
Academic Year:  2015/6 
Owning Department/School:  Department of Mathematical Sciences 
Credits:  6 
Level:  Masters UG & PG (FHEQ level 7) 
Period: 
Semester 1 
Assessment Summary:  CW 25%, EX 75% 
Assessment Detail: 

Supplementary Assessment: 
MA50125 Mandatory Extra Work (where allowed by programme regulations) 
Requisites:  
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 birthdeath process as explosive or nonexplosive; * Find the Qmatrix of a timereversed 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; * Demonstrate critical thinking and an indepth understanding of some aspects of Markov processes. Skills: Numeracy T/F A Problem Solving T/F A Written and Spoken Communication F (in tutorials) A (in coursework). 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: WrightFisher and Moran models. Kingman's coalescent process. Population models and branching processes: Birthdeath processes. Continuous time GaltonWatson 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: 
MA50125 is Optional on the following programmes:Department of Mathematical Sciences
