Problem 1:
Customers arrive at Jackie's Jelly Bean Joint according to a Poisson process at the rate of 1 customer every 3 minutes where {X(t); t ≥ 0} represents the number of arrivals by time t. Jackie's opens each day at 8 AM. (Please convert all times to hours)
(a) The interarrival times are distributed as with ______mean___
(b) The expected time of the arrival of the 12th customer is____
(c) The probability that one customer arrives before 9:00AM is____
(d) Given that 54 customers arrived from 8:00 AM to 11:00 AM, the expected number of arrivals before noon is_____
(e) The distribution of X(t) is____
(f) The distribution of the number of arrivals that occur between 3PM and 4:30PM is____
(g) The probability that 1 or more arrivals occur before 8:06AM is______
Note, to answer what a distribution is, you need to specify the name of the distribution and the value(s) of the parameter(s) of the distribution.
Problem 2
An athletic ticket office has two ticket agents answering incoming phone calls for ticket reser- vations. (Each agent has its own phone.) In addition, one caller can be put on "hold" until one of the agents is available to take the call. If both agent phones and the hold line are all busy, an arriving call gets a busy signal, and it is assumed that business is lost. The arriving calls occur according to a Poisson process with rate 12 calls/hour. Call lengths are exponentially distributed with average 4 minutes in length. Convert all rates to hourly rates.
(a) Find the long run probability that a caller gets an agent immediately?
(b) Find the long run probability that a caller is put on hold?
(c) Find the long run proportion of business that is lost.
Problem 3
Arrivals at a Jiffy Lube follow a Poisson process with rate 10/hour. Service time is exponentially distributed with mean 15 minutes per customer. There are 2 servers available. Since there is a competitor across the street, only 80% of arriving customers stay for service if they find one car WAITING in queue for service; only 60% stay if they find two cars waiting in queue for service; only 40% stay if they find three cars waiting in queue for service; none stay for service if they find four cars waiting in queue for service. Let X(t) represent the number of cars in the system at time. Convert all rates to hourly rates.
(a) What is the probability that an arriving customer will be served immediately?
(b) What is the long run expected number of cars in the system?
(c) What is the average amount of time a customer spends at the system?
(d) What is the arrival rate of customers INTO Jiffy Lube?
Problem 4
Cars pass a point on the highway according to a Poisson process at a rate of one car every two minutes. 20% of the cars are Toyotas and 10% are Hondas.
(a) What is the probability that at least one Toyota passes in an hour?
(b) Given that 10 Hondas have passed in 2 hours, what is the expected number of Toyotas to have passed in that time?
(c) Given that 60 cars pass in an hour, what is the probability that exactly 40 of them were neither Toyotas nor Hondas.
Problem 5
Players and spectators enter a ballpark according to independent Poisson processes having respective rates 5 and 20 per hour. Starting at an arbitrary time, compute the probability that at least 3 players arrive before 4 spectators.
Problem 6
Consider the M/M/6 queue with arrival rate λ, and service rate at each server µ. You arrive at this system and find 12 customers already in the system. What is the expected length of time until you leave the system?