MCMC and the Fibonacci distribution

A method to create a Markov transition matrix for Markov chain Monte Carlo studies is presented and applied to the Fibonacci probability distribution.     

Subexponential asymptotics of the waiting time distribution in a single-server queue with multiple Markovian arrival streams

This paper considers subexponential asymptotics of the tail distributions of waiting times in stationary work-conserving single-server queues with multiple Markovian arrival streams, where all arrival streams are modulated by the underlying Markov chain with finite states and service time distributions may differ for different arrival streams. Under the assumption that the equilibrium distribution of the overall (i.e., customer-average) service time distribution is subexponential, a subexponential asymptotic formula is first shown for the virtual waiting time distribution, using a closed formula recently found by the author. Further when customers are served on a FIFO basis, the actual waiting time and sojourn time distributions of customers from respective arrival streams are shown to have the same asymptotics as the virtual waiting time distribution.     

A relationship between a family of waiting time distributions and the asymptotic residual waiting time distribution

For a class of renewal process waiting time distributions defined herein, one may describe the distribution of asymptotic residual waiting times. The relationship between the two distributions characterizes the class, which includes the gamma distribution. Possible consequences for hypothesis testing are discussed.     

Asymptotic Inference for Waiting Times and Patiences in Queues with Abandonment

Motivated by applications in call center management, we propose a framework based on empirical process techniques for inference about waiting time and patience distributions in multiserver queues with abandonment. The framework rigorises heuristics based on survival analysis of independent and identically distributed observations by allowing correlated waiting times. Assuming a regenerative structure of offered waiting times, we establish asymptotic properties of estimators of limiting distribution functions and derived functionals. We discuss construction of bootstrap confidence intervals and statistical tests, including a simple bootstrap two-sample test for comparing patience distributions. A small simulation study and a real data example are presented.     

Analysis and Computation of the Joint Queue Length Distribution in a FIFO Single-Server Queue with Multiple Batch Markovian Arrival Streams

This paper considers a work-conserving FIFO single-server queue with multiple batch Markovian arrival streams governed by a continuous-time finite-state Markov chain. A particular feature of this queue is that service time distributions of customers may be different for different arrival streams. After briefly discussing the actual waiting time distributions of customers from respective arrival streams, we derive a formula for the vector generating function of the time-average joint queue length distribution in terms of the virtual waiting time distribution. Further assuming the discrete phase-type batch size distributions, we develop a numerically feasible procedure to compute the joint queue length distribution. Some numerical examples are provided also.     

ANALYSIS OF A GI/G Y /1 SYSTEM IN DISCRETE-TIME

ABSTRACT

We consider a class of single server queueing systems in which customers arrive singly and service is provided in batches, depending on the number of customers waiting when the server becomes free. Service is independent of the batch size. This system could also be considered as a batch service queue in which a server visits the queue at arbitrary times and collects a batch of waiting customers for service, or waits for a customer to arrive if there are no waiting customers. A waiting server immediately collects and processes the first arriving customer. The system is considered in discrete time. The interarrival times of customers and the inter-visit times of the server, which we call the service time, have general distributions and are represented as remaining time Markov chains. We analyze this system using the matrix-geometric method and show that the resulting R matrix can be determined explicitly in some special cases and the stationary distributions are known semi-explicitly in some other special cases.     

On k-match problems

Several waiting time random variables for a duplication within a memory window of size k in a sequence of {1,2,…,m}-valued random variables are investigated. The exact distributions of the waiting time random variables are derived by the method of conditional probability generating functions. In particular, the exact distribution of the waiting time for the first k-match is obtained when the underlying sequence is generated by higher order Markov dependent trials. Examples for numerical calculations are also given.     

Some classes of distributions and their application in queueing

New classes of distributions are defined in analogy to the properties NBU, NBUE known from reliability. They are applied to obtain bounds on certain parameters of the GI/G/1 queue, such as the mean and the variance of the stationary waiting time, the probability of waiting, and the covariances of waiting times.     

Stationary distributions of the R[X]/R/1 cross-correlated queue

We consider an infinite buffer single server queue wherein batch interarrival and service times are correlated having a bivariate mixture of rational (R) distributions, where R denotes the class of distributions with rational Laplace–Stieltjes transform (LST), i.e., ratio of a polynomial of degree at most n to a polynomial of degree n. The LST of actual waiting time distribution has been obtained using Wiener–Hopf factorization of the characteristic equation. The virtual waiting time, idle period (actual and virtual) distributions, as well as inter-departure time distribution between two successive customers have been presented. We derive an approximate stationary queue-length distribution at different time epochs using the Markovian assumption of the service time distribution. We also derive the exact steady-state queue-length distribution at an arbitrary epoch using distributional form of Little’s law. Finally, some numerical results have been presented in the form of tables and figures.     

On Level Crossing Analysis of Queues

In this note we introduce a new level crossing analysis and using it derive an integral equation for the steady state waiting time in the GI/G/1 Queue. For the GI/M/1 queue we derive the rates of up- and down-crossings of the virtual delay process and two integral equations, one for the steady state time spent in the system and the other for the steady state waiting time in the queue. Also, the steady state probability distributions of the time spent in the system and the waiting time in the queue are obtained by solving these integral equations.     

Multi-class M/PH/1 queues with deterministic impatience times

This article considers computational procedures for the waiting time and queue length distributions in stationary multi-class first-come, first-served single-server queues with deterministic impatience times. There are several classes of customers, which are distinguished by deterministic impatience times (i.e., maximum allowable waiting times). We assume that customers in each class arrive according to an independent Poisson process and a single server serves customers on a first-come, first-served basis. Service times of customers in each class are independent and identically distributed according to a phase-type distribution that may differ for different classes. We first consider the stationary distribution of the virtual waiting time and then derive numerically feasible formulas for the actual waiting time distribution and loss probability. We also analyze the joint queue length distribution and provide an algorithmic procedure for computing the probability mass function of the stationary joint queue length.     

Extremal limit theorems for observations separated by random power law waiting times

This paper develops extreme value theory for random observations separated by random waiting times whose exceedence probability falls off like a power law. In the case where the waiting times between observations have an infinite mean, a limit theorem is established, where the limit is comprised of an extremal process whose time index is randomized according to the non-Markovian hitting time process for a stable subordinator. The resulting limit distributions are shown to be solutions of fractional differential equations, where the order of the fractional time derivative coincides with the power law index of the waiting time. The probability that the limit process remains below a threshold is also computed. For waiting times with finite mean but infinite variance, a two-scale argument yields a fundamentally different limit process. The resulting limit is an extremal process whose time index is randomized according to the first passage time of a positively skewed stable Lévy motion with positive drift. This two-scale limit provides a second-order correction to the usual limit behavior.     

Flowgraph Models for Generalized Phase Type Distributions Having Non-Exponential Waiting Times

ABSTRACT. Aalen (1995) introduced phase type distributions based on Markov processes for modelling disease progression in survival analysis. For tractability and to maintain the Markov property, these use exponential waiting times for transitions between states. This article extends the work of Aalen (1995) by generalizing these models to semi-Markov processes with non-exponential waiting times. The generalization allows more realistic modelling of the stages of a disease where the Markov property and exponential waiting times may not hold. Flowgraph models are introduced to provide a closed form for the distributions in situations involving non-exponential waiting times. Flowgraph models work where traditional methods of stochastic processes are intractable. Saddlepoint approximations are used in the analysis. Together, generalized phase type distributions, flowgraphs, and saddlepoint approximations create exciting and innovative prospects for the analysis of survival data.     

Bayesian estimation for the M/G/1 queue using a phase-type approximation

This article deals with Bayesian inference and prediction for M/G/1 queueing systems. The general service time density is approximated with a class of Erlang mixtures which are phase-type distributions. Given this phase-type approximation, an explicit evaluation of measures such as the stationary queue size, waiting time and busy period distributions can be obtained. Given arrival and service data, a Bayesian procedure based on reversible jump Markov Chain Monte Carlo methods is proposed to estimate system parameters and predictive distributions.     

A characterization of stationary renewal processes and of memoryless distributions

We prove first that a renewal process is stationary if and only if the distributions of the age and the residual waiting time coincide for every t>0, and for 0≦x相似文献   

Erlang–Lindley distribution

In this paper, Erlang–Lindley distribution (ErLD) is proposed which offers a more flexible model for waiting time data. It has the property that it can accommodate increasing, bathtub, and inverted bathtub shapes. Several statistical and reliability properties are derived and studied. The moments, its associated measures, and the limiting distributions of order statistics are derived. The model parameters are estimated by maximum likelihood and method of moments. An application of the proposed distribution to some waiting time data shows that it can give a better fit than other important lifetime models.     

Multivariate inverse Gaussian distribution as a limit of multivariate waiting time distributions

Multivariate inverse Gaussian distribution proposed by Minami [2003. A multivariate extension of inverse Gaussian distribution derived from inverse relationship. Commun. Statist. Theory Methods 32(12), 2285–2304] was derived through multivariate inverse relationship with multivariate Gaussian distributions and characterized as the distribution of the location at a certain stopping time of a multivariate Brownian motion. In this paper, we show that the multivariate inverse Gaussian distribution is also a limiting distribution of multivariate Lagrange distributions, which is a family of waiting time distributions, under certain conditions.     

On discrete distributions generated from drawing balls with bivariate labels

In this article, we consider urn models under three types of sampling schemes in terms of the probability-generating functions. The tools are developed for the evaluation of the distributions arising from the urn models along with some examples. Furthermore, the distributions are investigated by making use of the Bell polynomials. The results presented here provide a wide framework for developing the theory of urn models. As examples, we propose new class of probability models, which are called multiple-player problems and examine their properties. Finally, we treat the parameter estimation problem in the waiting time distributions with a numerical example.     

A Two‐Queue Polling Model with Regularly Varying Service and/or Switchover Times

Abstract

We consider the cyclic polling system with two queues. One queue is severed according to the exhaustive discipline, and the other queue is served according to the 1‐limited discipline. At least one of the service and/or switchover times has a regularly varying tail. We obtain the tail behavior of the waiting time distributions. When one of the service and/or switchover times has an infinite second moment, we derive the heavy‐traffic behavior of the waiting time distribution at the 1‐limited queue.     

Comparing waiting times in a multi-stage model: A log-rank approach

A proper log-rank test for comparing two waiting (i.e. sojourn, gap) times under right censored data has been absent in the survival literature. The classical log-rank test provides a biased comparison even under independent right censoring since the censoring induced on the time since state entry depends on the entry time unless the hazards are semi-Markov. We develop test statistics for comparing K waiting time distributions from a multi-stage model in which censoring and waiting times may be dependent upon the transition history in the multi-stage model. To account for such dependent censoring, the proposed test statistics utilize an inverse probability of censoring weighted (IPCW) approach previously employed to define estimators for the cumulative hazard and survival function for waiting times in multi-stage models. We develop the test statistics as analogues to K-sample log-rank statistics for failure time data, and weak convergence to a Gaussian limit is demonstrated. A simulation study demonstrates the appropriateness of the test statistics in designs that violate typical independence assumptions for multi-stage models, under which naive test statistics for failure time data perform poorly, and illustrates the superiority of the test under proportional hazards alternatives to a Mann–Whitney type test. We apply the test statistics to an existing data set of burn patients.