On some probability distributions associated with random walks

The probability distribution of the total number of games to ruin in a gambler's ruin random walk with initial position n, the probability distribution of the total size of an epidemic starting with n cases and the probability distribution of the number of customers served during a busy period M/M/1 when the service starts with n waiting customers are identical. All these can be easily obtained by using Lagrangian expansions instead of long combinatorial methods. The binomial, trinomial, quadrinomial and polynomial random walks of a particle have been considered with an absorbing barrier at 0 when the particle starts its walks from a point n, and the pgfs. and the probability distributions of the total number of jumps (trials) before absorption at 0 have been obtained. The values for the mean and variance of such walks have also been given.     

Functionals of Markovian Branching D-BMAPS

Abstract

We consider a stochastic system in which Markovian customer attribute processes are initiated at customer arrivals in a discrete batch Markovian arrival process (D-BMAP). We call the aggregate a Markovian branching D-BMAP. Each customer attribute process is an absorbing discrete time Markov chain whose parameters depend both on the phase transition, of the driving D-BMAP, and the number of arrivals taking place at the customer's arrival instant. We investigate functionals of Markovian branching D-BMAPs that may be interpreted as cumulative rewards collected over time for the various customers that arrive to the system, in the transient and asymptotic cases. This is achieved through the derivation of recurrence relations for expected values and Laplace transforms in the former case, and Little's law in the latter case.     

Pareto Poisson–Lindley distribution with applications

A new lifetime distribution is introduced based on compounding Pareto and Poisson–Lindley distributions. Several statistical properties of the distribution are established, including behavior of the probability density function and the failure rate function, heavy- and long-right tailedness, moments, the Laplace transform, quantiles, order statistics, moments of residual lifetime, conditional moments, conditional moment generating function, stress–strength parameter, Rényi entropy and Song's measure. We get maximum-likelihood estimators of the distribution parameters and investigate the asymptotic distribution of the estimators via Fisher's information matrix. Applications of the distribution using three real data sets are presented and it is shown that the distribution fits better than other related distributions in practical uses.     

Large deviations of multiclass M/G/1 queues

Consider a multiclass M/G/1 queue where queued customers are served in their order of arrival at a rate which depends on the customer class. We model this system using a chain with states represented by a tree. Since the service time distribution depends on the customer class, the stationary distribution is not of product form so there is no simple expression for the stationary distribution. Nevertheless, we can find a harmonic function on this chain which provides information about the asymptotics of this stationary distribution. The associated h‐transformation produces a change of measure that increases the arrival rate of customers and decreases the departure rate thus making large deviations common. The Canadian Journal of Statistics 37: 327–346; 2009 © 2009 Statistical Society of Canada     

Acceptance Sampling Plans from Truncated Life Tests Based on the Birnbaum–Saunders Distribution for Percentiles

Time to failure due to fatigue is one of the common quality characteristics in material engineering applications. In this article, acceptance sampling plans are developed for the Birnbaum–Saunders distribution percentiles when the life test is truncated at a pre-specified time. The minimum sample size necessary to ensure the specified life percentile is obtained under a given customer's risk. The operating characteristic values (and curves) of the sampling plans as well as the producer's risk are presented. The R package named spbsq is developed to implement the developed sampling plans. Two examples with real data sets are also given as illustration.     

Sojourn time in an M/M/1 processor sharing queue with permanent customers

We explicitly compute the sojourn time distribution of an arbitrary customer in an M/M/1 processor sharing (PS) queue with permanent customers. We notably exhibit the orthogonal structure associated with this queuing system and we show how sieved Pollaczek polynomials and their associated orthogonality measure can be used to obtain an explicit representation for the complementary cumulative distribution function of the sojourn time of a customer. This explicit formula subsequently allows us to compute the two first moments of this random variable and to study the asymptotic behavior of its distribution. The most salient result is that the decay rate depends on the load of the system and the number K of permanent customers. When the load is above a certain threshold depending on K, the decay rate is identical to that of a regular M/M/1 PS queue.     

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.     

Coherent Frequentism: A Decision Theory Based on Confidence Sets

By representing fair betting odds according to one or more pairs of confidence set estimators, dual parameter distributions called confidence posteriors secure the coherence of actions without any prior distribution. This theory reduces to the maximization of expected utility when the pair of posteriors is induced by an exact or approximate confidence set estimator or when a reduction rule is applied to the pair. Unlike the p-value, the confidence posterior probability of an interval hypothesis is suitable as an estimator of the indicator of hypothesis truth since it converges to 1 if the hypothesis is true or to 0 otherwise.     

Probability models on horse-race outcomes

SUMMARY A number of models have been examined for modelling probability based on rankings. Most prominent among these are the gamma and normal probability models. The accuracy of these models in predicting the outcomes of horse races is investigated in this paper. The parameters of these models are estimated by the maximum likelihood method, using the information on win pool fractions. These models are used to estimate the probabilities that race entrants finish second or third in a race. These probabilities are then compared with the corresponding objective probabilities estimated from actual race outcomes. The data are obtained from over 15 000 races. it is found that all the models tend to overestimate the probability of a horse finishing second or third when the horse has a high probability of such a result, but underestimate the probability of a horse finishing second or third when this probability is low.     

Multivariate lagrange distributions and a subfamiliy

Second order moments about its means, i.e. the variances and covari-ances for multivariate Lagrange distributions are derived in a matrix form. A subfamily of multivariate Lagrange distributions which can be characterized as the distributions of customers served in a busy period in queues with some conditions are considered. Theorems about their probability functions, one of which is a multivariate generalization of a formula by Takà cs(1989). are given and the means and second order moments about its means are considered. As an example, a multivariate Borel-Tanner distribution is derived.     

Structural properties and estimation in m|EK| 1 queue

In this paper we study the distribution of the number of customers served in a busy period in the framework of modified power series distribution introduced by Gupta (197U) and obtain the moments and probability generating function of this distribution. We also study the maximum likelihood estimation of the parameter θand the variance and the asymptotic bias of the MLE are also obtained. The minimum variance unbiased estimate of θ^ris investigated and an estimate of the probabilities is given.     

AN INVENTORY PROBLEM APPLIED TO A RENTAL SITUATION

An inventory problem, applied to a rental situation business, has been considered. If no item is in stock when a demand occurs, the company borrows the units from other concerns in the same line of business. The profit function has been calculated and it has been shown how the inventory level increases with penalty cost. A review of the literature on inventory control reveals that not much work has been done in holding inventories of rental items. In a recent paper, Tainiter (1964) considered the situation in which a company rents out items such as cars, trucks, farm equipment, books, furniture, etc., and obtained the profit function by taking “rental-out” time as a negative exponential and demand as a general random variable. The model is equally applicable to companies which rent out “service personnel”, repair men, taxi cabs, etc. “We consider a company renting out items to customers. The company starts its business by purchasing a total number of M items in the inventory. The term inventory, defined by Arrow, Earlin and Scarf (1958) as the stock of goods which is kept for future sale or production, is applicable here. Whenever a demand occurs the item is rented out immediately, if it is available in the stock. But if the inventory is zero, i.e. all the items are rented out, the demand will be satisfied by borrowing items from other companies which are dealing in the same line of business. For example, a manufacturer of refrigerators maintains and repairs his product at the customer's house after sale. If a complaint arrives when no repairmen are available, the company will “borrow” repairmen from elsewhere and will attend to the complaint immediately. The borrowing cost may be negative or positive, representing a penalty or a profit. On the other hand if the company does not borrow and the customer has to wait (and such situations occur very often) the loss of the customer's goodwill may occur. It is also not possible to keep large numbers of items because of the storage costs and tied up capital. The problem is then to devise an optimal policy such that the profits of the company are maximized.     

Fiducial inference in combining expert judgements

In the expert use problem, hierarchical models provide an ideal perspective for classifying understanding and generalising the aggregative algoithms suitable to compose experts' opinions in a single synthesis distribution. After suggesting to look at Peter A. Morris' (1971, 1974, 1977) Bayesian model in such a light, this paper addresses the problem of modelling the multidimensional ‘performance function’, which encodes aggregator's beliefs about each expert's assessment ability and the degree of dependence among the experts. Whenever the aggregator has not an empirically founded probability distribution for the experts' performances, the proposed fiducial procedure provides a rational and very flexible tool for enabling the performance function to be specified with a relatively small number of assessments: moreover, it warrants aggregator's beliefs about the experts in terms of personal long run frequencies.     

Convergence of Monte Carlo distribution estimates from rival samplers

It is often necessary to make sampling-based statistical inference about many probability distributions in parallel. Given a finite computational resource, this article addresses how to optimally divide sampling effort between the samplers of the different distributions. Formally approaching this decision problem requires both the specification of an error criterion to assess how well each group of samples represent their underlying distribution, and a loss function to combine the errors into an overall performance score. For the first part, a new Monte Carlo divergence error criterion based on Jensen–Shannon divergence is proposed. Using results from information theory, approximations are derived for estimating this criterion for each target based on a single run, enabling adaptive sample size choices to be made during sampling.     

Bayesian Inference and Prediction in an M/G/1 with Optional Second Service

In this article, we exploit the Bayesian inference and prediction for an M/G/1 queuing model with optional second re-service. In this model, a service unit attends customers arriving following a Poisson process and demanding service according to a general distribution and some of customers need to re-service with probability “p”. First, we introduce a mixture of truncated Normal distributions on interval (? ∞, 0) to approximate the service and re-service time densities. Then, given observations of the system, we propose a Bayesian procedure based on birth-death MCMC methodology to estimate some performance measures. Finally, we apply the theories in practice by providing a numerical example based on real data which have been obtained from a hospital.     

ON THE INFINITE SERVER SHORTEST QUEUE PROBLEM: SYMMETRIC CASE

Abstract

We consider two identical parallel M/M/∞ queues. A new arrival is routed to the queue with the smaller number of customers. If both systems have equal occupancy, the arrival joins either with probability 1/2. These types of models have been used to describe CDMA (Code Division Multiple Access) cellular systems. We analyze this model both numerically and asymptotically. For the latter, we consider the limit ρ = λ/μ → ∞, where λ (resp., μ) is the arrival (resp., service) rate. An efficient numerical method is developed for computing the joint steady-state distribution of the number of customers in the two queues. We give several asymptotic formulas, valid for different ranges of the state variables, which show the qualitative structure of the joint distribution. The numerical accuracy of the asymptotic results is tested.     

Stakes and chips

Gambling has provided centuries of inspiration to probabilists and statisticians. The process continues. There also exist fundamental links between betting and a newer subject, Information Theory, which began with Claude Shannon and his ground-breaking 1948 paper A Mathematical Theory of Communication.
It is a result which arises very naturally. A successful gambler and a successful data compression algorithm must both accurately estimate probability distributions. There are practical results as well as theoretical: Shannon and colleagues, including the legendary gambler and mathematician Edward Thorp, actually attempted to apply results from information theory in Las Vegas casinos and stock market transactions. Oliver Johnson examines the connections.     

Markov's Inequality and Chebyshev's Inequality for Tail Probabilities: A Sharper Image

Markov's inequality gives an upper bound on the probability that a nonnegative random variable takes large values. For example, if the random variable is the lifetime of a person or a machine, Markov's inequality says that the probability that an individual survives more than three times the average lifetime in the population of such individuals cannot exceed one-third. Here we give a simple, intuitive geometric interpretation and derivation of Markov's inequality. These results lead to inequalities sharper than Markov's when information about conditional expectations is available, as in reliability theory, demography, and actuarial mathematics. We use these results to sharpen Chebyshev's tail inequality also.     

Tables of critical values of the normal probability paper test

The present paper relates to Holm's normal probability paper test for simple hypotheses. It is shown how to obtain the statistics for the one-sided and two-sided tests using normal probability paper. Tables of critical values of these test statistics are given for practical use of the test; the equations for calculating the distribution functions of the test statistics are listed. Two different methods lead to these equations; but details are beyond the scope of the present paper and the interested reader is referred to another paper.     

电子商务客户网络购物行为挖掘

电子商务已经成为越来越多的消费者购物的一个重要途径，分析网络购物客户的个人特征及其购物行为，对商业的成功有着至关重要的作用。然而电子商务还是一个崭新的商业领域，很多的业界人士仍忙于技术方面的考虑，却很少分析客户的网络购买行为。而使用真实网络购物KDD Cup 2000数据，分析Gazelle．com公司客户的个人特征和网络购物行为，并应用数据挖掘的购物篮模型对各商品之间的关联性进行分析，才能更确切地预测模型预测客户的忠诚度。