Two-lane totally asymmetric simple exclusion process with extended Langmuir kinetics

Multilane totally asymmetric simple exclusion processes with interactions between the lanes have recently been investigated actively. This paper proposes a two-lane model with extended Langmuir kinetics on a periodic lattice. Both bidirectional and unidirectional flows are investigated. In our model, the hopping, attachment, and detachment rates vary depending on the state of the corresponding site in the other lane. We obtain a theoretical expression for the global density of the system in the steady state from three kinds of mean-field analyses [($1\times 1$)-, ($2\times 1$)-, and ($2\times 2$)-cluster cases]. We verify that the ($2\times 2$)-cluster mean-field analysis reproduces the differences between the two directional flows and approximates well the results of computer simulations for some cases. We observe that ($2\times 1$)-cluster mean-field analyses are already good approximations of the simulation results for unidirectional flows; on the other hand, the accuracy of the approximations much improves by ($2\times 2$)-cluster one for bidirectional flows. We explain the phenomena in a qualitative manner by a simple analysis of correlations. We expect these findings to give informative suggestions for actual traffic systems.

Figure 1

Schematic illustration of the present model. The upper and lower panels represent unidirectional (U) and bidirectional (B) flows, respectively. Red (green) circles represent particles in lane 1 (2). The left and right boundaries are connected (periodic boundary conditions). The case $L=10$ is shown.

Figure 2

Simulation (circles) and mean-field values (curves) of $\rho$ for (a) unidirectional and (b) bidirectional flows as functions of $p$ with $({\omega }_{\mathrm{A}},{\omega }_{\mathrm{D}})\in \{(0.004,0.008)\left(\mathrm{red}\right),(0.005,0.005)\left(\mathrm{green}\right),(0.008,0.004)\left(\mathrm{blue}\right)\}$.

Figure 3

Values of $\rho$ from simulations (symbols) and mean-field calculations (blue solid, green dashed, red dash-dotted, and orange dotted curves) as functions of $p$ for (a) $({\omega }_{\mathrm{A}1},{\omega }_{\mathrm{A}2},{\omega }_{\mathrm{D}1},{\omega }_{\mathrm{D}2})=(0.008,0.004,0.004,0.004)$, (b) $({\omega }_{\mathrm{A}1},{\omega }_{\mathrm{A}2},{\omega }_{\mathrm{D}1},{\omega }_{\mathrm{D}2})=(0.002,0.004,0.004,0.004)$, (c) $({\omega }_{\mathrm{A}1},{\omega }_{\mathrm{A}2},{\omega }_{\mathrm{D}1},{\omega }_{\mathrm{D}2})=(0.004,0.008,0.004,0.004)$, (d) $({\omega }_{\mathrm{A}1},{\omega }_{\mathrm{A}2},{\omega }_{\mathrm{D}1},{\omega }_{\mathrm{D}2})=(0.004,0.002,0.004,0.004)$, (e) $({\omega }_{\mathrm{A}1},{\omega }_{\mathrm{A}2},{\omega }_{\mathrm{D}1},{\omega }_{\mathrm{D}2})=(0.004,0.004,0.008,0.004)$, (f) $({\omega }_{\mathrm{A}1},{\omega }_{\mathrm{A}2},{\omega }_{\mathrm{D}1},{\omega }_{\mathrm{D}2})=(0.004,0.004,0.002,0.004)$, (g) $({\omega }_{\mathrm{A}1},{\omega }_{\mathrm{A}2},{\omega }_{\mathrm{D}1},{\omega }_{\mathrm{D}2})=(0.004,0.004,0.004,0.008)$, and (h) $({\omega }_{\mathrm{A}1},{\omega }_{\mathrm{A}2},{\omega }_{\mathrm{D}1},{\omega }_{\mathrm{D}2})=(0.004,0.004,0.004,0.002)$. Black circles represent the simulation results for unidirectional flows (U), whereas black crosses represent those for bidirectional flows (B). The blue, green, red, and orange curves represent the ($1\times 1$)-cluster, ($2\times 1$)-cluster, ($2\times 2$)-cluster (U), and ($2\times 2$)-cluster (B) mean-field values, respectively. Error bars are almost within the size of the symbols.

Figure 4

Calculated values of 16 kinds of $\mu$ of unidirectional (blue) and bidirectional (red) flows for various $p\in \{0,0.01,0.1,0.2,0.5,1\}$, fixing $({\omega }_{\mathrm{A}1},{\omega }_{\mathrm{A}2},{\omega }_{\mathrm{D}1},{\omega }_{\mathrm{D}2})=(0,0,0,0)$ and $\rho =0.5$. We note (1) that those values are calculated by averaging 100 different initial configurations for $p=0$ and (2) that they are calculated for ${10}^9$ time steps after evolving the system with $p=0.01$ for ${10}^9$ time steps because it takes more time for the system to evolve into the steady state. Error bars represent one standard error. For depicting error bars, the evolving and calculated time is set to ${10}^8$ (for $p=0.01$) and ${10}^6$ (for $p\in \{0,0.1,0.2,0.5,1\}$), respectively.

Figure 5

Simulation results for steady-state values of $P\left(\begin{array}{c}1\\ 0\end{array}\right)+P\left(\begin{array}{c}0\\ 1\end{array}\right)$, and $P\left(\begin{array}{c}1\\ 1\end{array}\right)$ as functions of $p$, fixing $({\omega }_{\mathrm{A}1},{\omega }_{\mathrm{A}2},{\omega }_{\mathrm{D}1},{\omega }_{\mathrm{D}2})=(0,0,0,0)$ and $\rho =0.5$. The points between $p=0$ and $p=0.05$ are for $p=0.01$. We note (1) that the values for $p=0$ are calculated by averaging 100 different initial configurations and (2) that they are calculated for ${10}^9$ time steps after evolving the system with $p=0.01$ for ${10}^9$ time steps because it takes more time for the system to evolve into the steady state.

Figure 6

Schematic illustration of the trapping effect. Langmuir kinetics are not considered here; the case $L=10$ is shown.

Figure 7

Schematic illustration of the jamming effect. Langmuir kinetics are not considered here; the case $L=10$ is shown.

Figure 8

Schematic illustration of the blocking effect for unidirectional flows. Langmuir kinetics are not considered here; the case $L=10$ is shown.

Figure 9

State-transition diagram for $P\left(\begin{array}{c}0\\ 0\end{array}\right)$, $P(\begin{array}{c}1\\ 0\end{array})+P(\begin{array}{c}0\\ 1\end{array})$, and $P\left(\begin{array}{c}1\\ 1\end{array}\right)$.

Figure 10

Calculated values of $\frac{m}{\rho }$ for unidirectional (red solid) and bidirectional (blue dashed) flows at lane 1 as functions of $p$, fixing (a) $({\omega }_{\mathrm{A}1},{\omega }_{\mathrm{A}2},{\omega }_{\mathrm{D}1},{\omega }_{\mathrm{D}2})=(0,0,0,0)$ and $\rho =0.5$, and (b) $({\omega }_{\mathrm{A}1},{\omega }_{\mathrm{A}2},{\omega }_{\mathrm{D}1},{\omega }_{\mathrm{D}2})=(0.008,0.004,0.004,0.004)$. We note that for each trial $m$ is calculated for ${10}^6$ time steps after evolving the system for ${10}^6$ time steps, and for each $p$ the averages and standard errors are calculated over 10 trials. Error bars, representing one standard error, are almost the size of the symbols except for the case $p=0$ for (a).