--- title: Link Transmission Model (LTM) url: https://www.emergentmind.com/topics/link-transmission-model-ltm type: topic --- # Link Transmission Model (LTM) 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 [1405.7080]. 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 [1405.7080]. 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 [1405.7080]. 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 [1405.7080]. ## 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 $V > 0$ is free-flow speed, $W > 0$ is the absolute value of the backward wave speed, and $K$ is jam density. The critical density is $$ \bar{K} = \frac{W}{V+W}K, $$ and capacity is $$ C = V \bar{K}. $$ For a link of length $L$, the initial condition is $N(x) = A(x,0)$ on $x \in [0,L]$, and the boundary cumulative flows are $F(t)=A(0,t)$ at the upstream boundary and $G(t)=A(L,t)$ at the downstream boundary [1405.7080]. Specializing the Hopf–Lax formula to the triangular fundamental diagram yields $$ A(x,t) = \min_{(y,s) \in \partial\Omega(x,t)} B(y,s;x,t), $$ with $$ B(y,s;x,t) = A(y,s) + (t-s) C - (x-y)\bar{K}, $$ and contributing boundary points satisfying $$ u = \frac{x-y}{t-s} \in [-W, V], \qquad t>s. $$ This is the minimum principle underlying Newell’s solution [1405.7080]. 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: $$ A(x,t) = \min\{ N(x - V t), N(x + W t) + K W t \}; $$ - Region 2: $$ A(x,t) = \min\{ F(t - x/V), N(x + W t) + K W t \}; $$ - Region 3: $$ A(x,t) = \min\{ N(x - V t), G(t - (L-x)/W) + (L-x) K \}; $$ - Region 4: $$ A(x,t) = \min\{ F(t - x/V), G(t - (L-x)/W) + (L-x) K \}. $$ 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 $t-x/V$ and $t-(L-x)/W$ [1405.7080]. The same minimum structure also imposes feasibility conditions on Dirichlet boundary data. The data list, for example, $$ F(t) \le N(W t) + K W t, $$ $$ G(t) \le N(L - V t), $$ $$ F(t) \le G(t - L/W) + K L, $$ and $$ G(t) \le F(t - L/V). $$ 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 [1405.7080]. ## 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 $\Delta t$. Demand $d_a(t)$ is the maximal possible outflux at the downstream boundary of link $a$ if the downstream is empty; supply $s_a(t)$ is the maximal possible influx at the upstream boundary if the upstream is jammed. They are defined by $$ d_a(t) = \lim_{\Delta t \to 0^+} \frac{\hat{G}_a(t+\Delta t) - G_a(t)}{\Delta t}, $$ $$ s_a(t) = \lim_{\Delta t \to 0^+} \frac{\hat{F}_a(t+\Delta t) - F_a(t)}{\Delta t}, $$ where $\hat{G}_a$ and $\hat{F}_a$ are the downstream- and upstream-ideal cumulative flows with empty-downstream or jammed-upstream extensions [1405.7080]. Under a regular initial state, the closed-form formulas are $$ d_a(t) = \begin{cases} \min\{ k_a(L_a - V_a t,0) V_a + H(\lambda_a(t)), C_a \}, & t \le L_a/V_a,\\ \min\{ f_a(t - L_a/V_a) + H(\lambda_a(t)), C_a \}, & t > L_a/V_a, \end{cases} $$ and $$ s_a(t) = \begin{cases} \min\{ K_a W_a - k_a(W_a t,0) W_a + H(\gamma_a(t)), C_a \}, & t \le L_a/W_a,\\ \min\{ g_a(t - L_a/W_a) + H(\gamma_a(t)), C_a \}, & t > L_a/W_a. \end{cases} $$ Here the indicator function is $$ H(y)=\lim_{\Delta t\to 0^+}\frac{y}{\Delta t} = \begin{cases} 0, & y=0,\\ +\infty, & y>0. \end{cases} $$ The queue and vacancy functions are $$ \lambda_a(t) = \begin{cases} N_a(L_a - V_a t) - G_a(t), & t \le L_a/V_a,\\ F_a(t - L_a/V_a) - G_a(t), & t > L_a/V_a, \end{cases} $$ and $$ \gamma_a(t) = \begin{cases} N_a(W_a t) + K_a W_a t - F_a(t), & t \le L_a/W_a,\\ G_a(t - L_a/W_a) + K_a L_a - F_a(t), & t > L_a/W_a. \end{cases} $$ The interpretation given in the source is that $\lambda_a(t)$ is the stored queue and $\gamma_a(t)$ is the available storage. The indicator $H(\cdot)$ enforces that if a positive queue or vacancy exists, the link is demand- or supply-constrained at capacity [1405.7080]. For an initially empty link, the paper gives a particularly transparent specialization. Then $\lambda_a(t)=0$ and $d_a(t)=0$ for $t \le L_a/V_a$, while $\gamma_a(t)\ge (K_a W_a - C_a)t \ge 0$ and hence $s_a(t)\ge C_a$ for $t \le L_a/W_a$. The piecewise formulas become $$ d_a(t) = \begin{cases} 0, & t \le L_a/V_a,\\ f_a(t - L_a/V_a), & F_a(t - L_a/V_a)=G_a(t),\ t>L_a/V_a,\\ C_a, & F_a(t - L_a/V_a)>G_a(t),\ t>L_a/V_a, \end{cases} $$ and $$ s_a(t) = \begin{cases} C_a, & t \le L_a/W_a,\\ g_a(t - L_a/W_a), & F_a(t)=G_a(t - L_a/W_a)+K_aL_a,\ t>L_a/W_a,\\ C_a, & F_a(t)L_a/W_a. \end{cases} $$ These formulas show that LTM replaces internal link discretization by delayed boundary relations driven by characteristic travel times [1405.7080]. Two continuous formulations follow from these definitions. In the first, cumulative flows $F_a(t)$ and $G_a(t)$ are the state variables, with $$ \frac{dF_a(t)}{dt}=f_a(t), \qquad \frac{dG_a(t)}{dt}=g_a(t). $$ In the second, queue and vacancy are the state variables, satisfying the delay-ODEs $$ \frac{d\lambda_a(t)}{dt} = \begin{cases} k_a(L_a - V_a t,0) V_a - g_a(t), & t \le L_a/V_a,\\ f_a(t - L_a/V_a) - g_a(t), & t > L_a/V_a, \end{cases} $$ and $$ \frac{d\gamma_a(t)}{dt} = \begin{cases} -k_a(W_a t,0) W_a + K_a W_a - f_a(t), & t \le L_a/W_a,\\ g_a(t - L_a/W_a) - f_a(t), & t > L_a/W_a. \end{cases} $$ The paper also extends the model to commodities. If $F_{a,p}(t)$ and $G_{a,p}(t)$ are boundary cumulative flows for commodity $p$, then the upstream commodity share is $\eta_{a,p}(t)=f_{a,p}(t)/f_a(t)$, and FIFO implies $$ \xi_{a,p}(t)=\eta_{a,p}(t-\pi_a(t)), \qquad G_a(t)=F_a(t-\pi_a(t)). $$ Turning fractions at a node are $$ \xi_{a\to b}(t)=\sum_{p \in P_a \cap P_b}\xi_{a,p}(t). $$ This provides the disaggregation needed for multi-commodity network loading [1405.7080]. ## 4. Junction models, invariance, and well-posedness At each node $j$, upstream links $I_j$ and downstream links $O_j$ exchange flow subject to boundary demand, boundary supply, conservation, and FIFO-diverge consistency. The invariant macroscopic junction model requires $$ 0 \le g_a(t) \le d_a(t), \qquad a \in I_j, $$ $$ 0 \le f_b(t) \le s_b(t), \qquad b \in O_j, $$ $$ \sum_{a\in I_j} g_a(t) = \sum_{b\in O_j} f_b(t), $$ and $$ f_b(t)=\sum_{a\in I_j} g_a(t)\xi_{a\to b}(t). $$ To complete the model, the critical demand level is defined as $$ \theta_j(t) = \min\left\{ 1,\ \min_{b\in O_j}\max_{I_1(t)} \frac{s_b(t)-\sum_{\alpha\in I_j\setminus I_1(t)} d_\alpha(t)\xi_{\alpha\to b}(t)} {\sum_{a\in I_1(t)} C_a \xi_{a\to b}(t)} \right\}, $$ where $I_1(t)$ is any non-empty subset of $I_j$. Then $$ g_a(t)=\min\{d_a(t), \theta_j(t)C_a\}, \qquad a\in I_j, $$ and $$ f_b(t)=\sum_{a\in I_j} g_a(t)\xi_{a\to b}(t), \qquad b\in O_j. $$ Commodity fluxes satisfy $$ f_{b,p}(t)=g_{a,p}(t)=g_a(t)\xi_{a,p}(t). $$ The paper explicitly states that this model is invariant and ensures conservation, capacity, and FIFO [1405.7080]. A central analytical conclusion is that LTM is not well-defined with non-invariant junction models. The merge example given in the source uses $$ L_1=L_2=L_3=1,\quad C_1=C_2=C_3=1, $$ with demands $d_1^-=1$, $d_2^-=1/4$, and downstream supply $s_3^+=1$. Under the invariant fair-merge model, a consistent stationary solution exists with link 1 in SOC and link 2 in SUC: $$ q_1=3/4,\ \beta_1=1,\ d_1=1,\ s_1=3/4; \qquad q_2=1/4,\ \beta_2=0,\ d_2=1/4,\ s_2=1. $$ By contrast, the non-invariant fair-merge rule $$ q_1=\frac{d_1}{d_1+d_2}, \qquad q_2=\frac{d_2}{d_1+d_2} $$ is incompatible with LTM’s stationary demand/supply constraints and has no solution in this case [1405.7080]. 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 [1405.7080]. ## 5. Stationary states and stability analysis A link is stationary when $\partial k(x,t)/\partial t = 0$ for $x \in [0,L]$. Equivalently, the flux is constant in space and time, $q(x,t)=q$, and the cumulative flow has the form $$ A(x,t)=N(x)+qt, $$ with $N_x(x)\in\{-k_1,-k_2\}$, where $k_1 \le \bar{K} \le k_2$ satisfy $$ q = k_1V = (K-k_2)W. $$ The continuous LTM paper distinguishes four stationary regimes [1405.7080]. | Regime | Condition | Queue/vacancy pattern | |---|---|---| | Critical (C) | $q=C$ | $\lambda=\gamma=0$ | | Strictly under-critical (SUC) | $q0$ | | Strictly over-critical (SOC) | $q0,\ \gamma=0$ | | Zero-speed shock (ZS) | $q0,\ \gamma>0$ | In all stationary cases, $f(t)=g(t)=q$, and the stationary queue and vacancy are $$ \lambda = \beta (1-q/C)KL, \qquad \gamma = (1-\beta)(1-q/C)KL. $$ Thus $\lambda=\gamma=0$ at critical state, $\lambda=0$ and $\gamma>0$ at SUC, $\lambda>0$ and $\gamma=0$ at SOC, and both are positive at ZS. The stationary demand and supply are $$ d = \min\{q + H(\beta (1-q/C)KL), C\}, $$ $$ s = \min\{q + H((1-\beta)(1-q/C)KL), C\}. $$ These formulas characterize how a stationary link appears externally: the presence of stationary queues or vacancies triggers demand or supply at capacity [1405.7080]. The diverge–merge network in the source provides a closed-form network example. With capacities $(C_0,C_1,C_2,C_3)$, upstream split $\xi$ to link 1, origin demand $d_i=C_0$, and destination supply $s_o=C_3$, the SOC–SUC case on links 1–2 has necessary and sufficient conditions $$ C_3 < C_1 + C_2,\qquad C_3 < C_0,\qquad 1 - \frac{C_2}{C_3} < \xi < \frac{C_1}{C_3}, $$ with stationary flows $$ q_1=\xi C_3,\qquad q_2=(1-\xi)C_3. $$ This provides an explicit network-level stationary solution in terms of link capacities and diverge proportion [1405.7080]. The same network is used for a direct LTM derivation of a Poincaré map. Under the SOC–SUC stationary regime with $\beta_1=1$ and $\beta_2=0$, and perturbations that preserve SOC on link 1 and SUC on link 2, LTM implies $$ f_1(t)=g_1\!\left(t-\frac{L_1}{W_1}\right),\qquad g_2(t)=f_2\!\left(t-\frac{L_2}{V_2}\right). $$ At the diverge, $$ f_2(t)=\frac{1}{\mu}f_1(t),\qquad \mu=\frac{\xi}{1-\xi}, $$ and at the merge, $$ g_1(t)=C_3-g_2(t). $$ Combining these yields the Poincaré map $$ f_1(t)=C_3-\frac{1}{\mu}f_1(t-T), \qquad T=\frac{L_1}{W_1}+\frac{L_2}{V_2}. $$ Its fixed point is $f_1^*=\xi C_3$, and defining $\tilde{f}_1(t)=f_1(t)-\xi C_3$ gives $$ \tilde{f}_1(t)=-\frac{1}{\mu}\tilde{f}_1(t-T). $$ Therefore, the stationary state is stable if $|1/\mu|<1$, equivalently $\mu>1$ or $\xi>1/2$, and unstable if $\xi \le 1/2$ [1405.7080]. ## 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 $N_t^\omega(x)$ and time-varying link-policy incidence $\delta_{ab,t}^\omega$. Sending and receiving are written as $$ S_{a,t} = \min\!\left(N_{t'} (x_a^0) - N_{t-\Delta t} (x_a^{L_a}),\ Q_a\Delta t\right), $$ with $t'=t+\Delta t-\frac{L_a}{v_{f,a}}$, and $$ R_{b,t} = \min\!\left(N_{t'}(x_b^{L_b}) + k_bL_b - N_{t-\Delta t}(x_b^0),\ Q_b\Delta t\right), $$ with $t'=t+\Delta t-\frac{L_b}{w_{f,b}}$. 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 [1811.02474]. 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 $a$, the density ratio is $$ \rho_a=\frac{k_a}{k_a+k_{a'}}, $$ the effective jam density is $\hat{K}_{j,a}=\rho_a K_{j,a}$, and the effective free-flow speed can be modeled either by $$ \hat{v}_{f,a}=\frac{v_{f,a}}{\exp(1-\rho_a)} $$ or $$ \hat{v}_{f,a}=\rho_a^\gamma v_{f,a}. $$ The directional flow is then $$ q_a(k_a,k_{a'})= \begin{cases} \hat{v}_{f,a}k_a, & 0\le k_a\le K_{c,a},\\ w(\hat{K}_{j,a}-k_a), & K_{c,a}