Brooks’s Law and intercommunication: you talkin’ to me?

Surely everyone involved in developing software has heard of Brooks’s Law. First presented in the eponymous chapter of Frederick P. Brooks, Jr.’s classic The Mythical Man Month, it states: “Adding manpower to a late software project makes it later.” This “law” is much beloved by software developers as a handy bucket of cold water with which to cool the ardor of overly enthusiastic managers and executives. Lately, however, I’ve been thinking about Brooks’s Law and rereading The Mythical Man Month and I’m no longer as impressed with Brooks’s analysis as I once was. This is the third in a series of posts discussing some of the reasons why. The first post in the series discussed training costs and the second talked about sequential constraints.

In addition to training costs and sequential constraints, part of the justification for Brooks’s Law is the claim that as a team expands the cost of communication between team members grows faster than total productivity of the team. As with the issue of training costs, Brooks has a point — the number of possible pairwise communications paths between a team of n people is n(n – 1)/2; that’s a simple fact of math. If you then assume that every possible pair will need to spend a certain amount of time communicating or, equivalently, that the whole team will have to get together for meetings whose total length is determined by the number of people on the team, then it’s true that the number of person-hours spent communicating will grow as the square of the number of people while total productivity, again measured in person-hours, will only increase linearly. We can all see where that’s heading — pretty soon the amount of time spent on communication will be greater than the total amount of time available to work on anything at all and nothing else will get done. But how soon?

To take a concrete example, suppose we’ve got a six person team that we’re thinking of expanding to eight; should we be concerned that the increasing communication costs will eat up any additional productivity we might get from the two extra people? We can figure it out. Suppose that each pair on our team gets together for a pure-overhead, one-hour tête à tête every week. Assuming a week is five eight-hour work days, the whole team spends 30 person-hours on communication per week out of 240 person-hours worked, leaving 210 person-hours of productive work. What happens if we expand the team to eight? Each person will now spend seven hours a week in pairwise communication and the team as a whole will spend a total 56 person-hours a week communicating. But the team will also now be able to do a total of 320 person-hours of work, leaving 264 person-hours to be spent on productive work, or 54 more hours. This calculation does demonstrate Brooks’s larger point, that a “man month” is not a useful measure of productivity — if it were, then expanding a team from six to eight, a 33% increase in size, would likewise increase productivity by 33%, not the approximately 26% we actually get. But this example doesn’t justify, by itself anyway, a blanket claim that adding people to a project will always slow it down — the eight person team can, in fact, get more done than the six person team and therefore should finish the same amount of work sooner, all other things being equal. Of course all other things are not necessarily equal — training costs can reduce the initial productivity of new team members and it’s conceivable the sequential constraints introduce a long leg that can’t be reduced by adding people. In a later post I’ll discuss whether these three factors together might be enough to justify Brooks’s Law.

So, for this team, a jump from six to eight people probably won’t add more communication costs than the productivity of the new people. But obviously Brooks’s is right that eventually quadratic growth will outpace linear growth.1 With a bit of math, we can figure out exactly when the costs of communication start outpacing gains in the ability to do useful work for a given amount of per-pair communication overhead and a given number of hours worked per week. If we have a team of n people that work h-hour weeks and in which each pair spends c hours per week on communication overhead, then the cost to the existing team members of adding one person to the team is n×c because each of the n current team members will now be part of a pair with the newcomer and will have to spend c extra time tending to that pair’s communication needs. Meanwhile the newcomer will also spend n×c hours per week on pairwise communication which, when subtracted from the h total hours they’ll work each week, gives us the amount of non-communication work per week they’ll add to the team’s total capacity. When the amount of productivity lost from the existing team members is the same as the amount of productivity added by the newcomer there’s no point in expanding the team. So we can solve this equation for n:

to get this:

With this formula we can determine that for a team with one-hour per week of per-pair overhead, working 40-hour weeks, the the biggest the team can get without loosing more productivity than it gains, is 20.

But all of these computations may be beside the point, as they’re based on the assumption that communication has to be overhead. What if we had a team of six people that spend not one, but eight hours per day on pairwise communication because they spend all their time pair programing? If we add two people to that team, there is no change in time spent per person on pairwise communication — the only change is, assuming the team rotates partners, that each person will pair with each other person less often. But the communication time obviously can’t be all overhead — the only time anything gets done is when two people are communicating.

So how can this possibly square with Brooks’s analysis? One possibility is to join the ranks of the XP skeptics and simply deny that the pair programming team could possibly get anything done. I’ve had good experiences with pair programming though so I can’t buy that. I think the problem is with Brooks’s underlying assumptions. As I’ve mentioned previously, Brooks assumes that an n-person team will partition the task of writing whatever software they need to write into n pieces, each to be written by one person. To the extent that those pieces of software need to talk to each other, so do the people writing them and this communication is extra work on top of the base amount of work required to write the software. His arguments about training costs, intercommunication, and sequential constraints are all aimed at demonstrating that a task that a single developer could complete in x months will take n developers more than x/n months because the amount of work required for n developers is no longer simply x but x plus overhead.

But there’s another possibility. What if, as I suggested in my post about Sisyphus, n people don’t have more than x work to do but less than x because n people working together and communicating a lot are much more likely to discover a better solution than any one of them working alone? In that case, time spent communicating is not extra work but a way of reducing the total amount of work done.

1. One thing to note about the growth of intercommunication costs is that it is quadratic, not — as some writers have described it — exponential. Quadratic growth is faster than linear, for sure, but nowhere near as fast as exponential. Populations with no limits on their growth — bacteria in Petri dishes or rabbits in Australia — grow exponentially. If communication costs did grow exponentially with the size of the team, then a team would go from spending just slightly over half it’s time on communication to being able to do nothing but communicate, just by adding one person. One author who should certainly know better is Steve McConnell who described the growth of communication paths as “exponential” in Software Estimation, (p. 57). In fact he did know better — in his earlier book, Rapid Development, he described the growth, correctly, as “multiplicative” (p. 311).


Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: