Homework 7

Exercise 1 Use the alternating renewal theorem to show that under mild conditions, we have that the probability that the number of arrrivals for renewal process in the interval \([0,t]\) is even goes to \(1/2\) as \(t \to \infty\). Why do we need mild conditions? Why do we need \(t \to \infty\).

Exercise 2 Consider a \(M(3)/M(5)/1\) queue. Let \(t=323\), so the queue has had a chance to settle in.

• By running simulations, plot the probability mass function for the random number of customers in the system. Do you recognize it? Any guesses?

• Start keeping track of the departures and assume they proceed into a new shop. By simulations, produce a histogram of the inter-arrivals times of the items entering the new shop. You might be surprised.

Exercise 3 (Little’s law) There are many simple exercises on can do with Little’s law. See here

• Version: 01 December 2020
• Rmd Source