# Queuing Problems

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Queuing Problems
Problem 1 A gas station with only one gas pump employs the following policy: if a customer has to wait, the price is \$3.50 per gallon; if they don’t have to wait, the price is \$4.00 per gallon. Customers arrive according to a Poisson process with a mean rate of 20 per hour. Service times at the pump have an exponential distribution with a mean of 2 minutes. Arriving customers always wait until they can by gasoline. Determine the expected price of gasoline per gallon.

Problem 3 The Old Colony theme park has a new ride, the Double-Disgusting Cyclonic Twister. The ride holds 30 people in double roller coaster cars, and it takes 3.8 minutes to complete the ride circuit. It takes the ride attendants another 3 minutes (with virtually no variation) to load and unload passengers. Passengers arrive at the ride during peak park hours at a rate of 4 per minute (Poisson distributed). Determine the length of the waiting line for the ride, on average.

Problem 3 Customers arrive to check-in at the local (and very expensive) Royalé Treatment Hotel lobby at a rate of 40 per hour (Poisson distributed). The hotel normally has three clerks available at the desk to check-in guests. The average time for a clerk to check-in a guest is 4 minutes (exponentially distributed). Clerks at the Royale are paid \$24/hour, and the hotel assigns a goodwill cost of \$2/minute for the time any guest must wait in line. Determine whether the present check-in system is cost-effective. If not, recommend what the hotel management should do. Problem 4 Janos Romanesču builds custom furniture, primarily cabinets, bookcases, small tables, and chairs. He works on only one piece of furniture for a customer at a time. It takes him an average of 5 weeks (exponentially distributed) to finish a piece of furniture. An average of 14 customers approach Janos to order pieces of furniture each year (Poisson distributed) but Janos will take only a maximum of 8 advance orders. Determine the average time a customer

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