of modelling gene expression distribution on the basis of α-stable density. The proposed methodology uses very well known properties of α-stable distribution, such as the scale mixture of normals. A Bayesian log-posterior odds is calculated, which allows us to decide whether a gene is expressed differentially or not. The proposed methodology is illustrated using simulated and experimental data and the results are compared with other existing statistical approaches. The proposed heavy-tail model improves the performance of other distributions and is easily applicable to microarray gene data, specially if the dataset contains outliers or presents high variance between replicates.     

Executives’ perceived environmental uncertainty shortly after 9/11

Environmental uncertainty refers to situations when decision makers experience difficulty in predicting their organizations’ environments. Prediction difficulty is mapped by closeness of decision makers’ probability distributions of environmental variables to the uniform distribution. A few months after the 9/11 terrorist attacks, we solicited probabilities for three environmental variables from 93 business executives by a mail survey. Each executive assigned probabilities to the future state of the economy specified as categories of growth projected for a year after the 9/11 jolt, conditional probabilities of its effect on her/his organization, and conditional probabilities of her/his organizational response capability to each economic condition. Shannon entropy maps uncertainty, but the data do not provide trivariate state-effect-response distribution. We use maximum entropy method to impute the trivariate distributions from the data on state-effect and state-response bivariate probabilities. Uncertainty about each executive’s probability distribution is taken into account in two ways: using a Dirichlet model with each executive’s distribution as its mode, and using a Bayesian hierarchical model for the entropy. Both models reduce the observed heterogeneity among the executives’ environmental uncertainty. A Bayesian regression examines the effects of two organizational characteristics on uncertainty. Presentation of results includes uncertainty tableaux for visualizations of the joint and marginal entropies and mutual information between variables.     

New bounds on D-ary optimal codes

We propose a simple method that, given a symbol distribution, yields upper and lower bounds on the average code length of a D-ary optimal code over that distribution. Thanks to its simplicity, the method permits deriving analytical bounds for families of parametric distributions. We demonstrate this by obtaining new bounds, much better than the existing ones, for Zipf and exponential distributions when D>2.     

Computer generation of random variables with Lindley or Poisson-Lindley distribution via the Lambert W function

We provide procedures to generate random variables with Lindley distribution, and also with Poisson-Lindley or zero-truncated Poisson-Lindley distribution, as simple alternatives to the existing algorithms. Our procedures are based on the fact that the quantile functions of these probability distributions can be expressed in closed form in terms of the Lambert W function. As a consequence, the extreme order statistics from the above distributions can also be computer generated in a straightforward manner.     

The GI/M/1 queue and the GI/Geo/1 queue both with single working vacation

We first consider the continuous-time GI/M/1 queue with single working vacation (SWV). During the SWV, the server works at a different rate rather than completely stopping working. We derive the steady-state distributions for the number of customers in the system both at arrival and arbitrary epochs, and for the FIFO sojourn time for an arbitrary customer. We then consider the discrete-time GI/Geo/1/SWV queue by contrasting it with the GI/M/1/SWV queue.     

Class library ranlip for multivariate nonuniform random variate generation

This paper describes generation of nonuniform random variates from Lipschitz-continuous densities using acceptance/rejection, and the class library ranlip which implements this method. It is assumed that the required distribution has Lipschitz-continuous density, which is either given analytically or as a black box. The algorithm builds a piecewise constant upper approximation to the density (the hat function), using a large number of its values and subdivision of the domain into hyperrectangles.The class library ranlip provides very competitive preprocessing and generation times, and yields small rejection constant, which is a measure of efficiency of the generation step. It exhibits good performance for up to five variables, and provides the user with a black box nonuniform random variate generator for a large class of distributions, in particular, multimodal distributions. It will be valuable for researchers who frequently face the task of sampling from unusual distributions, for which specialized random variate generators are not available.

Program summary

Title of program: RanlipCatalogue number: ADVPProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADVPProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandComputers: IBM PC; DEC AlphaOperating systems under which the program has been tested: Windows XP, Linux, UnixProgramming language used: C++Memory required to execute with typical data: 32MNo. of bits in a word: 32No. of processors used: 1Has the code been vectorized?: NoNo. of lines in distributed program, including test data, etc.: 52 160No. of bytes in distributed program, including test data, etc.: 392 625Distribution format: tar.gzNature of physical problem:This program allows one to generate nonuniform random vectors from a variety of distributions (especially multimodal), using acceptance/rejection approach. Suitable for non-standard distributions for up to five variables.Method of solution:Assuming Lipschitz-continuous density, a tight overestimate (the hat function) is computed. Then random variates are generated using acceptance/rejection approach.Restrictions on the complexity of the problem:If the number of variables exceeds five, computation of the overestimate is inefficient.Typical running time:Preprocessing 1-5000 s, generation .