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All rights reserved. % Permission to copy this file and use after suitable modification is granted, % provided that this notice be retained on the copy and that no change is % made to the format of the text of the report number without the author's % permission. /$F2psDict 200 dict def F2psDictbeginF2psDict begin F2psDictbeginF2psDict /mtrx matrix put /l {lineto} bind def /m {moveto} bind def /s {stroke} bind def /n {newpath} bind def /gs {gsave} bind def /gr {grestore} bind def /clp {closepath} bind def /graycol {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul setrgbcolor} bind def /col-1 {} def /col0 {0 0 0 setrgbcolor} bind def /col1 {0 0 1 setrgbcolor} bind def /col2 {0 1 0 setrgbcolor} bind def /col3 {0 1 1 setrgbcolor} bind def /col4 {1 0 0 setrgbcolor} bind def /col5 {1 0 1 setrgbcolor} bind def /col6 {1 1 0 setrgbcolor} bind def /col7 {1 1 1 setrgbcolor} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def 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364 m 264 379 l gs col-1 s gr % Polyline n 264 404 m 264 419 l gs col-1 s gr % Polyline n 264 444 m 264 459 l gs col-1 s gr % Polyline n 264 484 m 264 499 l gs col-1 s gr 5.000 setlinewidth % Polyline n 119 319 m 199 319 l gs col-1 s gr % Polyline n 279 319 m 539 319 l gs col-1 s gr % Polyline n 499 319 m 499 274 l gs col-1 s gr 0.500 setlinewidth % Polyline n 499 259 m 499 274 l gs col-1 s gr 1.000 setlinewidth % Polyline n 434 429 m 479 414 l 534 439 l gs col-1 s gr 0.500 setlinewidth % Polyline n 279 179 m 279 454 l gs col-1 s gr % Polyline n 279 484 m 279 509 l gs col-1 s gr % Polyline n 279 129 m 279 174 l gs col-1 s gr % Polyline n 199 129 m 199 174 l gs col-1 s gr % Polyline n 199 179 m 199 439 l gs col-1 s gr % Polyline n 504 334 m 279 334 l gs col-1 s gr % Polyline n 199 334 m 119 334 l gs col-1 s gr % Polyline n 119 579 m 119 599 l 154 599 l 134 594 l 124 584 l 119 569 l clp gs 0.80 setgray fill gr gs col-1 s gr % Polyline n 619 459 m 619 494 l 614 489 l 609 484 l 614 479 l 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464.000 534.000 469.000 DrawSplineSection 534.000 469.000 529.000 474.000 536.500 476.500 DrawSplineSection 536.500 476.500 544.000 479.000 551.500 474.000 DrawSplineSection 551.500 474.000 559.000 469.000 561.500 464.000 DrawSplineSection 561.500 464.000 564.000 459.000 574.000 456.500 DrawSplineSection 574.000 456.500 584.000 454.000 586.500 456.500 DrawSplineSection 586.500 456.500 589.000 459.000 594.000 451.500 DrawSplineSection 594.000 451.500 599.000 444.000 599.000 436.500 DrawSplineSection 599.000 436.500 599.000 429.000 599.000 426.500 DrawSplineSection 599.000 426.500 599.000 424.000 601.500 421.500 DrawSplineSection 601.500 421.500 604.000 419.000 601.500 411.500 DrawSplineSection 601.500 411.500 599.000 404.000 594.000 406.500 DrawSplineSection 594.000 406.500 589.000 409.000 586.500 406.500 DrawSplineSection 586.500 406.500 584.000 404.000 584.000 399.000 DrawSplineSection 584.000 399.000 584.000 394.000 576.500 396.500 DrawSplineSection closepath gs 0.80 setgray fill gr gs col-1 s gr % Closed spline n 614.000 406.500 m 614.000 406.500 609.000 404.000 606.500 406.500 DrawSplineSection 606.500 406.500 604.000 409.000 601.500 409.000 DrawSplineSection 601.500 409.000 599.000 409.000 599.000 414.000 DrawSplineSection 599.000 414.000 599.000 419.000 596.500 424.000 DrawSplineSection 596.500 424.000 594.000 429.000 596.500 434.000 DrawSplineSection 596.500 434.000 599.000 439.000 596.500 444.000 DrawSplineSection 596.500 444.000 594.000 449.000 591.500 454.000 DrawSplineSection 591.500 454.000 589.000 459.000 586.500 459.000 DrawSplineSection 586.500 459.000 584.000 459.000 581.500 456.500 DrawSplineSection 581.500 456.500 579.000 454.000 576.500 456.500 DrawSplineSection 576.500 456.500 574.000 459.000 571.500 459.000 DrawSplineSection 571.500 459.000 569.000 459.000 564.000 461.500 DrawSplineSection 564.000 461.500 559.000 464.000 554.000 469.000 DrawSplineSection 554.000 469.000 549.000 474.000 546.500 476.500 DrawSplineSection 546.500 476.500 544.000 479.000 549.000 481.500 DrawSplineSection 549.000 481.500 554.000 484.000 571.500 486.500 DrawSplineSection 571.500 486.500 589.000 489.000 604.000 489.000 DrawSplineSection 604.000 489.000 619.000 489.000 619.000 449.000 DrawSplineSection 619.000 449.000 619.000 409.000 614.000 406.500 DrawSplineSection closepath gs 0.80 setgray fill gr gs col-1 s gr % Closed spline n 501.500 471.500 m 501.500 471.500 474.000 469.000 471.500 459.000 DrawSplineSection 471.500 459.000 469.000 449.000 466.500 446.500 DrawSplineSection 466.500 446.500 464.000 444.000 451.500 441.500 DrawSplineSection 451.500 441.500 439.000 439.000 436.500 434.000 DrawSplineSection 436.500 434.000 434.000 429.000 424.000 429.000 DrawSplineSection 424.000 429.000 414.000 429.000 406.500 431.500 DrawSplineSection 406.500 431.500 399.000 434.000 399.000 444.000 DrawSplineSection 399.000 444.000 399.000 454.000 404.000 459.000 DrawSplineSection 404.000 459.000 409.000 464.000 409.000 471.500 DrawSplineSection 409.000 471.500 409.000 479.000 419.000 474.000 DrawSplineSection 419.000 474.000 429.000 469.000 436.500 469.000 DrawSplineSection 436.500 469.000 444.000 469.000 449.000 471.500 DrawSplineSection 449.000 471.500 454.000 474.000 464.000 471.500 DrawSplineSection 464.000 471.500 474.000 469.000 501.500 471.500 DrawSplineSection 501.500 471.500 529.000 474.000 501.500 471.500 DrawSplineSection closepath gs 0.80 setgray fill gr gs col-1 s gr % Closed spline n 506.500 469.000 m 506.500 469.000 479.000 469.000 466.500 469.000 DrawSplineSection 466.500 469.000 454.000 469.000 444.000 469.000 DrawSplineSection 444.000 469.000 434.000 469.000 424.000 471.500 DrawSplineSection 424.000 471.500 414.000 474.000 404.000 479.000 DrawSplineSection 404.000 479.000 394.000 484.000 394.000 489.000 DrawSplineSection 394.000 489.000 394.000 494.000 401.500 504.000 DrawSplineSection 401.500 504.000 409.000 514.000 419.000 511.500 DrawSplineSection 419.000 511.500 429.000 509.000 449.000 506.500 DrawSplineSection 449.000 506.500 469.000 504.000 484.000 506.500 DrawSplineSection 484.000 506.500 499.000 509.000 516.500 511.500 DrawSplineSection 516.500 511.500 534.000 514.000 556.500 516.500 DrawSplineSection 556.500 516.500 579.000 519.000 596.500 519.000 DrawSplineSection 596.500 519.000 614.000 519.000 616.500 516.500 DrawSplineSection 616.500 516.500 619.000 514.000 619.000 501.500 DrawSplineSection 619.000 501.500 619.000 489.000 614.000 489.000 DrawSplineSection 614.000 489.000 609.000 489.000 586.500 486.500 DrawSplineSection 586.500 486.500 564.000 484.000 556.500 481.500 DrawSplineSection 556.500 481.500 549.000 479.000 539.000 476.500 DrawSplineSection 539.000 476.500 529.000 474.000 531.500 471.500 DrawSplineSection 531.500 471.500 534.000 469.000 506.500 469.000 DrawSplineSection closepath gs 0.80 setgray fill gr gs col-1 s gr % Closed spline n 409.000 469.000 m 409.000 469.000 409.000 464.000 404.000 456.500 DrawSplineSection 404.000 456.500 399.000 449.000 399.000 446.500 DrawSplineSection 399.000 446.500 399.000 444.000 391.500 436.500 DrawSplineSection 391.500 436.500 384.000 429.000 369.000 429.000 DrawSplineSection 369.000 429.000 354.000 429.000 341.500 431.500 DrawSplineSection 341.500 431.500 329.000 434.000 316.500 434.000 DrawSplineSection 316.500 434.000 304.000 434.000 294.000 441.500 DrawSplineSection 294.000 441.500 284.000 449.000 279.000 454.000 DrawSplineSection 279.000 454.000 274.000 459.000 274.000 469.000 DrawSplineSection 274.000 469.000 274.000 479.000 281.500 484.000 DrawSplineSection 281.500 484.000 289.000 489.000 304.000 484.000 DrawSplineSection 304.000 484.000 319.000 479.000 329.000 486.500 DrawSplineSection 329.000 486.500 339.000 494.000 349.000 494.000 DrawSplineSection 349.000 494.000 359.000 494.000 366.500 496.500 DrawSplineSection 366.500 496.500 374.000 499.000 379.000 494.000 DrawSplineSection 379.000 494.000 384.000 489.000 386.500 491.500 DrawSplineSection 386.500 491.500 389.000 494.000 391.500 496.500 DrawSplineSection 391.500 496.500 394.000 499.000 394.000 491.500 DrawSplineSection 394.000 491.500 394.000 484.000 401.500 479.000 DrawSplineSection 401.500 479.000 409.000 474.000 409.000 474.000 DrawSplineSection 409.000 474.000 409.000 474.000 409.000 469.000 DrawSplineSection closepath gs 0.80 setgray fill gr gs col-1 s gr % Closed spline n 401.500 501.500 m 401.500 501.500 394.000 494.000 389.000 491.500 DrawSplineSection 389.000 491.500 384.000 489.000 379.000 494.000 DrawSplineSection 379.000 494.000 374.000 499.000 364.000 494.000 DrawSplineSection 364.000 494.000 354.000 489.000 346.500 491.500 DrawSplineSection 346.500 491.500 339.000 494.000 329.000 486.500 DrawSplineSection 329.000 486.500 319.000 479.000 309.000 494.000 DrawSplineSection 309.000 494.000 299.000 509.000 291.500 509.000 DrawSplineSection 291.500 509.000 284.000 509.000 279.000 511.500 DrawSplineSection 279.000 511.500 274.000 514.000 266.500 514.000 DrawSplineSection 266.500 514.000 259.000 514.000 251.500 519.000 DrawSplineSection 251.500 519.000 244.000 524.000 244.000 531.500 DrawSplineSection 244.000 531.500 244.000 539.000 249.000 541.500 DrawSplineSection 249.000 541.500 254.000 544.000 261.500 549.000 DrawSplineSection 261.500 549.000 269.000 554.000 279.000 554.000 DrawSplineSection 279.000 554.000 289.000 554.000 301.500 549.000 DrawSplineSection 301.500 549.000 314.000 544.000 324.000 549.000 DrawSplineSection 324.000 549.000 334.000 554.000 346.500 549.000 DrawSplineSection 346.500 549.000 359.000 544.000 369.000 544.000 DrawSplineSection 369.000 544.000 379.000 544.000 389.000 541.500 DrawSplineSection 389.000 541.500 399.000 539.000 404.000 531.500 DrawSplineSection 404.000 531.500 409.000 524.000 411.500 519.000 DrawSplineSection 411.500 519.000 414.000 514.000 411.500 511.500 DrawSplineSection 411.500 511.500 409.000 509.000 409.000 509.000 DrawSplineSection 409.000 509.000 409.000 509.000 401.500 501.500 DrawSplineSection closepath gs 0.80 setgray fill gr gs col-1 s gr % Closed spline n 284.000 566.500 m 284.000 566.500 274.000 559.000 271.500 554.000 DrawSplineSection 271.500 554.000 269.000 549.000 256.500 544.000 DrawSplineSection 256.500 544.000 244.000 539.000 244.000 531.500 DrawSplineSection 244.000 531.500 244.000 524.000 234.000 519.000 DrawSplineSection 234.000 519.000 224.000 514.000 214.000 511.500 DrawSplineSection 214.000 511.500 204.000 509.000 199.000 511.500 DrawSplineSection 199.000 511.500 194.000 514.000 184.000 519.000 DrawSplineSection 184.000 519.000 174.000 524.000 166.500 529.000 DrawSplineSection 166.500 529.000 159.000 534.000 156.500 546.500 DrawSplineSection 156.500 546.500 154.000 559.000 154.000 569.000 DrawSplineSection 154.000 569.000 154.000 579.000 194.000 584.000 DrawSplineSection 194.000 584.000 234.000 589.000 251.500 586.500 DrawSplineSection 251.500 586.500 269.000 584.000 281.500 579.000 DrawSplineSection 281.500 579.000 294.000 574.000 294.000 574.000 DrawSplineSection 294.000 574.000 294.000 574.000 284.000 566.500 DrawSplineSection closepath gs 0.80 setgray fill gr gs col-1 s gr % Closed spline n 279.000 564.000 m 279.000 564.000 289.000 574.000 306.500 576.500 DrawSplineSection 306.500 576.500 324.000 579.000 339.000 581.500 DrawSplineSection 339.000 581.500 354.000 584.000 361.500 576.500 DrawSplineSection 361.500 576.500 369.000 569.000 366.500 559.000 DrawSplineSection 366.500 559.000 364.000 549.000 359.000 546.500 DrawSplineSection 359.000 546.500 354.000 544.000 341.500 549.000 DrawSplineSection 341.500 549.000 329.000 554.000 321.500 549.000 DrawSplineSection 321.500 549.000 314.000 544.000 301.500 549.000 DrawSplineSection 301.500 549.000 289.000 554.000 279.000 554.000 DrawSplineSection 279.000 554.000 269.000 554.000 279.000 564.000 DrawSplineSection closepath gs 0.80 setgray fill gr gs col-1 s gr % Closed spline n 399.000 534.000 m 399.000 534.000 394.000 539.000 386.500 541.500 DrawSplineSection 386.500 541.500 379.000 544.000 371.500 544.000 DrawSplineSection 371.500 544.000 364.000 544.000 361.500 546.500 DrawSplineSection 361.500 546.500 359.000 549.000 364.000 554.000 DrawSplineSection 364.000 554.000 369.000 559.000 366.500 566.500 DrawSplineSection 366.500 566.500 364.000 574.000 356.500 576.500 DrawSplineSection 356.500 576.500 349.000 579.000 361.500 586.500 DrawSplineSection 361.500 586.500 374.000 594.000 386.500 596.500 DrawSplineSection 386.500 596.500 399.000 599.000 409.000 594.000 DrawSplineSection 409.000 594.000 419.000 589.000 424.000 584.000 DrawSplineSection 424.000 584.000 429.000 579.000 429.000 566.500 DrawSplineSection 429.000 566.500 429.000 554.000 434.000 551.500 DrawSplineSection 434.000 551.500 439.000 549.000 429.000 541.500 DrawSplineSection 429.000 541.500 419.000 534.000 414.000 531.500 DrawSplineSection 414.000 531.500 409.000 529.000 406.500 529.000 DrawSplineSection 406.500 529.000 404.000 529.000 399.000 534.000 DrawSplineSection closepath gs 0.80 setgray fill gr gs col-1 s gr % Closed spline n 446.500 536.500 m 446.500 536.500 459.000 529.000 466.500 521.500 DrawSplineSection 466.500 521.500 474.000 514.000 471.500 511.500 DrawSplineSection 471.500 511.500 469.000 509.000 459.000 506.500 DrawSplineSection 459.000 506.500 449.000 504.000 439.000 506.500 DrawSplineSection 439.000 506.500 429.000 509.000 421.500 509.000 DrawSplineSection 421.500 509.000 414.000 509.000 411.500 509.000 DrawSplineSection 411.500 509.000 409.000 509.000 409.000 516.500 DrawSplineSection 409.000 516.500 409.000 524.000 406.500 526.500 DrawSplineSection 406.500 526.500 404.000 529.000 414.000 534.000 DrawSplineSection 414.000 534.000 424.000 539.000 429.000 541.500 DrawSplineSection 429.000 541.500 434.000 544.000 446.500 536.500 DrawSplineSection closepath gs 0.80 setgray fill gr gs col-1 s gr % Closed spline n 424.000 556.500 m 424.000 556.500 419.000 564.000 424.000 566.500 DrawSplineSection 424.000 566.500 429.000 569.000 436.500 571.500 DrawSplineSection 436.500 571.500 444.000 574.000 446.500 566.500 DrawSplineSection 446.500 566.500 449.000 559.000 444.000 551.500 DrawSplineSection 444.000 551.500 439.000 544.000 434.000 546.500 DrawSplineSection 434.000 546.500 429.000 549.000 424.000 556.500 DrawSplineSection closepath gs 0.80 setgray fill gr gs col-1 s gr % Closed spline n 419.000 599.000 m 419.000 599.000 394.000 599.000 274.000 599.000 DrawSplineSection 274.000 599.000 154.000 599.000 154.000 589.000 DrawSplineSection 154.000 589.000 154.000 579.000 156.500 576.500 DrawSplineSection 156.500 576.500 159.000 574.000 176.500 579.000 DrawSplineSection 176.500 579.000 194.000 584.000 216.500 586.500 DrawSplineSection 216.500 586.500 239.000 589.000 254.000 586.500 DrawSplineSection 254.000 586.500 269.000 584.000 284.000 576.500 DrawSplineSection 284.000 576.500 299.000 569.000 309.000 574.000 DrawSplineSection 309.000 574.000 319.000 579.000 326.500 579.000 DrawSplineSection 326.500 579.000 334.000 579.000 344.000 579.000 DrawSplineSection 344.000 579.000 354.000 579.000 364.000 586.500 DrawSplineSection 364.000 586.500 374.000 594.000 381.500 596.500 DrawSplineSection 381.500 596.500 389.000 599.000 401.500 596.500 DrawSplineSection 401.500 596.500 414.000 594.000 419.000 586.500 DrawSplineSection 419.000 586.500 424.000 579.000 426.500 586.500 DrawSplineSection 426.500 586.500 429.000 594.000 436.500 596.500 DrawSplineSection 436.500 596.500 444.000 599.000 419.000 599.000 DrawSplineSection closepath gs 0.80 setgray fill gr gs col-1 s gr % Closed spline n 184.000 504.000 m 184.000 504.000 179.000 494.000 161.500 484.000 DrawSplineSection 161.500 484.000 144.000 474.000 136.500 476.500 DrawSplineSection 136.500 476.500 129.000 479.000 124.000 481.500 DrawSplineSection 124.000 481.500 119.000 484.000 119.000 496.500 DrawSplineSection 119.000 496.500 119.000 509.000 119.000 521.500 DrawSplineSection 119.000 521.500 119.000 534.000 121.500 541.500 DrawSplineSection 121.500 541.500 124.000 549.000 134.000 549.000 DrawSplineSection 134.000 549.000 144.000 549.000 149.000 549.000 DrawSplineSection 149.000 549.000 154.000 549.000 159.000 539.000 DrawSplineSection 159.000 539.000 164.000 529.000 171.500 524.000 DrawSplineSection 171.500 524.000 179.000 519.000 184.000 516.500 DrawSplineSection 184.000 516.500 189.000 514.000 184.000 504.000 DrawSplineSection closepath gs 0.80 setgray fill gr gs col-1 s gr % Closed spline n 119.000 451.500 m 119.000 451.500 119.000 419.000 126.500 419.000 DrawSplineSection 126.500 419.000 134.000 419.000 139.000 421.500 DrawSplineSection 139.000 421.500 144.000 424.000 146.500 429.000 DrawSplineSection 146.500 429.000 149.000 434.000 159.000 436.500 DrawSplineSection 159.000 436.500 169.000 439.000 174.000 441.500 DrawSplineSection 174.000 441.500 179.000 444.000 184.000 444.000 DrawSplineSection 184.000 444.000 189.000 444.000 194.000 441.500 DrawSplineSection 194.000 441.500 199.000 439.000 204.000 441.500 DrawSplineSection 204.000 441.500 209.000 444.000 214.000 444.000 DrawSplineSection 214.000 444.000 219.000 444.000 224.000 449.000 DrawSplineSection 224.000 449.000 229.000 454.000 231.500 459.000 DrawSplineSection 231.500 459.000 234.000 464.000 229.000 471.500 DrawSplineSection 229.000 471.500 224.000 479.000 216.500 479.000 DrawSplineSection 216.500 479.000 209.000 479.000 204.000 484.000 DrawSplineSection 204.000 484.000 199.000 489.000 191.500 494.000 DrawSplineSection 191.500 494.000 184.000 499.000 179.000 499.000 DrawSplineSection 179.000 499.000 174.000 499.000 169.000 491.500 DrawSplineSection 169.000 491.500 164.000 484.000 156.500 481.500 DrawSplineSection 156.500 481.500 149.000 479.000 144.000 479.000 DrawSplineSection 144.000 479.000 139.000 479.000 131.500 481.500 DrawSplineSection 131.500 481.500 124.000 484.000 121.500 484.000 DrawSplineSection 121.500 484.000 119.000 484.000 119.000 451.500 DrawSplineSection closepath gs 0.80 setgray fill gr gs col-1 s gr % Closed spline n 119.000 561.500 m 119.000 561.500 119.000 599.000 161.500 599.000 DrawSplineSection 161.500 599.000 204.000 599.000 184.000 596.500 DrawSplineSection 184.000 596.500 164.000 594.000 159.000 591.500 DrawSplineSection 159.000 591.500 154.000 589.000 159.000 581.500 DrawSplineSection 159.000 581.500 164.000 574.000 161.500 564.000 DrawSplineSection 161.500 564.000 159.000 554.000 161.500 541.500 DrawSplineSection 161.500 541.500 164.000 529.000 141.500 526.500 DrawSplineSection 141.500 526.500 119.000 524.000 119.000 561.500 DrawSplineSection closepath gs 0.80 setgray fill gr gs col-1 s gr % Interpolated spline n 199 314 m 190.330 314.255 186.580 314.255 184 314 curveto 178.152 313.421 164.624 310.672 159 309 curveto 153.061 307.234 139.499 301.893 134 299 curveto 131.239 297.547 127.489 295.047 119 289 curveto gs col-1 s gr % Interpolated spline n 540 314 m 543.318 307.577 555.928 305.882 560 302 curveto 561.888 300.199 564.137 294.809 566 293 curveto 569.251 289.844 577.027 283.882 582 283 curveto 587.117 282.092 596.735 284.934 601 287 curveto 604.431 288.662 610.712 293.998 613 297 curveto 614.770 299.322 617.354 305.186 618 308 curveto 620.191 317.550 620.434 339.427 619 349 curveto 618.889 349.738 618.764 351.373 618 352 curveto 616.052 353.597 611.829 352.820 610 353 curveto 606.815 353.313 599.233 354.086 596 354 curveto 592.286 353.901 583.707 352.128 580 352 curveto 575.855 351.857 567.113 356.638 562 353 curveto 559.935 351.531 561.180 347.538 560 346 curveto 559.055 344.768 556.321 343.456 555 343 curveto 553.173 342.370 548.827 342.630 547 342 curveto 545.679 341.544 543.375 338.221 542 339 curveto 541.764 339.134 542.021 339.999 542 340 curveto 541.450 340.021 536.291 321.181 540 314 curveto clp gs 0.80 setgray fill gr gs col-1 s gr % Interpolated spline n 601 597 m 604.646 596.935 612.600 601.400 617 597 curveto 621.675 592.325 616.916 583.874 617 580 curveto 617.114 574.754 618.105 562.269 618 557 curveto 617.945 554.244 617.323 547.728 617 545 curveto 616.647 542.020 616.517 534.993 615 532 curveto 614.348 530.713 611.802 529.041 611 528 curveto 609.029 525.443 607.449 518.137 604 516 curveto 600.268 513.688 592.427 515.114 589 515 curveto 584.440 514.848 573.657 515.844 569 515 curveto 567.011 514.639 562.919 512.478 561 512 curveto 555.792 510.702 543.231 509.754 538 509 curveto 533.662 508.375 523.508 506.009 519 506 curveto 515.917 505.994 509.207 508.315 506 508 curveto 503.146 507.719 497.722 504.480 495 504 curveto 492.674 503.590 487.331 504.433 485 504 curveto 481.754 503.397 475.087 499.866 472 499 curveto 469.522 498.305 463.979 496.297 461 497 curveto 459.668 497.315 457.718 498.919 457 500 curveto 456.408 500.891 456.181 503.075 456 504 curveto 455.689 505.586 453.330 509.075 455 511 curveto 455.944 512.089 458.072 510.849 459 511 curveto 459.733 511.119 461.267 511.881 462 512 curveto 462.232 512.038 462.771 511.981 463 512 curveto 464.392 512.115 468.278 511.333 469 513 curveto 469.428 513.988 467.462 515.320 467 516 curveto 464.976 518.980 461.421 526.634 458 529 curveto 455.324 530.851 448.674 530.926 446 532 curveto 444.257 532.700 440.696 535.242 439 536 curveto 437.004 536.893 431.983 538.030 430 539 curveto 429.246 539.369 427.415 539.871 427 541 curveto 426.162 543.279 427.031 546.515 429 548 curveto 429.000 548.000 429.000 548.000 429 548 curveto 430.914 548.693 433.839 551.534 435 553 curveto 435.332 553.420 435.582 554.598 436 555 curveto 438.257 557.174 445.759 555.822 447 560 curveto 448.339 564.509 443.379 569.145 440 571 curveto 438.704 571.711 436.291 571.691 435 571 curveto 434.069 570.502 433.931 568.498 433 568 curveto 432.742 567.862 432.241 567.927 432 568 curveto 430.265 568.525 427.100 570.439 426 572 curveto 424.742 573.785 424.846 578.223 424 580 curveto 423.487 581.078 421.513 582.922 421 584 curveto 420.577 584.889 419.620 586.786 420 588 curveto 420.980 591.129 425.799 593.739 428 595 curveto 428.655 595.375 430.342 595.655 431 596 curveto 432.040 596.545 435.275 598.175 435 599 curveto 434.945 599.165 434.000 599.000 434 599 curveto 434.000 599.000 437.861 599.000 439 599 curveto 439.000 599.000 439.000 599.000 439 599 curveto 440.170 599.116 442.571 600.433 444 600 curveto 444.396 599.880 444.648 599.146 445 599 curveto 445.997 598.587 448.089 599.000 449 599 curveto 450.367 599.000 453.584 599.334 455 599 curveto 455.528 598.875 456.437 598.085 457 598 curveto 458.532 597.770 461.608 598.885 463 599 curveto 464.373 599.114 467.619 598.830 469 599 curveto 469.949 599.117 472.051 599.883 473 600 curveto 474.611 600.198 478.405 600.000 480 600 curveto 481.367 600.000 484.609 599.774 486 600 curveto 486.733 600.119 488.260 600.893 489 601 curveto 515.242 604.779 575.463 597.456 601 597 curveto clp gs 0.80 setgray fill gr gs col-1 s gr gr gs 2 setlinewidth n 66 80 moveto 500 0 rlineto 0 650 rlineto -500 0 rlineto closepath stroke 282 80 moveto 0 180 rlineto stroke gr F2psBegin−64−736translategs0setlinecap0setlinejoinF2psEnd showpage %%EndDocument @endspecial 7 425 a FD(Bounding)27 b(Dela)n(ys)f(in)h(P)n(ac)n(k)n (et-Routing)g(Net)n(w)n(orks)e(with)i(Ligh)n(t)926 516 y(T)-7 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Fq(2,)g(the)i(pac)o(k)o(et)g(has)f(the)h(path)f Fp(a)g Fo(!)f Fp(b)h Fo(!)f Fp(c)h Fo(!)p Fq(;)h(with)262 2253 y(probabilit)o(y)c(1)p Fp(=)p Fq(4)i(the)h(pac)o(k)o(et)g(has)g (the)g(path)f Fp(a)g Fo(!)f Fp(b)h Fo(!)f Fp(a)h Fo(!)g Fp(b)f Fo(!)p Fq(;)i(with)f(probabilit)o(y)262 2302 y(1)p Fp(=)p Fq(4)d(the)h(pac)o(k)o(et)g(has)g(the)h(path)f Fp(b)d Fo(!)g Fp(c)h Fo(!)p Fq(.)p 262 2339 573 2 v 307 2366 a Fn(\003)325 2378 y Fm(Supp)q(orted)c(b)o(y)i(National)f(Ph)o (ysical)h(Science)e(Consortium)h(\(NPSC\))i(F)m(ello)o(wship.)j(Also)d (supp)q(orted)262 2417 y(b)o(y)g(NSF)g(gran)o(t)f(n)o(um)o(b)q(er)g (CCR-9201092.)k(harc)o(hol@cs.b)q(e)o(rk)o(e)o(ley)m(.)o(edu)967 2574 y Fq(1)p eop %%Page: 2 3 2 2 bop 528 266 a 14208860 14350948 10327736 15919185 29207101 35127459 startTexFig 528 266 a %%BeginDocument: queueingnet.ps /arrowhead { 0 begin transform originalCTM itransform /taily exch def /tailx exch def transform originalCTM itransform /tipy exch def /tipx exch def /dy tipy taily sub def /dx tipx tailx sub def /angle dx 0 ne dy 0 ne or { dy dx atan } { 90 } ifelse def gsave originalCTM setmatrix tipx tipy translate angle rotate newpath arrowHeight neg arrowWidth 2 div moveto 0 0 lineto arrowHeight neg arrowWidth 2 div neg lineto patternNone not { originalCTM setmatrix /padtip arrowHeight 2 exp 0.25 arrowWidth 2 exp mul add sqrt brushWidth mul arrowWidth div def /padtail brushWidth 2 div def tipx tipy translate angle rotate padtip 0 translate arrowHeight padtip add padtail add arrowHeight div dup scale arrowheadpath ifill } if brushNone not { originalCTM setmatrix tipx tipy translate angle rotate arrowheadpath istroke } if grestore end } dup 0 9 dict put def /arrowheadpath { newpath arrowHeight neg arrowWidth 2 div moveto 0 0 lineto arrowHeight neg arrowWidth 2 div neg lineto } def /leftarrow { 0 begin y exch get /taily exch def x exch get /tailx exch def y exch get /tipy exch def x exch get /tipx exch def brushLeftArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /rightarrow { 0 begin y exch get /tipy exch def x exch get /tipx exch def y exch get /taily exch def x exch get /tailx exch def brushRightArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /arrowHeight 11 def /arrowWidth 5 def /IdrawDict 52 dict def IdrawDict begin /reencodeISO { dup dup findfont dup length dict begin { 1 index /FID ne { def }{ pop pop } ifelse } forall /Encoding ISOLatin1Encoding def currentdict end definefont } def /ISOLatin1Encoding [ /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /space/exclam/quotedbl/numbersign/dollar/percent/ampersand/quoteright /parenleft/parenright/asterisk/plus/comma/minus/period/slash /zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon /less/equal/greater/question/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N /O/P/Q/R/S/T/U/V/W/X/Y/Z/bracketleft/backslash/bracketright /asciicircum/underscore/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m /n/o/p/q/r/s/t/u/v/w/x/y/z/braceleft/bar/braceright/asciitilde /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/dotlessi/grave/acute/circumflex/tilde/macron/breve /dotaccent/dieresis/.notdef/ring/cedilla/.notdef/hungarumlaut /ogonek/caron/space/exclamdown/cent/sterling/currency/yen/brokenbar /section/dieresis/copyright/ordfeminine/guillemotleft/logicalnot /hyphen/registered/macron/degree/plusminus/twosuperior/threesuperior /acute/mu/paragraph/periodcentered/cedilla/onesuperior/ordmasculine /guillemotright/onequarter/onehalf/threequarters/questiondown /Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla /Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex /Idieresis/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis /multiply/Oslash/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute /Thorn/germandbls/agrave/aacute/acircumflex/atilde/adieresis /aring/ae/ccedilla/egrave/eacute/ecircumflex/edieresis/igrave /iacute/icircumflex/idieresis/eth/ntilde/ograve/oacute/ocircumflex /otilde/odieresis/divide/oslash/ugrave/uacute/ucircumflex/udieresis /yacute/thorn/ydieresis ] def /Helvetica-Bold reencodeISO def /Helvetica reencodeISO def /none null def /numGraphicParameters 17 def /stringLimit 65535 def /Begin { save numGraphicParameters dict begin } def /End { end restore } def /SetB { dup type /nulltype eq { pop false /brushRightArrow idef false /brushLeftArrow idef true /brushNone idef } { /brushDashOffset idef /brushDashArray idef 0 ne /brushRightArrow idef 0 ne /brushLeftArrow idef /brushWidth idef false /brushNone idef } ifelse } def /SetCFg { /fgblue idef /fggreen idef /fgred idef } def /SetCBg { /bgblue idef /bggreen idef /bgred idef } def /SetF { /printSize idef /printFont idef } def /SetP { dup type /nulltype eq { pop true /patternNone idef } { dup -1 eq { /patternGrayLevel idef /patternString idef } { /patternGrayLevel idef } ifelse false /patternNone idef } ifelse } def /BSpl { 0 begin storexyn newpath n 1 gt { 0 0 0 0 0 0 1 1 true subspline n 2 gt { 0 0 0 0 1 1 2 2 false subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 2 copy false subspline } if n 2 sub dup n 1 sub dup 2 copy 2 copy false subspline patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Circ { newpath 0 360 arc patternNone not { ifill } if brushNone not { istroke } if } def /CBSpl { 0 begin dup 2 gt { storexyn newpath n 1 sub dup 0 0 1 1 2 2 true subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 0 0 false subspline n 2 sub dup n 1 sub dup 0 0 1 1 false subspline patternNone not { ifill } if brushNone not { istroke } if } { Poly } ifelse end } dup 0 4 dict put def /Elli { 0 begin newpath 4 2 roll translate scale 0 0 1 0 360 arc patternNone not { ifill } if brushNone not { istroke } if end } dup 0 1 dict put def /Line { 0 begin 2 storexyn newpath x 0 get y 0 get moveto x 1 get y 1 get lineto brushNone not { istroke } if 0 0 1 1 leftarrow 0 0 1 1 rightarrow end } dup 0 4 dict put def /MLine { 0 begin storexyn newpath n 1 gt { x 0 get y 0 get moveto 1 1 n 1 sub { /i exch def x i get y i get lineto } for patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Poly { 3 1 roll newpath moveto -1 add { lineto } repeat closepath patternNone not { ifill } if brushNone not { istroke } if } def /Rect { 0 begin /t exch def /r exch def /b exch def /l exch def newpath l b moveto l t lineto r t lineto r b lineto closepath patternNone not { ifill } if brushNone not { istroke } if end } dup 0 4 dict put def /Text { ishow } def /idef { dup where { pop pop pop } { exch def } ifelse } def /ifill { 0 begin gsave patternGrayLevel -1 ne { fgred bgred fgred sub patternGrayLevel mul add fggreen bggreen fggreen sub patternGrayLevel mul add fgblue bgblue fgblue sub patternGrayLevel mul add setrgbcolor eofill } { eoclip originalCTM setmatrix pathbbox /t exch def /r exch def /b exch def /l exch def /w r l sub ceiling cvi def /h t b sub ceiling cvi def /imageByteWidth w 8 div ceiling cvi def /imageHeight h def bgred bggreen bgblue setrgbcolor eofill fgred fggreen fgblue setrgbcolor w 0 gt h 0 gt and { l w add b translate w neg h scale w h true [w 0 0 h neg 0 h] { patternproc } imagemask } if } ifelse grestore end } dup 0 8 dict put def /istroke { gsave brushDashOffset -1 eq { [] 0 setdash 1 setgray } { brushDashArray brushDashOffset setdash fgred fggreen fgblue setrgbcolor } ifelse brushWidth setlinewidth originalCTM setmatrix stroke grestore } def /ishow { 0 begin gsave fgred fggreen fgblue setrgbcolor /fontDict printFont printSize scalefont dup setfont def /descender fontDict begin 0 [FontBBox] 1 get FontMatrix end transform exch pop def /vertoffset 1 printSize sub descender sub def { 0 vertoffset moveto show /vertoffset vertoffset printSize sub def } forall grestore end } dup 0 3 dict put def /patternproc { 0 begin /patternByteLength patternString length def /patternHeight patternByteLength 8 mul sqrt cvi def /patternWidth patternHeight def /patternByteWidth patternWidth 8 idiv def /imageByteMaxLength imageByteWidth imageHeight mul stringLimit patternByteWidth sub min def /imageMaxHeight imageByteMaxLength imageByteWidth idiv patternHeight idiv patternHeight mul patternHeight max def /imageHeight imageHeight imageMaxHeight sub store /imageString imageByteWidth imageMaxHeight mul patternByteWidth add string def 0 1 imageMaxHeight 1 sub { /y exch def /patternRow y patternByteWidth mul patternByteLength mod def /patternRowString patternString patternRow patternByteWidth getinterval def /imageRow y imageByteWidth mul def 0 patternByteWidth imageByteWidth 1 sub { /x exch def imageString imageRow x add patternRowString putinterval } for } for imageString end } dup 0 12 dict put def /min { dup 3 2 roll dup 4 3 roll lt { exch } if pop } def /max { dup 3 2 roll dup 4 3 roll gt { exch } if pop } def /midpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 x1 add 2 div y0 y1 add 2 div end } dup 0 4 dict put def /thirdpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 2 mul x1 add 3 div y0 2 mul y1 add 3 div end } dup 0 4 dict put def /subspline { 0 begin /movetoNeeded exch def y exch get /y3 exch def x exch get /x3 exch def y exch get /y2 exch def x exch get /x2 exch def y exch get /y1 exch def x exch get /x1 exch def y exch get /y0 exch def x exch get /x0 exch def x1 y1 x2 y2 thirdpoint /p1y exch def /p1x exch def x2 y2 x1 y1 thirdpoint /p2y exch def /p2x exch def x1 y1 x0 y0 thirdpoint p1x p1y midpoint /p0y exch def /p0x exch def x2 y2 x3 y3 thirdpoint p2x p2y midpoint /p3y exch def /p3x exch def movetoNeeded { p0x p0y moveto } if p1x p1y p2x p2y p3x p3y curveto end } dup 0 17 dict put def /storexyn { /n exch def /y n array def /x n array def n 1 sub -1 0 { /i exch def y i 3 2 roll put x i 3 2 roll put } for } def /SSten { fgred fggreen fgblue setrgbcolor dup true exch 1 0 0 -1 0 6 -1 roll matrix astore } def /FSten { dup 3 -1 roll dup 4 1 roll exch newpath 0 0 moveto dup 0 exch lineto exch dup 3 1 roll exch lineto 0 lineto closepath bgred bggreen bgblue setrgbcolor eofill SSten } def /Rast { exch dup 3 1 roll 1 0 0 -1 0 6 -1 roll matrix astore } def %I Idraw 10 Grid 8 8 Begin %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 0.754552 0 0 0.754552 0 0 ] concat /originalCTM matrix currentmatrix def Begin %I Line %I b 65535 2 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.239697 -0.239697 0.239697 0.239697 391.839 628.529 ] concat %I 136 365 136 270 Line %I 1 End Begin %I Line %I b 65535 2 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.239697 -0.239697 0.239697 0.239697 399.509 621.338 ] concat %I 136 365 136 270 Line %I 1 End Begin %I Line %I b 65535 2 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.239697 -0.239697 0.239697 0.239697 391.36 629.967 ] concat %I 104 298 155 251 Line %I 1 End Begin %I Line %I b 65535 2 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ -0.239697 -0.239697 0.239697 -0.239697 465.186 750.295 ] concat %I 104 298 155 251 Line %I 1 End Begin %I Elli %I b 65535 2 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 1 0 0 1 199 151 ] concat %I 251 483 50 24 Elli End Begin %I Pict %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 0.239697 0.239697 -0.239697 0.239697 308.634 366.318 ] concat Begin %I Line %I b 65535 2 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 1 0 0 1 201 258 ] concat %I 136 365 136 270 Line %I 1 End Begin %I Line %I b 65535 2 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 1 0 0 1 232 259 ] concat %I 136 365 136 270 Line %I 1 End Begin %I Line %I b 65535 2 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 1 0 0 1 197 260 ] concat %I 104 298 155 251 Line %I 1 End Begin %I Line %I b 65535 2 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ -4.37114e-08 -1 1 -4.37114e-08 100 665 ] concat %I 104 298 155 251 Line %I 1 End End %I eop Begin %I Elli %I b 65535 2 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 1 0 0 1 231 38 ] concat %I 251 483 50 24 Elli End Begin %I Elli %I b 65535 2 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 1 0 0 1 63 74 ] concat %I 251 483 50 24 Elli End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-bold-r-normal-*-14-*-*-*-*-*-*-* Helvetica-Bold 14 SetF %I t [ 1 0 0 1 419 638 ] concat %I [ (Server a) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-bold-r-normal-*-14-*-*-*-*-*-*-* Helvetica-Bold 14 SetF %I t [ 1 0 0 1 284 560 ] concat %I [ (Server b) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-bold-r-normal-*-14-*-*-*-*-*-*-* Helvetica-Bold 14 SetF %I t [ 1 0 0 1 457 525 ] concat %I [ (Server c) ] Text End Begin %I BSpl %I b 65535 2 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 1 0 0 1 199 151 ] concat %I 5 208 469 192 473 154 460 142 437 142 429 5 BSpl %I 1 End Begin %I BSpl %I b 65535 2 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 1 0 0 1 199 151 ] concat %I 4 159 420 187 428 206 444 218 463 4 BSpl %I 1 End Begin %I BSpl %I b 65535 2 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 1 0 0 1 199 151 ] concat %I 5 275 461 285 448 288 427 287 405 282 395 5 BSpl %I 1 End Begin %I BSpl %I b 65535 2 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 1 0 0 1 199 151 ] concat %I 5 128 382 143 357 180 350 218 353 233 364 5 BSpl %I 1 End Begin %I BSpl %I b 65535 2 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 1 0 0 1 197 260 ] concat %I 3 286 357 297 347 314 349 3 BSpl %I 1 End Begin %I BSpl %I b 65535 2 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 1 0 0 1 197 260 ] concat %I 3 254 241 237 231 235 217 3 BSpl %I 1 End Begin %I BSpl %I b 65535 2 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 1 0 0 1 197 260 ] concat %I 3 115 273 111 254 92 246 3 BSpl %I 1 End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-14-*-*-*-*-*-*-* Helvetica 14 SetF %I t [ 1 0 0 1 516 703 ] concat %I [ (Outside) (Arrivals) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-14-*-*-*-*-*-*-* Helvetica 14 SetF %I t [ 1 0 0 1 512 620 ] concat %I [ (Departures) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-14-*-*-*-*-*-*-* Helvetica 14 SetF %I t [ 1 0 0 1 212 633 ] concat %I [ (Outside) (Arrivals) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-14-*-*-*-*-*-*-* Helvetica 14 SetF %I t [ 1 0 0 1 229 447 ] concat %I [ (ROUTING SCHEME:) () (average rate of arrival from outside = 1/5) () (with probability 1/2, the packet has path ) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-14-*-*-*-*-*-*-* Helvetica 14 SetF %I t [ 1 0 0 1 483 391 ] concat %I [ ( a->b->c) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-14-*-*-*-*-*-*-* Helvetica 14 SetF %I t [ 1 0 0 1 228 365 ] concat %I [ (with probability 1/4, the packet has path a->b->a->b->) () (with probability 1/4, the packet has path b->c->) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-helvetica-medium-r-normal-*-14-*-*-*-*-*-*-* Helvetica 14 SetF %I t [ 1 0 0 1 546 391 ] concat %I [ (->) ] Text End End %I eop showpage end %%EndDocument endTexFig 262 1266 a Fq(Figure)9 b(1:)16 b Fl(In)c(this)f(p)n(ap)n(er,)g(a)h (queueing)g(network)e(denotes)i(a)g(network)e(of)i(servers)e(to)n (gether)262 1316 y(with)k(a)h(r)n(outing)f(scheme.)324 1448 y Fq(A)k(queueing)h(net)o(w)o(ork)f(is)g(also)g(c)o(haracterized)i (b)o(y)e(the)h(service)g(time)e(distribution)262 1498 y(asso)q(ciated)k(with)f(eac)o(h)g(serv)o(er)i(and)e(the)h(order)g(in)f (whic)o(h)g(pac)o(k)o(ets)h(are)g(serv)o(ed)h(at)e(a)262 1548 y(serv)o(er)13 b(\(the)f(con)o(ten)o(tion)g(resolution)f(proto)q (col\).)17 b(If)12 b Fo(N)17 b Fq(is)12 b(a)f(queueing)h(net)o(w)o(ork) g(and)f Fp(c)1636 1554 y Fk(s)1666 1548 y Fq(is)262 1598 y(the)i(mean)e(service)k(time)c(at)i(serv)o(er)h Fp(s)f Fq(of)f Fo(N)6 b Fq(,)13 b(de\014ne)h Fo(N)1134 1604 y Fs(C,F)o(CFS)1271 1598 y Fq(\(resp)q(ectiv)o(ely)m(,)f Fo(N)1554 1604 y Fs(E,F)o(CFS)1678 1598 y Fq(\))262 1648 y(to)h(b)q(e)h(the)h(queueing)e(net)o(w)o(ork)h Fo(N)21 b Fq(where)16 b(the)f(service)h(time)d(at)i(serv)o(er)h Fp(s)f Fq(is)g(a)f(constan)o(t)262 1697 y Fp(c)280 1703 y Fk(s)309 1697 y Fq(\(resp)q(ectiv)o(ely)m(,)e(an)g(indep)q(enden)o(t) h(exp)q(onen)o(tially)d(distributed)i(random)e(v)n(ariable)h(with)262 1747 y(mean)g Fp(c)389 1753 y Fk(s)406 1747 y Fq(\))i(and)f(the)h(pac)o (k)o(ets)g(are)f(serv)o(ed)i(in)e(a)g(\014rst-come-\014rst-serv)o(ed)i (order.)k(Lik)o(ewise,)262 1797 y(de\014ne)d Fo(N)416 1803 y Fs(C,PS)510 1797 y Fq(to)f(b)q(e)h(the)g(queueing)g(net)o(w)o (ork)g Fo(N)20 b Fq(where)c(the)f(service)h(time)d(at)h(serv)o(er)i Fp(s)262 1847 y Fq(is)d(a)h(constan)o(t)g Fp(c)523 1853 y Fk(s)555 1847 y Fq(and)g(the)g(con)o(ten)o(tion)g(resolution)g(proto) q(col)f(is)h(pro)q(cessor-sharing.)324 1897 y(Harc)o(hol-Balter)19 b(and)g(W)m(olfe,)g([3)o(],)g(demonstrate)h(that)f(man)o(y)e(real-w)o (orld)h(pac)o(k)o(et-)262 1947 y(routing)13 b(net)o(w)o(orks)i(can)g(b) q(e)g(mo)q(deled)e(b)o(y)h(queueing)h(net)o(w)o(orks)g(of)f(t)o(yp)q(e) h Fo(N)1465 1953 y Fs(C,F)o(CFS)1589 1947 y Fq(.)k(It)c(is)262 1996 y(therefore)f(desirable)g(to)f(b)q(e)h(able)f(to)g(compute)g(the)h (steady-state)g(a)o(v)o(erage)f(pac)o(k)o(et)h(dela)o(y)262 2046 y(of)d(net)o(w)o(orks)h(of)g(t)o(yp)q(e)g Fo(N)650 2052 y Fs(C,F)o(CFS)775 2046 y Fq(.)17 b(\(The)12 b(dela)o(y)g(of)f(a)h (pac)o(k)o(et)g(is)g(de\014ned)h(as)f(the)h(total)e(time)262 2096 y(the)17 b(pac)o(k)o(et)h(sp)q(ends)h(w)o(aiting)d(in)h(queues)h (at)f(serv)o(ers)i(from)d(the)i(time)e(it)h(is)g(b)q(orn)g(un)o(til)262 2146 y(it)d(reac)o(hes)i(its)f(destination.\))21 b(Unfortunately)m(,)14 b(it)g(is)h(not)f(kno)o(wn)h(ho)o(w)f(to)h(compute)f(the)262 2196 y(a)o(v)o(erage)f(pac)o(k)o(et)g(dela)o(y)g(for)g(all)f(but)h(the) h(simplest)e Fo(N)1108 2202 y Fs(C,F)o(CFS)1246 2196 y Fq(t)o(yp)q(e)h(net)o(w)o(orks.)19 b(Ho)o(w)o(ev)o(er,)262 2245 y(the)e(corresp)q(onding)h(net)o(w)o(ork)f(of)f(t)o(yp)q(e)h Fo(N)947 2251 y Fs(E,F)o(CFS)1087 2245 y Fq(is)g(a)g(pro)q(duct-form)e (net)o(w)o(ork)i(\(more)262 2295 y(sp)q(eci\014cally)e(it)g(can)g(b)q (e)h(mo)q(deled)f(as)g(a)g(classed)h(Jac)o(kson)g(queueing)f(net)o(w)o (ork\))h(and)f(the)262 2345 y(a)o(v)o(erage)i(pac)o(k)o(et)h(dela)o(y)g (is)f(easy)h(to)g(determine)f(for)h(net)o(w)o(orks)g(of)f(this)h(t)o (yp)q(e)g(\([8)o(],)g([2)o(]\).)262 2395 y(Harc)o(hol-Balter)c(and)f(W) m(olfe)g(therefore)i(p)q(ose)g(the)f(follo)o(wing)d(question:)967 2574 y(2)p eop %%Page: 3 4 3 3 bop 365 307 a Fl(Is)15 b(it)g(p)n(ossible)f(to)h(b)n(ound)h(the)f (aver)n(age)g(delay)g(of)g Fo(N)1183 313 y Fs(C,F)o(CFS)1322 307 y Fl(\(which)g(we)f(c)n(ar)n(e)365 357 y(ab)n(out\))f(by)g(the)g (aver)n(age)g(delay)g(of)g Fo(N)942 363 y Fs(E,F)o(CFS)1078 357 y Fl(\(which)f(we)g(know)h(how)g(to)g(c)n(om-)365 407 y(pute\))p Fq(?)262 478 y(Let)h Fo(S)j Fq(denote)e(the)f(set)h(of)e (queueing)h(net)o(w)o(orks)h Fo(N)20 b Fq(for)13 b(whic)o(h)536 555 y(AvgDela)o(y\()p Fo(N)764 561 y Fs(C,F)o(CFS)887 555 y Fq(\))f Fo(\024)g Fq(AvgDela)o(y\()p Fo(N)1187 561 y Fs(E,F)o(CFS)1309 555 y Fq(\))p Fp(:)304 b Fq(\(1\))262 631 y(Harc)o(hol-Balter)15 b(and)g(W)m(olfe)e(giv)o(e)i(a)g(simple)e (pro)q(of)i(that)g(ev)o(ery)h(net)o(w)o(ork)f(with)g(Mark)o(o-)262 681 y(vian)g(routing)h(is)g(con)o(tained)g(in)g Fo(S)s Fq(.)26 b(\(In)16 b(Mark)o(o)o(vian)f(routing)h(a)g(pac)o(k)o(et's)h (route)g(is)f(not)262 731 y(con)o(tained)j(within)g(the)g(pac)o(k)o (et,)i(but)f(rather)g(there)g(are)g(probabilities)f(on)g(the)h(edges) 262 781 y(lea)o(ving)11 b(a)g(serv)o(er)j(whic)o(h)e(determine)h(all)e (pac)o(k)o(ets')h(routes.)19 b(In)12 b(other)h(w)o(ords,)f(Mark)o(o)o (vian)262 830 y(routing)j(is)g(classless.\))25 b(They)16 b(also)f(demonstrate)h(a)f(net)o(w)o(ork)h(whic)o(h)f(is)h(not)g(con)o (tained)262 880 y(in)11 b Fo(S)s Fq(.)18 b(They)13 b(lea)o(v)o(e)f(as)g (an)h(op)q(en)f(problem)f(the)i(question)g(of)f(determining)f(whether)j (more)262 930 y(net)o(w)o(orks)g(are)g(con)o(tained)g(in)g Fo(S)s Fq(.)324 980 y(In)f(this)h(pap)q(er)g(w)o(e)g(approac)o(h)g(the) g(problem)e(of)h(determining)g Fo(S)k Fq(b)o(y)c(restricting)i(our-)262 1030 y(selv)o(es)k(to)f(only)f(net)o(w)o(orks)i(with)f(ligh)o(t)g (tra\016c.)31 b(Let)19 b Fp(\025)1163 1036 y Fj(N)1215 1030 y Fq(denote)g(the)g(outside)g(arriv)n(al)262 1080 y(rate)13 b(in)o(to)f(queueing)h(net)o(w)o(ork)g Fo(N)6 b Fq(.)17 b Fo(S)f Fq(is)d(the)g(set)h(of)e(queueing)h(net)o(w)o(orks)g Fo(N)19 b Fq(that)13 b(satisfy)262 1129 y(equation\()d(1\))h(for)g(all) f(\(stable\))h(v)n(alues)g(of)f Fp(\025)946 1135 y Fj(N)980 1129 y Fq(.)17 b(De\014ne)12 b Fo(S)1160 1135 y Fk(Lig)q(ht)1257 1129 y Fq(to)f(b)q(e)g(the)h(set)g(of)e(queueing)262 1179 y(net)o(w)o(orks)k Fo(N)20 b Fq(that)14 b(satisfy)g(equation\()f (1\))h(in)f(the)i(case)g(of)e(ligh)o(t)g(tra\016c,)g(i.e.,)g(small)e Fp(\025)1639 1185 y Fj(N)1673 1179 y Fq(.)324 1229 y(W)m(e)17 b(giv)o(e)g(a)g(simple)f(su\016cien)o(t)i(criterion)f(for)h(whether)g (a)f(queueing)h(net)o(w)o(ork)g(is)f(in)262 1279 y Fo(S)287 1285 y Fk(Lig)q(ht)372 1279 y Fq(.)37 b(This)21 b(simple)d(criterion)j (enables)g(us)f(to)g(pro)o(v)o(e)h(man)o(y)d(net)o(w)o(orks)j(b)q (elong)f(to)262 1329 y Fo(S)287 1335 y Fk(Lig)q(ht)386 1329 y Fq(whic)o(h)14 b(ha)o(v)o(en't)g(y)o(et)g(b)q(een)h(sho)o(wn)f (to)f(b)q(elong)h(to)g Fo(S)s Fq(.)324 1378 y(By)h(de\014nition)h Fo(S)i Fq(is)d(con)o(tained)h(in)f Fo(S)928 1384 y Fk(Lig)q(ht)1014 1378 y Fq(.)23 b(Ho)o(w)o(ev)o(er)15 b(it)g(seems)h(lik)o(ely)e(that)i Fo(S)1608 1384 y Fk(Lig)q(ht)262 1428 y Fq(is)d(also)g(con)o(tained)h (in)f Fo(S)s Fq(,)g(since)i(it)e(seems)h(probable)f(that)h(AvgDela)o (y\()p Fo(N)1424 1434 y Fs(E,F)o(CFS)1546 1428 y Fq(\))g(should)262 1478 y(increase)d(at)g(a)f(faster)h(rate)g(than)f(AvgDela)o(y\()p Fo(N)1008 1484 y Fs(C,F)o(CFS)1132 1478 y Fq(\))h(as)f(the)h(tra\016c)g (load)e(is)i(increased.)262 1528 y(Therefore,)h(the)f(signi\014cance)h (of)e(the)h(ab)q(o)o(v)o(e)g(result)g(is)g(that)g(it)g(suggests)h(that) f(man)o(y)e(more)262 1578 y(net)o(w)o(orks)14 b(are)g(con)o(tained)g (in)g Fo(S)j Fq(than)d(has)g(already)f(b)q(een)i(pro)o(v)o(en.)324 1627 y(Reiman)f(and)i(Simon)f(\([6)o(],)h([7]\))g(pro)o(v)o(e)g(a)h (ligh)o(t)e(tra\016c)i(result)g(whic)o(h)f(is)h(somewhat)262 1677 y(similar)g(to)j(ours,)h(as)f(w)o(ell)f(as)h(man)o(y)e(other)j (ligh)o(t)d(tra\016c)i(results.)38 b(Their)20 b(tec)o(hnique)262 1727 y(in)o(v)o(olv)o(es)d(computing)g(deriv)n(ativ)o(es)i(of)f(the)h (dela)o(y)f(at)h(the)g(p)q(oin)o(t)g(of)f(zero)h(arriv)n(al)f(rate.)262 1777 y(The)e(ligh)o(t)e(tra\016c)i(pro)q(of)g(tec)o(hniques)h(w)o(e)f (use)g(are)g(m)o(uc)o(h)f(simpler,)g(and)g(will)f(hop)q(efully)262 1827 y(lead)f(to)h(other)g(simple)f(ligh)o(t)f(tra\016c)j(analysis.)324 1877 y(Section)h(2)f(states)i(the)g(su\016cien)o(t)f(criterion)g (theorem)f(precisely)i(and)e(pro)o(v)o(es)h(it.)24 b(In)262 1926 y(Section)15 b(3)f(w)o(e)h(discuss)h(whic)o(h)e(queueing)h(net)o (w)o(orks)g(can)g(easily)f(b)q(e)h(seen)h(to)f(satisfy)f(the)262 1976 y(su\016cien)o(t)g(criterion.)262 2111 y Fr(2)69 b(Main)23 b(Theorem)262 2202 y Fq(By)14 b([1)o(])g(and)g([4)o(],)g(w)o (e)g(kno)o(w)g(that)g(the)h(a)o(v)o(erage)f(pac)o(k)o(et)h(dela)o(y)f (in)f Fo(N)1339 2208 y Fs(C,PS)1433 2202 y Fq(is)h(equal)g(to)g(the)262 2252 y(a)o(v)o(erage)f(pac)o(k)o(et)h(dela)o(y)f(in)g Fo(N)730 2258 y Fs(E,F)o(CFS)866 2252 y Fq(for)h(all)e Fo(N)6 b Fq(.)1039 2237 y Fi(1)1076 2252 y Fq(Therefore)14 b(it)g(is)f(equiv)n(alen)o(t)g(to)g(study)262 2301 y(for)g(whic)o(h)h (queueing)g(net)o(w)o(orks)g Fo(N)6 b Fq(,)599 2378 y(AvgDela)o(y\()p Fo(N)827 2384 y Fs(C,F)o(CFS)950 2378 y Fq(\))12 b Fo(\024)g Fq(AvgDela)o(y\()p Fo(N)1250 2384 y Fs(C,PS)1329 2378 y Fq(\))p Fp(:)p 262 2411 573 2 v 308 2437 a Fh(1)325 2449 y Fm(This)g(p)q(o)o(w)o(erful)e(theorem)f(is)i(also)g(describ)q (ed)e(more)i(recen)o(tly)e(in)i([8)o(])h(and)f([5)o(].)967 2574 y Fq(3)p eop %%Page: 4 5 4 4 bop 262 307 a Fq(W)m(e)9 b(will)g(assume)h(this)g(form)o(ulation)d (of)i(the)i(problem)e(throughout)h(the)h(rest)g(of)f(the)g(pap)q(er,) 262 357 y(since)k(it)g(simpli\014es)e(our)i(analysis.)324 407 y(In)19 b(this)g(section)h(w)o(e)g(see)h(that,)f(sp)q(eaking)f(lo)q (osely)m(,)g(to)g(test)i(whether)f(a)f(queueing)262 457 y(net)o(w)o(ork)14 b Fo(N)21 b Fq(b)q(elongs)15 b(to)f Fo(S)701 463 y Fk(Lig)q(ht)801 457 y Fq(it)h(is)f(enough)h(to)f(c)o (hec)o(k)h(whether)h(the)f(exp)q(ected)i(dela)o(y)262 506 y(created)j(b)o(y)f(exactly)g(2)g(pac)o(k)o(ets)h(in)f Fo(N)903 512 y Fs(C,F)o(CFS)1046 506 y Fq(is)g(smaller)f(than)h(the)h 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Fm(\))p Fy(\000)p Fm(AvgDela)o(y\()p Fy(N)1251 1582 y Fs(C,PS)1328 1573 y Fm(\))f Fe(<)g(P)1411 1578 y Fh(1)1429 1573 y Fm(\()p Fe(D)1473 1559 y Ft(F)t(C)r(F)t(S)1472 1582 y Fh(1)1565 1573 y Fy(\000)p Fe(D)1622 1559 y Ft(P)t(S)1621 1582 y Fh(1)1668 1573 y Fm(\)+)p Fe(P)1731 1578 y Fh(1)1755 1550 y Fm(1)p 1754 1566 V 1754 1598 a Fe(k)1778 1573 y(;)276 1656 y Fl(wher)n(e)15 b Fp(D)429 1641 y Fk(F)t(C)r(F)t(S)428 1667 y Fi(1)530 1656 y Fl(,)g Fp(D)593 1641 y Fk(P)t(S)592 1667 y Fi(1)643 1656 y Fl(,)f Fp(\025)694 1662 y Fj(N)728 1656 y Fl(,)h Fp(P)783 1662 y Fi(1)801 1656 y Fl(,)g Fp(k)q Fl(,)f(and)i Fp(m)f Fl(ar)n(e)g(as)g(de\014ne)n(d)h(in)f(the)g (ab)n(ove)g(the)n(or)n(em.)262 1740 y Fg(Corollary)f(2)21 b Fl(Given)15 b(a)g(queueing)h(network,)e Fo(N)6 b Fl(,)15 b(if)f Fp(\025)1149 1746 y Fj(N)1183 1740 y Fp(<)1280 1724 y Fi(1)p 1232 1731 113 2 v 1232 1755 a(8)p Fk(e)1265 1746 y Ff(2)1280 1755 y Fk(k)q(m)1328 1746 y Ff(2)1349 1740 y Fl(,)725 1848 y(if)g Fp(D)800 1831 y Fk(F)t(C)r(F)t(S)799 1858 y Fi(1)913 1848 y Fp(<)e(D)992 1831 y Fk(P)t(S)991 1858 y Fi(1)1051 1848 y Fo(\000)1099 1820 y Fq(1)p 1098 1838 23 2 v 1098 1876 a Fp(k)1126 1848 y Fl(,)i(then)610 1946 y(A)o(vgDelay\()p Fo(N)831 1952 y Fs(C,F)o(CFS)955 1946 y Fl(\))e Fp(<)g Fl(A)o(vgDelay\()p Fo(N)1249 1952 y Fs(C,PS)1329 1946 y Fl(\))262 2016 y(wher)n(e)i Fp(D)414 2001 y Fk(F)t(C)r(F)t(S)413 2027 y Fi(1)515 2016 y Fl(,)h Fp(D)578 2001 y Fk(P)t(S)577 2027 y Fi(1)628 2016 y Fl(,)g Fp(\025)680 2022 y Fj(N)713 2016 y Fl(,)g Fp(P)768 2022 y Fi(1)786 2016 y Fl(,)g Fp(k)q Fl(,)f(and)i Fp(m)f Fl(ar)n(e)g(as)g (de\014ne)n(d)h(in)f(the)g(ab)n(ove)g(the)n(or)n(em.)324 2100 y Fq(A)i(few)g(commen)o(ts)e(on)i(the)g(ab)q(o)o(v)o(e)g (corollary)f(b)q(efore)i(w)o(e)f(b)q(egin)g(the)h(pro)q(of.)27 b(First,)262 2150 y(observ)o(e)17 b(that)f Fp(k)h Fq(is)f(a)g(free)h (parameter)f(of)g Fp(\025)981 2156 y Fj(N)1014 2150 y Fq(.)25 b(Therefore)1248 2134 y Fi(1)p 1247 2141 19 2 v 1247 2165 a Fk(k)1286 2150 y Fq(ab)q(o)o(v)o(e)16 b(can)h(b)q(e)g (made)e(as)262 2200 y(small)f(as)i(w)o(e)h(wish)f(b)o(y)h(decreasing)g Fp(\025)875 2206 y Fj(N)909 2200 y Fq(.)26 b(Second,)17 b(note)g(that)g Fp(D)1325 2185 y Fk(F)t(C)r(F)t(S)1324 2210 y Fi(1)1443 2200 y Fq(is)f(an)g(a)o(v)o(erage.)262 2250 y(Therefore,)e(it)g(includes)g(the)h(case)g(where)g(the)g(t)o(w)o (o)e(pac)o(k)o(ets)i(happ)q(en)f(to)g(ha)o(v)o(e)g(the)g(same)262 2300 y(route)e(and)f(b)q(oth)h(pac)o(k)o(ets)g(start)g(within)f(one)h (service)h(time)d(unit)h(of)g(eac)o(h)h(other.)18 b(F)m(or)12 b(this)262 2350 y(particular)g(case)h(PS)f(clearly)g(do)q(es)h(w)o (orse)g(than)f(F)o(CFS;)g(call)g(the)g(di\013erence)i(in)e(dela)o(y)g (for)262 2399 y(this)i(case)h Fp(\016)r Fq(.)k(Since)c(w)o(e)g(can)f (alw)o(a)o(ys)f(c)o(ho)q(ose)i Fp(k)h Fq(in)d Fp(\025)1104 2405 y Fj(N)1153 2399 y Fq(suc)o(h)i(that)f Fp(\016)g(>)1419 2383 y Fi(1)p 1419 2390 V 1419 2414 a Fk(k)1442 2399 y Fq(,)g(w)o(e)g(can)h(state)262 2449 y(a)e(simple)f(consequence)17 b(of)c(Corollary)g(2,)g(namely:)967 2574 y(4)p eop %%Page: 5 6 5 5 bop 365 307 a Fq(F)m(or)18 b(ligh)o(t)f(enough)i(tra\016c,)g (AvgDela)o(y\()p Fo(N)1057 313 y Fs(C,F)o(CFS)1181 307 y Fq(\))g Fp(<)g Fq(AvgDela)o(y\()p Fo(N)1495 313 y Fs(C,PS)1574 307 y Fq(\))365 357 y(whenev)o(er)12 b Fp(D)579 342 y Fk(F)t(C)r(F)t(S)578 367 y Fi(1)692 357 y Fp(<)g(D)771 342 y Fk(P)t(S)770 367 y Fi(1)821 357 y Fq(,)e(giv)o(en)g(that)g(the)h (t)o(w)o(o)f(pac)o(k)o(ets)h(are)g(on)f(di\013eren)o(t)365 407 y(paths.)262 520 y Fg(Pro)q(of)k(of)i(Theorem:)324 570 y Fq(By)9 b(P)m(AST)m(A)h(\(P)o(oisson)f(Arriv)n(als)g(See)h(Time)e (Av)o(erages\),)j(the)f(exp)q(ected)i(dela)o(y)d(a)g(newly)262 619 y(arriving)h(pac)o(k)o(et)i(exp)q(eriences)j(is)c(equal)h(to)f(the) i(a)o(v)o(erage)f(pac)o(k)o(et)g(dela)o(y)f(for)g(the)i(net)o(w)o(ork.) 262 669 y(Let)f Fo(N)18 b Fq(b)q(e)13 b(an)o(y)e(queueing)i(net)o(w)o (ork.)k(F)m(or)12 b(the)g(case)h(of)f(ligh)o(t)f(tra\016c)h(\(i.e.,)f Fp(\025)1466 675 y Fj(N)1500 669 y Fp(<)1596 653 y Fi(1)p 1548 660 113 2 v 1548 684 a(8)p Fk(e)1581 675 y Ff(2)1597 684 y Fk(k)q(m)1645 675 y Ff(2)1666 669 y Fq(\),)262 719 y(w)o(e)i(will)f(compute)h(upp)q(er)h(and)f(lo)o(w)o(er)g(b)q (ounds)h(on)f(the)g(dela)o(y)g(an)g(arriv)n(al)f(exp)q(eriences)k(in) 262 769 y Fo(N)296 775 y Fs(C,F)o(CFS)420 769 y Fq(.)i(The)c(pro)q(of)g (for)f Fo(N)741 775 y Fs(C,PS)834 769 y Fq(is)h(iden)o(tical.)324 819 y(T)m(o)j(compute)g(an)g(upp)q(er)h(b)q(ound)g(on)f(the)i(dela)o(y) e(in)g Fo(N)1211 825 y Fs(C,F)o(CFS)1335 819 y Fq(,)h(let)g Fp(p)f Fq(represen)o(t)j(an)262 869 y(arriving)14 b(pac)o(k)o(et)i(in)g Fo(N)636 875 y Fs(C,F)o(CFS)760 869 y Fq(.)24 b(Clearly)m(,)14 b Fp(p)i Fq(ma)o(y)e(only)h(b)q(e)h(dela)o(y)o(ed)g(b)o(y)f(pac)o(k)o (ets)i(whic)o(h)262 918 y(are)e(in)g Fo(N)416 924 y Fs(C,F)o(CFS)555 918 y Fq(during)g(the)h(time)e Fp(p)h Fq(is)g(in)g Fo(N)1020 924 y Fs(C,F)o(CFS)1144 918 y Fq(.)22 b(Note)16 b(that)f(if)g Fp(i)g Fq(pac)o(k)o(ets)h(are)g(in)262 968 y Fo(N)296 974 y Fs(C,F)o(CFS)420 968 y Fq(,)g(they)h(ma)o(y)e(tak)o(e)h(up)h(to)f (time)f Fp(im)i Fq(to)f(clear)h(the)g(system.)25 b(So,)17 b(denoting)f Fp(p)p Fq('s)262 1018 y(arriv)n(al)10 b(time)g(b)o(y)i(0,) g(if)e(pac)o(k)o(et)j Fp(p)e Fq(is)h(dela)o(y)o(ed,)g(at)f(least)h(one) g(of)g(the)g(follo)o(wing)d(m)o(ust)i(o)q(ccur:)324 1101 y Fo(\017)20 b Fq(at)14 b(least)g(1)g(other)g(pac)o(k)o(et)h(arriv)o (es)f(during)f(\()p Fo(\000)p Fp(m;)7 b(m)p Fq(\).)324 1184 y Fo(\017)20 b Fq(at)14 b(least)g(2)g(other)g(pac)o(k)o(ets)h (arriv)o(e)f(during)f(\()p Fo(\000)p Fq(2)p Fp(m;)7 b Fq(2)p Fp(m)p Fq(\).)324 1267 y Fo(\017)20 b Fq(at)14 b(least)g(3)g(other)g(pac)o(k)o(ets)h(arriv)o(e)f(during)f(\()p Fo(\000)p Fq(3)p Fp(m;)7 b Fq(3)p Fp(m)p Fq(\).)324 1350 y Fo(\017)20 b Fq(etc.)324 1433 y(De\014ne)422 1524 y Fp(E)455 1509 y Fj(0)453 1535 y Fk(i)480 1524 y Fq(:)e(the)d(ev)o(en)o (t)f(that)g Fl(at)h(le)n(ast)f Fp(i)g Fq(pac)o(k)o(ets)g(arriv)o(e)g (during)g(\()p Fo(\000)p Fp(im;)7 b(im)p Fq(\))425 1616 y Fp(E)456 1622 y Fk(i)483 1616 y Fq(:)18 b(the)d(ev)o(en)o(t)f(that)g Fl(exactly)g Fp(i)h Fq(pac)o(k)o(ets)f(arriv)o(e)g(during)g(\()p Fo(\000)p Fp(im;)7 b(im)p Fq(\))324 1690 y(No)o(w)17 b Fp(p)g Fq(can)g(only)f(b)q(e)i(dela)o(y)o(ed)f(is)g(at)g(least)h(one) f(of)f(the)i Fp(E)1282 1675 y Fj(0)1280 1701 y Fk(i)1311 1690 y Fq(o)q(ccur,)h(i.e.,)d(if)1562 1659 y Fd(S)1604 1690 y Fp(E)1637 1675 y Fj(0)1635 1701 y Fk(i)1666 1690 y Fq(is)262 1740 y(true.)j(Ho)o(w)o(ev)o(er)534 1709 y Fd(S)575 1740 y Fp(E)608 1725 y Fj(0)606 1751 y Fk(i)634 1740 y Fq(can)14 b(only)f(o)q(ccur)i(if)952 1709 y Fd(S)993 1740 y Fp(E)1024 1746 y Fk(i)1052 1740 y Fq(o)q(ccurs)g(\(See)g(fo)q (otnote)1420 1725 y Fi(2)1453 1740 y Fq(for)f(a)g(pro)q(of)f(of)262 1790 y(this)e(subtle)i(p)q(oin)o(t\),)e(so)h Fp(p)f Fq(can)h(only)f(b)q (e)i(dela)o(y)o(ed)e(if)g(at)h(least)g(one)g(of)f(the)h(follo)o(wing)e (ev)o(en)o(ts)262 1840 y(o)q(ccurs:)324 1931 y Fo(\017)20 b Fq(exactly)14 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Fm(,)h(with)g(probabilit)o(y)c(1)k(it)g(m)o(ust)e (ev)o(en)o(tually)f(cross)i Fy(L)i Fm(and)e(come)f(b)q(elo)o(w)h Fy(L)i Fm(\(b)o(y)262 2364 y(the)e(La)o(w)i(of)f(Large)g(Num)o(b)q (ers\).)j(Th)o(us)d(if)836 2335 y Fd(S)877 2364 y Fe(E)905 2352 y Fn(0)903 2375 y Ft(i)928 2364 y Fm(is)g(true,)f(then)h(so)g(is) 1207 2335 y Fd(S)1247 2364 y Fe(E)1273 2369 y Ft(i)1298 2364 y Fm(.)967 2574 y Fq(5)p eop %%Page: 6 7 6 6 bop 324 307 a Fq(W)m(e)15 b(will)f(compute)h(the)h(exp)q(ected)i (dela)o(y)d(on)g Fp(p)h Fq(due)g(to)f(eac)o(h)i(of)e(the)h(ab)q(o)o(v)o (e)f(ev)o(en)o(ts,)262 357 y(and)h(then)h(w)o(e'll)e(sum)h(these.)27 b(This)16 b(will)f(b)q(e)j(an)e(o)o(v)o(ercoun)o(t,)h(but)f(that's)h (o.k.)25 b(b)q(ecause)262 407 y(w)o(e're)14 b(just)g(upp)q(erb)q (ounding.)262 457 y(Let)508 506 y Fp(P)535 512 y Fk(i)560 506 y Fq(=)26 b Fg(Pr)6 b Fo(f)p Fq(exactly)14 b Fp(i)g Fq(arriv)n(als)f(during)g(time)g(\()p Fo(\000)p Fp(im;)7 b(im)p Fq(\))q Fo(g)262 579 y Fq(Let)307 668 y Fp(D)342 651 y Fk(F)t(C)r(F)t(S)341 678 y(i)455 668 y Fq(=)26 b Fg(E)7 b Fo(f)p Fq(dela)o(y)13 b(on)h Fp(p)g Fq(due)g(to)g Fp(i)g Fq(arriv)n(als)f(during)g(\()p Fo(\000)p Fp(im;)7 b(im)p Fq(\))16 b(in)d Fo(N)1489 674 y Fs(C,F)o(CFS)1613 668 y Fo(g)262 756 y Fq(So)300 844 y Fg(E)7 b Fo(f)p Fq(dela)o(y)13 b(on)h Fp(p)g Fq(in)f Fo(N)642 850 y Fs(C,F)o(CFS)766 844 y Fo(g)42 b(\024)g Fp(P)930 850 y Fi(1)948 844 y Fp(D)983 827 y Fk(F)t(C)r(F)t(S)982 855 y Fi(1)1093 844 y Fq(+)10 b Fp(P)1162 850 y Fi(2)1180 844 y Fp(D)1215 827 y Fk(F)t(C)r(F)t(S)1214 855 y Fi(2)1326 844 y Fq(+)f Fp(P)1394 850 y Fi(3)1413 844 y Fp(D)1448 827 y Fk(F)t(C)r(F)t(S)1447 855 y Fi(3)1558 844 y Fq(+)h Fp(:)d(:)g(:)829 907 y Fo(\024)42 b Fp(P)930 913 y Fi(1)948 907 y Fp(D)983 890 y Fk(F)t(C)r(F)t(S)982 917 y Fi(1)1093 907 y Fq(+)10 b Fp(P)1162 913 y Fi(2)1180 907 y Fq(\(2)p Fp(m)p Fq(\))g(+)f Fp(P)1347 913 y Fi(3)1366 907 y Fq(\(3)p Fp(m)p Fq(\))g(+)h Fp(:)d(:)g(:)262 995 y Fq(where)16 b(the)g(last)g(inequalit)o(y)e(is)i(an)f(o)o(v)o (er-estimate,)g(since)h(w)o(e)g(are)g(assuming)e(the)i(w)o(orst)262 1045 y(case)g(where)g(all)e(the)i(pac)o(k)o(ets)g(con)o(tin)o(ually)e (run)h(in)o(to)g(eac)o(h)g(other)h(o)o(v)o(er)f(and)h(o)o(v)o(er)f (again)262 1095 y(during)e(their)h(en)o(tire)h(time)d(in)i(the)g(net)o (w)o(ork.)19 b(By)14 b(de\014nition)f(of)h(the)g(P)o(oisson)g(Pro)q (cess,)732 1217 y Fp(P)759 1223 y Fk(i)784 1217 y Fq(=)833 1189 y Fp(e)852 1174 y Fj(\000)p Fp(\025)902 1180 y Fj(N)936 1174 y(\001)p Fi(2)p Fk(im)1013 1189 y Fq(\()p Fp(\025)1053 1195 y Fj(N)1096 1189 y Fo(\001)9 b Fq(2)p Fp(im)p Fq(\))1204 1168 y Fk(i)p 833 1208 386 2 v 1013 1246 a Fp(i)p Fq(!)262 1317 y(F)m(or)k Fp(i)f Fo(\025)g Fq(2,)h(w)o(e)h(can)g(express)i Fp(P)761 1323 y Fk(i)788 1317 y Fq(in)d(terms)h(of)f Fp(P)1026 1323 y Fi(1)1058 1317 y Fq(as)h(follo)o(ws:)615 1440 y Fp(P)642 1446 y Fk(i)656 1440 y Fq(\()p Fp(i)e Fo(\025)g Fq(2\))41 b(=)899 1412 y Fp(e)918 1397 y Fj(\000)p Fp(\025)968 1403 y Fj(N)1002 1397 y(\001)p Fi(2)p Fk(im)1079 1412 y Fq(\()p Fp(\025)1119 1418 y Fj(N)1162 1412 y Fo(\001)9 b Fq(2)p Fp(im)p Fq(\))1270 1391 y Fk(i)p 899 1431 V 1079 1469 a Fp(i)p Fq(!)820 1548 y(=)899 1520 y Fp(i)913 1504 y Fk(i)p 899 1538 29 2 v 900 1576 a Fp(i)p Fq(!)941 1548 y Fo(\001)g Fp(e)981 1530 y Fj(\000)p Fp(\025)1031 1536 y Fj(N)1065 1530 y(\001)p Fi(2)p Fk(im)1142 1548 y Fq(\()p Fp(\025)1182 1554 y Fj(N)1225 1548 y Fo(\001)g Fq(2)p Fp(m)p Fq(\))1319 1527 y Fk(i)820 1639 y Fp(<)42 b(e)913 1622 y Fk(i)936 1639 y Fo(\001)9 b Fp(e)976 1622 y Fj(\000)p Fp(\025)1026 1628 y Fj(N)1060 1622 y(\001)p Fi(2)p Fk(m)1125 1639 y Fq(\()p Fp(\025)1165 1645 y Fj(N)1208 1639 y Fo(\001)g Fq(2)p Fp(m)p Fq(\))1302 1618 y Fk(i)820 1701 y Fq(=)42 b Fp(P)921 1707 y Fi(1)948 1701 y Fo(\001)9 b Fq(\()p Fp(\025)1009 1707 y Fj(N)1052 1701 y Fo(\001)g Fq(2)p Fp(m)p Fq(\))1146 1684 y Fk(i)p Fj(\000)p Fi(1)1212 1701 y Fo(\001)g Fp(e)1252 1684 y Fk(i)262 1789 y Fq(Substituting)k Fp(\025)523 1795 y Fj(N)557 1789 y Fq(=)654 1773 y Fi(1)p 606 1780 113 2 v 606 1804 a(8)p Fk(e)639 1795 y Ff(2)654 1804 y Fk(k)q(m)702 1795 y Ff(2)723 1789 y Fq(,)h(w)o(e)g(ha)o(v)o(e:) 652 1883 y Fp(P)679 1889 y Fk(i)693 1883 y Fq(\()p Fp(i)e Fo(\025)g Fq(2\))41 b Fp(<)h(P)958 1889 y Fi(1)985 1883 y Fo(\001)9 b Fq(\()p Fp(\025)1046 1889 y Fj(N)1089 1883 y Fo(\001)g Fq(2)p Fp(m)p Fq(\))1183 1866 y Fk(i)p Fj(\000)p Fi(1)1249 1883 y Fo(\001)g Fp(e)1289 1866 y Fk(i)857 1964 y Fq(=)42 b Fp(P)958 1970 y Fi(1)985 1964 y Fo(\001)9 b Fq(\()1076 1936 y(1)p 1027 1955 119 2 v 1027 1993 a(4)p Fp(e)1067 1981 y Fi(2)1086 1993 y Fp(k)q(m)1150 1964 y Fq(\))1166 1947 y Fk(i)p Fj(\000)p Fi(1)1232 1964 y Fo(\001)g Fp(e)1272 1947 y Fk(i)857 2064 y Fp(<)42 b(P)958 2070 y Fi(1)985 2064 y Fo(\001)1069 2036 y Fq(1)p 1011 2055 137 2 v 1011 2093 a Fp(k)q Fq(4)1055 2081 y Fk(i)p Fj(\000)p Fi(1)1111 2093 y Fp(m)262 2164 y Fq(No)o(w,)16 b(substituting)h Fp(P)632 2170 y Fk(i)646 2164 y Fq(,)g Fp(i)f Fo(\025)h Fq(2)f(in)o(to)g(the)i(form)o(ula)c(for)i(the)i(exp)q (ected)h(dela)o(y)d(on)h Fp(p)p Fq(,)f(w)o(e)262 2214 y(ha)o(v)o(e:)347 2352 y Fg(E)7 b Fo(f)p Fq(dela)o(y)13 b(on)h Fp(p)g Fq(in)f Fo(N)689 2358 y Fs(C,F)o(CFS)813 2352 y Fo(g)42 b(\024)f Fp(P)976 2358 y Fi(1)995 2352 y Fp(D)1030 2335 y Fk(F)t(C)r(F)t(S)1029 2363 y Fi(1)1140 2352 y Fq(+)10 b Fp(P)1209 2358 y Fi(2)1227 2352 y Fq(\(2)p Fp(m)p Fq(\))g(+)f Fp(P)1394 2358 y Fi(3)1413 2352 y Fq(\(3)p Fp(m)p Fq(\))g(+)h Fp(:)d(:)g(:)876 2437 y(<)41 b(P)976 2443 y Fi(1)995 2437 y Fp(D)1030 2420 y Fk(F)t(C)r(F)t(S)1029 2448 y Fi(1)1140 2437 y Fq(+)1187 2409 y Fp(P)1214 2415 y Fi(1)p 1187 2428 46 2 v 1188 2466 a Fq(2)p Fp(k)1246 2437 y Fq(+)1301 2409 y Fp(P)1328 2415 y Fi(1)p 1293 2428 63 2 v 1293 2466 a Fq(2)1314 2454 y Fi(2)1332 2466 y Fp(k)1369 2437 y Fq(+)1424 2409 y Fp(P)1451 2415 y Fi(1)p 1416 2428 V 1416 2466 a Fq(2)1437 2454 y Fi(3)1455 2466 y Fp(k)1492 2437 y Fq(+)10 b Fp(:)d(:)g(:)967 2574 y Fq(6)p eop %%Page: 7 8 7 7 bop 876 321 a Fq(=)41 b Fp(P)976 327 y Fi(1)995 321 y Fp(D)1030 303 y Fk(F)t(C)r(F)t(S)1029 331 y Fi(1)1140 321 y Fq(+)10 b Fp(P)1209 327 y Fi(1)1233 292 y Fq(1)p 1232 311 23 2 v 1232 349 a Fp(k)876 426 y Fq(=)41 b Fp(P)976 432 y Fi(1)1002 367 y Fd(\022)1032 426 y Fp(D)1067 409 y Fk(F)t(C)r(F)t(S)1066 436 y Fi(1)1178 426 y Fq(+)1225 398 y(1)p 1224 416 V 1224 454 a Fp(k)1252 367 y Fd(\023)324 539 y Fq(T)m(o)16 b(deriv)o(e)i(a)f(simple)e(lo)o(w)o(er)i(b)q(ound)h (for)e(the)i(exp)q(ected)i(dela)o(y)c(in)h Fo(N)1443 545 y Fs(C,F)o(CFS)1567 539 y Fq(,)h(again)262 588 y(let)f Fp(p)g Fq(represen)o(t)i(an)e(arriving)f(pac)o(k)o(et)i(in)f Fo(N)985 594 y Fs(C,F)o(CFS)1109 588 y Fq(.)28 b(Assume)17 b Fp(p)g Fq(arriv)o(es)g(at)g Fo(N)1569 594 y Fs(C,F)o(CFS)262 638 y Fq(at)12 b(time)f(0.)17 b(T)m(o)12 b(lo)o(w)o(erb)q(ound)h(the)g Fg(E)7 b Fo(f)p Fq(Dela)o(y)13 b(on)g Fp(p)h Fq(in)g Fo(N)1157 644 y Fs(C,F)o(CFS)1281 638 y Fo(g)p Fq(,)e(w)o(e)h(consider) g(only)f(the)262 688 y(dela)o(y)h(on)h Fp(p)f Fq(caused)i(b)o(y)f(1)g (pac)o(k)o(et)g(arriving)f(during)g(\()p Fo(\000)p Fp(m;)7 b(m)p Fq(\).)615 788 y Fg(E)g Fo(f)p Fq(dela)o(y)14 b(on)f Fp(p)h Fq(in)g Fo(N)958 794 y Fs(C,F)o(CFS)1082 788 y Fo(g)d(\025)h Fp(P)1185 794 y Fi(1)1203 788 y Fp(D)1238 771 y Fk(F)t(C)r(F)t(S)1237 799 y Fi(1)324 879 y Fb(2)262 1016 y Fr(3)69 b(Characterizing)21 b Fa(S)865 1025 y Fp(Lig)q(ht)262 1107 y Fq(In)d(Section)g(2,)g(w)o(e)h(found)e(that)h (to)g(c)o(hec)o(k)h(whether)h Fo(N)k Fq(is)18 b(in)f Fo(S)1314 1113 y Fk(Lig)q(ht)1418 1107 y Fq(it)h(is)g(enough)g(to)262 1156 y(c)o(hec)o(k)f(whether)h(the)f(exp)q(ected)i(dela)o(y)d(created)i (b)o(y)e(exactly)h(2)f(pac)o(k)o(ets)i(in)e Fo(N)1525 1162 y Fs(C,F)o(CFS)1666 1156 y Fq(is)262 1206 y(smaller)9 b(than)h(the)h(exp)q(ected)i(dela)o(y)d(created)i(b)o(y)f(exactly)g(2)f (pac)o(k)o(ets)h(in)g Fo(N)1429 1212 y Fs(C,PS)1508 1206 y Fq(,)g(when)g(the)262 1256 y(pac)o(k)o(ets)17 b(tak)o(e)f(di\013eren) o(t)h(paths.)26 b(In)17 b(this)f(section)h(w)o(e)g(discuss)g(whic)o(h)g (net)o(w)o(orks)g(whic)o(h)262 1306 y(satisfy)c(this)h(easy)g(test.)19 b(W)m(e)14 b(mak)o(e)e(t)o(w)o(o)i(p)q(oin)o(ts:)312 1388 y(1.)20 b(W)m(e)d(demonstrate)h(a)f(class)g(of)g(net)o(w)o(orks)h (with)f Fl(non-Markovian)i Fq(routing)e(whic)o(h)365 1438 y(satis\014es)g(the)f(test)g(for)f(b)q(eing)g(in)g Fo(S)928 1444 y Fk(Lig)q(ht)1014 1438 y Fq(.)23 b(\(Because)17 b(the)f(routing)f(is)g(not)g(Mark)o(o-)365 1488 y(vian,)d(the)h(result) h(in)e([3)o(])g(on)h(net)o(w)o(orks)g(in)f Fo(S)k Fq(do)q(es)d(not)g (apply)f(to)g(these)i(net)o(w)o(orks.\))312 1570 y(2.)20 b(In)e([3],)f(a)h(net)o(w)o(ork)g Fo(N)24 b Fq(is)18 b(constructed)i(for)e(whic)o(h)g(the)g(exp)q(ected)i(dela)o(y)e(in)f (the)365 1620 y(case)k(of)e(t)o(w)o(o)g(pac)o(k)o(ets)i(is)e (signi\014can)o(tly)g(greater)i(in)e Fo(N)1268 1626 y Fs(C,F)o(CFS)1412 1620 y Fq(than)g(in)h Fo(N)1603 1626 y Fs(C,PS)1682 1620 y Fq(.)365 1670 y(This)c(prop)q(ert)o(y)h(is)e (highly)g(dep)q(enden)o(t)i(on)f(the)g(fact)g Fo(N)6 b Fq('s)15 b(serv)o(ers)j(ha)o(v)o(e)e(di\013eren)o(t)365 1720 y(service)21 b(times.)33 b(W)m(e)19 b(therefore)i(p)q(ose)f(the)g (question)f(of)g(whether)h Fl(al)r(l)f Fq(net)o(w)o(orks)365 1769 y(where)e(the)f(serv)o(ers)h(ha)o(v)o(e)e(the)h Fl(same)g Fq(service)h(time)d(are)i(in)f Fo(S)1354 1775 y Fk(Lig)q(ht)1455 1769 y Fq(\(or)h(almost)d(in)365 1819 y Fo(S)390 1825 y Fk(Lig)q(ht)476 1819 y Fq(\).)24 b(W)m(e)15 b(giv)o(e)h(some)f(argumen)o(ts)g(for)g(wh)o(y)h(it)f(is)h(lik)o(ely)e (that)i(the)g(answ)o(er)h(to)365 1869 y(this)d(question)g(is)g(y)o(es.) 324 1951 y(Consider)i(the)h(class)f(of)g(net)o(w)o(orks)g(\(with)g (non-Mark)o(o)o(vian)f(routing\))g(with)h(the)h(fol-)262 2001 y(lo)o(wing)e(prop)q(ert)o(y:)27 b(an)o(y)17 b(t)o(w)o(o)g(routes) h(whic)o(h)g(in)o(tersect)h(and)e(then)h(split)f(up)h(can)g(nev)o(er) 262 2051 y(subsequen)o(tly)e(rejoin.)23 b(It's)16 b(easy)f(to)h(see)h (that)e(these)i(net)o(w)o(orks)f(are)g(all)f(in)g Fo(S)1517 2057 y Fk(Lig)q(ht)1603 2051 y Fq(:)21 b(W)m(e)262 2100 y(only)14 b(need)j(to)f(lo)q(ok)e(at)i(the)g(case)h(of)e(t)o(w)o(o)g (pac)o(k)o(ets)h(in)g(the)g(net)o(w)o(ork.)24 b(The)16 b(t)o(w)o(o)f(pac)o(k)o(ets)262 2150 y(don't)f(a\013ect)i(eac)o(h)f (other)h(at)f(all)f(un)o(til)g(they)h(bump)f(in)o(to)g(eac)o(h)i (other.)22 b(Th)o(us,)15 b(un)o(til)f(the)262 2200 y(pac)o(k)o(ets)e (bump)f(in)o(to)g(eac)o(h)i(other,)g Fo(N)849 2206 y Fs(C,F)o(CFS)985 2200 y Fq(and)f Fo(N)1098 2206 y Fs(C,PS)1189 2200 y Fq(b)q(eha)o(v)o(e)h(iden)o(tically)m(.)i(The)e(\014rst)262 2250 y(time)c(the)j(t)o(w)o(o)f(pac)o(k)o(ets)h(do)f(bump)f(in)o(to)h (eac)o(h)h(other)g(will)d(therefore)k(o)q(ccur)f(at)g(exactly)f(the)262 2300 y(same)g(time)f(and)i(will)f(b)q(e)h(at)g(exactly)g(the)h(same)e (lo)q(cation)g(in)g Fo(N)1264 2306 y Fs(C,F)o(CFS)1401 2300 y Fq(and)g Fo(N)1513 2306 y Fs(C,PS)1593 2300 y Fq(.)17 b(The)262 2350 y(dela)o(y)d(incurred)j(on)e(the)h(pac)o(k)o (ets)g(during)f(the)h(p)q(erio)q(d)f(of)g(in)o(tersection)h(will)e(b)q (e)i(smaller)262 2399 y(in)c Fo(N)343 2405 y Fs(C,F)o(CFS)479 2399 y Fq(than)h(in)f Fo(N)656 2405 y Fs(C,PS)735 2399 y Fq(.)18 b(After)13 b(the)g(routes)h(split)e(up,)g(the)h(t)o(w)o(o)f (pac)o(k)o(ets)i(will)d(nev)o(er)262 2449 y(again)17 b(see)i(eac)o(h)g(other)g(in)e(either)i(net)o(w)o(ork.)32 b(Th)o(us)18 b(the)h(dela)o(y)f(in)f(ev)o(ery)i(case)g(of)f(t)o(w)o(o) 967 2574 y(7)p eop %%Page: 8 9 8 8 bop 858 266 a 3789043 11442909 13814169 -197345 30983208 51770245 startTexFig 858 266 a %%BeginDocument: twopack.ps /arrowhead { 0 begin transform originalCTM itransform /taily exch def /tailx exch def transform originalCTM itransform /tipy exch def /tipx exch def /dy tipy taily sub def /dx tipx tailx sub def /angle dx 0 ne dy 0 ne or { dy dx atan } { 90 } ifelse def gsave originalCTM setmatrix tipx tipy translate angle rotate newpath arrowHeight neg arrowWidth 2 div moveto 0 0 lineto arrowHeight neg arrowWidth 2 div neg lineto patternNone not { originalCTM setmatrix /padtip arrowHeight 2 exp 0.25 arrowWidth 2 exp mul add sqrt brushWidth mul arrowWidth div def /padtail brushWidth 2 div def tipx tipy translate angle rotate padtip 0 translate arrowHeight padtip add padtail add arrowHeight div dup scale arrowheadpath ifill } if brushNone not { originalCTM setmatrix tipx tipy translate angle rotate arrowheadpath istroke } if grestore end } dup 0 9 dict put def /arrowheadpath { newpath arrowHeight neg arrowWidth 2 div moveto 0 0 lineto arrowHeight neg arrowWidth 2 div neg lineto } def /leftarrow { 0 begin y exch get /taily exch def x exch get /tailx exch def y exch get /tipy exch def x exch get /tipx exch def brushLeftArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /rightarrow { 0 begin y exch get /tipy exch def x exch get /tipx exch def y exch get /taily exch def x exch get /tailx exch def brushRightArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /arrowHeight 11 def /arrowWidth 5 def /IdrawDict 50 dict def IdrawDict begin /none null def /numGraphicParameters 17 def /stringLimit 65535 def /Begin { save numGraphicParameters dict begin } def /End { end restore } def /SetB { dup type /nulltype eq { pop false /brushRightArrow idef false /brushLeftArrow idef true /brushNone idef } { /brushDashOffset idef /brushDashArray idef 0 ne /brushRightArrow idef 0 ne /brushLeftArrow idef /brushWidth idef false /brushNone idef } ifelse } def /SetCFg { /fgblue idef /fggreen idef /fgred idef } def /SetCBg { /bgblue idef /bggreen idef /bgred idef } def /SetF { /printSize idef /printFont idef } def /SetP { dup type /nulltype eq { pop true /patternNone idef } { dup -1 eq { /patternGrayLevel idef /patternString idef } { /patternGrayLevel idef } ifelse false /patternNone idef } ifelse } def /BSpl { 0 begin storexyn newpath n 1 gt { 0 0 0 0 0 0 1 1 true subspline n 2 gt { 0 0 0 0 1 1 2 2 false subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 2 copy false subspline } if n 2 sub dup n 1 sub dup 2 copy 2 copy false subspline patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Circ { newpath 0 360 arc patternNone not { ifill } if brushNone not { istroke } if } def /CBSpl { 0 begin dup 2 gt { storexyn newpath n 1 sub dup 0 0 1 1 2 2 true subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 0 0 false subspline n 2 sub dup n 1 sub dup 0 0 1 1 false subspline patternNone not { ifill } if brushNone not { istroke } if } { Poly } ifelse end } dup 0 4 dict put def /Elli { 0 begin newpath 4 2 roll translate scale 0 0 1 0 360 arc patternNone not { ifill } if brushNone not { istroke } if end } dup 0 1 dict put def /Line { 0 begin 2 storexyn newpath x 0 get y 0 get moveto x 1 get y 1 get lineto brushNone not { istroke } if 0 0 1 1 leftarrow 0 0 1 1 rightarrow end } dup 0 4 dict put def /MLine { 0 begin storexyn newpath n 1 gt { x 0 get y 0 get moveto 1 1 n 1 sub { /i exch def x i get y i get lineto } for patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Poly { 3 1 roll newpath moveto -1 add { lineto } repeat closepath patternNone not { ifill } if brushNone not { istroke } if } def /Rect { 0 begin /t exch def /r exch def /b exch def /l exch def newpath l b moveto l t lineto r t lineto r b lineto closepath patternNone not { ifill } if brushNone not { istroke } if end } dup 0 4 dict put def /Text { ishow } def /idef { dup where { pop pop pop } { exch def } ifelse } def /ifill { 0 begin gsave patternGrayLevel -1 ne { fgred bgred fgred sub patternGrayLevel mul add fggreen bggreen fggreen sub patternGrayLevel mul add fgblue bgblue fgblue sub patternGrayLevel mul add setrgbcolor eofill } { eoclip originalCTM setmatrix pathbbox /t exch def /r exch def /b exch def /l exch def /w r l sub ceiling cvi def /h t b sub ceiling cvi def /imageByteWidth w 8 div ceiling cvi def /imageHeight h def bgred bggreen bgblue setrgbcolor eofill fgred fggreen fgblue setrgbcolor w 0 gt h 0 gt and { l w add b translate w neg h scale w h true [w 0 0 h neg 0 h] { patternproc } imagemask } if } ifelse grestore end } dup 0 8 dict put def /istroke { gsave brushDashOffset -1 eq { [] 0 setdash 1 setgray } { brushDashArray brushDashOffset setdash fgred fggreen fgblue setrgbcolor } ifelse brushWidth setlinewidth originalCTM setmatrix stroke grestore } def /ishow { 0 begin gsave fgred fggreen fgblue setrgbcolor /fontDict printFont printSize scalefont dup setfont def /descender fontDict begin 0 [FontBBox] 1 get FontMatrix end transform exch pop def /vertoffset 1 printSize sub descender sub def { 0 vertoffset moveto show /vertoffset vertoffset printSize sub def } forall grestore end } dup 0 3 dict put def /patternproc { 0 begin /patternByteLength patternString length def /patternHeight patternByteLength 8 mul sqrt cvi def /patternWidth patternHeight def /patternByteWidth patternWidth 8 idiv def /imageByteMaxLength imageByteWidth imageHeight mul stringLimit patternByteWidth sub min def /imageMaxHeight imageByteMaxLength imageByteWidth idiv patternHeight idiv patternHeight mul patternHeight max def /imageHeight imageHeight imageMaxHeight sub store /imageString imageByteWidth imageMaxHeight mul patternByteWidth add string def 0 1 imageMaxHeight 1 sub { /y exch def /patternRow y patternByteWidth mul patternByteLength mod def /patternRowString patternString patternRow patternByteWidth getinterval def /imageRow y imageByteWidth mul def 0 patternByteWidth imageByteWidth 1 sub { /x exch def imageString imageRow x add patternRowString putinterval } for } for imageString end } dup 0 12 dict put def /min { dup 3 2 roll dup 4 3 roll lt { exch } if pop } def /max { dup 3 2 roll dup 4 3 roll gt { exch } if pop } def /midpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 x1 add 2 div y0 y1 add 2 div end } dup 0 4 dict put def /thirdpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 2 mul x1 add 3 div y0 2 mul y1 add 3 div end } dup 0 4 dict put def /subspline { 0 begin /movetoNeeded exch def y exch get /y3 exch def x exch get /x3 exch def y exch get /y2 exch def x exch get /x2 exch def y exch get /y1 exch def x exch get /x1 exch def y exch get /y0 exch def x exch get /x0 exch def x1 y1 x2 y2 thirdpoint /p1y exch def /p1x exch def x2 y2 x1 y1 thirdpoint /p2y exch def /p2x exch def x1 y1 x0 y0 thirdpoint p1x p1y midpoint /p0y exch def /p0x exch def x2 y2 x3 y3 thirdpoint p2x p2y midpoint /p3y exch def /p3x exch def movetoNeeded { p0x p0y moveto } if p1x p1y p2x p2y p3x p3y curveto end } dup 0 17 dict put def /storexyn { /n exch def /y n array def /x n array def n 1 sub -1 0 { /i exch def y i 3 2 roll put x i 3 2 roll put } for } def /SSten { fgred fggreen fgblue setrgbcolor dup true exch 1 0 0 -1 0 6 -1 roll matrix astore } def /FSten { dup 3 -1 roll dup 4 1 roll exch newpath 0 0 moveto dup 0 exch lineto exch dup 3 1 roll exch lineto 0 lineto closepath bgred bggreen bgblue setrgbcolor eofill SSten } def /Rast { exch dup 3 1 roll 1 0 0 -1 0 6 -1 roll matrix astore } def %I Idraw 10 Grid 191.881 6.50092e+37 Begin %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 0.754552 0 0 0.754552 0 0 ] concat /originalCTM matrix currentmatrix def Begin %I Pict %I b u %I cfg u %I cbg u %I f u %I p u %I t u Begin %I Elli %I b 65535 3 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 210.347 -364.84 ] concat %I 298 522 67 29 Elli End Begin %I Elli %I b 65535 3 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 95.6667 -242.84 ] concat %I 298 522 67 29 Elli End Begin %I Elli %I b 65535 3 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 322.587 -295.707 ] concat %I 298 522 67 29 Elli End Begin %I Elli %I b 65535 3 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 322.587 -213.56 ] concat %I 298 522 67 29 Elli End Begin %I BSpl %I b 65535 3 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 79.4 -28.1201 ] concat %I 5 495 364 524 360 553 355 569 343 585 326 5 BSpl %I 1 End Begin %I BSpl %I b 65535 3 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 79.4 -28.1201 ] concat %I 3 595 264 595 245 594 223 3 BSpl %I 1 End Begin %I BSpl %I b 65535 3 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 79.4 -28.1201 ] concat %I 5 584 165 572 152 536 144 504 136 481 136 5 BSpl %I 1 End Begin %I BSpl %I b 65520 3 0 1 [12 4] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 79.4 -28.1201 ] concat %I 5 418 364 379 360 351 346 323 313 321 288 5 BSpl %I 1 End Begin %I BSpl %I b 65520 3 0 1 [12 4] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 79.4 -28.1201 ] concat %I 6 310 229 321 193 348 158 378 146 411 136 434 136 6 BSpl %I 1 End Begin %I Elli %I b 65535 3 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813334 0 0 0.813334 208.72 319.987 ] concat %I 298 522 67 29 Elli End Begin %I Elli %I b 65535 3 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813334 0 0 0.813334 94.04 441.987 ] concat %I 298 522 67 29 Elli End Begin %I Elli %I b 65535 3 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813334 0 0 0.813334 320.96 389.12 ] concat %I 298 522 67 29 Elli End Begin %I Elli %I b 65535 3 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813334 0 0 0.813334 320.96 471.267 ] concat %I 298 522 67 29 Elli End Begin %I Elli %I b 65535 3 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813334 0 0 0.813334 208.72 547.72 ] concat %I 298 522 67 29 Elli End Begin %I BSpl %I b 65535 3 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813334 0 0 0.813334 77.7733 656.707 ] concat %I 5 495 364 524 360 553 355 569 343 585 326 5 BSpl %I 1 End Begin %I BSpl %I b 65535 3 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813334 0 0 0.813334 77.7733 656.707 ] concat %I 3 595 264 595 245 594 223 3 BSpl %I 1 End Begin %I BSpl %I b 65535 3 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813334 0 0 0.813334 77.7733 656.707 ] concat %I 5 584 165 572 152 536 144 504 136 481 136 5 BSpl %I 1 End Begin %I BSpl %I b 65520 3 0 1 [12 4] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813334 0 0 0.813334 77.7733 656.707 ] concat %I 5 418 364 379 360 351 346 323 313 321 288 5 BSpl %I 1 End Begin %I BSpl %I b 65520 3 0 1 [12 4] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813334 0 0 0.813334 77.7733 656.707 ] concat %I 6 310 229 321 193 348 158 378 146 411 136 434 136 6 BSpl %I 1 End Begin %I Elli %I b 65535 3 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 211.973 -138.733 ] concat %I 298 522 67 29 Elli End Begin %I Elli %I b 65535 3 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 97.2933 -16.7333 ] concat %I 298 522 67 29 Elli End Begin %I Elli %I b 65535 3 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 324.213 -69.6 ] concat %I 298 522 67 29 Elli End Begin %I Elli %I b 65535 3 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 324.213 12.5467 ] concat %I 298 522 67 29 Elli End Begin %I Elli %I b 65535 3 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 211.973 89 ] concat %I 298 522 67 29 Elli End Begin %I BSpl %I b 65535 3 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 81.0266 197.987 ] concat %I 5 495 364 524 360 553 355 569 343 585 326 5 BSpl %I 1 End Begin %I BSpl %I b 65535 3 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 81.0266 197.987 ] concat %I 3 595 264 595 245 594 223 3 BSpl %I 1 End Begin %I BSpl %I b 65535 3 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 81.0266 197.987 ] concat %I 5 584 165 572 152 536 144 504 136 481 136 5 BSpl %I 1 End Begin %I BSpl %I b 65520 3 0 1 [12 4] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 81.0266 197.987 ] concat %I 5 418 364 379 360 351 346 323 313 321 288 5 BSpl %I 1 End Begin %I BSpl %I b 65520 3 0 1 [12 4] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 81.0266 197.987 ] concat %I 6 310 229 321 193 348 158 378 146 411 136 434 136 6 BSpl %I 1 End Begin %I Elli %I b 65535 3 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 95.6666 212.627 ] concat %I 298 522 67 29 Elli End Begin %I Elli %I b 65535 3 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 322.587 159.76 ] concat %I 298 522 67 29 Elli End Begin %I Elli %I b 65535 3 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 322.587 241.907 ] concat %I 298 522 67 29 Elli End Begin %I BSpl %I b 65535 3 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 79.4 427.347 ] concat %I 5 495 364 524 360 553 355 569 343 585 326 5 BSpl %I 1 End Begin %I BSpl %I b 65535 3 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 79.4 427.347 ] concat %I 3 595 264 595 245 594 223 3 BSpl %I 1 End Begin %I BSpl %I b 65535 3 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 79.4 427.347 ] concat %I 5 584 165 572 152 536 144 504 136 481 136 5 BSpl %I 1 End Begin %I BSpl %I b 65520 3 0 1 [12 4] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 79.4 427.347 ] concat %I 5 418 364 379 360 351 346 323 313 321 288 5 BSpl %I 1 End Begin %I BSpl %I b 65520 3 0 1 [12 4] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 0.813333 0 0 0.813333 79.4 427.347 ] concat %I 6 310 229 321 193 348 158 378 146 411 136 434 136 6 BSpl %I 1 End Begin %I BSpl %I b 65520 3 0 1 [12 4] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 4 0 0 4 -1327 -723 ] concat %I 2 424 440 437 430 2 BSpl %I 0.25 End Begin %I BSpl %I b 65535 3 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 4 0 0 4 -1327 -723 ] concat %I 2 468 441 454 430 2 BSpl %I 0.25 End Begin %I Line %I b 65520 3 0 1 [12 4] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 2 0 0 2 -461 -99 ] concat %I 451 67 416 51 Line %I 0.5 End Begin %I Line %I b 65535 3 0 1 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg none SetP %I p n %I t [ 2 0 0 2 -461 -99 ] concat %I 467 68 501 50 Line %I 0.5 End End %I eop End %I eop showpage end %%EndDocument endTexFig 262 1082 a Fq(Figure)9 b(2:)16 b Fl(Example)11 b(wher)n(e)f Fo(N)748 1088 y Fs(C,F)o(CFS)883 1082 y Fl(b)n(ehaves)i(slightly)e (worse)g(on)h(aver)n(age)g(than)g Fo(N)1601 1088 y Fs(C,PS)1681 1082 y Fl(,)262 1132 y(given)18 b(exactly)g(two)g(p)n(ackets)h(ar)n(e)f (in)g(the)g(network,)h(and)g(al)r(l)e(servic)n(e)h(times)g(ar)n(e)g(e)n (qual.)262 1181 y(Packets)13 b(fol)r(low)g(the)h(dashe)n(d)g(r)n(oute)f (or)g(the)h(solid)f(r)n(oute)h(with)e(e)n(qual)i(pr)n(ob)n(ability.)k (In)13 b(b)n(oth)262 1231 y(networks)j(two)g(p)n(ackets)h(on)g(opp)n (osite)g(r)n(outes)f(ar)n(e)g(likely)g(to)h(me)n(et)f(b)n(e)n(c)n(ause) h(the)g(dashe)n(d-)262 1281 y(r)n(oute)11 b(p)n(acket)g(wil)r(l)f (likely)g(c)n(atch)i(up)g(to)f(the)g(solid-r)n(oute)g(p)n(acket)g(r)n (e)n(gar)n(d)r(less)g(of)g(the)g(original)262 1331 y(arrival)k(times)h (of)h(the)g(p)n(ackets.)24 b(When)18 b(the)f(two)f(p)n(ackets)h(do)g(c) n(atch)g(up,)h(in)e Fo(N)1543 1337 y Fs(C,PS)1640 1331 y Fl(the)262 1381 y(p)n(ackets)e(wil)r(l)f(interfer)n(e)g(with)h(e)n (ach)g(other)g(at)h(most)f(for)g(the)g(dur)n(ation)g(of)h(one)f (server.)k(In)262 1430 y Fo(N)296 1436 y Fs(C,F)o(CFS)420 1430 y Fl(,)12 b(with)f(pr)n(ob)n(ability)734 1414 y Fi(1)p 734 1421 17 2 v 734 1445 a(2)756 1430 y Fl(,)h(the)g(the)g(p)n (ackets)h(me)n(et)e(onc)n(e)i(mor)n(e)f(and)g(with)g(pr)n(ob)n(ability) 267 1464 y Fi(1)p 267 1471 V 267 1495 a(4)307 1480 y Fl(they)19 b(me)n(et)g(twic)n(e)f(mor)n(e,)h(etc.,)g(thus)g(the)g(exp)n (e)n(cte)n(d)g(delay)h(for)e Fo(N)1377 1486 y Fs(C,F)o(CFS)1520 1480 y Fl(is)g(slightly)262 1530 y(higher)c(than)i(for)e Fo(N)582 1536 y Fs(C,PS)661 1530 y Fl(.)262 1667 y Fq(pac)o(k)o(ets)e (is)f(smaller)g(in)g Fo(N)667 1673 y Fs(C,F)o(CFS)803 1667 y Fq(than)g Fo(N)931 1673 y Fs(C,PS)1011 1667 y Fq(,)g(so)h(certainly)g(the)g(a)o(v)o(erage)g(dela)o(y)f(for)g(the)262 1716 y(case)j(of)g(t)o(w)o(o)f(pac)o(k)o(ets)i(is)e(smaller)g(in)g Fo(N)889 1722 y Fs(C,F)o(CFS)1027 1716 y Fq(than)h Fo(N)1158 1722 y Fs(C,PS)1238 1716 y Fq(.)324 1766 y(If)i(in)g(the)h(ab)q(o)o(v)o (e)f(pro)q(of)h(the)g(routes)g(had)g(crossed)h(rep)q(eatedly)m(,)f(it)f (is)h(p)q(ossible)g(that)262 1816 y(the)f(t)o(w)o(o)g(pac)o(k)o(ets)h (in)f Fo(N)650 1822 y Fs(C,F)o(CFS)790 1816 y Fq(migh)o(t)e(meet)i(rep) q(eatedly)i(with)e(high)f(probabilit)o(y)g(and)262 1866 y(the)g(t)o(w)o(o)f(pac)o(k)o(ets)i(in)e Fo(N)644 1872 y Fs(C,PS)738 1866 y Fq(migh)o(t)f(nev)o(er)j(meet)e(again,)g (resulting)g(in)h(greater)h(a)o(v)o(erage)262 1916 y(dela)o(y)f(for)g Fo(N)471 1922 y Fs(C,F)o(CFS)595 1916 y Fq(.)23 b([3])15 b(sho)o(ws)h(suc)o(h)g(an)f(example)g(in)o(v)o(olving)e(serv)o(ers)k (with)f(di\013eren)o(t)262 1965 y(service)h(times,)d(where)j(the)g(exp) q(ected)h(dela)o(y)d(in)g(the)h(case)h(of)e(t)o(w)o(o)h(pac)o(k)o(ets)g (is)g(\002\()p Fp(n)p Fq(\))g(for)262 2015 y Fo(N)296 2021 y Fs(C,F)o(CFS)434 2015 y Fq(and)e(only)f(\002\(1\))h(for)g Fo(N)803 2021 y Fs(C,PS)883 2015 y Fq(,)f(where)i Fp(n)f Fq(is)g(the)h(n)o(um)o(b)q(er)e(of)h(serv)o(ers)i(in)d Fo(N)6 b Fq(.)19 b(W)m(e)262 2065 y(no)o(w)13 b(question)h(whether)h (this)f(could)f(still)g(happ)q(en)h(in)g(the)g(case)h(where)g(all)d (serv)o(ers)k(ha)o(v)o(e)262 2115 y(the)e(same)f(mean)g(service)i (time,)d(1.)324 2165 y(F)m(or)18 b(net)o(w)o(orks)h(where)h(all)d(serv) o(ers)k(ha)o(v)o(e)d(service)i(time)d(1,)i(ev)o(en)g(if)f(the)h(routes) h(do)262 2214 y(cross)d(rep)q(eatedly)m(,)f(it's)g(tough)g(to)f(\014nd) i(examples)e(where)i Fo(N)1246 2220 y Fs(C,F)o(CFS)1386 2214 y Fq(b)q(eha)o(v)o(es)g(badly)e(in)262 2264 y(the)f(case)h(of)e(t) o(w)o(o)h(pac)o(k)o(ets.)k(The)d(reason)f(is)g(t)o(w)o(ofold:)312 2347 y(1.)20 b(T)m(o)13 b(guaran)o(tee)h(that)g(the)g Fl(aver)n(age)g Fq(dela)o(y)f(is)g(high)g(in)g(the)h(case)h(of)e(t)o(w) o(o)g(pac)o(k)o(ets)h(for)365 2397 y Fo(N)399 2403 y Fs(C,F)o(CFS)524 2397 y Fq(,)f(w)o(e)i(need)g(to)f(b)q(e)h(able)f(to)g (guaran)o(tee)h(that)f(with)g(high)f(probabilit)o(y)g(the)365 2447 y(t)o(w)o(o)h(pac)o(k)o(ets)g(will)f(meet)g(at)h(least)g(once.)967 2574 y(8)p eop %%Page: 9 10 9 9 bop 312 307 a Fq(2.)20 b(If)15 b(t)o(w)o(o)g(pac)o(k)o(ets)g(in)o (terfere)h(with)f(eac)o(h)h(other)f(at)g(all)f(in)h Fo(N)1295 313 y Fs(C,F)o(CFS)1419 307 y Fq(,)g(the)g(next)h(time)365 357 y(that)j(they)g(meet)f(\(if)g(ev)o(er\))i(will)d(b)q(e)i(suc)o(h)g (that)g(they)g(b)q(oth)g(arriv)o(e)f(at)h(a)f(serv)o(er)365 407 y(at)e(the)g(exact)g(same)e(time.)22 b(When)15 b(this)h(happ)q (ens,)g(either)g(pac)o(k)o(et)g(could)g(end)g(up)365 457 y(serving)c(\014rst)g(with)e(equal)h(probabilit)o(y)m(.)k(Since)d (w)o(e)f(don't)g(kno)o(w)g(whic)o(h)g(pac)o(k)o(et)g(will)365 506 y(go)k(\014rst)h(it)f(is)g(di\016cult)g(to)g(construct)i(routes)f (whic)o(h)f(force)h(the)g(pac)o(k)o(ets)g(to)f(meet)365 556 y(again.)k(A)14 b(bad)h(instance)g(for)f Fo(N)881 562 y Fs(C,F)o(CFS)1020 556 y Fq(m)o(ust)f(force)i(the)g(pac)o(k)o(ets) g(to)g(meet)f(again)365 606 y(rep)q(eatedly)g(regardless)f(of)e(whic)o (h)h(pac)o(k)o(et)h(serv)o(ed)g(\014rst)g(in)f(the)g(previous)h (collision.)324 689 y(W)m(e)f(w)o(ere)i(able)f(to)g(come)f(up)i(with)e (a)h(net)o(w)o(ork)g(whic)o(h)g(addressed)i(issue)f(1)f(ab)q(o)o(v)o(e) f(but)262 739 y(didn't)j(address)j(issue)g(2)e(w)o(ell,)g(see)i(Figure) f(2.)26 b(W)m(e)16 b(w)o(ere)i(also)e(able)g(to)g(come)g(up)h(with)262 789 y(a)d(net)o(w)o(ork)i(whic)o(h)f(addressed)i(issue)f(2)f(ab)q(o)o (v)o(e,)f(but)i(not)f(issue)h(1)f(\(i.e.,)f(w)o(e)h(constructed)262 839 y(an)f Fo(N)22 b Fq(suc)o(h)16 b(that,)f(giv)o(en)f(a)h(collision,) f(the)h(exp)q(ected)j(dela)o(y)c(in)h Fo(N)1332 845 y Fs(C,F)o(CFS)1471 839 y Fq(w)o(as)g Fp(O)q Fq(\(lg)7 b Fp(n)p Fq(\),)262 888 y(where)18 b Fp(n)g Fq(w)o(as)g(the)g(n)o(um)o (b)q(er)f(of)g(serv)o(ers)j(in)d Fo(N)6 b Fq(,)18 b(but)g(the)g (probabilit)o(y)e(of)i(t)o(w)o(o)f(pac)o(k)o(ets)262 938 y(colliding)10 b(in)h(the)i(\014rst)g(place)f(w)o(as)g(v)o(ery)g (lo)o(w\).)17 b(W)m(e)12 b(h)o(yp)q(othesize)h(that)f(in)g(the)g(case)h (where)262 988 y(all)8 b(pac)o(k)o(ets)j(ha)o(v)o(e)f(the)h(same)e (service)i(time)e Fo(N)974 994 y Fs(C,F)o(CFS)1108 988 y Fq(nev)o(er)i(b)q(eha)o(v)o(es)g(signi\014can)o(tly)e(w)o(orse)262 1038 y(than)k Fo(N)392 1044 y Fs(C,PS)472 1038 y Fq(.)262 1175 y Fr(4)69 b(F)-6 b(uture)23 b(W)-6 b(ork)262 1266 y Fq(It)12 b(w)o(ould)g(b)q(e)h(useful)f(to)h(c)o(haracterize)h(more)d (precisely)j(exactly)e(whic)o(h)h(net)o(w)o(orks)g(satisfy)262 1316 y(the)h(criterion)g(from)e(Section)j(2.)262 1453 y Fr(5)69 b(Ac)n(kno)n(wledgemen)n(ts)262 1544 y Fq(W)m(e)13 b(thank)g(Da)o(vid)f(W)m(olfe)g(for)h(his)h(help)f(in)g(clarifying)f (some)g(subtle)i(p)q(oin)o(ts)g(in)f(the)h(anal-)262 1594 y(ysis.)k(Thanks)c(also)f(to)h(Micah)f(Adler)i(for)e(man)o(y)f (useful)i(discussions.)262 1731 y Fr(References)262 1822 y Fq([1])19 b(F.)h(Bask)o(ett,)k(K.M.)c(Chandy)m(,)h(R.R.)e(Mun)o(tz,)k (and)d(F.)h(P)o(alacios-Gomez.)36 b(Op)q(en,)326 1872 y(closed,)19 b(and)f(mixed)f(net)o(w)o(orks)h(of)g(queues)h(with)f (di\013eren)o(t)h(classes)g(of)f(customers.)326 1922 y 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b(Systems)f(V)m(olume)g(II:)f(Computer)h(Applic)n (ations)p Fq(.)326 357 y(John)d(Wiley)f(and)h(Sons,)f(New)i(Y)m(ork,)e (1976.)262 440 y([6])19 b(Martin)e(I.)g(Reiman)e(and)j(Burton)g(Simon.) 26 b(Ligh)o(t)17 b(tra\016c)g(limits)f(of)g(so)r(journ)i(time)326 490 y(distributions)10 b(in)h(mark)o(o)o(vian)c(queueing)k(net)o(w)o (orks.)i Fl(Communic)n(ations)f(in)g(Statistics)326 540 y(-)j(Sto)n(chastic)g(Mo)n(dels)p Fq(,)f(4\(2\):191{233,)d(1988.)262 623 y([7])19 b(Martin)11 b(I.)f(Reiman)f(and)i(Burton)h(Simon.)f(Op)q (en)h(queueing)f(systems)h(in)e(ligh)o(t)g(tra\016c.)326 672 y Fl(Mathematics)15 b(of)g(Op)n(er)n(ations)g(R)n(ese)n(ar)n(ch)p Fq(,)e(14\(1\):26{59,)e(1989.)262 756 y([8])19 b(Jean)i(W)m(alrand.)35 b Fl(Intr)n(o)n(duction)21 b(to)g(Queueing)g(Networks)p Fq(.)36 b(Pren)o(tice)22 b(Hall,)e(New)326 805 y(Jersey)m(,)15 b(1989.)957 2574 y(10)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF