4 - Queueing.pdf OPMGT 301 Prof. Leela Nageswaran | Winter 2022 1 HOMEWORK 4: QUEUEING NOTE: To receive partial/full credit all calculations must be explained and shown. Question XXXXXXXXXXpoints...

Assignment is attached below


4 - Queueing.pdf OPMGT 301 Prof. Leela Nageswaran | Winter 2022 1 HOMEWORK 4: QUEUEING NOTE: To receive partial/full credit all calculations must be explained and shown. Question 1. 30 points Wenting runs a small coffee shop and works there by herself every day. During a busy day, an average of 20 customers arrive requesting coffees every hour, following a Poisson process. It takes about 2 minutes to take down and fulfill each customer’s order, and this time follows an exponential distribution. a. What’s the percentage of time that Wenting is busy checking out customers? b. What is the probability of three or more customers in total in the cafe? c. What is the average total number of customers waiting in line to place their coffee orders (not including those currently placing their orders with Wenting)? Question 2. 20 points Trucks are required to pass through a weighing station so that they can be checked for weight violations by an inspector. Trucks arrive at the station according to a Poisson distribution at the rate of one every 1.5 minutes on average. Currently there are two inspectors on duty, each of whom takes 2.4 minutes to inspect a truck on average. The time to inspect a truck can be assumed to be Exponential. If a truck was just arriving at the station, about how many minutes could the driver expect to be at the station (including the time waiting to be weighed and getting weighed)? Question 3. 30 points Machines at a production facility break down at the rate of three per hour, and the breakdowns are distributed in a Poisson manner. Downtime costs the company $25 per hour per machine, and each maintenance worker receives $15 per hour. One worker can service machines at an average rate of five per hour distributed exponentially. What is the optimal maintenance crew size for servicing the machines in order to minimize the total cost? Question 4. 20 points Wenting (from Question 1) has just invested in a self-serve-and-mobile-pay system so that some customers can avoid having to wait for Wenting to process their coffee drink order. The machine processes the drink order through a sophisticated display panel, dispenses the drink, and also allows for mobile pay using a kiosk, i.e., these customers skip the barista entirely! Wenting noticed that around 25% of the cafe’s customers use the self-service system. Customer arrivals to the cafe have since increased and now follow a Poisson distribution at the rate of 30 persons per hour, and Wenting takes about 2 minutes to take down and fulfill each non-self-serve customer’s order, and this time is exponentially distributed as before. What is the average total number of non-self-serve customers waiting in line to place their coffee orders with Wenting (not including those currently placing their orders with Wenting)?