Solve the problem:
Waiting Lines and Queuing Theory Models
Q1 Bill First, general manager of Worthmore Department Store, has estimated that every hour of customer time spent waiting in line for the sales clerk to become available costs the store $100 in lost sales and goodwill. Customers arrive at the checkout counter at the rate 30 per hour, and the average service time is 3 minutes. The Poisson distribution describes the arrivals and the service times are exponentially distributed. The number of sales clerks can be 2, 3 or 4, with each one working at the same rate. Bill estimates the salary and benefits for each clerk to be $10 per hour. The store opens 10 hours per day.
(a) Find the average time in the line if 2, 3 and 4 clerks are used.
(b) What is the total time spent waiting in line each day if 2, 3 and 4 clerks are used?
(c) Calculate the total of the daily waiting cost and the service cost 2, 3 and 4 clerks are used. What is the minimum total daily cost?
Q2 Customers arrive at an automated coffee ending machine at a rate of 4 per minute, following a Poisson distribution. The coffee machine dispenses a cup of coffee in exactly 10 seconds.
(a) What is the average number of people waiting in line?
(b) What is the average number in the system?
(c) How long does the average person wait in line before receiving service?
Q3 One mechanic services 5 drilling machines for a steel plate manufacturer. Machines break down on an average of once every 6 working days, and breakdowns tend to follow a Poisson distribution. The mechanic can handle an average of one repair job per day. Repairs follow an exponential distribution.
(a) How many machines are waiting for service, on average?
(b) How many are in the system, on average?
(c) How many drills are in running order, on average?
(d) What Is the average waiting time in the queue.
(e) What is the average wait in the system?
Q4 The Clear Brook High School band is holding a car wash as a fundraiser to buy new equipment. The average time to wash a car is 4 minutes, and the time is exponentially distributed. Cars arrive at a rate of one every 5 minutes (or 12 per hour), and the number of arrivals per time period is described by the Poisson distribution.
(a) What is the average time for cars waiting in line?
(b) What is the average number of cars in the line?
(c) What is the average time in the system?
(d) What is the probability there are more than three cars in the system?
Q5 When additional band members arrived to help at the car wash it was decided that two cars should be washed at a time instead of just the one. Both work crews would work at the same rate.
(a) What is the average time for cars waiting in the line?
(b) What is the average number of cars in the line?
(c) What is the average number of cars in the system?
(d) What is the average time in the system