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Fig 1: A Markov chain with 6 states, with arrows indicating allowed transitions between states.
Stochastic Modeling is a research area within Operations Research that focuses on developing probabilistic models for real-life systems having an element of uncertainty. The work involves constructing useful models, analyzing them analytically, deriving mathematical expressions for various important performance measures, and building efficient algorithms for their numerical evaluations. Markov Chains is the most important class of stochastic models due to their powerful modeling features, numerical tractability, and applicability to a wide range of real-life systems of great engineering or environmental significance. A Markov Chain, named for Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the sequence of events that preceded it. This specific kind of "memorylessness" is called the Markov property.
Markovian-modulated models are built on the concept of Markov Chains. Their state space is two-dimensional and consists of the phase variable i(t) and the level variable X(t). Phase is used to model the state of some real-life environment, while level is used to model its performance measure at time t. We assume that the phase changes according to some underlying Markov Chain. Further, we assume that the rate at which the level changes at time t depends on the current phase i(t). For example, in the modeling of a telecommunications buffer, the phase may represent the operating switch, while the level may be the amount of data in the buffer. The rate at which the buffer empties or fills in, depends on which switch is on. See the diagram of a simple two-phase model (with on and off phase) and a graph illustrating the evolution of X(t) in time below.
Markovian-modulated models have attracted a lot of interest due to their applicability to a wide range of real-life systems of great engineering or environmental significance, well beyond applications in high-speed telecommunications systems, for which they were originally derived. Very interesting results for this class of models have been obtained in recent years. The recent developments are in 2-dimensional models. The applications of these include ad-hoc mobile networks, the process of coral bleaching, and operation of hydro-power generation, amongst other examples.
This simulation shows the evolution in time of a Stochastic 2-Dimensional Fluid model (i(t),X(t),Y(t)) with a lower boundary.
We assume that {i(t)} is a continuous-time Markov Chain (CTMC). Further, the rate of change in X(t) is a constant c(i) given the current phase of the CTMC is i. Similarly, the rate of change in Y(t) is a constant r(i) given the current phase of the CTMC is i.
In this example, there is a clear drift of both X(t) and Y(t) towards zero. Here, both X(t) and Y(t) are bounded from below by level zero.
If you are interested in post-graduate study in Operations Research, feel free to contact Dr O'Reilly to discuss possibilities. Also, see the examples of topics below.
Dr Małgorzata O'Reilly
School of Mathematics & Physics, University of Tasmania
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