Introduction to Industrial Engineering

By Jane M. Fraser

Chapter 10

Operations research and other mathematical methods

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10.5 Stochastic models

Consider the following problem:

Assume customers arrive at a bank randomly at rate 15 per hour. Also assume service times average 3 minutes and are modeled as a random variable with an exponential distribution. If we have one teller available, answer the following questions.

  1. What proportion of time is the teller idle?
  2. What is the expected number of people waiting for the teller?
  3. What is the expected time spent a person waits for the teller?
  4. What is the probability a person waits more than 10 minutes for a teller?

The arrival rate of customers is called &lambda and in this problem &lambda is 15 persons/hour. We can take the average service time (3 minutes) and compute the service rate of customers, called &mu, as 20 persons/hour (60 minutes divided by 3 minutes per person).

  1. The proportion of time the teller is busy can be computed as (&lambda/&mu) = 15/20 = .75, so the teller is idle 25% of the time.
  2. The expected number of people in the queue is &lambda2/(&lambda - &mu) = 152/20(5) = 2.25 people.
  3. The average waiting time in line is &lambda/&mu(&lambda – &mu) = 15/20(5) = .15 hour = 9 minutes.
  4. The probability that some one waits more than 10 minutes is

As an industrial engineer you can look at these answers and decide, for your organization, if these levels of service are sufficient. The teller is idle 25% of the time, so the teller can be doing other tasks during that time. The average number of people waiting is 2.25 and the average waiting time is 9 minutes. Finally the probability of waiting more than 10 minutes is xxx. Do we think we will lose customers with that amount of wait?

The industrial engineer might redo the analysis with two tellers. What will happen to each answer? The answer to question 1 will go up, while the answers to questions 2, 3, and 4 will go down. Again, the industrial engineer has to determine which solution is best for the organization.

Obviously, if the arrival rate, &lambda, is bigger than the service rate, &mu, the queue will explode in length. The ratio of these numbers &rho=&lambda/&mu is called the utilization and is a measure of how much the system is being utilized. When &rho=0, no customers are arriving and the queue length is zero. As &rho increases the average queue length will also increase. The following graph shows how the average queue length grows as utilization increases.

The average queue length stays below 5 people until &rho reaches about .84; the average queue length stays below 10 people until &rho reaches about .91. Above that point the average queue length grows quite rapidly as utilization increases. The problem is randomness: if the arrivals occurred at regularly intervals and if every service time were the same, the system could function well at high utilization. But sometimes customers arrive quickly and sometimes slowly; sometimes service takes a shorter time, but sometimes it takes longer. At high utilization, that randomly long service time has a big effect. Some managers don't account for the effect of randomness. They think that if an employee can handle, on average, 10 customers per hour, that only one employee is needed to serve an average arrival rate of 10 customers per hour, but this graph shows that randomness will make the queue grow unacceptably long.

This problem is an example of an M/M/1 queuing model. More generally it is a stochastic model. “Stochastic” means that probabilities are involved. In deterministic optimization we gave a mathematical formula for the objective function, but in most stochastic models, we have several goals (minimize the time the teller is idle, minimize the average number in the queue), so we want to compute the values of our various goals and then make trade offs.

Where people are the items in the queue, the IE also has to consider the psychology of waiting lines. Watts describes Disneyland’s mastery of making waiting in line seem acceptable.

“Disney planners came up with a unique system: first, a snakelike pattern masked the length of a line by running it back and forth in parallel lines; then a variety of visual and audio images kept those in line entertained; and finally, a cleverly engineered schedule kept visitors steadily embarking on the ride so the line would always appear to be moving forward.” (page 389)

Some stochastic models allow for a sequence of choices that occur over time. A decision tree, with choices and chance events, can represent such situations.

Some real world situations are so complicated that we may not be able to write down formulas to calculate the measures we want to compare. For example, a sequence of queues can be used to represent the flow of products through a manufacturing system or the flow of customers through a service system. Such a system can be modeled in a computer simulation. The computer program actually simulates the operation of the production system, including randomness in the system. The simulation can be run many times so we can observe the average performance, and gather data to determine how often the system performs at different desired levels.

As we discussed in Section 5.5, Six Sigma seeks to reduce the variation in any production process. Once the variation in the system has been brought to an acceptable level (recall the analogy with keeping the arrows in a tight cluster), statistical quality control (SQC) is used to monitor the process to determine whether it is still under control.

[insert SQC example]