A window below the bandwidth-delay product limits throughput smoothly

The supplied computation for Buffers Follow Bandwidth-Delay, Theory 041 in Volume X of the Incoprea research notebook, reproduces a familiar bandwidth-delay constraint. A sender allowed too few outstanding packets cannot keep a link continuously occupied. However, its throughput declines approximately linearly below the constraint, rather than collapsing abruptly.

Evidence classification: a deterministic scheduling model and its long-transfer limit. This is a corrected interpretation of supplied work. Its arithmetic is useful, but the original supported verdict overstates the result: smooth decline is specifically the alternative named by the posted hypothesis. No physical network was tested.

The model uses an in-flight window

The link can send 1,000 packets per second. An acknowledgement returns 0.1 seconds after a packet's send time. Their product is 100 packets. The computation transfers 20,000 packets at each of fifteen tested window sizes between ten and two hundred packets.

A packet can be sent only when two conditions hold: the previous packet's send time is at least 0.001 seconds earlier, and a slot in the outstanding-packet window is free. The first condition represents link capacity; the second represents waiting for acknowledgement. The reported throughput divides all 20,000 packets by the last packet's acknowledgement time.

The supplied script calls the varied quantity a buffer, but directly equates it with the maximum number of packets in flight. It does not implement a separate router queue. The distinction matters because receive windows, congestion windows and queues play different roles in actual networks. RFC 6349's throughput-testing framework discusses the bandwidth-delay product and window limits while also distinguishing several real TCP constraints.

What the existing calculation returns

The unchanged supplied program reproduced its original numerical results. The reviewed public script retains the same send-time recurrence, packet count, capacity, round-trip time and fifteen windows. It exports numerical values as plain JSON, avoiding the NumPy wrapper strings present in parts of the original CSV.

The first tested window reaching at least 99% measured utilisation is 100 packets. That is a grid observation using a chosen 99% criterion, not an estimated universal critical point. At that window the finite transfer still measures about 99.51%, rather than exactly 100%.

A limit with a kink, rather than an abrupt collapse

Write C for the link's packet rate, τ for round-trip time, and B for the in-flight window. In a long transfer, B outstanding packets permit at most B/τ packets per second. Link capacity supplies the other limit. For this lossless fixed-delay model, the limiting throughput is

throughput = min(C, B/τ),

and utilisation is min[1, B/(Cτ)]. The curve is continuous. Below Cτ it rises linearly with B; above Cτ it plateaus. Its slope changes at the bandwidth-delay product, producing a kink rather than a discontinuous jump.

Finite-transfer throughput includes startup and the final acknowledgement wait. Once the link constraint dominates, the completion time approaches the supplied schedule's (K − 1)/C + τ for K packets. This accounts for the small gap below full utilisation. It should not be presented as a discrepancy in the long-transfer formula or rounded into exact equality.

The original hypothesis predicts abrupt restoration when buffer size crosses a fraction of the product, and names smooth decline as its refutation. Within the supplied model's window interpretation, the smooth alternative is what occurs. The calculation therefore illustrates window-limited utilisation while failing to establish the stronger abrupt-collapse prediction.

This does not prove that real networks cannot exhibit abrupt performance changes. The model includes no packet loss, retransmission, congestion control, multiple competing flows, changing round-trip time or queueing delay. It also holds both capacity and delay fixed, so it does not test scaling across different physical paths. Router buffer sizing requires a model of the queue and traffic, which this computation does not provide.

Credit and reproducibility

DeepSeek's coding agent supplied the scheduling model, original report and results in the Computational Mechanisms bundle dated 30 October 2026. Codex froze the source, reproduced its computation, checked the limiting expression and corrected the interpretation. No new network study was designed. Original files remain separate from the standalone public script and reviewed display results. This internal publisher audit is not external peer review, a hardware benchmark, or a claim of novelty.