1. (10 points).Queuing Analysis Basics (15 minutes) 1.1 A company in the year 20
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Question
1. (10 points).Queuing Analysis Basics (15 minutes) 1.1 A company in the year 2030 has developed a robotic dry cleaning service. The shirts are placed in arn assembly line such that they arrive to the dry-clean robot every 5 minutes exactly. It takes exactly4 minutes for the robot to clean each shirt. . What is the proportion of time that the dry-clean robot is busy? 2. After eight hours of operation how much time did the dry-clean robot spend 3. After eight hours of operation how much time was the robot idle? 4 How many shirts are expected to be cleaned during an eight-hour time period? 5. What is the probability that the dry-clean robot will be idle? dry cleaning? 2 Given an M/M/L/K Queuing System, with K - 4 1. Write the balance equations for states 0 and 4. 2. (10 points). Queuing System Analysis-Morning Energy Cafe Inc. (One Server) (15 minutes) The Morning Energy Cafe Inc. has a service process as provided in Problem 1. The customers arrive according to an exponential PDF with an average of one customer per 3 minutes Table 1. Probability Density Function of the Service Time Cumulative Time (min) Price Service Coffee Probability Probability VarlS Coffee EIS] Type Regular Espresso Americano 0 0.4 0.6 0.7 1.2 2.5 2.9 3.6 $1.90 $2.25 $2.75 $3.75 0.1 0.3 Latte 1. Estimate the Expected Service Rate in customers per minute 2. Estimate the Corresponding Variance of the Service time. 3. Estimate the corresponding Standard Deviation of the service time (in minutes). 4. Estimate the average Arrival Rate in customers per minute. 5. What type of queuing system is it? 6. What is the fraction of time that the server is idle? 7. Assume that you are the owner of the Cafe and you wculd like to examine various potentia! optic to improve the operation of the Cafe. What type of analysis wouid you do to examine various options? (10 points). Linear Programing- Graphical Method (20 minutes) u are given the following maximization problem: Max z = 4x + 5y ject to:Explanation / Answer
Solution
Given exactly one arrival every 5 minutes, number of arrivals per hour = 60/5 = 12
Given robot takes exactly 4 minutes per shirt, number of shirts cleaned by robot per hour = 60/4 = 15
1.1.1
In one hour 12 shirts arrive which will be cleaned by robot in (12 x 4) = 48 minutes. Thus, in one hour or 60 minutes, the robot is busy for 48 minutes and hence, the proportion of time the robot is busy = 48/60 = 0.8 or 80% ANSWER
1.1.2
In 8 hours (12 x 8) = 96 shirts arrive which will be cleaned by robot in (96 x 4) = 384 minutes. Thus, after 8 hours of operation, the robot spends 384 minutes or 6 hours 24 minutes dry cleaning. ANSWER
1.1.3
In 8 hours (12 x 8) = 96 shirts arrive which will be cleaned by robot in (96 x 4) = 384 minutes. Thus, after 8 hours of operation, the robot spends 384 minutes dry cleaning implying it is idle for 96 minutes or 1 hour 36 minutes. ANSWER
1.1.4
In 8 hours (12 x 8) = 96 shirts arrive which will be cleaned by robot. Thus, the expected number of shirts dry cleaned by the robot in 8 hour period = 96 ANSWER
1.1.5
In 8 hours (12 x 8) = 96 shirts arrive which will be cleaned by robot in (96 x 4) = 384 minutes. Thus, after 8 hours of operation, the robot spends 384 minutes dry cleaning implying it is idle for 96 minutes or 1 hour 36 minutes. =>
P(robot is idle) = 96/480 = 0.2 ANSWER
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