# Why bucket brigades are self-balancing

## An informal explanation

For simplicity, consider a bucket brigade with just two workers and suppose that the assembly line is initially balanced. Now imagine what happens if the first, slower worker is delayed in his work so that when his item is taken over by the faster worker it is some distance X from the balanced handoff point. This means that the faster worker has taken over an item with X more work remaining than expected and so will require longer to finish assembling it. And thus in this extra time the first, slower worker moves past the balance point with his next item. But the extra time alloted to this slower worker to move beyond the balance point is the time required for the faster worker to accomplish X amount of work. And thus the slower worker will move beyond the balance point with his next item some amount less than X. In fact, any time a shock or disruption causes the handoff point to deviate from the balanced handoff point by some distance, the next handoff point will be strictly closer to the balance point. In this way handoffs will tend toward the balance point after any shock or disruption, reestablishing balance to the line.

## A more formal mathematical explanation

### The model

The simplest model of the dynamics of bucket brigades is based on the following assumptions.

• Insignificant Walking Time: The total time to assemble a product is significantly greater than the time to walk the length of the assembly line.
• Total Ordering of Workers by Velocity: Each worker i is characterized by a distinct, constant work velocity vi.
• Smoothness and Predictability of Work: The nominal work content of the product is a constant (which we normalize to 1); and the work content is spread continuously and uniformly along the assembly line.

We call this the "Normative Model" because it represents the ideal conditions for bucket brigades to work well.

### The theorem

For notational convenience, let V(i) = v1 + ... + vi. Then:

When workers are sequenced from slowest to fastest the behavior of a bucket brigade line converges exponentially fast to the following pattern:

• Worker i repeatedly executes the interval of work content from V(i-1) / V(n) to V(i) / V(n).
• The production rate is V(n), the largest possible.

### The proof

Because of the Assumption of Smoothness and Predictability of Work, we can model the work content as the unit interval [0,1]. Consider the moment at which worker i takes over the item being assembled by worker i-1. Let xi(k) be the fraction completed of that item at the moment of hand-off, after a total of k items have been completed. Then the clock time separating workers i and i+1 is

``` (1) ti(k) = (xi+1(k) - xi(k)) / vi ```

and the next item will be completed after time

``` (2) tn(k) = (1 - xn(k)) / vn. ```

After completion of the (k+1)-st item the clock-time separating adjacent workers becomes

``` (3) ti(k+1) = (xi+1(k+1) - xi(k+1)) / vi. ```

But we know the new positions of the workers: It is just their previous position plus the work they just accomplished, so that:

``` (4) xi+1(k+1) = xi(k) + vi tn(k) ```

and

``` (5) xi(k+1) = xi-1(k) + vi-1 tn(k). ```

Substituting Expressions (4) and (5) into (1) and then simplifying by substituting (1), (2), and (3) gives:

``` (6) ti(k+1) = (vi-1 / vi) ti-1(k) + (1 - vi-1 / vi) tn(k). ```

The workers are sequenced from slowest to fastest, so we may interpret these equations as describing a finite state Markov Chain with transition matrix A with all transition probabilities 0 except for the following:

``` P1,n = 1; Pi,i-1 = vi-1 / vi; Pi,n = 1 - vi-1 / vi ```

We can now write Equation (6) more succinctly as t(k+1) = A t(k) = Ak+1 t(0). This Markov Chain is irreducible and, because vi-1 < vi for all i, aperiodic. Therefore, by basic results about Markov chains, Ak converges exponentially fast to a matrix, each row of which is

``` ( v1 / V(n), v2 / V(n), ..., vn / V(n) ) ```

In other words, the clock time invested by each worker converges to a common value: The assembly line is perfectly coordinated. The convergence of the xi(k), the points of hand-off, and the specific claims follow by simple algebra. QED

## Extensions

The Normative Model can be extended in many different directions. The analysis generally becomes more complicated but results are similar. You can find examples of such analysis in the academic papers listed under Resources.