Overview
Problem Scenario --
A discount clothing store (like Ross or TJ Maxx) is trying to decide whether to configure its lines for checkout as a bank or a grocery store.In the case of the bank model, one line serves all the registers.In the case of the grocery store model, lines form in front of every register.
These questions will walk you through a common analysis for queueing systems.
Assume that we have 5 register stations.Each worker is identical in terms of how fast they process customers – this time is distributed as an exponential random variable with an average time of 6 minutes (or 12 per hour).
Customers arrive according to a Poisson process, which means that the time between arrivals is distributed as an exponential random variable.45 customers arrive per hour, on average.
Questions to answer
In the bank model, we assume that all the customers join one line and then go to the next available register.In the sheet named “Question 1”, copy the template for the M/M/s system and enter the values for the M/M/5 system.
In the grocery store model, we assume that the customers randomly select one of the 5 available lines.In this case, the arrival rate to each M/M/1 queueing system is 45/5 = 9 customers per hour, and the time between arrivals is still distributed as an exponential random variable (this is known as “splitting a Poisson process” – another reason we love the exponential random variable).In the sheet named “Question 2”, copy the template for the M/M/s system and enter the values for the M/M/1 system with an arrival rate of 45/5 per hour.
When a person walks up to a line, regardless of how many registers there are, a long line is greeted with a groan.Let’s compare the two systems: M/M/5 and 5 M/M/1 queues.For the 5 M/M/1 queues, how many people are expected to be in all the lines?How does this compare to the single M/M/5 queue?(Do not make this problem too hard! Thinking is better than lots of computation! First, think about one M/M/1 system, then what would expect for five of them?)
Based on Little’s Law, you should be able to guess the answer to this question, or at least the general idea.In which system will a customer wait longer?We have the average queue length for each system, and can use the same arrival rate, 45 per hour, for each.So, what is the average time in queue for each system?
- A company may be interested in how often a customer can arrive to a register without waiting for a line.You should have a value for the probability in the M/M/1 system that the register is idle.Let X be the binomial distributed number of registers that are busy.What is the probability, with 5 M/M/1 queues, that a person walks up and no registers are idle?How does this compare to the value for the M/M/5 queue?
- Now, you should have an opinion of which system is more “efficient”.Which one is it?But, I think that most consumers prefer the other system.What makes the more efficient system more efficient, and why don’t consumers like it more?Perhaps, look at the probability of a long line, like 10 customers in the queue, to help you explore this answer.
- Now we will use your answers to 1 and 2 to see which system is “better” – one line for all registers or one line per register. The situation for the M/M/5 is pretty clear, so we will walk through the 5 M/M/1 case now.Answer these questions in a sheet named “Questions 3 to 6”, with some computation and your answers in a text box or several text boxes.
- Note: IE 851 is now linked through the IE 852 Module page.But, if you are clever, you can do this without the binomial distribution.This is the hardest of questions 3 to 5.