In the previous installment, we discussed the dynamics of a (very) simple network of queues, and showed how much extra capacity was required to accommodate the build-up of population inside the queue, based on two factors: the rate at which people enter and leave the queue.

Today, we will look at a related question. Last time we determined the expected queue size at equilibrium, given the flow of people into the queue. This time, we want to consider the reverse problem: if you knew how many people are in the queue at equilibrium, what population breakdown would you expect between the two queues?

The question may sound theoretical – it isn’t. If you knew the total size of a market, the relative preferences of consumers between the products, and how long it takes them to replace their product, then determining how many consumers would be using each product at any given time is equivalent to the question we are considering.

Let’s illustrate on a fictional example. Imagine there is a disease, which can be treated two ways – using a blue pill, or a red pill. Doctors prescribe the blue pill to 25% of the patients, and the red one to 75%. The blue pill treatment takes 5 weeks, and the red pill treatment 8 (which we convert to average rates of exit of 0.2 and 0.125 per week). Suppose you knew that currently, 1000 people were under treatment: how many patients would you expect to be treated with a blue pill?

(picture from www.hackthematrix.org)

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I am currently prototyping an application, which brought up some fun modeling questions.

Imagine the following situation: there are 2 products on the market. Customers use either of them, but not both. In each time period (we consider a discrete time model), new customers come on the market, and select one of the 2 products, with probability p and (1-p). At the end of each period, some existing customers stop using their product, and leave the market, with a rate of exit specific to the product.

Suppose that you knew p and the rates of exit for each product. If the size of the total market size was stable, what market share would you expect to see for each product?

Before tackling that question, let’s start with an easier problem: if you knew how many new customers were coming in each period, what would you expect the product shares to be?

Let’s illustrate with an example. You want to open the Awesome Bar & Restaurant, an Awesome place with a large bar and dining room. You expect that 100 customers will show up at the door every hour. A large majority of the customers (70%) head straight to the bar, but one the other hand, people who come for dinner stay for much longer. How many seats should you have in the bar and the restaurant so that no one has to wait to be seated?

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I have been using test-driven development since I read Kent Beck’s book on the topic. I loved the book, I tried it, and adopted it, simply because it makes my life easier. It helps me think through what I am trying to achieve, and I like the safety net of having a test suite which tells me exactly what I have broken. It also fits well with the type of code I write, which is usually math-oriented, with nice, tight domain objects.

So when I decided recently to write a C# implementation of the Simplex algorithm, a classic method to resolve linear programming optimization problems, I expected a walk in the park.

(Side note:I am aware that re-implementing the Simplex is pointless, I am doing this as an exercise/experiment)

Turns out, I was mistaken. I have been struggling with this project pretty much from the beginning, and unit testing hasn’t really helped so far. Unfortunately, I didn’t reach a point where I fully understand what it is that is not flowing, but I decided I would share some of the problems I encountered. Maybe brighter minds than me can help me see what I am doing wrong!More...

I was recently inspired by an article in OR/MS mag to write 2 posts on using Excel data tables to resolve optimization problems. As it turns out, these puzzles are not only published in the magazine: they also have their online home at Puzzlor.com. So if you like optimization and math problems in general, and like puzzles, check it out!

In my last post, I illustrated how to quickly to pick the best value from a selection to get the optimal result, by using Excel Data Tables. This time, we will see how to pick the best possible pair of values.

We are trying to figure out which 2 bridges we should build, in order to minimize the overall travel time for the inhabitants of the island.

I worked out the math for one bridge last time. We will start we a similar setup, but adjust our spreadsheet so that for each islander, we compute the travelling distance for 2 bridges, and select the shortest route.

The ranges B1 and B2 are named Bridge1 and Bridge2. Column I now contains the formula computing the shortest route for each islander. For row 5 for instance, the formula is

=MIN(D5+E5-H5,F5+G5-H5)

Cell I10 is the total of the vertical distances travelled by each individual.

We can select from 4 bridge locations: 2, 4, 7 and 12. What we need is to find out which 2 numbers give us the lowest total travel. Let’s build our data table, this time using 2 bridge positions.

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