Stochastic fluid models and their applications

Stochastic Fluids Models (SFMs) are key processes in Matrix-Analytic Methods (MAM). This thesis by publication contains three papers on theoretical applications of SFMs. The first application considered is deteriorating systems. In Paper 1, we construct a stochastic model for maintenance that is suitable for deteriorating systems. The level representing the deterioration is continuous and is impacted by both the operating mode and the current level of deterioration. Previous models assumed continuous observation and, therefore, perfect knowledge of the level of deterioration, which is unrealistic in a practical system. We address this deficiency by expanding the state space to record the most recently observed deterioration level. This feature addresses known issues, and corrects subsequent modelling errors, in previous models. Key performance measures for the stochastic model for maintenance can be evaluated using existing methods. The theory for both the multi-layer model and the stochastic model for maintenance is illustrated throughout the paper through the use of numerical examples. Through these examples, we discuss the ways in which this model could be used to perform a comparison of different maintenance strategies. The second application considered is the accumulation of reward. In Paper 2, we construct a generalised reward generator, an extension of the fluid generator \\(Q\\)(s) of a SFM, to enable the total reward generated to be deconstructed into the reward generated in each individual state. We also consider projections of the generalised reward generator and discuss the applications of the resulting generators in various contexts. We use one particular projection to define the matrix \\(M\\) which records the expected number of crossings of level 0. We use the matrix \\(M\\) to create an explicit expression for the key matrix ˜í¬Æ(s). The development of algorithmic techniques to efficiently compute \\(M\\) is in progress, but outside the scope of this thesis. The third application considered in this thesis is that of using algorithms constructed for SFMs to construct algorithms for discrete-time quasi-birth-and-death processes (dtQBDs). In Paper 3, we develop a general methodology, in which we first write a summation expression for \\(G\\)(z) by considering a physical interpretation analogous to that of an algorithm for ˜í¬Æ(s). Next, we construct the corresponding iterative scheme and prove its convergence to \\(G\\)(z). We construct, in detail, two algorithms for \\(G\\)(z) in dtQBDs, one of which is Newton's Method. We then use a similar decomposition of sample paths to generate a comprehensive set of algorithms, an additional one of which turns out to be quadratically convergent. By symmetry arguments we also construct two novel, quadratically convergent algorithms for the key matrix \\(R\\)(z), one of which is equivalent to the application of the general Newton's method to \\(R\\)(z). This thesis encompasses SFMs and their applications, specifically addressing: accurately representing the uncertainty of deteriorating systems for systems which are not continuously observed; the reward generated by a system to be able to be deconstructed into the reward generated in each individual state; and the construction of algorithms in dtQBDs by considering the decomposition of relevant sample paths from SFMs.