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# Introduction to performance modelling

 Course Code : 2001WETSMT Study domain: Computer Science Academic year: 2019-2020 Semester: 1st semester Contact hours: 45 Credits: 6 Study load (hours): 168 Contract restrictions: No contract restriction Language of instruction: English Exam period: exam in the 1st semester Lecturer(s) Benny Van Houdt

### 3. Course contents *

This course introduces various fundamental concepts when developing stochastic models, such as the Bernoulli/Poisson process,  Markov chains,  Erlang loss models, etc. A table of contents of the course notes is given below:

BERNOULLI AND POISSON PROCESS
1 Bernoulli Process
2 The Poisson Process
3 Superposition, Random Split, Random Selection
4 Exercise

DISCRETE-TIME MARKOV CHAINS
1 De nition and Basic Properties
2 Communicating States and Classes
3 A Fast Algorithm to check the Irreducibility of a Finite Markov Chain
4 Hitting Probabilities and Hitting Times
5 Transient and Recurrent States
6 Invariant Vectors and Distributions
7 Convergence to the Steady State
8 A Fast Algorithm to determine the Period of a Finite Markov Chain
9 Lemma of Pakes and Kaplan
10 Birth-and-Death Markov chains
11 Summary
12 Exercises

CONTINUOUS-TIME MARKOV CHAINS
1 De nition and Basic Properties
2 Limiting Behavior
3 Uniformization and Embedded Markov Chain
4 Birth-and-Death Markov Chains
5 PASTA Property
6 Exercises

APPLICATIONS
1 Some Fundamental Queueing Systems
1.1 The M/M/1 queue
1.2 The M/M/1 queue
1.3 Insensitivity
2 Dimensioning the Plain Old Telephone System
2.1 Erlang B Formula (M/M/C/C queue)
2.2 Engset Formula
2.3 Erlang C Formula (M/M/C/C+Q queue)
3 Jackson Networks
4 Bianchi's 802.11 Model
4.1 802.11 DCF Operation
4.2 A Markovian Model for the 802.11 Saturation Throughput
4.2.1 Packet Transmission Probability
4.2.2 Throughput
5 Blocking Probability in an OPS/OBS Switching Element
6 Exercises