______________________________________________________________________________ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The On Queue Example Question Set ______________________________________________________________________________ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1) Customers arrive at Big Burger, on average, every 10 minutes. Big Bob, the owner of Big Burger, serves all the customers at an average rate of 8 minutes per customer. What is the steady state: A) Expected number of customers in Big Burger? How long a line do they form? With what variance? B) Expected time customers take for service? How long do they wait on line? With what variance? C) Probability that Big Bob is busy? Probability that Big Bob is idle? D) Percent of time that Big Bob is busy? 2) During lunch time customers arrive at Big Burger every 5 minutes. A) What is the minimum number of (8 minutes/customer) servers required during lunch time to maintain a service rate that is higher than the arrival rate. B) For this number of servers answer questions 1A through 1D. 3) Big Bob believes that he loses about $15/hour in profits (transitional effect) for the time that customers stand in line, because some regular customers switch to a competitor. Big Burger employees earn $4.25/hour. A) How many servers should Big Burger employ during lunch time. B) For this number of employees, solve 1A through 1D. 4) Big Burger's customers arrive, during a typical lunch hour, as follows: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 36, 39, 42, 45, 48, 51, 55, 60 } Big Bob was working alone during this time serving pre-packaged "meals" that he "nuked". Every second and third customer asked for salt, pepper and ketchup. Big Bob took an extra minute to serve these customers. This resulted in the following service distribution: { 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2 } (Distributions are shown, as patterns, for clarity -- assume Poisson/ Exponential) Big Burger, eventually, lost one customer, valued at $15.00, to a competitor, for each customer arrival that ocurred when the queue length exceeded 4 customers. A) How many customers will Big Burger lose if customers must request condiments? (identify by customer number and time that queue size was reset) At what cost? (Hint: see the transaction record) B) Is the loss of customers, due to a long queue, expected? Is customer loss possible within 1 standard deviation of the expected queue length? If Big Burger provides condiments, to all customers, profits will decline by $0.10 per three customers, however, service time will decrease, on average, by approximately 1 minute. This will result in the following service time distribution: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} C) Would it be better for Big Bob to provide condiments to all customers whether they ask for them or not? At what cost (Include the cost of lost customers)? D) If Big Burger made $2.00 profit, for all the customers Big Bob was theoretically capable of serving, how much profit would Big Burger make when requiring customers to ask for condiments? How much profit if all customers were given condiments without having to ask for them? E) If customer time waiting in line is valued at $4.25 hour, and Big Bob earns $4.25 hour, how many servers, like Big Bob, would be optimal when requiring customers to ask for condiments? How many servers are optimal when customers are given condiments without having to ask for them? F) How well does the arrival distribution to Big Burger, shown above, fit a Poisson distribution? What is the Chi-Square statistic ? (Ignore the shareware limitation on population size) G) How well does the service pattern, shown above, fit an exponential distribution? (Ignore the shareware limitation on population size) H) Should the interarrival times fit an exponential distribution? 5) Big Bob wants to find out if he should hire an assistant. He also wants to find out if the assistant should work in parallel with him or in series. The following is a typical customer arrival pattern for a two hour period: { 5, 10, 15, 20, 25, 30, 35, 37, 40, 42, 44, 48, 50, 51, 55, 60, 68, 70, 75, 80, 85, 90, 95,100,101,105,110,115,117,120 } A typical service pattern for Big Bob, working alone, is: { 8, 7, 6, 9, 7, 8, 7, 9, 8, 7, 9, 8, 7, 8, 7, 9, 9, 8, 7, 7, 8, 8, 9, 7, 8, 9, 7, 8, 8, 8`} A typical service pattern for Big Bobs assistant, working alone, is: { 8, 6, 9, 5, 11, 6, 7, 8, 12, 5, 5, 6, 7, 12, 11, 6, 6, 6, 5, 5, 5, 7, 9, 12, 9, 10, 5, 7, 8, 6 } If Big Bob and his assistant work together in series, with Big Bob collecting the cash and his assistant filling the orders, Big Bob's service time is: { 2, 3, 2, 3, 3, 3, 2, 2, 3, 1, 3, 2, 1, 2, 3, 1, 4, 3, 2, 2, 3, 1, 4, 2, 3, 3, 2, 2, 2, 3 } The assistant's service time is: { 2, 3, 3, 3, 5, 4, 5, 4, 6, 3, 6, 3, 4, 2, 3, 3, 3, 4, 4, 2, 2, 2, 3, 3, 3, 4, 4, 2, 2, 2 } A) For each of the three service models -- single server, multiple server in series and in parallel: What are the steady state characteristics for each model? What is the maximum number of people in line for each model? What is the server utilization for each server for each model? What are the service distribution characteristics for each server? (Tip -- arrival random numbers can be input once for all service models, however, service random numbers change for each service model in this problem. Also, Big Bob is always server number one -- first to receive a customer if both servers are idle) In parallel service how many customers does Big Bob serve? How many are served by his assistant? B) Which service model is best for Big Burger? Why? 6) Jim's 3 person work group, typically, has .845037 widgets waiting .00845 hours for a worker, for a throughput time of .028252 hours. Widgets arrive every .01 hours. A) What is the service rate for this work group? B) How much "free time" per hour do these employees have? C) What is the total cost of a three person work group? D) How many employees are optimal for Jim's group if the holding cost of work in process is $2.00 per hour and, employee plus machine time costs, are $196.50 per hour? E) What is the total cost for the optimal work group size? F) What is the service rate for this work group? G) What are the steady state characteristics of the optimal work group? H) How much "free time" per hour do these employees have? I) If the company makes $2.00 profit per hour, per worker, by selling them lunch, what is the optimal work group size? J) If costs were reversed, with CW = $196.50 and CS = $2.00 would you change the work group size from 3 persons? What is the new size? G) If widgets take up 1 square foot of floor space, how many square feet of floor space will the optimal service level require? H) How many square feet will the original 3 person work group require? I) If space were limited how would this affect your system evaluation? 7) The DMV has three employees at its "Start Here" counter. Typically, it was found that, with an arrival rate of 25 drivers per hour, .730279 drivers stand in line for .029211 hours. and they spend .106134 hours total in the facility. A) What is each servers service rate. B) If the DMV assumes a $1.50 cost for the drivers to wait, due to their use of paper and pencils, and a $35.00 cost of service for each DMV employee, what is the optimal number of servers for the DMV. C) If drivers need to fill out 1 extra form per hour, bringing the total cost of waiting to $1.60, what is the optimal number of servers for the DMV. 8) Pushy Electronics Inc. wants to hire more salespersons. Salespersons are paid nothing unless they earn a comission. Customers arrive every half minute and take up 6 minutes of a salespersons time. A) Assuming the customers merely browse, how many salespersons should Pushy employ? B) If it's unlucky to have 13 or more customers in the store, how many salespersons should Pushy employ. C) On special sale days Pushy gives out free drinks -- cash value $0.25 each. If salespersons have 5 drinks per hour, and customers have 1 drink per hour, how many salespersons should Pushy employ? Is there a lesson to be learned? Pushy wants customers in and out of the store in less than 6.13 minutes, total, otherwise they have too many drinks and become unable to sign their checks. How many salespersons should Pushy hire on special sale days? D) If the salesperson gets the customer a drink, this adds 2 minutes to each service time. Now how many salespersons are required? If salespersons have 5 drinks per hour, and customers have 1 drink per hour, as before, how many salespersons should Pushy employ? Now can the salespersons outsmart their Pushy boss? ______________________________________________________________________________ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Copyright, Sept. 1993, Victor Elias -- all rights reserved ______________________________________________________________________________ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~