Mind Your Pizz’s & Queues

Most Friday nights, Marika and I like ordering pizza for dinner from our favorite Gainesville spot, Satchel's Pizza. Being a Friday night though, many people have the same idea, and we can often get stuck with long wait times. This past Friday, we decided to get our order in earlier, ended up getting in before the rush, and our pizza was ready sooner than we would have liked. This situation, strategizing our moves relative to others trying to do the same thing, reminded me of a project I was part of when I first started at the University of Michigan. It was in a parallel/sub-field to Physics called Complex Systems, where we try to apply techniques from Physics to other systems. In this case, we were looking at co-adaptation and co-evolution. Our model involved simulating a group of agents that would act on an environment with different strategies. Based on the strategy, they would change the environment, and get some benefit or penalty, then try to change their strategy for more benefit.

How does this relate to pizza ordering? The other customers and I are the agents, acting on the pizza place. We choose when to place our orders, and based on how many orders are ahead of us, it takes the pizza place a certain amount of time to prepare the order. We then change our ordering time according to how close to dinnertime our pizza finished. The adjustment I settled on was

This says we adjust our ordering time by 10% of the time between when we want to have dinner, and when we actually got our pizza. I put the 10% in there to make the transitions a bit smoother.

The pizza place I modeled as a set of ovens and a queue. When a customer orders, they're added to the queue, and each empty oven will serve the first customer in the queue, and take 20 minutes to cook the pizza before becoming empty again. I was curious how the customers would change their order times, and what kind of wait times would be involved. I decided to run the simulation with a couple different numbers of ovens, and see how things changed. I decided all the customers would try to eat at time t=0, and use t=-20 minutes as their first order time. Here are the average order times used by the customers at each iteration:

After some initial transients, each of the oven cases settles into a sinusoidal pattern, but with different amplitude and frequency. We can also look at how far off the customer was from their dinnertime:

A couple weeks ago was Homecoming weekend, and the wait times did indeed get into the 2 hour range! Where these results get really interesting is if we combine the previous two plots into one showing the relationship between the order time and the wait time:

This has the appearance of an attractor, a common feature of complex systems, where the state will trend toward an equilibrium point, but not necessarily reach it. This is the average over all the customers though, so we can also look at the individuals over time:

It seems the other Satchel's customers and I are trapped forever in a cycle of never getting our pizza quite when we want it!

Blade Sprinter

Shortly after last week's post, I got a followup from my brother Nate: I like your latest blog post about how you feel the need for speed-mowing. However, I think you did not follow your curve far enough. Can you make a follow-up post where you show when travel-speed becomes a problem because the blades don’t cut all the grass?

What Nate is suggesting is that as the speed of the mower across the grass increases, the speed of the blade relative to the grass during part of its rotation will slow down, since it's moving in the opposite direction:

We can write an expression for the ends of the blades using a parametric equation:

where v is the forward speed of the mower, and ω is the angular velocity of the blades. The manual for the mower gives the blade speed as 2800 RPM, so we can look at how the coverage changes as we approach the speed of the tips, which works out to about 200 MPH. I tried a couple different steps in the range, along with the mower's default maximum, 3 MPH:

Around 60 MPH, we can see some gaps appearing, and if we went for the full 200 MPH, we'd be missing about half the width of the mower. On the other hand, I could finish the lawn in 17.3 seconds, so maybe there's something to Nate's idea! In fact, he's far from the first person to consider supercharged lawnmowers, though that example only went up to ~27 MPH. Call a physicist when you really want to exceed the limits of reality!

That's Mow Like It!

[Title shamelessly stolen from The Simpsons.]

Yesterday, I mowed the lawn with our fancy electric mower, which has a nice speed control for its self-propelling feature. It got me wondering though: What's the most efficient speed to mow the lawn? The blade is constantly running, so it seems like finishing the lawn as quickly as possible would be best, but then the drive is using more power. Is it better to use a lower drive speed to save energy?

I couldn't find many details on the battery/motor specifications, so we'll have to make some assumptions: I'm not sure how the variable drive speed/constant blade speed would work with a single motor, so I'm betting there's two. That means we can split up the power use like this:

The manual gives a couple different cases for battery life, but since it applies to both the self-propelled and meatbag-propelled models, I'm guessing it's assuming the drive is off, so that only tells us about the blade power. For the drive power, I decided the closest analogy would be an electric golf cart. We can look at its power use, then scale it by the ratio of weights and speeds. This model uses 2.6 kW to haul 270 kg at 15 mph. The mower weighs 28.4 kg with a top speed of 3.1 mph, so a rough approximation for power would be 57 Watts.

Now there's the question of efficiency: When we put power into an electric motor, how well does it put that power back out? Luckily, I found a great DoE paper on how the efficiency changes with percent of maximum power. I grabbed a couple points from Figure 1 and did a rough curve fit:

where e is the output efficiency and L is the fractional load.

For the "light load" case, the manual says the battery, which holds 7.5 Amp-hours of charge and runs at 56 Volts, will last for 60 minutes. After a bit of computation, that means the blade is using 420 Watts. Now we need to know how long it takes to mow the lawn, depending on the speed. Using a tool like this, we can find the total area of the property, 1065 m^2, and the total area of the house, 246 m^2, to find the approximate area of the lawn, 819 m^2. The mower is 0.53 m wide, so the minimum total distance it would have to travel is 1545 m.

We now have a long chain of relationships we can put together. The total energy used is

where P is the power, and t is the total time. The time it takes to mow is

where d is the distance, and v the velocity, which in turn is

The power, from earlier, is

The equation we get from putting all these together is a bit involved, so I just made a plot instead:

The battery's total capacity is about 1,500 kJ, so even at the slowest speed we're well within a single charge. I was hoping using the motor efficiency curve would give a more interesting result than "go as fast as you can," but since the maximum load is most efficient, that's the result. Maybe my next step will be figuring out how to balance the exertion of following the mower with the time out in the hot sun!