ADR-016: ltm — Yperman (2007) link transmission model

Status: accepted (implemented) Date: 2026-07-09 Deciders: DNL-models track — the second DNL link model (the Newell-Daganzo method) File: docs/design/adr-016-ltm.md

Context

The Link Transmission Model (Yperman 2007) is the second DNL link sprint after ctm (adr-015). It is the Newell-Daganzo cumulative-curve method: instead of discretising the link interior into cells, it evaluates Newell’s two shifted cumulative curves directly at the link’s two ends, trading interior state for a need to remember the boundary cumulative curves back L/vf and L/w in time — which the LinkModel base already retains in full.

Decision

1. LTMLink(LinkModel) in src/tabench/dnl/ltm.py, additive on the frozen sending/receiving interface. It is stateless beyond the base cumulative curves n_in (upstream) / n_out (downstream): no cells, _advance_state is the inherited no-op, and it carries no turning logic (node models handle junctions). It requires a finite jam density (the kappa·L receiving term) — the mirror image of the point-queue reference’s kappa = inf.

2. Newell-Daganzo sending/receiving (Yperman eq. 4.31/4.35 ≡ Boyles eq. 9.65/9.67):

   • sending(k)   = min(N_up(t_{k+1} - L/vf) - N_dn(t_k), q_max·dt) — byte- identical to the point queue’s;

   • receiving(k) = min(N_dn(t_{k+1} - L/w) + kappa·L - N_up(t_k), q_max·dt) — the kappa·L storage term is exactly what turns the point queue’s unconstrained receiving into a finite backward wave. The shifted terms use the base class’s exact linear interp_curve; n_out[k] / n_in[k] are read at grid edges directly. assert_wave_resolved (dt <= min(L/vf, L/w), already enforced at scenario construction) is both the stability and the causality guarantee — the look-ahead never reads a future value.

3. No CFL=1 cell alignment (the LTM advantage). LTM has no cells, so — unlike CTM — a link length need not be an integer multiple of vf·dt; LTM runs on any wave-resolved grid, including coarser / non-cell-aligned ones CTMLink rejects at construction. This grid flexibility is LTM’s concrete, testable edge over CTM (anchor d).

On numerical diffusion (honest scope)

Boyles §9.5.4 states the Newell-Daganzo values are exact, with no backward-shock spreading, “which does happen in the cell transmission model.” That advantage is real in principle (LTM never discretises the interior), but on the small single-shock anchors here LTM and CTM agree to machine precision — CTM’s O((w/vf)^n_cells) spreading stays below the certificate tolerance at that scale, so the harness’s backward-wave residual (C5) is ~0 for both. The sprint therefore does not assert an “LTM exact / CTM diffuse” gap it cannot demonstrate; the demonstrable distinction is grid flexibility (anchor d).

Analytic anchors (hand-derived from the read primaries, machine-verified — test_dnl_ltm.py)

• (a) Free-flow translation: L=4, vf=w=1, kappa=4, cap=2, inflow 1.0. n_out(t)=n_in(t-4) exactly, TSTT=16, zero delay — bit-identical to CTM (a).

• (b) Symmetric bottleneck (CTM cross-check): the L=4 link feeds a 0.5 bottleneck at inflow 1.5. LTM reproduces CTM’s curves byte-for-byte — n_in=1.5t, n_out=max(0,0.5(t-4)), storage 3.5·4=14, RH shock speed -0.5.

• (c) Asymmetric wave (w<vf): vf=2, w=1, kappa=3, cap=2, L=4, inflow 1.0, 0.5 bottleneck. RH speed s=(1-0.5)/(0.5-2.5)=-0.25, shock reaches x=0 at t=18; n_out=max(0,0.5(t-2)), storage k_B·L=2.5·4=10 (verified via the Yperman receiving recursion).

• (d) Grid flexibility: L=3, vf=2, dt=1 gives L/vf=1.5 (non-integer) — CTMLink raises, LTM free-flow-translates by the 1.5 lag exactly.

Alternatives considered

• Reusing PointQueueLink with a finite-kappa FD: rejected — the point queue’s receiving = q_max·dt (unbounded) has no backward wave; LTM’s whole content is the finite kappa·L receiving. But the shared sending confirms the “LTM = point queue + finite receiving” framing.

• Cell-based like CTM: rejected — the interior-free formulation is LTM’s point (exactness + grid flexibility), and it reuses the base cumulative curves with no new state.

Consequences

The benchmark gains its second DNL link model and a diffusion-free, grid-flexible alternative to CTM, sharing the same interface, certifier, node models, and loader. All changes are additive (a new module + tests + exports), so the 562-test suite, every road/DNL hash, and the golden Braess content hash are byte-untouched.

Sourcing

Both primaries are open and were read: Yperman (2007) PhD thesis (KU Leuven, mech.kuleuven.be mirror) §4.6 eq. 4.31/4.35, and Boyles/Lownes/Unnikrishnan Transportation Network Analysis Vol. I (2025) §9.5.2 eq. 9.65/9.67 (worked example Table 9.6). Sign-convention note: Yperman typesets the backward term +L/w because his w is the signed (negative) backward-wave velocity; this repo (and Boyles) use wave_speed > 0 as a magnitude, so the equivalent form is - L/w — which this module uses, verified against Boyles’ R(10)=5 example.