Tail Probability of Low-Priority Queue Length in a Discrete-Time Priority BMAP/PH/1 Queue

ABSTRACT

We investigate the tail probability of the queue length of low-priority class for a discrete-time priority BMAP/PH/1 queue that consists of two priority classes, with BMAP (Batch Markovian Arrival Process) arrivals of high-priority class and MAP (Markovian Arrival Process) arrivals of low-priority class. A sufficient condition under which this tail probability has the asymptotically geometric property is derived. A method is designed to compute the asymptotic decay rate if the asymptotically geometric property holds. For the case when the BMAP for high-priority class is the superposition of a number of MAP's, though the parameter matrices representing the BMAP is huge in dimension, the sufficient condition is numerically easy to verify and the asymptotic decay rate can be computed efficiently.     

On the adequacy of variational lower bound functions for likelihood-based inference in Markovian models with missing values

Summary. Variational methods have been proposed for obtaining deterministic lower bounds for log-likelihoods within missing data problems, but with little formal justification or investigation of the worth of the lower bound surfaces as tools for inference. We provide, within a general Markovian context, sufficient conditions under which estimators from the variational approximations are asymptotically equivalent to maximum likelihood estimators, and we show empirically, for the simple example of a first-order autoregressive model with missing values, that the lower bound surface can be very similar in shape to the true log-likelihood in non-asymptotic situations.     

Strong approximations for time-varying infinite-server queues with non-renewal arrival and service processes

In real stochastic systems, the arrival and service processes may not be renewal processes. For example, in many telecommunication systems such as internet traffic where data traffic is bursty, the sequence of inter-arrival times and service times are often correlated and dependent. One way to model this non-renewal behavior is to use Markovian Arrival Processes (MAPs) and Markovian Service Processes (MSPs). MAPs and MSPs allow for inter-arrival and service times to be dependent, while providing the analytical tractability of simple Markov processes. To this end, we prove fluid and diffusion limits for MAP_t/MSP_t/∞ queues by constructing a new Poisson process representation for the queueing dynamics and leveraging strong approximations for Poisson processes. As a result, the fluid and diffusion limit theorems illuminate how the dependence structure of the arrival or service processes can affect the sample path behavior of the queueing process. Finally, our Poisson representation for MAPs and MSPs is useful for simulation purposes and may be of independent interest.     

Application of a Markovian process to the calculation of mean time equilibrium in a genetic drift model

The most common phenomena in the evolution process are natural selection and genetic drift. In this article, we propose a probabilistic method to calculate the mean and variance time for random genetic drift equilibrium, measured as number of generations, based on Markov process and a complex probabilistic model. We studied the case of a constant, panmictic population of diploid organisms, which had a demonstrated lack of mutation, selection or migration for a determined autonomic locus, and two possible alleles, H and h. The calculations presented in this article were based on a Markov process. They explain how genetic and genotypic frequencies changed in different generations and how the heterozygote alleles became extinguished after many generations. This calculation could be used in more evolutionary applications. Finally, some simulations are presented to illustrate the theoretical calculations presented using different basal situations.     

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.     

A Profit‐Maximizing Approach to Disposition Decisions for Product Returns*

Motivated by the asset recovery process at IBM, we analyze the optimal disposition decision for product returns in electronic products industries. Returns may be either remanufactured for reselling or dismantled for spare parts. Reselling a remanufactured unit typically yields higher unit margins. However, demand is uncertain. A common policy in many firms is to rank disposition alternatives by unit margins. We propose a profit‐maximization approach that considers demand uncertainty. We develop single period and multiperiod stochastic optimization models for the disposition problem. Analyzing these models, we show that the optimal allocation balances expected marginal profits across the disposition alternatives. A detailed numerical study reveals that our approach to the disposition problem outperforms the current practice of focusing exclusively on high‐margin options, and we identify conditions under which this improvement is the highest. In addition, we show that a simple myopic heuristic in the multiperiod problem performs well.     

Large Sample Inference for Markovian Exponential Families with Application to Branching Processes with Immigration

This paper presents limit distributions for the score and likelihood-ratio (L.R.) statistic for testing a composite hypothesis involving the mean of the offspring distribution of the Bienaymé-Galton-Watson branching process with immigration (BPWI) when the process is subcritical, critical or supercritical. The BPWI is shown to be a member of a certain Markovian exponential family.     

Random coefficient models for binary longitudinal responses with attrition

We extend the approach introduced by Aitkin and Alfò (1998, Statistics and Computing, 4, pp. 289–307) to the general framework of random coefficient models and propose a class of conditional models to deal with binary longitudinal responses, including unknown sources of heterogeneity in the regression parameters as well as serial dependence of Markovian form.Furthermore, we discuss the extension of the proposed approach to the analysis of informative drop-outs, which represent a central problem in longitudinal studies, and define, as suggested by Follmann and Wu (1995, Biometrics, 51, pp. 151–168), a conditional specification of the full shared parameter model for the primary response and the missingness indicator. The model is applied to a dataset from a methadone maintenance treatment programme held in Sydney in 1986 and previously analysed by Chan et al. (1998, Australian & New Zealand Journal of Statistics, 40, pp. 1–10).All of the proposed models are estimated by means of an EM algorithm for nonparametric maximum likelihood, without assuming any specific parametric distribution for the random coefficients and for the drop-out process.A small scale simulation work is described to explore the behaviour of the extended approach in a number of different situations where informative drop-outs are present.     

Four general multivariate stationary extremal Markovian processes

Two general multivariate stationary Markovian process with maximization structure (denoted by Max-AR(1) and MaxI-AR(1)) are developed respectively. Max-AR(1) is a subclass of MaxI-AR(1). The characterization of the Max-AR(1) and MaxI-AR(1) to be stationary is studied. Some properties of the two maximization processes are derived. Two more related general multivariate stochastic Markovian process with minification structure are analogously constructed (denoted by Min-AR(1) and MinI-AR(1)). Some well known maximization and minification processes are special cases of these four extermal Markovian processes. Two of them are simulated and some point estimations are provided as an illustration of the wide application of these four processes.     

Bayesian estimation of traffic intensity based on queue length in a multi-server M/M/s queue

In this article, we focus on multi-server queueing systems in which inter-arrival and service times are exponentially distributed (Markovian). We use a Bayesian technique, the sampling/importance resampling method (SIR), to estimate the parameters of these queueing systems, making possible the determination of performance measures that are essential to the evaluation of important practical applications such as computer and telecommunication networks, manufacturing and service systems, health care, and other similar real-life problems. Extensive numerical results are presented to demonstrate the accuracy and efficiency of the technique, as well as some of its limitations.