"Span of Control" and the Inefficiency of Hierarchy

"Span of Control" is one term for the management concept that one person can only effectively control a limited number of subordinates. As a hierarchal organization grows, more and more intermediary layers must be created to keep this span of control within reasonable bounds. Let's explore the (quite obvious) ramifications of this, as a means of better understanding RA Wilson's SNAFU principle: As hierarchy grows, the increasing number of relays that information must cross, and the self-interested distortion of information at each relay ensures the inefficiency of information processing within hierarchy.

The US Federal Government's National Incident Management System (NIMS) is based upon the Incident Control System (ICS) methodology developed by wildfire fighters to create a standard for command and control systems (hierarchy) as government agencies respond to incidents. NIMS and ICS both state that the maximum desirable span of control is 5, meaning that one supervisor should control no more than 5 subordinates. The US Military follows a similar formula: one commander controls three subordinate units, as well as a staff function, which results in a span of control of roughly 5. This military formula is virtually identical around the world--a time-tested formula for maximum span of control. The military formula, however, is more revealing, for while it uses a 5:1 span of control, the operational span of control is only 3:1 (that is, the number of subordinate units that actually carry out the fundamental mission of the organization). The remaining two (roughly) staff positions under each commander are actually information processing assistants necessary to make even the 3:1 span of control effective. Without getting in to two much details, those staff positions are normally broken down to an executive officer, who is in turn responsible for the commander's administrative staff, and a deputy commander, who is in turn responsible for the commander's non-administrative staff (Intelligence, Logistics, Human Resources, etc.). As a result of the executive officer and deputy commander concept, the non-operational tail actually extends down two layers from each "operational" commander at the higher levels.

Let's flesh out this formula for a moment, and see what happens. I've used military (US Air Force) terminology for the organizational levels:

2-Layer organization (1 information relay): 1 Element leader : 5 subordinates (3 operational)
50% of a 2-layer organization is directly involved in the "operational execution" of the organization's mission--in otherwords, they are the actual tip of the spear.*

*In the simple case of rifle-infantry, there is more tip and less tail. We actually see exactly the opposite in the case of more complex functions: For example, the support tail required to put a single F-16 and Pilot in play can become almost humorously large.

3-Layer organization (2 information relays): 1 Flight Leader: 3 Element leaders (+ staff): 9 "front-line" operational personnel (out of 21 total personnel).
43% of a 3-layer organization is at the "tip" of the spear.

4-Layer organization (3 information relays): 1 Squadron Commander : 3 Flight Leaders (+ two-tiered staff) : 9 Element Leaders (+ staff) : 27 "front-line" operational personnel (out of 106 total personnel)
25% of a 4-layer organization is at the "tip" of the spear.

5-Layer organization (4 information relays): 1 Group Commander: 3 Squadron Commanders (+ two-tiered staff): 9 Flight Leader (+ two-tiered staff) : 27 Element leaders (+ staff) : 81 "front-line" operational personnel (out of 331 total personnel)
24% of a 5-layer organization is at the "tip" of the spear.

6-Layer ("Wing"): 243 "operational" personnel of 1006 total personnel = 24%

In reality, the number of staff tiers keeps increasing (for example, I've never seen an Air Force "wing" with only 12 wing-staff personnel, as the two-tiered staff formula would suggest). Wilson's SNAFU principle would suggest that as the number of layers (and hence relays) increases, the number of personnel involved in information processing functions will keep increasing beyond the 76% suggested in the 6-layer organization above. In reality, this does in fact happen, as at each higher level there are additional staff functions that must be added (e.g. at the Flight level, the staff doesn't include medical, but at the Wing level it may include an entire hospital). Additionally, the degree of autonomy is increased from the Group to Wing level, as necessitated by the sheer impossibility of maintaining effective communications through 5 hierarchal relays.

Question: How can the efficiency gains through task specialization be accurately quantified, so that the efficiency gains (specialization) and losses (information processing burden) of hierarchy can be accurately compared, both internally, and between hierarchal and non-hierarchal structures?