Academic Year:  2022/23 
Owning Department/School:  Department of Mathematical Sciences 
Credits:  6 [equivalent to 12 CATS credits] 
Notional Study Hours:  120 
Level:  Intermediate (FHEQ level 5) 
Period: 
 Semester 2

Assessment Summary:  EX 100% 
Assessment Detail:  
Supplementary Assessment: 
 Likeforlike reassessment (where allowed by programme regulations)

Requisites: 
Before taking this module you must take MA20224

Learning Outcomes:
 After taking this unit, students should be able to:
* Classify the states of a Markov chain;
* Find hitting probabilities, expected hitting times and invariant distributions;
* Calculate waiting time distributions, transition probabilities and limiting behaviour of various Markov processes.

Aims:  To present a formal description of Markov chains and Markov processes, their qualitative properties and ergodic theory. To apply results in modelling real life phenomena, such as biological processes, queuing systems, renewal problems and machine repair problems.

Skills:  Numeracy T/F A
Problem Solving T/F A
Written and Spoken Communication F (in tutorials).

Content:  Markov chains with discrete states in discrete time: Examples, including random walks. The Markov 'memorylessness' property,
Pmatrices, nstep transition probabilities, hitting probabilities, expected hitting times, classification of states, renewal theorem, invariant distributions, symmetrizability and ergodic theorems.
Markov processes with discrete states in continuous time: Examples, including Poisson processes, birth & death processes and various types of Markovian queues. Qmatrices, waiting time distributions, equilibrium distributions and ergodicity.

Programme availability: 
MA20225 is Compulsory on the following programmes:
Department of Mathematical Sciences
 USMAAFB01 : BSc(Hons) Mathematics and Statistics (Year 2)
 USMAAAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 2)
 USMAAKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 2)
MA20225 is Optional on the following programmes:
Department of Economics
 UHESAFB04 : BSc(Hons) Economics and Mathematics (Year 2)
 UHESAFB04 : BSc(Hons) Economics and Mathematics (Year 3)
 UHESAAB04 : BSc(Hons) Economics and Mathematics with Study year abroad (Year 2)
 UHESAAB04 : BSc(Hons) Economics and Mathematics with Study year abroad (Year 4)
 UHESAKB04 : BSc(Hons) Economics and Mathematics with Year long work placement (Year 2)
 UHESAKB04 : BSc(Hons) Economics and Mathematics with Year long work placement (Year 4)
 UHESACB04 : BSc(Hons) Economics and Mathematics with Combined Placement and Study Abroad (Year 2)
 UHESACB04 : BSc(Hons) Economics and Mathematics with Combined Placement and Study Abroad (Year 4)
Department of Mathematical Sciences
 USMAAFB15 : BSc(Hons) Mathematical Sciences (Year 2)
 USMAAFB15 : BSc(Hons) Mathematical Sciences (Year 3)
 USMAAAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 2)
 USMAAAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
 USMAAKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 2)
 USMAAKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
 USMAAFB13 : BSc(Hons) Mathematics (Year 2)
 USMAAFB13 : BSc(Hons) Mathematics (Year 3)
 USMAAAB14 : BSc(Hons) Mathematics with Study year abroad (Year 2)
 USMAAAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
 USMAAKB14 : BSc(Hons) Mathematics with Year long work placement (Year 2)
 USMAAKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
 USMAAFB20 : BSc(Hons) Mathematics, Statistics, and Data Science (Year 2)
 USMAAAB20 : BSc(Hons) Mathematics, Statistics, and Data Science with Study year abroad (Year 2)
 USMAAKB20 : BSc(Hons) Mathematics, Statistics, and Data Science with Industrial Placement (Year 2)
 USMAAFM14 : MMath(Hons) Mathematics (Year 2)
 USMAAFM14 : MMath(Hons) Mathematics (Year 3)
 USMAAAM15 : MMath(Hons) Mathematics with Study year abroad (Year 2)
 USMAAKM15 : MMath(Hons) Mathematics with Year long work placement (Year 2)
 USMAAKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)
