The Reggae Rhythm Electronics Corporation retains a service crew to repair machine breakdowns that occur on an average of 3 per day (approximately Poisson in nature). The crew can service an average of 8 machines per day, with a repair time distribution that resembles the exponential distribution. a. What is the utilization rate for this service? b. What is the average downtime for a machine that is broken? c. How many machines are waiting to be serviced at any given time? d. What is the probability that more than one machine is broken and waiting to be repaired or being serviced? that is the probability of more than one machine being in the system) 2. ‘Fridaz Car Wash at the UTech Barn’ estimates that dirty cars arrive at the rate of 10 per hour all day Friday. With the Hotters’ Girl crew working the single wash line, it is estimated that cars can be cleaned at the rate of one every 5 minutes. One car at a time is cleaned. Assuming Poisson arrivals and exponential service times, find the a. average number of cars in line. b. average time a car waits before being washed. c. average time a ca spends in the service system. d. tilization rate of the car wash. e. probability that no cars are in the system. 3. Jucier Foods has just opened a new cafeteria in Papine square just for university students. This is a self-serve facility in which the students select the food items they want and then form a single line to pay the cashier. Students arrive at a rate of about 4 per minute according to a Poisson distribution. The single cashier ringing up sales takes about 12 seconds per customer, following an exponential distribution. a. What is the probability that the system is empty? b.
How long will the average student have to wait before reaching the cashier? c. What is the expected number of students in the queue? d. What is the average number in the system? 4 Black Beard’s Barber Shop has one barber. Customers arrive at the rate of 2. 2 customers per hour, and haircuts are given at the average rate of 1 per 20 minutes. Use the Poisson arrivals and exponential service times model to answer the following questions. a. What is the probability that no units are in the system? b. What is the probability that 1 customer is receiving a haircut and no one is waiting? c.
What is the probability that 1 customer is receiving a haircut and 1 customer is waiting? d. What is the probability that 1 customer is receiving a haircut and 2 customers are waiting? e. What is the probability that more than 2 customers are waiting? f. What is the average time a customer waits for service? 5. Avid Interior Design provides home and office decorating assistance to its customers. In normal operation, an average of 2. 5 customers arrive each hour. One design consultant is available to answer customer questions and make product recommendations. The consultant averages 10 minutes with each customer. . Compute the operating characteristics of the customer waiting line, assuming Poisson arrivals and exponential service times. b. Service goals dictate that an arriving customer should not wait for service more than an average of 5 minutes. Is this goal being met? If not, what action do you recommend? c. If the consultant can reduce the average time spent per customer to 8 minutes, what is the mean service rate? Will the service goal be met? 6. Cars arrive at the drive-through teller at National Quick Service Bank Ltd at the rate of 4 every 10 minutes. The average service time is 2 minutes.
The Poisson distribution is appropriate for the arrival rate and service times are exponentially distributed. a. What is the average time a car is in the system? b. What is the average number of cars in the system? c. What is the average time cars spend waiting to receive service? d. What is the average number of cars in line BEHIND the customer receiving service? e. What is the probability that there are no cars at the teller window? f. What is the percentage of time the teller is busy? g. What is the probability that there are exactly 2 cars in the system? =============================
