Quiz 9: The Impact of Variability on Process Performance: Throughput Losses

Business

(Loss System) Flow units arrive at a demand rate of 55 units per hour. It takes, on average, six minutes to serve a flow unit. Service is provided by seven servers. a. What is the probability that all seven servers are utilized b. How many units are served every hour c. How many units are lost every hour

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Essay

In order to compute the probability of m resources being utilized and a newly arrived unit is rejected, the Erlang Loss Formula table has to be used. a. The average arrival rate is 55 units per hour. So, the average interarrival time, The average processing time, The number of servers, m = 7 Look up the Erlang Loss Formula table for So, the probability that all the seven servers are busy is b. Compute the flow rate, R using the following formula: So, the flow rate i.e. the customers served per hour is c. Compute the customer lost customer per hour by subtracting the flow rate from the average arrival rate. So, So, customers lost per hour is

Q 2

(Home Security) A friend of yours approaches you with the business idea of a private home security service. This private home security service guarantees to either dispatch one of their own five guards immediately if one of their customers sends in an alarm or, in the case that all five guards are responding to other calls, direct the alarm to the local police. The company receives 12 calls per hour, evenly distributed over the course of the day. The local police charges the home security company $500 for every call that the police responds to. It takes a guard, on average, 90 minutes to respond to an alarm. a. What fraction of the time are incoming alarms directed to the police b. How much does the home security company have to pay the local police every month

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Essay

Consider the following information: Number of guards = 5 Calls per hour = 12 calls Therefore, inter-arrival time = 60/12 = 5 minutes Activity time = 90 minutes a. Determine the fraction of time the incoming alarms are directed to the police as shown below: Initially determine the ratio of activity time to inter-arrival time using given formula: Now, use the Erlang loss table to find the probability that that all five servers are utilized. For the table it is found that for = 5 and the probability that all servers are busy is, Hence, the fraction of time the incoming alarms are directed to the police is 0.7402 or 74.02%. b. Determine the total amount the home security has to pay the local police every month as shown below: Determine the lost customer using given formula: The local police charges $5000 for every call. Therefore, the total amount that home security pays every month to local police is calculated as shown below: Hence, the home security has to pay to the local police.

Q 3

(Video Store) A small video store has nine copies of the DVD Captain Underpants, The Movie in its store. There are 15 customers every day who request this movie for their children. If the movie is not on the shelf, they leave and go to a competing store. Customers arrive evenly distributed over 24 hours. The average rental duration is 36 hours. a. What is the likelihood that a customer going to the video store will find the movie available b. Assume each rental is $5. How much revenue does the store make per day from the movie c. Assume each child that is not able to obtain the movie will receive a $1 bill. How much money would the store have to give out to children requesting Captain Underpants every day d. Assume the demand for the movie will stay the same for another six months. What would be the payback time (not considering interest rates) for purchasing an additional copy of the movie at $50 Consider the extra revenues related to question b and the potential cost savings (part (c)).

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Essay

The given information is shown below: Number of copies = 9 Arrival rate, A Average Rental duration, P = 36 hours (a) Calculate the likelihood that a customer going to the video store would find the movie available as shown below : Use "Erlang Loss" Table to find the probability with all utilized servers Probability of r (22.5) = 62.44% Thus, people going to video store would find the movie available with the probability of (100-62.44) = 37.56% (b) Calculate the revenue of the store per day as shown below: Flow rate = 0.235 customers per hour. The Revenue of the store Thus, the Revenue of the store per day is $28.16 (c) The loss rate is 0.39 customers per hour and each of them get $1. Calculate the loss for the day as shown below: Thus, loss for the day is $9.36 (d) Number of copies = 9 Arrival rate, A Average Rental duration, P = 36 hours Calculate the revenue of the store per day as shown below : Flow rate = 0.259 customers per hour. The Revenue of the store Thus, the Revenue of the store per day is $31.19 Thus, the Increased profit is (31.19 - 28.16) = $3.62

Q 4

(Gas Station) Consider the situation of Mr. R. B. Cheney, who owns a large gas station on a highway in Vermont. In the afternoon hours, there are, on average, 1,000 cars per hour passing by the gas station, of whicH₂ percent would be willing to stop for refueling. However, since there are several other gas stations with similar prices on the highway, potential customers are not willing to wait and bypass Cheney's gas station. The gas station has six spots that can be used for filling up vehicles and it takes a car, on average, five minutes to free up the spot again (includes filling up and any potential delay caused by the customer going inside the gas station). a. What is the probability that all six spots are taken b. How many customers are served every hour

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Q 5

(Two Workstations) Suppose a process contains two workstations that operate with no buffer between them. Now consider the three possible scenarios below: a. Which of the three scenarios will have, on average, the highest flow rate b. Which of the three scenarios will have, on average, the lowest flow time

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Q 6

(XTremely Fast Service Inc.) XTremely Fast Service Inc. is a call center with several business units. One of its business units, Fabulous 4, currently staffs four operators who work eight hours per day Monday through Friday. They provide customer support for a mail-order catalog company. Assume customers call Fabulous 4 during business hours and that-on average-a call arrives every 3 minutes (standard deviation of the interarrival time is equal to 3 minutes). You do NOT have to consider any seasonality in this call arrival pattern. If all four staff members are busy, the customer is rerouted to another business unit instead of being put on hold. Suppose the processing time for each call is 5 minutes on average. a. What is the probability that an incoming call is not processed by Fabulous 4 b. Suppose that Fabulous 4 receives $1 for each customer that it processes. What is Fabulous 4's daily revenue c. Suppose Fabulous 4 pays $5 for every call that gets routed to another business unit. What is its daily transfer payment to the other business unit

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Q 7

(Gotham City Ambulance Services) Gotham City Ambulance Services (GCAS) owns eight ambulances. On average, emergencies are reported to GCAS every 15 minutes (with a coefficient of variation of 1, no seasonality exists). If GCAS has available ambulances, it immediately dispatches one. If there are no ambulances available, the incident is served by the emergency services at a neighboring community. You can assume that in the neighboring community, there is always an ambulance available. On average, an ambulance and its crew are engaged for 1.5 hours (with a coefficient of variation of 1.5) on every call. GCAS operates 24 hours a day. a. What fraction of the emergencies reported to GCAS are handled by the emergency services at the neighboring community b. How many emergencies are served by GCAS during an average 24-hour day c. GCAS updated the operating procedures for its staff. This led to a reduction in the coefficient of variation of the time spent on each trip by its staff from 1.5 to 1.25. How will this training program affect the number of emergencies attended to by the GCAS d. New regulations require that every emergency service respond to at least 95 percent of all incidents reported in its area of service. Does GCAS need to buy more ambulances to meet this requirement If yes, how many ambulances will be required (Assume that the mean time spent on each trip cannot be changed.)

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