## Abstract

Markov processes have found widespread use in the analysis of computer systems and beyond. Over time the size of the systems under consideration has grown considerably, e.g., Google has hundreds of thousands of servers located in its various data centers. This growth in the system size has made conventional methods to analyse these Markov processes infeasible.
As such, deterministic approximations, also known as mean field or fluid models, have been introduced to analyse such large scale systems. Interestingly, these deterministic models have been shown to correspond to the limit of a sequence of appropriately scaled Markov processes showing that the systems behaviour becomes deterministic as the system size tends to infinity.
These Markov processes typically have a countable state space and the limiting system is described by a set of ordinary differential equations. However, in order to analyse large scale computer systems with general job size distributions, one needs to keep track of the age or residual service time of each job. This makes the state space uncountable and the natural candidate for the limiting system becomes a set of partial differential equations (PDEs).
The aim of this project is to develop PDE mean field models for large scale computer systems, to establish convergence results and to use these models to gain insight into the system behaviour. The project combines techniques from stochastic modelling, probability, numerical analysis and simulation.

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