# Computer Systems Modelling

**Principal lecturer:** Prof Srinivasan Keshav

**Taken by:** Part II CST 75%

**Term:** Lent

**Hours:** 16 (online lectures and 8 examples classes)

**Format:** Video lectures and in-person examples classes

**Class limit:** max. 20 students

**Prerequisites:** Data Science, Introduction to Probability

**timetable**

- Overview of computer systems modeling using both analytic techniques and simulation.
- Stochastic processes (builds on starred material in
Foundations of Data Science)
- Discrete and continuous stochastic processes.
- Markov processes and Chapman-Kolmogorov equations.
- Discrete time Markov chains.
- Ergodicity and the stationary distribution.
- Continuous time Markov chains.
- Birth-death processes, flow balance equations.
- The Poisson process.

- Queueing theory
- The M/M/1 queue in detail.
- The equilibrium distribution with conditions for existence and common performance metrics.
- Extensions of the M/M/1 queue: the M/M/k queue, the M/M/infinity queue.
- Queueing networks. Jacksonian networks.
- The M/G/1 queue.

- Signals, systems, and transforms
- Discrete- and continuous-time convolution.
- Signals. The complex exponential signal.
- Linear Time-Invariant Systems. Modeling practical systems as an LTI system.
- Fourier and Laplace transforms.

- Control theory.
- Controlled systems. Modeling controlled systems.
- State variables. The transfer function model.
- First-order and second-order systems.
- Feedback control. PID control.
- Stability. BIBO stability. Lyapunov stability.
- Introduction to Model Predictive Control.

- Introduction to discrete event simulation.
- Simulation techniques
- Random number generation methods
- Statistical aspects: confidence intervals, stopping criteria
- Variance reduction techniques.

Objectives

At the end of the course students should

- Be aware of different approaches to modeling a computer system; their pros and cons

- Understand the concept of a stochastic process and how they arise in practice
- Be able to build simple Markov models and understand the critical modelling assumptions
- Be able to solve simple birth-death processes
- Understand and use M/M/1 queues to model computer systems
- Be able to model a computer system as a linear time-invariant system
- Understand the dynamics of a second-order controlled system
- Design a PID control for an LTI system
- Understand what is meant by BIBO and Lyapunov stability

- Be aware of the issues in building a simulation of a computer system and analysing

the results obtained

Assessment

There will be three in-person assessments for the course: Stochastic processes and Queueing theory: 45%; Signals, Systems, Transforms, and Control Theory: 45%; Simulation: 10%.

Reference books

- Keshav, S. (2012)*. Mathematical Foundations of Computer Networking. Addison-Wesley.
- Kleinrock, L. (1975). Queueing systems, vol. 1. Theory. Wiley.
- Kraniauskas, Peter. Transforms in signals and systems. Addison-Wesley Longman, 1992.
- Jain, R. (1991). The art of computer systems performance analysis. Wiley.