A large ski mountain has a fleet of 4 snowmobiles it uses during the winter season. It believes that the time between consecutive repair operations for a single snowmobile is an exponential random variable with a mean of .75 weeks. There are two dedicated technicians that are capable of repairing the snowmobiles. A broken down snowmobile will be repaired by one of these technicians (when one becomes available) and the repair time of a snowmobile can be modeled as an exponential random variable with a mean of .4 weeks. (a) Describe a queueing system model of the repair operations of the snowmobiles by answering the following questions: What are the customers in the queueing system? What are the servers and what service is being provided? b) Provide the rate transition diagram of the birth and death process that captures the queueing system. Be sure to provide λn and µn for each relevant value of n and your logic behind any calculations used to obtain them (c) Provide the balance equations (where appropriate numbers are plugged in for λn and µn) that can be used to calculate the Pn values for all relevant n in this queueing system. Solve these equations to determine the Pn values. (d) Determine the average number of snowmobiles that are unavailable to the ski mountain and the average length of time that a broken down snowmobile is unavailable to the ski mountain.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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A large ski mountain has a fleet of 4 snowmobiles it uses during the winter season. It believes that the time between consecutive repair operations for a single snowmobile is an exponential random variable with a mean of .75 weeks. There are two dedicated technicians that are capable of repairing the snowmobiles. A broken down snowmobile will be repaired by one of these technicians (when one becomes available) and the repair time of a snowmobile can be modeled as an exponential random variable with a mean of .4 weeks. (a) Describe a queueing system model of the repair operations of the snowmobiles by answering the following questions: What are the customers in the queueing system? What are the servers and what service is being provided? b) Provide the rate transition diagram of the birth and death process that captures the queueing system. Be sure to provide λn and µn for each relevant value of n and your logic behind any calculations used to obtain them (c) Provide the balance equations (where appropriate numbers are plugged in for λn and µn) that can be used to calculate the Pn values for all relevant n in this queueing system. Solve these equations to determine the Pn values. (d) Determine the average number of snowmobiles that are unavailable to the ski mountain and the average length of time that a broken down snowmobile is unavailable to the ski mountain.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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