Towards Improving Stochastic Awareness

It is possible for a student to pass a course on stochastic analysis without actually understanding that W = 1/(μ λ) is not the same sort of equation as F = ma. That is a student might grossly underestimate the role of variability in stochastic systems. Failure to grasp this concept early can cause a student to mischaracterize much of the presented information. This is especially an issue in distance courses because students do not interact as much as in residence courses. This paper describes a collection of exercises intended to determine the level of students’ understanding of stochastic behavior and build their stochastic awareness early in a course so that they will better understand the role of randomness and correlation in such systems. Introduction Engineering students are highly adept in mathematical analysis. This ability has been refined through countless instances of using relationships and formulas to predict results. For example is a relationship that is so consistent that it is a basis for commerce around the world. However, when the same students encounter expressions such as 1 , many do not appreciate that this is not a predictor of a quantity, but a parameter of a random variable. Consequently, they tend to misunderstand the nature of these expressions and the nature of the results they imply. This situation is made worse by the fact that most books on queuing systems and stochastic processes focus mainly on expectations and do not dwell on the variability of these systems. I have been teaching undergraduate and graduate courses in stochastic modeling and analysis for almost 30 years. It would seem that a straight forward presentation of the facts would cause students to draw correct conclusions, but I have discovered that some students are incredibly resistant to stochastic concepts. Many will readily accept factual statements that imply stochastic behavior while simultaneously insisting such behavior does not exist. For example a student may acknowledge the correctness of for a (M/M/1) queueing system, while at the same time insisting that if λ is less than μ, it is not possible for a waiting line to form. Other students may simply deny evidence that contradicts their preconceptions. For example when I asked one student to explain a waiting line formed in his simulation of a (M/M/1) queuing system, he replied that there must be something wrong with the simulation because such a waiting line could not occur. In face-to-face classes, I frequently check the students’ level of awareness. Confused looks clearly indicate the need to look at an idea from a different perspective. However, lately I’ve been teaching mostly online courses. As a result, I found the need to take explicit steps to assess and address conceptual lapses. P ge 23255.2 The remainder of this paper is a series of exercises I have used to determine students’ degree of sophistication with respect to stochastic behavior and to improve their understanding of how these systems work. I have used them in face-to-face and distance courses with some success. The paper also presents some limited assessment of efficacy. Exercise 1: Sam’s Newsstand This first exercise is one that can be used early in a course. I typically use this in some sort of interactive session so I can see students’ initial analysis and promote a discussion. Sam sells the monthly magazine Fantastic Fireflies. The demand for this magazine has a Poisson distribution with a mean of three per month. At the start of each month, Sam sends any unsold copies back to the publisher and gets four new copies for the coming month. On the average, how many copies of Fantastic Fireflies will Sam sell per month? a) Four copies b) Between three and four copies c) Three copies d) Fewer than three copies Typically, very few, if any, students initially select the right answer (d). Students are guided to the correct answer through an interactive discussion. Two arguments I often follow up with are: Argument 1: A characteristic of the Poisson distribution is that the demand in any month can be any non-negative integer value, so in some months the demand will be greater than four copies. However, Sam can sell no more than four, so in those months, the number Sam sells will be less than the demand and that means the average number sold must be less than the average demand. Argument 2: We can calculate the expectation of the number sold (Y) using the fact that and pY(y) can be derived from pX(x). Together, the class and I develop a MS Excel spreadsheet similar to that below. Notes: I try to develop arguments in steps as part of the discussion. For example, in Argument 1, I might ask the class, “Suppose the demand in a particular month is five. How many magazines x p(x) y yp(y) 0 0.050 0 0.000 1 0.149 1 0.149 2 0.224 2 0.448 3 0.224 3 0.672 4 0.168 4 0.672 5 or more 0.185 4 0.739 Total 2.681 P ge 23255.3 will Sam sell?” If students are not familiar with the Poisson random variable, I use a simple discrete distribution in which it is possible for the demand to exceed the inventory. Also, we discuss the situation in which the beginning inventory is less than the average demand to discover that now the expectation of the number sold is less than the beginning inventory. The main point here is to get students to think a little more carefully about what is going on and to lay a foundation for later discussions. Exercise 2: A Very Simple Queueing System While the first exercise can be used in any stochastic course, this one is intended specifically for queueing system courses. It serves two purposes. Firstly, I use it to clarify the definitions of the many parameters and measures of performance associated with queuing systems. Secondly, it demonstrates that λ and μ alone are not sufficient to predict behavior. Situation 1: A semiautomatic machine finishes castings. One casting is delivered to the machine every 30 seconds and the machine takes exactly 20 seconds to process the part. Thus, the arrival rate (λ) is two castings per minute and the service rate (μ) is three per minute. Also, the utilization ratio is ρ = λ/μ = 2/3. Since each casting will be finished before the next casting arrives, the average waiting time (Wq) is zero, implying the average time in the system (W) is 20 seconds. Results are derived with the aid of a plot like this: This is a plot of N(t), the number of casting in the work center, verses time. The pattern repeats indefinitely. As in the first exercise, I elicit interaction from students to develop this graph. All results are developed from the visual record and MOP definitions. For example,