# (solution) Boilermaker Land theme park conducts free tours of the engineering department that designs its…

Boilermaker Land theme park conducts free tours of the engineering department that designs its rides. The tour times are not scheduled; they begin whenever there are a sufficient number of patrons wanting to take a tour. The arrival of patrons wanting a tour is well modeled as a renewalarrival process with expected time between arrivals of 1 minute. It costs Boilermaker Land \$10 each time it conducts a tour, regardless of how many people are in the tour group. But there is also a cost to Boilermaker Land of having patrons waiting for a tour, because if they are waiting, then they are not spending money in the park. Accountants have estimated that patrons in the park spend money at the rate of \$0.50 per minute. (a) What should the size of each tour group be to minimize long-run cost to Boilermaker Land? (Hint: Use the renewal-reward theorem to solve this problem, with waiting and tour costs being the rewards and the beginning of each tour being the renewal process. Let n be the size of each tour group, and notice that the nth patron to arrive for a tour incurs no waiting cost, the (n — 1)st incurs an expected waiting cost of (1 minute)(\$0.50 per minute), the (n — 2)d incurs an expected waiting cost of (2 minutes)(\$0.50 per minute), and so on.) (b) What is the expected time between the departure of tours for your optimal-size tour group?