Waiting Lines and Queuing Theory Models
Q14-23 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?
Q14-27 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?
Q14-29 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?
Q14-32 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?
Q14-33 When additional band members arrived to help at the car wash (See problem 14-32), 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
This posting contains detailed solutions to following waiting line problems.
Queuing: M/M/s model: Comparing no of servers based on total costs
1. Department Store has approximately 300 customers shopping in its store between 9 A.M. and 5 P.M. on Saturdays. In deciding how many cash registers to keep open each Saturday, Schmedley's manager considers two factors: customer waiting time (and the associated waiting cost) and the service costs of employing additional checkout clerks. Checkout clerks are paid an average of $8 per hour. When only one is on duty, the waiting time per customer is about 10 minutes (or 1/6 of an hour); when two clerks are on duty, the average checkout time is 6 minutes per person; 4 minutes when three clerks are working; and 3 minutes when four clerks are on duty. Schmedley's management has conducted customer satisfaction surveys and has been able to estimate that the store suffers approximately $10 in lost sales and goodwill for every hour of customer time spent waiting in checkout lines. Using the information provided, determine the optimal number of clerks to have on duty each Saturday to minimize the store's total expected cost.
2. Zimmerman's Bank is the only bank in the small town of St. Thomas. On a typical Friday, an average of 10 customers per hour arrive at the bank to transact business. There is one teller at the bank, and the average time required to transact business is 4 minutes. It is assumed that service times may be described by the exponential distribution. A single line would be used, and the customer at the front of the line would go to the first available bank teller. The salary and the benefits fir a teller would be $12 per hour. The bank is open 8 hours each day. It has been estimated that the waiting time cost per hour is $25 per hour in the line.
a) How many customers would enter the bank in a typical day
b) How much total time would a customer spend waiting in line during the entire day if one teller were used? What is the total daily waiting time cost?
c) How much total time would the customer spend waiting in line during the entire day if 2 tellers were used? What is the total waiting cost?
d) If the bank wishes to minimize the total waiting time and personnel cost, how many tellers should be used?