A message can tell one computer what another clock displayed, but it arrives later than it was sent. Synchronization therefore involves an inference about transmission delays as well as a comparison of readings. Four timestamps make the inference explicit—and reveal what those timestamps cannot determine by themselves.
Four recorded events
Computer A sends a request at its local time T₁. Computer B records receipt at T₂ and sends a reply at T₃. A records the reply's arrival at T₄. The middle pair uses B's clock, while the first and last use A's.
Assume both clocks run at the same rate during this short exchange. Let B's clock be ahead of A's by a constant offset θ. Let the outward and return transmission delays be df and dr, and let B spend p units of time processing the request.
What the estimator assumes
The familiar four-timestamp calculation used in NTP forms an offset estimate θ̂ = [(T₂ − T₁) + (T₃ − T₄)]/2 and a round-trip network delay δ = (T₄ − T₁) − (T₃ − T₂). Substituting the model gives
Equal outward and return delays make the estimated offset equal to the actual offset in this idealization. Unequal delays create a bias equal to half their difference. Subtracting the processing interval removes that interval, but does not reveal how the remaining delay is divided between the two directions.
Try the exchange
The demonstration fixes T₁ = 0 and p = 10 milliseconds. The actual offset and both delays are known because we set them in the model. A real observer would initially have the timestamps rather than privileged access to those three inputs.
Actual offset: 20 ms. Outward delay: 30 ms. Return delay: 70 ms.
| Event | Clock | Reading |
|---|---|---|
| T₁: request sent | A | 0 |
| T₂: request received | B | 50 |
| T₃: reply sent | B | 60 |
| T₄: reply received | A | 110 |
Estimated offset: 0 ms; actual offset: 20 ms; bias: −20 ms. Round-trip network delay: 100 ms. Offsets consistent with nonnegative one-way delays: −50 to 50 ms.
Exact arithmetic in an illustrative constant-rate model. No network traffic is sent and no computer clock is changed.
What remains undetermined?
In the starting example, T₁ = 0, T₂ = 50, T₃ = 60 and T₄ = 110 milliseconds. The same readings also fit θ = 0 with delays of 50 and 50 milliseconds. They fit θ = −20 with delays of 70 and 30 milliseconds. The record alone does not select one of these histories.
Requiring both one-way delays to be nonnegative gives T₃ − T₄ ≤ θ ≤ T₂ − T₁. Here the compatible interval is −50 to +50 milliseconds. The symmetric-delay estimate is its midpoint; extra information can narrow the interval.
This is an identifiability problem. More decimal places in the timestamps do not resolve it. Independent knowledge of the link, bounds on directional delay or a different comparison arrangement can supply information that numerical precision alone cannot.
From one exchange to a protocol
Real synchronization systems also handle changing delays, oscillator drift, timestamp placement and unreliable observations. RFC 5905 specifies a protocol and algorithms beyond the single-exchange formulas shown here. The demonstration isolates delay asymmetry; it is not an implementation or performance model of the complete protocol.
Repeating exchanges can help characterize variation. It does not automatically eliminate a persistent directional asymmetry. A carefully stated model distinguishes a random error that averages down from a systematic effect that remains.
Records and physical time
The example connects naturally to SMM/DTT's distinction between an underlying process and a completed record. Several process histories can produce the same recorded data. To infer the hidden timing, one needs justified constraints, not simply a label asserting that the records share a universal present.
This is an operational lesson, not a derivation of a particular ontology of time. It helps formulate what a proposed temporal account should predict and how a comparison could distinguish it from alternatives.
Sources and further reading
- D. Mills et al., RFC 5905: Network Time Protocol Version 4 (2010), especially the four-timestamp offset and delay formulas. The compatible-offset interval and numerical examples are derived here.
- Atomic clock principles; precision and accuracy.
- Inner generation and outer completion; Ultra-Fine Timing.