Two Workers Are Quadratically Better Than One

I don't think this is true, unless I'm misunderstanding your description. Do you mean if the work comes in faster than the worker can process it?

If work arrives at the same rate the worker can process it, the depth of the queue is a random walk starting at zero, with the restriction that it can never go below zero. I'm not sure how that behaves, but it definitely grows much less than linearly.

Python simulation code below:

    import random
    
    def random_walk(steps):
        count = 0
        max_count = 0
        for step in range(steps):
            if count > 0:
                count -= 1
            if random.random() > .5:
                count += 2
            max_count = max(count, max_count)
            if step in [2 ** i for i in range(14)]:
                print(f"step={step}, count={count}, max_count={max_count}")
    
    for i in range(10):
        random_walk(2**13 + 1)

* Workload distribution (heavy tailed, exponential, uniform etc.)

* Scheduling policies (FirstComeFirstServe, ShortestRemainingProcessingTime etc.)

* Arrival rate of the jobs (poisson, uniform etc.) in comparison to their "service" rate

One of the biggest issues with drive thru lines is that often very few people have actually placed their order at a time, so the kitchen isn’t actually running at the pace needed to handle all the orders that are about to come in. Get the orders earlier and increase the pace in the kitchen, everything else can keep up because the kitchen is the bottleneck in a place like that.

This is the base hypothesis that doesn’t match semi-realistic situations.

As long as the demand exceeds your lining capacity (let’s say no more than 20 cars because of sheer space constraints, or when the last in queue would be served past 9AM, which is an actually inflexible limit), consuming the queue twice or three times as fast would bring more revenue.

It stops being interesting expanding mostly when your processing ability match the demand or it the cost/benefit balance changes (e.g. you have to buy the surrounding buildings)

The lines are indeed short there, but I only discovered this discount scheme by accident - they don't advertise it too heavily.

The food is average for the price and they're always right next to other options which leads me to believe that it's the short lines that are their product.

Yes this is an assumption. This isn't important though - you use a queue when you think tasks will have some level of build-up anyway i.e. the chance of the task being above your processing speed is non-zero.

> Because two ppl working on tasks coming in will be able to respond in the minimal time, rather than them banking up

No - that's not what the article is suggesting at all. It's saying that when you have two workers, the relative probability that both workers are queued up at the same time, causes the loss of the quadratic relationship between latency and N (see formula at the end).

EDIT:

Clarification on the first part. It is not that they will come in faster than the processing speed, but they that they can.

It's my version of not reading specifications for perpetual motion machines.

They are coming in at exactly the speed that one person can deal with them. Which means that in the long run the worker is always working, and lines can get arbitrarily long. That is what causes his "quadratic".

A second worker makes the odds of a long line drop dramatically.

I don't think this article answered that question.

Since the problem is really that the first worker is overburdened, by being asked to work at maximum capacity plus randomness.

Not everything that grows, not ever everything that grows fast, grows exponentially.

I currently work at a place that prioritizes employee utilization over process speed, so my team has a foreverqueue to ensure that time on task (TOT on my metrics dashboard) is maximized to its fullest. I'm just glad it is tickets and not phone calls.