Statistical modelling of queueing systems
Date
2022
Authors
James, Sarah Ellen
Editors
Advisors
Tuke, Simon
Bean, Nigel
Glonek, Garique
Bean, Nigel
Glonek, Garique
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Thesis
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Abstract
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.
School/Discipline
School of Mathematical Sciences
Dissertation Note
Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 2022
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