|
 Academic Year:
| 2013/4 |
| Owning Department/School:
| Department of Mathematical Sciences |
| Credits:
| 6 |
| Level:
| Honours (FHEQ level 6) |
| Period: |
Semester 1 |
| Assessment:
| EX100 |
| Supplementary Assessment: |
Supplementary assessment information not currently available (this will be added shortly) |
| Requisites:
| Before taking this unit you must take MA20225 |
| Description:
| Aims: To study a variety of Markov processes in both discrete and continuous time, including simple Lévy processes. To give a basic introduction to martingales and demonstrate their use. To apply results in areas such as genetics, biological processes, queues, telecommunication networks, insurance, electrical networks, resource management, random walks and elsewhere. Learning Outcomes: On completing the course, students should be able to: * Formulate appropriate Markovian models for a variety of real world problems and apply suitable theoretical results to obtain solutions; * Classify a birth-death process as explosive or non-explosive; * Find the Q-matrix of a time-reversed chain and make effective use of time reversal to describe departures from queues; * Compute basic properties of branching processes; * Perform standard computations for simple Lévy processes; * Verify the martingale property and use martingales effectively in applications. Skills: Numeracy T/F A Problem Solving T/F A Written and Spoken Communication F (in tutorials) Content: Topics covering discrete and continuous time Markov chains, simple Lévy processes and some applications of martingales will be chosen from: Resource management: warehouse restocking, reservoir model. Telecommunication models: blocking probabilities for land line and mobile networks. Queuing networks: M/M/s queue, departure process. Series of M/M/s queues. M/G/1 queue. Reflecting random walks as queuing models in one or more dimensions. Open and closed migration networks. Simple genetics models: Wright-Fisher and Moran models. Kingman's coalescent process. Population models and branching processes: Birth-death processes. Continuous time Galton-Watson processes. Extinction probabilities. Population growth. Epidemics. Markov processes: Stopping times. Strong Markov property. Explosions. Ergodic theorem. Stationarity. Time reversal. Reversibility. Basic martingales and applications: statement of convergence theorem, discussion of optional stopping theorem. Change of measure. Simple Lévy processes: Compound Poisson Processes with drift. Stationary independent increments. Characteristic exponents. First passage problems. Applications to continuous time workload model and ruin problems in insurance. Markov random fields. Electrical networks. Other applications. |
| Programme availability: |
MA30125 is Optional on the following programmes:Department of Mathematical Sciences
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