Problems on Poisson processes, Markov chains, Brownian motion, martingales, and queueing theory
Instructions
Customers arrive at a service counter according to a Poisson process with rate λ = 3 per hour.
(1) What is the probability that exactly 5 customers arrive in the next 2 hours?
(2) Given that 5 customers arrived in 2 hours, what is the probability that exactly 2 arrived in the first hour?
(3) What is the expected waiting time until the 4th customer arrives?
A Markov chain has state space {1, 2, 3} with transition matrix:
(1) Find the stationary distribution π.
(2) If X₀ = 1, what is P(X₂ = 3)?
(3) Is this chain irreducible? Is it aperiodic?
Consider a birth-death process with birth rates λₙ = nλ and death rates μₙ = nμ for n ≥ 1, μ₀ = 0.
(1) Write the generator matrix Q for states {0, 1, 2} when λ = 2, μ = 1.
(2) Find the stationary distribution if it exists.
(3) What is the physical interpretation of this process?
A symmetric random walk on integers starts at X₀ = 0. At each step, the walk moves +1 or -1 with equal probability.
(1) Find P(X₄ = 0).
(2) Find E[X₁₀].
(3) What is the probability that the walk first returns to 0 at time n = 4?
Let {B(t), t ≥ 0} be a standard Brownian motion.
(1) Find P(B(2) > 1 | B(1) = 0.5).
(2) Calculate Cov(B(2), B(5)).
(3) What is the distribution of M = max{B(t): 0 ≤ t ≤ 1}?
Let {Xₙ, n ≥ 0} be a sequence of i.i.d. random variables with E[Xᵢ] = 0 and Var(Xᵢ) = σ².
Define Sₙ = X₁ + X₂ + ... + Xₙ with S₀ = 0.
(1) Show that {Sₙ} is a martingale.
(2) Show that {Sₙ² - nσ²} is a martingale.
(3) Use the optional stopping theorem to find E[Sₜ] where T is a stopping time with E[T] < ∞.
A machine breaks down, and the time until breakdown follows an Exponential(λ) distribution. After each breakdown, repair takes a random time with Exponential(μ) distribution, independent of breakdown times.
(1) What is the long-run proportion of time the machine is operating?
(2) If λ = 0.5 failures/hour and μ = 2 repairs/hour, find the expected number of failures in 100 hours.
Consider an M/M/1 queue with arrival rate λ = 4 customers/hour and service rate μ = 5 customers/hour.
(1) Find the utilization ρ and verify stability.
(2) Find the average number of customers in the system L.
(3) Find the average time a customer spends in the system W.
(4) What is the probability that an arriving customer finds the system empty?