A Computational Framework for the Mixing Times in the QBD Processes with Infinitely-Many Levels

In this paper, we develop some matrix Poisson's equations satisfied by the mean and variance of the mixing time in an irreducible positive-recurrent discrete-time Markov chain with infinitely-many levels, and provide a computational framework for the solution to the matrix Poisson's equations by means of the UL-type of $RG$-factorization as well as the generalized inverses. In an important special case: the level-dependent QBD processes, we provide a detailed computation for the mean and variance of the mixing time. Based on this, we give new highlight on computation of the mixing time in the block-structured Markov chains with infinitely-many levels through the matrix-analytic method

A Retrial Queueing Model With Thresholds and Phase Type Retrial Times

There is an extensive literature on retrial queueing models. While a majority of the literature on retrial queueing models focuses on the retrial times to be exponentially distributed (so as to keep the state space to be of a reasonable size), a few papers deal with nonexponential retrial times but with some additional restrictions such as constant retrial rate, only the customer at the head of the retrial queue will attempt to capture a free server, 2-state phase type distribution, and finite retrial orbit. Generally, the retrial queueing models are analyzed as level-dependent queues and hence one has to use some type of a truncation method in performing the analysis of the model. In this paper we study a retrial queueing model with threshold-type policy for orbiting customers in the context of nonexponential retrial times. Using matrix-analytic methods we analyze the model and compare with the classical retrial queueing model through a few illustrative numerical examples. We also compare numerically our threshold retrial queueing model with a previously published retrial queueing model that uses a truncation method

Statistical modelling of queueing systems

Queueing models are mathematical models used to describe queueing systems, such as healthcare systems or telecommunication systems. Standard queueing models such as the M/M/. and M/PH/. queueing models assume independence between the arrival process and the distribution of service times. For some queueing systems, this is a reasonable assumption. However, it is possible for some queueing systems to have dependence between how often customers arrive and how long each customer spends in service. Intensive care units are generally described as complex queueing systems, in that a server is not clearly de ned and patient admissions vary considerably and often depend on resource availability. In addition to this, studies have found evidence of a dependence between the patient admission process and the distribution of patient length of stay. Given that standard queueing models assume independence between the arrival process and the distribution of service times, such models are invalid for modelling the bed occupancy of an intensive care unit. An alternative to modelling the bed occupancy of an intensive care unit is to use quasi-birth-and-death (QBD) processes, which not only allow for dependence between the patient admission process and the distribution of patient length of stay but also provide freedom in the distribution of time spent at each bed occupancy. However, limited research exists on the statistical modelling of queueing systems using QBD processes. Therefore, in this thesis we focus on developing statistical methods to t various types of QBD processes to queueing system data, as well as a goodness of t method to assess the t of QBD processes to observed queueing system data. Firstly, we develop two statistical fitting methods for level-dependent and level-independent QBD processes, respectively. These methods are based on the EM algorithm, since all that is observed while watching the evolution of a QBD process are the changes in level and the times at which those changes occurred. That is, the phase process remains hidden. We assess the accuracy of our methods by using simulated data from known QBD processes. In particular, we compare the stationary and transient behaviour of a known QBD process to what is expected under the fitted QBD process. The statistical fitting of level-dependent QBD processes to queueing system data is advantageous in that we potentially gain valuable insight into the operation of a queueing system. However, such models can be over-parameterised and therefore cannot be used for prediction. We therefore develop a new class of QBD process called structured QBD processes which offer a reduction in the number of parameters through using observable behaviours of the queueing system. We then extend our statistical fitting method to structured QBD processes and assess the accuracy of our method by comparing the stationary and transient behaviour of several known structured QBD processes to what is expected under the respective fitted structured QBD process. Since we often do not know the true QBD process for a queueing system, we develop a goodness of t test which statistically determines if data observed from a queueing system is modelled as a realisation of a particular type of QBD process. We also develop methods to visually assess the t of a QBD process to queueing system data, which is insightful in situations where the fitted QBD process does not capture the stationary and transient behaviour observed in the queueing system data. We then consider several numerical examples to demonstrate the application and performance of the goodness of t test and usefulness of the diagnostic plots. A bene t of modelling an intensive care unit using a level-dependent QBD process is that we gain valuable insight into the patient ow of the intensive care unit. However, structured QBD processes are more useful in that they have fewer parameters than a level-dependent QBD process, and hence can be used for prediction. Through the use of our goodness of t test, we identify the best fitting structured QBD process which is then used to predict future behaviours under various scenarios. The statistical methods developed in this thesis enable the fitting and analysis of QBD processes to provide meaningful insight and reliable predictions for queueing systems, including those with dependence between the arrival process and the distribution of service times. Therefore, our statistical methods for QBD processes provide an alternative to modelling intensive care units, particularly when there exists a dependence between the patient admission process and the distribution of patient length of stay.Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 202

Markov modeling and performance analysis of infectious diseases with asymptomatic patients

After over three years of COVID-19, it has become clear that infectious diseases are difficult to eradicate, and humans remain vulnerable under their influence in a long period. The presence of presymptomatic and asymptomatic patients is a significant obstacle to preventing and eliminating infectious diseases. However, the long-term transmission of infectious diseases involving asymptomatic patients still remains unclear. To address this issue, this paper develops a novel Markov process for infectious diseases with asymptomatic patients by means of a continuous-time level-dependent quasi-birth-and-death (QBD) process. The model accurately captures the transmission of infectious diseases by specifying several key parameters (or factors). To analyze the role of asymptomatic and symptomatic patients in the infectious disease transmission process, a simple sufficient condition for the stability of the Markov process of infectious diseases is derived using the mean drift technique. Then, the stationary probability vector of the QBD process is obtained by using RG-factorizations. A method of using the stationary probability vector is provided to obtain important performance measures of the model. Finally, some numerical experiments are presented to demonstrate the model's feasibility through analyzing COVID-19 as an example. The impact of key parameters on the system performance evaluation and the infectious disease transmission process are analyzed. The methodology and results of this paper can provide theoretical and technical support for the scientific control of the long-term transmission of infectious diseases, and we believe that they can serve as a foundation for developing more general models of infectious disease transmission