Link Transmission Model (LTM)

• Link Transmission Model (LTM) is a discrete formulation of the network kinematic wave model that computes boundary cumulative flows from demand and supply using Newell’s variational solution.
• LTM enhances efficiency over cell-based methods by advancing only link-level flows with delay-based updates and invariant junction models to ensure conservation and well-posedness.
• LTM’s analytical framework supports studying stationary states, stability, and extends to applications like stochastic dynamic traffic assignment and pedestrian flow modeling.

The Link Transmission Model (LTM) is a discrete version of the network kinematic wave model that computes boundary fluxes using link demand and supply defined from cumulative flows based on Newell’s variational solution. In the formulation developed in “Continuous formulations and analytical properties of the link transmission model,” LTM is positioned as an alternative formulation of the network kinematic wave model that complements Newell’s simplified kinematic wave model: LTM determines boundary cumulative flows via junction models, while Newell’s model recovers interior fields $A(x,t)$ from those boundary cumulative flows (Jin, 2014). Within this framework, LTM is intended for efficient and accurate simulation of large-scale traffic networks, with analytical treatment of stationary states, stability, and junction well-posedness.

1. Historical and theoretical placement

LTM arises within the Lighthill–Whitham–Richards kinematic wave framework, in which traffic evolution is governed by the conservation law $k_t + q_x = 0$ with $q = Q(k)$. In this setting, the Cell Transmission Model (CTM) is a discrete Godunov method that divides links into cells and time into steps. The data state that, because each time-step requires updating all cells and solving junction Riemann problems, “the computational cost is inversely proportional to the square of the time-step size.” By contrast, LTM advances only in time and works with link-level boundary cumulative flows and demand/supply relations, so its computational cost is only inversely proportional to the time-step size. For the same time-step, LTM is therefore described as both more efficient and more accurate than CTM (Jin, 2014).

The continuous formulation of LTM is built from the Hamilton–Jacobi representation of the LWR model. Let $A(x,t)$ denote the cumulative count, or Moskowitz function, of vehicles that have passed position $x$ by time $t$. Then $k = -A_x$, $q = A_t$, and the Hamilton–Jacobi equation is

$A_t - Q(-A_x) = 0.$

The key theoretical step in the 2014 formulation is to derive Newell’s simplified kinematic wave model from the Hopf–Lax formula for this Hamilton–Jacobi equation under a triangular fundamental diagram and given boundary cumulative flows (Jin, 2014).

This theoretical placement matters because Newell’s model and LTM solve different parts of the network problem. Newell’s model solves the Hamilton–Jacobi equation with Dirichlet boundary data given as cumulative flows. LTM, in turn, uses demand/supply and invariant junction models to solve the BLN-type network boundary problem and to generate those boundary cumulative flows consistently over the network. This suggests that the principal novelty of the continuous formulation is not merely computational economy, but a reorganization of the network kinematic wave problem into boundary resolution plus interior reconstruction (Jin, 2014).

2. Fundamental diagram, cumulative flows, and Hopf–Lax structure

The baseline continuous LTM uses the triangular fundamental diagram

$Q(k) = \min\{V k, (K - k) W\},$

where $k_t + q_x = 0$0 is free-flow speed, $k_t + q_x = 0$1 is the absolute value of the backward wave speed, and $k_t + q_x = 0$2 is jam density. The critical density is

$k_t + q_x = 0$3

and capacity is

$k_t + q_x = 0$4

For a link of length $k_t + q_x = 0$5, the initial condition is $k_t + q_x = 0$6 on $k_t + q_x = 0$7, and the boundary cumulative flows are $k_t + q_x = 0$8 at the upstream boundary and $k_t + q_x = 0$9 at the downstream boundary (Jin, 2014).

Specializing the Hopf–Lax formula to the triangular fundamental diagram yields

$q = Q(k)$0

with

$q = Q(k)$1

and contributing boundary points satisfying

$q = Q(k)$2

This is the minimum principle underlying Newell’s solution (Jin, 2014).

Under the no-transonic-rarefaction assumption in the initial condition, the Hopf–Lax minimum principle yields region-wise explicit formulas over the U-shaped domain. The “simplified Newell” formulas given in the data are:

• Region 1:

$q = Q(k)$3

• Region 2:

$q = Q(k)$4

• Region 3:

$q = Q(k)$5

• Region 4:

$q = Q(k)$6

The region-4 expression is often referred to as Newell’s simplified model. These formulas reproduce shock and rarefaction propagation under the triangular fundamental diagram and encode link travel times and queue dynamics through the arguments $q = Q(k)$7 and $q = Q(k)$8 (Jin, 2014).

The same minimum structure also imposes feasibility conditions on Dirichlet boundary data. The data list, for example,

$q = Q(k)$9

$A(x,t)$0

$A(x,t)$1

and

$A(x,t)$2

Violations render the problem ill-posed. In the continuous LTM interpretation, this is precisely why boundary cumulative flows cannot be prescribed arbitrarily and must instead be generated through admissible demand, supply, and junction relations (Jin, 2014).

3. Link demand, supply, queue, and vacancy

In continuous LTM, link demand and supply are defined from “ideal” boundary cumulative flows obtained via Hopf–Lax in pan-shaped domains over a small horizon $A(x,t)$3. Demand $A(x,t)$4 is the maximal possible outflux at the downstream boundary of link $A(x,t)$5 if the downstream is empty; supply $A(x,t)$6 is the maximal possible influx at the upstream boundary if the upstream is jammed. They are defined by

$A(x,t)$7

$A(x,t)$8

where $A(x,t)$9 and $x$0 are the downstream- and upstream-ideal cumulative flows with empty-downstream or jammed-upstream extensions (Jin, 2014).

Under a regular initial state, the closed-form formulas are

$x$1

and

$x$2

Here the indicator function is

$x$3

The queue and vacancy functions are

$x$4

and

$x$5

The interpretation given in the source is that $x$6 is the stored queue and $x$7 is the available storage. The indicator $x$8 enforces that if a positive queue or vacancy exists, the link is demand- or supply-constrained at capacity (Jin, 2014).

For an initially empty link, the paper gives a particularly transparent specialization. Then $x$9 and $t$0 for $t$1, while $t$2 and hence $t$3 for $t$4. The piecewise formulas become

$t$5

and

$t$6

These formulas show that LTM replaces internal link discretization by delayed boundary relations driven by characteristic travel times (Jin, 2014).

Two continuous formulations follow from these definitions. In the first, cumulative flows $t$7 and $t$8 are the state variables, with

$t$9

In the second, queue and vacancy are the state variables, satisfying the delay-ODEs

$k = -A_x$0

and

$k = -A_x$1

The paper also extends the model to commodities. If $k = -A_x$2 and $k = -A_x$3 are boundary cumulative flows for commodity $k = -A_x$4, then the upstream commodity share is $k = -A_x$5, and FIFO implies

$k = -A_x$6

Turning fractions at a node are

$k = -A_x$7

This provides the disaggregation needed for multi-commodity network loading (Jin, 2014).

4. Junction models, invariance, and well-posedness

At each node $k = -A_x$8, upstream links $k = -A_x$9 and downstream links $q = A_t$0 exchange flow subject to boundary demand, boundary supply, conservation, and FIFO-diverge consistency. The invariant macroscopic junction model requires

$q = A_t$1

$q = A_t$2

$q = A_t$3

and

$q = A_t$4

To complete the model, the critical demand level is defined as

$q = A_t$5

where $q = A_t$6 is any non-empty subset of $q = A_t$7. Then

$q = A_t$8

and

$q = A_t$9

Commodity fluxes satisfy

$A_t - Q(-A_x) = 0.$0

The paper explicitly states that this model is invariant and ensures conservation, capacity, and FIFO (Jin, 2014).

A central analytical conclusion is that LTM is not well-defined with non-invariant junction models. The merge example given in the source uses

$A_t - Q(-A_x) = 0.$1

with demands $A_t - Q(-A_x) = 0.$2, $A_t - Q(-A_x) = 0.$3, and downstream supply $A_t - Q(-A_x) = 0.$4. Under the invariant fair-merge model, a consistent stationary solution exists with link 1 in SOC and link 2 in SUC:

$A_t - Q(-A_x) = 0.$5

By contrast, the non-invariant fair-merge rule

$A_t - Q(-A_x) = 0.$6

is incompatible with LTM’s stationary demand/supply constraints and has no solution in this case (Jin, 2014).

This point has broader methodological significance. The source states that the invariant node model guarantees well-posedness of the traffic statics problem and robustness in simulation, while non-invariant models can fail. A plausible implication is that continuous LTM requires invariance not merely as a modeling preference, but as a structural condition linking boundary flows, storage constraints, and stationary feasibility (Jin, 2014).

5. Stationary states and stability analysis

A link is stationary when $A_t - Q(-A_x) = 0.$7 for $A_t - Q(-A_x) = 0.$8. Equivalently, the flux is constant in space and time, $A_t - Q(-A_x) = 0.$9, and the cumulative flow has the form

$Q(k) = \min\{V k, (K - k) W\},$0

with $Q(k) = \min\{V k, (K - k) W\},$1, where $Q(k) = \min\{V k, (K - k) W\},$2 satisfy

$Q(k) = \min\{V k, (K - k) W\},$3

The continuous LTM paper distinguishes four stationary regimes (Jin, 2014).

Regime	Condition	Queue/vacancy pattern
Critical (C)	$Q(k) = \min\{V k, (K - k) W\},$4	$Q(k) = \min\{V k, (K - k) W\},$5
Strictly under-critical (SUC)	$Q(k) = \min\{V k, (K - k) W\},$6 and $Q(k) = \min\{V k, (K - k) W\},$7 on $Q(k) = \min\{V k, (K - k) W\},$8	$Q(k) = \min\{V k, (K - k) W\},$9
Strictly over-critical (SOC)	$k_t + q_x = 0$00 and $k_t + q_x = 0$01 on $k_t + q_x = 0$02	$k_t + q_x = 0$03
Zero-speed shock (ZS)	$k_t + q_x = 0$04 with interface at $k_t + q_x = 0$05	$k_t + q_x = 0$06

In all stationary cases, $k_t + q_x = 0$07, and the stationary queue and vacancy are

$k_t + q_x = 0$08

Thus $k_t + q_x = 0$09 at critical state, $k_t + q_x = 0$10 and $k_t + q_x = 0$11 at SUC, $k_t + q_x = 0$12 and $k_t + q_x = 0$13 at SOC, and both are positive at ZS. The stationary demand and supply are

$k_t + q_x = 0$14

$k_t + q_x = 0$15

These formulas characterize how a stationary link appears externally: the presence of stationary queues or vacancies triggers demand or supply at capacity (Jin, 2014).

The diverge–merge network in the source provides a closed-form network example. With capacities $k_t + q_x = 0$16, upstream split $k_t + q_x = 0$17 to link 1, origin demand $k_t + q_x = 0$18, and destination supply $k_t + q_x = 0$19, the SOC–SUC case on links 1–2 has necessary and sufficient conditions

$k_t + q_x = 0$20

with stationary flows

$k_t + q_x = 0$21

This provides an explicit network-level stationary solution in terms of link capacities and diverge proportion (Jin, 2014).

The same network is used for a direct LTM derivation of a Poincaré map. Under the SOC–SUC stationary regime with $k_t + q_x = 0$22 and $k_t + q_x = 0$23, and perturbations that preserve SOC on link 1 and SUC on link 2, LTM implies

$k_t + q_x = 0$24

At the diverge,

$k_t + q_x = 0$25

and at the merge,

$k_t + q_x = 0$26

Combining these yields the Poincaré map

$k_t + q_x = 0$27

Its fixed point is $k_t + q_x = 0$28, and defining $k_t + q_x = 0$29 gives

$k_t + q_x = 0$30

Therefore, the stationary state is stable if $k_t + q_x = 0$31, equivalently $k_t + q_x = 0$32 or $k_t + q_x = 0$33, and unstable if $k_t + q_x = 0$34 (Jin, 2014).

6. Extensions, variants, and later applications

Subsequent work has adapted the LTM structure to settings that preserve the cumulative-count logic while changing routing, node, or link physics.

In stochastic time-dependent traffic assignment, a policy-based LTM replaces fixed paths by policies that adapt next-link decisions based on current information. The 2018 formulation retains cumulative boundary counts, sending and receiving flows, and standard node types, but introduces policy-disaggregate cumulative counts $k_t + q_x = 0$35 and time-varying link-policy incidence $k_t + q_x = 0$36. Sending and receiving are written as

$k_t + q_x = 0$37

with $k_t + q_x = 0$38, and

$k_t + q_x = 0$39

with $k_t + q_x = 0$40. The paper formulates a policy-based stochastic dynamic equilibrium as a fixed-point problem and solves it with the method of successive averages. It reports that the policy-based chronological loader is substantially faster than iterative path-based loading, with TwoLinks at 62.0 s versus 6.8 s and Diamond at 1949.5 s versus 52.4 s (Gehlot et al., 2018).

In bidirectional sidewalk networks, the LTM has been coupled to a three-dimensional triangular bidirectional fundamental diagram and a generalized first-order node model. For direction $k_t + q_x = 0$41, the density ratio is

$k_t + q_x = 0$42

the effective jam density is $k_t + q_x = 0$43, and the effective free-flow speed can be modeled either by

$k_t + q_x = 0$44

or

$k_t + q_x = 0$45

The directional flow is then

$k_t + q_x = 0$46

The associated bidirectional LTM uses

$k_t + q_x = 0$47

$k_t + q_x = 0$48

and a node model with a bidirectional “look-ahead” supply term

$k_t + q_x = 0$49

This framework is coupled to a dynamic user equilibrium with pedestrian volume–delay functions and is reported to reproduce formation and propagation of shockwaves in walking corridors and networks due to bidirectional effects (Lilasathapornkit et al., 2024).

A more implementation-oriented pedestrian variant, PedNStream, extends LTM-based pedestrian models with stochastic link dynamics. In this formulation, link sending uses a time-varying travel time boundary,

$k_t + q_x = 0$50

a width-scaled link capacity,

$k_t + q_x = 0$51

and final sending

$k_t + q_x = 0$52

Receiving uses area-based storage,

$k_t + q_x = 0$53

The model also introduces congestion-aware blending

$k_t + q_x = 0$54

and

$k_t + q_x = 0$55

as well as diffusion and stochastic release. The paper reports runtime complexity $k_t + q_x = 0$56, space complexity $k_t + q_x = 0$57, and a runtime of approximately 45 s for the Delft network with 298 nodes, 818 links, 500 steps, and 46,501 pedestrians (Mai et al., 1 Jul 2026).

Urban road-network variants have also relaxed the homogeneous-link assumption of classical LTM. A 2023 link-based flow model introduces turn-level queue inflow as an additional state variable and makes outflow explicitly dependent on time-varying free-flow speed. Each road segment is decomposed into turn links $k_t + q_x = 0$58, with cumulative counts $k_t + q_x = 0$59, $k_t + q_x = 0$60, and $k_t + q_x = 0$61. Critical density and critical flow become

$k_t + q_x = 0$62

$k_t + q_x = 0$63

and movement-level sending is represented by

$k_t + q_x = 0$64

The paper reports that the proposed extended LTM reduces average difference in cumulative flows from 0.12% for a baseline LTM to 0.04% against calibrated SUMO in a field experiment, while preserving computational tractability (Wei et al., 2023).

Across these extensions, the recurring structure is unchanged: cumulative counts at link boundaries, delayed sending and receiving relations, and node models that enforce conservation and admissibility. What changes are the effective fundamental diagram, the link state variables, or the routing/control layer. This suggests that the central abstraction of LTM is the transmission of admissible cumulative flows across links and nodes, rather than any single vehicular interpretation (Jin, 2014).

7. Interpretation, assumptions, and scope

The continuous LTM formulation in the 2014 analysis assumes a triangular fundamental diagram and no initial transonic rarefaction wave on links, although the Hopf–Lax/Newell derivations can treat general initial data with feasibility conditions for Dirichlet data (Jin, 2014). Its implementation logic is delay-based rather than cell-based: delayed terms at $k_t + q_x = 0$65 and $k_t + q_x = 0$66 must be resolved in time-stepping, and the step size should resolve the shortest relevant delay and junction dynamics. The source explicitly notes that a CFL-like restriction is not invoked in the same way as in CTM (Jin, 2014).

A common misconception is to treat LTM as merely a faster CTM. The material summarized here does not support that reduction. LTM is grounded in Newell’s variational theory and the Hopf–Lax formula, defines link demand and supply directly from cumulative flows, admits continuous formulations in terms of cumulative flows or queue/vacancy states, and supports direct analytical work on stationary states and stability (Jin, 2014). Another misconception is that any reasonable node rule can be inserted into the model. The merge counterexample shows otherwise: non-invariant junction models can make the model ill-defined (Jin, 2014).

Within the scope documented here, LTM serves at least three roles. First, it is a computationally efficient network loading model for first-order traffic dynamics. Second, it is a continuous analytical framework for studying statics, admissibility, and stability in network kinematic wave systems. Third, it is a transferable link-based architecture that has been adapted to stochastic dynamic traffic assignment, bidirectional pedestrian loading, controller-integrated pedestrian management, and turn-level urban intersection modeling (Gehlot et al., 2018, Lilasathapornkit et al., 2024, Mai et al., 1 Jul 2026, Wei et al., 2023).

The 2014 paper concludes that Newell’s model and LTM complement each other and provide an alternative formulation of the network kinematic wave model. In the terms of the material presented here, that complementarity is the defining feature of LTM: boundary cumulative flows are generated by demand, supply, and invariant junction models; interior states are then recovered variationally from those cumulative flows (Jin, 2014).