Computer Systems Modelling Sam Haines Clare College, University of Cambridge 1 st March 2015

About the Course 12lectures 3supervisions 2exam questions (Papers 8 and 9)

Course Overview Introduction to modelling Simulation techniques – Random number generation – Monte Carlo simulation techniques – Statistical analysis of results from simulation and measurements Queueing theory – Applications of Markov Chains – Single/multiple servers – Queues with finite/infinite buffers – Queueing networks

Why model? Fundamental design decisions to help quantify a cost/benefit analysis A system is performing poorly – which problem should be tackled first? How long will a database request wait for before receiving CPU service? What is the utilization of a resource?

Deciding on the type of model Techniques Measurement Simulation Queueing theory Operational analysis (not covered) How to choose Stage of development Time available Resources Desired accuracy Credibility

Little’s Result Relates number of jobs in a system with the time they spend there

Little’s Result (contd.) λ(t) = α(t)/tAverage arrival rate T(t) = γ(t)/α(t)System time per customer N(t) = γ(t)/tAvg num customers in system N(t) = λ(t)T(t) In the limit t →∞:N = λ T

Recap from MMfCS Coefficient of variation: C x = Std Dev / Mean Exponential distribution f x (x) = λ e −λ x for x > 0, 0 otherwise

Memoryless Property Exponential distribution is the only distribution with the Memoryless property P(X > t+s | X > t) = P(X > s) Intuitively, its used to model the inter-event times in which the time until the next event does not depend on the time that has already elapsed If inter-event times are IID RVs with Exp(λ), then λ is the mean event rate

Poisson Process A process of events occuring at random points of time, let N(t) be the number of events in the interval [0,t]. A Poisson process at rate λ is: N(0) = 0 # of events in disjoint intervals is independent

Poisson Process (contd.) Consider number of events N(t) in interval t Divide into n non-overlapping subintervals each of leangth h = t/n Each interval contains single event with probability λt/n Number of such intervals follows Binomial distribution, parameters n and p = λt/n As N →∞, the distribution is a Poisson RV with parameter λt

Poisson Process (contd.) For a Poisson process, let X n be the time between the (n-1) st and n th events The sequence X 1, X 2,... gives the sequence of inter-event times P(X 1 > t) = P(N(t) = 0) = e -λt So X 1 (and hence the inter-event times) are random variables with Exp(λ)

Exam Question Structure Typically broken down into many smaller sections, each worth between 2 and 5 marks Typically one question each on: – Queueing – Simulation

Any Questions ?

Pub?