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.     

Linearity of regression for non-adjacent record values

Let X₁,X₂,… be a sequence of iid random variables having a continuous distribution; by R₁,R₂,… denote the corresponding record values. All the distributions allowing linearity of regressions either E(R_m+k|R_m) or E(R_m|R_m+k) are identified.     

A transient analysis of M/G/1 queues with N-policy

We consider an M/G/1 queueing model with N-policy operating. This means, that the server will start up only if a queue of a prescribed length has built up. For this model the time dependent distribution of the queue length is given by simple renewal arguments without resorting to integral transform techniques.     

On the non-existence of generalised Hadamard matrices

The determinant of a generalized Hadamard matrix over its group ring factored out by the relation Σ_gεG G = 0 is shown to have certain number theoretic properties. These are exploited to prove the non-existence of many generalised Hadamard matrices for groups whose orders are divisible by 3, 5 or 7. For example the GH(15, C₁₅), GH(15, C₃) and GH(15, C₅) do not exist. Also for certain n and G we find the set of determinants of the GH(n, G) matrices.     

A Remark on the Zhang Omnibus Test for Normality

Zhang (1999) proposed a novel test statistic Q for testing normality based on the ratio of two unbiased standard deviation estimators, q₁ and q₂, for the true population standard deviation σ. Mingoti & Neves (2003) discussed some properties of q₁ and q₂ and showed that the variance of q₁ increases as the true population variance increases. In this paper, we show that the distribution of q₁ is not normal. As a result, normality percentage points for Q are not appropriate. In this paper, percentage points of Q are obtained using simulations. Monte Carlo simulations are provided to evaluate the performance of the new method and Zhang's method.     

Bayesian analysis of Er/M/1 and Er/M/c queues

Bayesian inference and prediction tasks for Er/M/1 and Er/M/c queues are undertaken. Equilibrium probabilities of the queue size and waiting time distributions are estimated using conditional Monte-Carlo simulation methods. We illustrate that some standard queueing measures do not exist when independent priors are used for the arrival and service rates of a G/M/1 queue.     

Likelihood ratio ordering of order statistics

This paper provides an improvement on the work of Bapat and Kochar (1994, Linear Algebra Appl., 199, 281–291) and strengthens the literature on the likelihood ratio ordering of order statistics. For independent (but possibly nonidentically distributed) absolutely continuous random variables X₁,…,X_n, it is shown under some weak conditions that

X_1:n^lrlrX_n:n,

where ^lr stands for the likelihood ratio ordering and X_k:n represents the kth-order statistic.     

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     

ON THE CONVERGENCE OF EMPIRICAL PROCESSES OF MIXING VARIABLES1

Conditions are given for the weak convergence of (t—t²)L_N(a_N^-1( t )) to a Gaussian process where v<1/2, a _N is a cdf and L _N is the normalized weighted empirical cumulative distribution function (cdf) for an α-mixing sample of random variables in R which may be non-stationary with discontinuous marginals.     

Commuting Matrices in the Queue Length and Sojourn Time Analysis of MAP/MAP/1 Queues

Queues with Markovian arrival and service processes, i.e., MAP/MAP/1 queues, have been useful in the analysis of computer and communication systems and different representations for their stationary sojourn time and queue length distribution have been derived. More specifically, the class of MAP/MAP/1 queues lies at the intersection of the class of QBD queues and the class of semi-Markovian queues. While QBD queues have a matrix exponential representation for their queue length and sojourn time distribution of order N and N², respectively, where N is the size of the background continuous time Markov chain, the reverse is true for a semi-Markovian queue. As the class of MAP/MAP/1 queues lies at the intersection, both the queue length and sojourn time distribution of a MAP/MAP/1 queue has an order N matrix exponential representation. The aim of this article is to understand why the order N² distributions of the sojourn time of a QBD queue and the queue length of a semi-Markovian queue can be reduced to an order N distribution in the specific case of a MAP/MAP/1 queue. We show that the key observation exists in establishing the commutativity of some fundamental matrices involved in the analysis of the MAP/MAP/1 queue.     

Optimal on-line detection of outside observations

Suppose that a Markov process X₁,X₂,… appears consecutively. There are two random moments of time θ₁ and θ₂, independent of (X_n). The distribution of the process (X_n) changes for the first time at instant θ₁ and for the second time at instant θ₂. Our objective is to find a stopping rule based only upon the former values of (X_n) which maximizes the probability that the moment of stopping is between θ₁ and θ₂. A sufficient condition is given under which an optimal stopping time is finite. The maximal probability corresponding to the optimal rule is found.     

Testing equality of proportions with incomplete correlated data

Let (ψ_i,φ_i) be independent, identically distributed pairs of zero-one random variables with (possible) dependence of ψ_i and φ_i within the pair. For n pairs, both variables are observed, but for m₁ additional pairs only ψ_i is observed and for m₂ others φ_i is observed. If π_1· = P{ψ_i = 1} and π_·1=P{φ_i, the problem is to test π_1·=π_·1. Maximum likelihood estimates of π_1· and π_·1 are obtained via the EM algorithm. A test statistic is developed whose null distribution is asymptotically chi-square with one degree of freedom (as n and either m₁ or m₂ tend to infinity). If m₁ = m₂ = 0 the statistic reduces to that of McNemar's test; if n = 0, it is equivalent to the statistic for testing equality of two independent proportions. This test is compared with other tests by means of Pitman efficiency. Examples are presented.     

Computation of the Generalized F Distribution

Exact expressions for the cumulative distribution function of a random variable of the form ( α ₁ X ₁+ α ₂ X ₂)/ Y are given where X ₁, X ₂ and Y are independent chi-squared random variables. The expressions are applied to the detection of joint outliers and Hotelling's mis-specified T ² distribution.     

Estimation of Measures in M/M/1 Queue

Maximum likelihood and uniform minimum variance unbiased estimators of steady-state probability distribution of system size, probability of at least ? customers in the system in steady state, and certain steady-state measures of effectiveness in the M/M/1 queue are obtained/derived based on observations on X, the number of customer arrivals during a service time. The estimators are compared using Asympotic Expected Deficiency (AED) criterion leading to recommendation of uniform minimum variance unbiased estimators over maximum likelihood estimators for some measures.     

Likelihood Ratio Tests Under Local and Fixed Alternatives in Monotone Function Problems

Abstract.  We focus on a class of non-standard problems involving non-parametric estimation of a monotone function that is characterized by n ^1/3 rate of convergence of the maximum likelihood estimator, non-Gaussian limit distributions and the non-existence of     -regular estimators. We have shown elsewhere that under a null hypothesis of the type ψ ( z ₀) =  θ ₀ ( ψ being the monotone function of interest) in non-standard problems of the above kind, the likelihood ratio statistic has a 'universal' limit distribution that is free of the underlying parameters in the model. In this paper, we illustrate its limiting behaviour under local alternatives of the form ψ _n ( z ), where ψ _n (·) and ψ (·) vary in O ( n ^−1/3) neighbourhoods around z ₀ and ψ _n converges to ψ at rate n ^1/3 in an appropriate metric. Apart from local alternatives, we also consider the behaviour of the likelihood ratio statistic under fixed alternatives and establish the convergence in probability of an appropriately scaled version of the same to a constant involving a Kullback–Leibler distance.     

Bayesian model selection for join point regression with application to age-adjusted cancer rates

Summary.  The method of Bayesian model selection for join point regression models is developed. Given a set of K +1 join point models M ₀,  M ₁, …,  M _K with 0, 1, …,  K join points respec-tively, the posterior distributions of the parameters and competing models M _k are computed by Markov chain Monte Carlo simulations. The Bayes information criterion BIC is used to select the model M _k with the smallest value of BIC as the best model. Another approach based on the Bayes factor selects the model M _k with the largest posterior probability as the best model when the prior distribution of M _k is discrete uniform. Both methods are applied to analyse the observed US cancer incidence rates for some selected cancer sites. The graphs of the join point models fitted to the data are produced by using the methods proposed and compared with the method of Kim and co-workers that is based on a series of permutation tests. The analyses show that the Bayes factor is sensitive to the prior specification of the variance σ ², and that the model which is selected by BIC fits the data as well as the model that is selected by the permutation test and has the advantage of producing the posterior distribution for the join points. The Bayesian join point model and model selection method that are presented here will be integrated in the National Cancer Institute's join point software ( http://www.srab.cancer.gov/joinpoint/ ) and will be available to the public.     

Large deviations for the L1-distance in kernel density estimation

We establish a large deviation limit theorem of Chernoff type for the L₁-distance between the nonparametric kernel density estimator and the underlying density. The estimation is based on a sequence of independent and identically distributed random variables. The rate function is well identified, distribution-free and independent of the choice of the kernel.     

A PROBABILISTIC DERIVATION OF THE NON-CENTRAL χ2 DISTRIBUTION

A simple derivation of the non-central χ² distribution is presented. This requires no advanced mathematical knowledge, and is suitable for use in elementary courses.
The non-central χ² distribution is of great importance in statistical theory, both in its own right, and as a step in the derivation of other distributions; however, it tends to be neglected in statistical courses largely, it seems, because the standard derivations are too difficult. A recent review of derivations by Guenther (1964) shows that most require some knowledge of n -dimensional geometry or the equivalent matrix theory. The alternative is the use of generating functions, which is straightforward, apart from the inversion back to the density function. An objection to such methods is that they give no insight into the probabilistic nature of the proble.