Operational Research

Problem Sheet

Question 1) Consider the M/M/s/K queueing model with s=2,K=100,λ=99 and μ=50.

Provide a sketch to describe this birth-and-death process and describe in plain words the meaning of the parameters involved.

Compute the probability that, in a steady state scenario, the queueing system is empty.

Question 2) Consider the function f:R→R defined by f(x)?exp?(cos?(2x) ). The objective of this exercise is to minimize (at least approximately) the function f.

Determine all stationary points of f in R.

Perform one step of Newton's method starting from x_0=2.

Perform one step of the steepest descent method starting from x_0=2 and using the smallest positive optimal step size.

Question 3) For a fixed number n∈N, consider the function f:R^n→R defined by f(x)?(Ax-b)^T??(Ax-b)?, where A∈R^(n,n) is an invertible matrix and b∈R^n is a given vector. Perform one step of Newton's method using a generic starting point x_0∈R^n.