Incoprea Research · 30 October 2026
Evidence status: Exact conditional support for serial multiplication; counterexample to depth as a sufficient predictor. Reliability model and synthetic checks, not measured outage evidence.
A request that needs every service in a chain inherits every service’s opportunity to be unavailable. That mechanism in Depth Beats Count is correct under independence. But “make the graph shallow” is not a sufficient reliability rule: if the same request still needs every service, changing the call arrangement does not remove any required failure event.
The original hypothesis proposes that dependency depth controls availability rather than service count, and that a shallow system with many services can outperform a deep one with few. The second statement can be true, but only with additional information about which services each request actually needs.
Before calculation, we fixed a point-in-time model. Each component is up independently with probability a. A request succeeds exactly when the required components are up; there are no retries, timeouts, queueing, degraded responses or load effects. We varied a over 0.99, 0.999 and 0.9999 and total component count N over 3, 10, 30 and 100. We retained all four graph/request patterns at every setting.
Point-in-time uptime is the outcome. It does not determine the frequency or length of outages, so the report does not convert these probabilities into observed downtime histories.
If all N independent components are necessary, their joint uptime probability is aᴺ. This remains true whether they call one another in a long chain or a root calls N−1 leaves simultaneously and requires every response. The call graphs have different depths, but both success conditions are “every one of these N components is up.”
If a request instead selects just one of the leaves, it needs two components: the root and that selected leaf. Its success probability is a², regardless of how many other leaves exist. If the leaves are genuine alternatives and any one is sufficient, the success probability is a[1−(1−a)ᴺ⁻¹]. Here redundancy changes the success logic rather than merely the drawing.
The standard series and redundant-system distinction is established background; see the NIST series model and parallel model. Our calculation applies those probability rules to explicitly defined request requirements.
With a=99% and N=30, the four models give:
| Request requirements | Depth including root | Exact uptime |
|---|---|---|
| 30-component mandatory chain | 30 | 73.970037% |
| Root plus 29 mandatory leaves | 2 | 73.970037% |
| Root plus one selected leaf | 2 | 98.01% |
| Root plus any available leaf | 2 | Approximately 99% |
The two mandatory designs have equal availability despite very different depths. Meanwhile a chain of only three components has uptime 0.99³=97.0299%, far above the shallow 30-component mandatory fanout. A shallow graph can therefore be worse than a deeper graph without violating any component assumption.
The positive version of the original intuition also appears. A 30-component system in which each request needs only the root and one selected leaf has uptime 98.01%, higher than the three-component mandatory chain. Service count is not enough either. What decides these examples is the required set and its success condition.
The code exhaustively enumerated every up/down state for 3- and 10-component systems and agreed with the closed forms to within floating-point tolerance. A separate simulation used one million snapshots at each of N=3, 10 and 30 with a=0.99. All 12 estimates were within 1.81 Monte Carlo standard errors of their exact probabilities. Synthetic sampling is an implementation check; it supplies no field evidence about real microservices.
A declared common-mode event, unavailable with probability 0.005 and otherwise leaving component states independent, multiplies every conditional success probability by 0.995. Even ideal leaf redundancy cannot remove the root or that shared failure. Actual failure dependence needs measurement; the chosen 0.5% is an illustrative parameter.
The narrower hypothesis is supported: every additional independent mandatory dependency reduces request availability. The broader claim that depth sets availability is not established and has a counterexample within this elementary model. NIST’s discussion of complex systems likewise makes operational success logic central when simple reductions are inadequate.
Reproduce the full grid, enumeration and simulations with the accompanying Python 3 script, frozen plan and seed 20261030. Original hypothesis: DeepSeek; model analysis and report: Codex. This internal check is not external peer review. The original remains unchanged and linked from Incoprea Research.