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.     

Quality Control Chart for the Mean using Double Ranked Set Sampling

In this paper, an attempt is made to develop Quality Control Charts for monitoring the process mean based on Double Ranked Set Sampling (DRSS) rather than the traditional Simple Random Sampling (SRS). Considering a normal population and several shift values, the performance of the Average Run Length (ARL) of these new charts was compared with the control charts based on Ranked Set Sampling (RSS) and SRS with the same number of observations. It is shown that the new charts do a better job of detecting changes in process mean compared with SRS and RSS.     

A computational algorithm for the loss probability in the M/G/1+PH queue

This article develops a computational algorithm for the loss probability in the stationary M/G/1 queue with impatient customers whose impatience times follow a phase-type distribution (M/G/1+PH). The algorithm outputs the loss probability, along with an upper-bound of its numerical error due to truncation, and it is readily applicable to the M/D/1+PH, M/PH/1+PH, and M/Pareto/1+PH queues.     

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.     

The variance constant for continuous-time level dependent quasi-birth-and-death processes

We derive the variance constant of continuous-time level dependent quasi-birth-and-death processes by investigating the expected integral functionals of the first return times. As an application, we consider the variance constant for the M/M/c retrial queue with non-persistent customers. For this model, analytical expressions and numerical results are obtained for the cases of single server and multiple servers, respectively. We also apply the obtained result to test the M/M/c vacation model for airport security pre-board screening checkpoint services by constructing a confidence interval for the mean queue length.     

Estimation of the mean of the exponential distribution using moving extremes ranked set sampling

Moving Extremes Ranked Set Sampling (MERSS) is a useful modification of Ranked Set Sampling (RSS). Unlike RSS, MERSS allows for an increase of set size without introducing too much ranking error. The method is considered parametrically under exponential distribution. Maximum likelihood estimator (MLE), and a modified MLE are considered and their properties are studied. The method is studied under both perfect and imperfect ranking (with error in ranking). It appears that these estimators can be real competitors to the MLE using the usual simple random sampling (SRS).     

An RG-Factorization Approach for a BMAP/M/1 Generalized Processor-Sharing Queue

ABSTRACT

In this paper, we study a BMAP/M/1 generalized processor-sharing queue. We propose an RG-factorization approach, which can be applied to a wider class of Markovian block-structured processor-sharing queues. We obtain the expressions for both the distribution of the stationary queue length and the Laplace transform of the sojourn time distribution. From these two expressions, we develop an algorithm to compute the mean and variance of the sojourn time approximately.     

Performance measures of M/G/1 retrial queues with recurrent customers,breakdowns, and general delays

In this paper, an M/G/1 retrial system with two classes of customers: transit and recurrent customers is studied. After service completion, recurrent customers always return to the orbit and transit customers leave the system forever. The server is subject to breakdowns and delayed repairs. The customer whose service is interrupted stays in the service, waiting for delay and repair of the server. After repair this customer completes his service. The study of the system concerns the joint generating function of the server state and the queue length in steady state. Some performance measures of the system are then derived and some numerical results are presented to illustrate the effect of the system parameters on the developed performance measures.     

Generalized Systematic Sampling

In this paper, we introduce a new systematic sampling design, called a Generalized Systematic Sampling (GSS), for estimation of finite population mean. The proposed design is found to be better than Simple Random Sampling (SRS) and the generalization of the several existing systematic sampling schemes such as, Linear Systematic Sampling (LSS), Diagonal Systematic Sampling (DSS), and Generalized Diagonal Systematic Sampling (GDSS). All of these designs become special cases of the proposed design.     

Valid estimation of odds ratio using two types of moving extreme ranked set sampling

The paper provides estimation of the odds ratio between two independent groups using two types of Moving Extreme Ranked Set Sampling (MERSS). Theoretical properties of the suggested estimator are derived and compared with its counterpart estimator using simple random sampling (SRS). It is found that the estimator based on MERSS is always valid and has some advantages over that based on SRS. Real data from a level I Trauma center are used to illustrate the procedures developed in this paper.     

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.     

Filtering recursions for calculating likelihoods for queues based on inter-departure time data

We consider inference for queues based on inter-departure time data. Calculating the likelihood for such models is difficult, as the likelihood involves summing up over the (exponentially-large) space of realisations of the arrival process. We demonstrate how a likelihood recursion can be used to calculate this likelihood efficiently for the specific cases of M/G/1 and Er/G/1 queues. We compare the sampling properties of the mles to the sampling properties of estimators, based on indirect inference, which have previously been suggested for this problem.     

Unbiased variance estimation in a simple exponential population using ranked set samples

The problem considered in this paper is that of unbiased estimation of the variance of an exponential distribution using a ranked set sample (RSS). We propose some unbiased estimators each of which is better than the non-parametric minimum variance quadratic unbiased estimator based on a balanced ranked set sample as well as the uniformly minimum variance unbiased estimator based on a simple random sample (SRS) of the same size. Relative performances of the proposed estimators and a few other properties of the estimators including their robustness under imperfect ranking have also been studied.     

The analysis of MX/M/1 queue with two-stage vacations policy

Abstract

In this article, we consider a batch arrival M^X/M/1 queue with two-stage vacations policy that comprises of single working vacation and multiple vacations, denoted by M^X/M/1/SWV?+?MV. Using the matrix analytic method, we derive the probability generating function (PGF) of the stationary system size and investigate the stochastic decomposition structure of stationary system size. Further, we obtain the Laplace–Stieltjes transform (LST) of stationary sojourn time of a customer by the first passage time analysis. At last, we illustrate the effects of various parameters on the performance measures numerically and graphically by some numerical examples.     

A note on Bayesian estimation of traffic intensity in single-server Markovian queues

ABSTRACT

In queuing theory, a major interest of researchers is studying the behavior and formation process and analyzing the performance characteristics of queues, particularly the traffic intensity, which is defined as the ratio between the arrival rate and the service rate. How these parameters can be estimated using some statistical inferential method is the mathematical problem treated here. This article aims to obtain better Bayesian estimates for the traffic intensity of M/M/1 queues, which, in Kendall notation, stand for Markovian single-server infinity queues. The Jeffreys prior is proposed to obtain the posterior and predictive distributions of some parameters of interest. Samples are obtained through simulation and some performance characteristics are analyzed. It is observed from the Bayes factor that Jeffreys prior is competitive, among informative and non-informative prior distributions, and presents the best performance in many of the cases tested.     

Estimation of system reliability for exponential distributions based on L ranked set sampling

Abstract

In the case where strength and stress both follow exponential distributions, this paper considers the maximum likelihood estimator (MLE) of the system reliability based on L ranked set sampling (LRSS). The proposed MLE is shown to have existence, uniqueness and asymptotic normality, and its asymptotic variance is obtained by the Fisher information matrix of LRSS. The values of asymptotic relative efficiencies show that the proposed MLE is always more efficient than the MLE using simple random sampling (SRS). However, the MLE using LRSS cannot be written in closed form. Therefore, the modified MLE is proposed using the technique replaced some terms in the maximum likelihood equations by their expectations. The newly modified MLE using LRSS is shown to be superior to the MLE using SRS. Finally, the proposed method is applied to a real data set on metastatic renal carcinoma study.     

Estimation of the distribution function under hybrid ranked set sampling

Recently, a hybrid ranked set sampling (HRSS) scheme has been proposed in the literature. The HRSS scheme encompasses several existing ranked set sampling (RSS) schemes, and it is a cost-effective alternative to the classical RSS and double RSS schemes. In this paper, we propose an improved estimator for estimating the cumulative distribution function (CDF) using HRSS. It is shown, both theoretically and numerically, that the CDF estimator under HRSS scheme is unbiased and its variance is always less than the variance of the CDF estimator with simple random sampling (SRS). An unbiased estimator of the variance of CDF estimator using HRSS is also derived. Using Monte Carlo simulations, we also study the performances of the proposed and existing CDF estimators under both perfect and imperfect rankings. It turns out that the proposed CDF estimator is by far a superior alternative to the existing CDF estimators with SRS, RSS and L-RSS schemes. For a practical application, a real data set is considered on the bilirubin level of babies in neonatal intensive care.