Analysis of Traffic Evolution on Road Networks of a Roundabout

<table class="figure-group"><tr class="fig-image" id="a"><td><object data="https://static-preview.hindawi.com/articles/jam/volume-2020/3959613/figures/3959613.fig.008a.svgz" name="3959613.fig.008a" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(a) </b>Density between the merging and diverging main lane of the roundabout</td></tr><tr class="fig-image" id="b"><td><object data="https://static-preview.hindawi.com/articles/jam/volume-2020/3959613/figures/3959613.fig.008b.svgz" name="3959613.fig.008b" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(b) </b>Flux between the merging and diverging main lane of the roundabout</td></tr><tr class="fig-image" id="c"><td><object data="https://static-preview.hindawi.com/articles/jam/volume-2020/3959613/figures/3959613.fig.008c.svgz" name="3959613.fig.008c" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(c) </b>Density on the exits road</td></tr><tr class="fig-image" id="d"><td><object data="https://static-preview.hindawi.com/articles/jam/volume-2020/3959613/figures/3959613.fig.008d.svgz" name="3959613.fig.008d" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(d) </b>Flux on the exits road</td></tr></table>

<div>Density and flux profiles between the merging and diverging main lane of the roundabout under consideration and on the existing roads. In the case (a), shocks are produced on roads <span class="nowrap"><svg height="12.599pt" id="M304" style="vertical-align:-3.2911pt" version="1.1" viewbox="-0.0498162 -9.3079 65.2901 12.599" width="65.2901pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M449 634C442 637 425 643 405 650C376 660 341 666 307 666C181 666 98 590 98 485C98 400 170 343 215 310L246 288C307 243 343 204 343 147C343 67 291 18 219 18C104 18 61 124 51 202L23 199C28 124 27 71 27 47C47 22 122 -16 204 -16C324 -16 428 60 428 174C428 256 379 309 307 360L276 382C223 419 179 455 179 516C179 576 221 632 293 632C379 632 410 564 418 487L448 490C446 536 446 592 449 634Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.071,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"></path></g><g transform="matrix(.013,0,0,-0.013,11.017,0)"><path d="M95 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62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z"></path></g><g transform="matrix(.013,0,0,-0.013,27.178,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"></path></g><g transform="matrix(.013,0,0,-0.013,32.321,0)"><path d="M433 39L423 65C413 59 399 54 387 54C370 54 352 69 352 114V299C352 352 342 392 307 422C285 440 255 449 225 449C168 437 102 399 75 379C56 365 44 353 44 339C44 315 69 296 87 296C101 296 111 303 116 319C124 349 133 371 145 385C156 397 171 404 190 404C241 404 275 364 275 291V274C253 256 180 229 120 209C65 190 39 159 39 110C39 47 88 -12 159 -12C189 -12 237 25 277 52C282 35 288 21 301 8C312 -3 333 -12 348 -12L433 39ZM275 84C256 65 221 48 195 48C164 48 124 73 124 124C124 161 146 180 185 198C206 208 254 229 275 240V84Z"></path></g><g transform="matrix(.013,0,0,-0.013,37.911,0)"><path d="M524 0V26C466 32 460 36 460 104V297C460 393 411 449 331 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179 516C179 576 221 632 293 632C379 632 410 564 418 487L448 490C446 536 446 592 449 634Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,60.154,3.132)"><path d="M158 548H390L417 615L410 623H122L83 318C105 326 143 337 185 337C296 337 350 275 350 188C350 116 308 42 225 42C164 42 122 74 100 93C90 101 82 99 72 92C60 82 51 68 50 59C48 46 52 38 66 24C82 9 125 -12 172 -12C225 -11 292 15 346 59C408 108 437 166 437 226C437 309 371 397 242 397C214 397 170 382 133 369L158 548Z"></path></g></svg>.</span> Further, rarefaction waves are produced on roads <svg height="12.599pt" id="M305" style="vertical-align:-3.2911pt" version="1.1" viewbox="-0.0498162 -9.3079 61.2189 12.599" width="61.2189pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M405 650H141L135 622C222 616 230 610 215 535L133 116C118 41 113 33 29 28L23 0H289L295 28C209 33 205 40 219 116L298 535C312 609 317 616 399 622L405 650Z"></path></g><g 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37 239C37 86 126 -12 245 -12C367 -12 454 89 454 210C454 314 380 397 272 397C252 397 230 390 208 380L141 347ZM231 338C323 338 366 257 366 174C366 108 340 27 263 27C180 27 129 123 129 244C129 268 130 292 135 310C158 323 193 338 231 338Z"></path></g></svg> as shown in (c). Corresponding traffic fluxes are shown in (b) and (d), respectively.</div>

Journal of Applied Mathematics

Analysis of Traffic Evolution on Road Networks of a Roundabout

Figure 8