Advanced Computational Techniques in Engineering: Markov Processes and Maintenance

UNIVERSITY AFFILIATION
FACULTY OR DEPARTMENT
COURSE ID & NAME
TITLE: ADVANCED COMPUTATIONAL TECHNIQUES IN ENGINEERING
STUDENT NAME
STUDENT REGISTRATION NUMBER
PROFESSOR (TUTOR)
DATE OF SUBMISSION
2018

REPORT
A researcher and scientist, Andrei Andreevich Markov, introduced a model into the
statistical and logistics field that would provide a simple generalization of the probability model
of independent trials. The expected outcomes of the successive trials are based on previously
done trials whose data is fed into the model. The random variables based on time form the
stochastic process. These processes are also known as Markov processes when they include the
property states. The attributes follow that the probability of a system having a transition from one
state to another is highly based on the current state and not on previous states. The system tends
to exhibit the memory-less property. The case study of the UQ equipment maintenance uses the
continuous-time Markov chain. The homogenous mode allows for easy calculation of the state
probabilities by using a transition matrix. It has transition rates matrix Q where the elements in
the matrix transition from state i→ state j such that,
qii=−∑
i ≠ j
qij
To compute the steady-state probabilities using the Gauss-Jordan elimination method,
{ Πe P=Π e
∑
kεs
πk
e=1
The steady-state distribution is obtained as,
Πe=e . ( 1+ E−P )−1
πi=E ( hi ) . πi
e
∑ π k
e E ( hi )
The semi-Markov process are much more suitable in the design of aging processes and
maintenance of equipment. The model introduces the sojourn time to enable modeling of
complex stochastic processes. Its accuracy impacts the entire model’s accuracy.

METHODOLOGY
1. The number of schools in the University of Queensland, n
2. Vector of n elements, with each element indicating the parameter for the arrival process
of the machines in each school.
3. Number of states in the Markov chain that represent the health of machines in the
different schools, 1... n.
4. Transition matrix for the Markov chain
The transition rate is defined as,
The state probabilities through the transition rate is given as,