## MA50125: Markov processes & applications

[Page last updated: 05 August 2021]

Owning Department/School: Department of Mathematical Sciences
Credits: 6 [equivalent to 12 CATS credits]
Notional Study Hours: 120
Level: Masters UG & PG (FHEQ level 7)
Period:
Semester 1
Assessment Summary: CW 25%, EX 75%
Assessment Detail:
• Coursework (CW 25%)
• Examination (EX 75%)
Supplementary Assessment:
Like-for-like reassessment (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 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;
* Demonstrate critical thinking and an in-depth 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: 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:

#### MA50125 is Optional on the following programmes:

Department of Mathematical Sciences

 Notes: This unit catalogue is applicable for the 2021/22 academic year only. Students continuing their studies into 2022/23 and beyond should not assume that this unit will be available in future years in the format displayed here for 2021/22. Programmes and units are subject to change in accordance with normal University procedures. Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules. Find out more about these and other important University terms and conditions here.