1017:
28:
are equations that describe the mean arrival rate of traffic, allowing the arrival rates at individual nodes to be determined. Mitrani notes "if the network is stable, the traffic equations are valid and can be solved."
220:
448:
275:
307:
66:
334:
113:
568:
135:
857:
376:
534:
489:
774:
309:
to this equation, so the mean arrival rates at each of the nodes can be determined given knowledge of the external arrival rates
633:
845:
561:
926:
815:
888:
731:
357:
893:
698:
1039:
1020:
916:
703:
554:
235:
1004:
799:
999:
789:
638:
830:
693:
820:
741:
660:
672:
989:
969:
964:
736:
285:
44:
994:
979:
946:
840:
312:
91:
21:
611:
984:
883:
794:
784:
530:
518:
485:
835:
779:
665:
477:
623:
852:
719:
577:
505:
38:
17:
862:
825:
921:
523:
1033:
974:
959:
936:
748:
348:
is surely non-singular as otherwise in the long run the network would become empty.
931:
758:
84:
arrivals from each of the other nodes on the network. If external arrivals at node
481:
360:
there are no external arrivals, so the traffic equations take the form (for
954:
911:
878:
655:
650:
645:
628:
618:
606:
601:
596:
591:
76:
arrivals (that is, arrivals from outside the network directly placed onto node
677:
753:
215:{\displaystyle \lambda _{i}=\gamma _{i}+\sum _{j=1}^{m}p_{ji}\lambda _{j}.}
525:
Performance
Modelling of Communication Networks and Computer Architectures
508:
article, jobs travel among the nodes following a fixed routing matrix.
546:
550:
443:{\displaystyle \lambda _{i}=\sum _{j=1}^{m}p_{ji}\lambda _{j}.}
379:
315:
288:
238:
138:
94:
47:
945:
904:
871:
808:
767:
712:
686:
584:
522:
442:
328:
301:
269:
214:
107:
60:
562:
8:
282:and there is a unique solution of unknowns
569:
555:
547:
472:Mitrani, I. (1997). "Queueing networks".
467:
465:
431:
418:
408:
397:
384:
378:
320:
314:
293:
287:
263:
237:
203:
190:
180:
169:
156:
143:
137:
99:
93:
52:
46:
270:{\displaystyle \lambda (I-P)=\gamma \,,}
461:
364: = 1, 2, ...,
123: = 1, 2, ...,
119:, the traffic equations are, (for
20:, a discipline within the mathematical
227:This can be written in matrix form as
72:in the network is given by the sum of
7:
14:
1016:
1015:
254:
242:
1:
846:Flow-equivalent server method
927:Adversarial queueing network
816:Continuous-time Markov chain
482:10.1017/CBO9781139173087.005
302:{\displaystyle \lambda _{i}}
115:, and the routing matrix is
61:{\displaystyle \lambda _{i}}
889:Heavy traffic approximation
634:Pollaczek–Khinchine formula
521:; Patel, Naresh M. (1992).
329:{\displaystyle \gamma _{i}}
108:{\displaystyle \gamma _{i}}
1056:
1013:
894:Reflected Brownian motion
699:Markovian arrival process
917:Layered queueing network
704:Rational arrival process
41:, the mean arrival rate
1005:Teletraffic engineering
800:Shortest remaining time
474:Probabilistic Modelling
1000:Scheduling (computing)
639:Matrix analytic method
444:
413:
330:
303:
271:
216:
185:
109:
62:
831:Product-form solution
732:Gordon–Newell theorem
694:Poisson point process
585:Single queueing nodes
445:
393:
358:Gordon–Newell network
352:Gordon–Newell network
331:
304:
272:
217:
165:
110:
63:
22:theory of probability
858:Decomposition method
504:As explained in the
476:. pp. 122–155.
377:
313:
286:
236:
136:
92:
45:
990:Pipeline (software)
970:Flow control (data)
965:Erlang distribution
947:Information systems
737:Mean value analysis
344: −
995:Quality of service
980:Network congestion
841:Quasireversibility
821:Kendall's notation
529:. Addison-Wesley.
519:Harrison, Peter G.
440:
326:
299:
267:
212:
105:
58:
1027:
1026:
985:Network scheduler
884:Mean-field theory
795:Shortest job next
785:Processor sharing
742:Buzen's algorithm
725:Traffic equations
713:Queueing networks
687:Arrival processes
661:Kingman's formula
26:traffic equations
1047:
1019:
1018:
836:Balance equation
768:Service policies
666:Lindley equation
571:
564:
557:
548:
541:
540:
528:
515:
509:
502:
496:
495:
469:
449:
447:
446:
441:
436:
435:
426:
425:
412:
407:
389:
388:
335:
333:
332:
327:
325:
324:
308:
306:
305:
300:
298:
297:
276:
274:
273:
268:
221:
219:
218:
213:
208:
207:
198:
197:
184:
179:
161:
160:
148:
147:
114:
112:
111:
106:
104:
103:
67:
65:
64:
59:
57:
56:
1055:
1054:
1050:
1049:
1048:
1046:
1045:
1044:
1040:Queueing theory
1030:
1029:
1028:
1023:
1009:
941:
900:
867:
853:Arrival theorem
804:
763:
720:Jackson network
708:
682:
673:Fork–join queue
612:Burke's theorem
580:
578:Queueing theory
575:
545:
544:
537:
517:
516:
512:
506:Jackson network
503:
499:
492:
471:
470:
463:
458:
427:
414:
380:
375:
374:
354:
336:and the matrix
316:
311:
310:
289:
284:
283:
234:
233:
199:
186:
152:
139:
134:
133:
95:
90:
89:
80:, if any), and
48:
43:
42:
39:Jackson network
35:
33:Jackson network
18:queueing theory
12:
11:
5:
1053:
1051:
1043:
1042:
1032:
1031:
1025:
1024:
1014:
1011:
1010:
1008:
1007:
1002:
997:
992:
987:
982:
977:
972:
967:
962:
957:
951:
949:
943:
942:
940:
939:
934:
929:
924:
922:Polling system
919:
914:
908:
906:
902:
901:
899:
898:
897:
896:
886:
881:
875:
873:
872:Limit theorems
869:
868:
866:
865:
860:
855:
850:
849:
848:
843:
838:
828:
823:
818:
812:
810:
806:
805:
803:
802:
797:
792:
787:
782:
777:
771:
769:
765:
764:
762:
761:
756:
751:
746:
745:
744:
739:
729:
728:
727:
716:
714:
710:
709:
707:
706:
701:
696:
690:
688:
684:
683:
681:
680:
675:
670:
669:
668:
663:
653:
648:
643:
642:
641:
636:
626:
621:
616:
615:
614:
604:
599:
594:
588:
586:
582:
581:
576:
574:
573:
566:
559:
551:
543:
542:
535:
510:
497:
490:
460:
459:
457:
454:
453:
452:
451:
450:
439:
434:
430:
424:
421:
417:
411:
406:
403:
400:
396:
392:
387:
383:
353:
350:
323:
319:
296:
292:
280:
279:
278:
277:
266:
262:
259:
256:
253:
250:
247:
244:
241:
225:
224:
223:
222:
211:
206:
202:
196:
193:
189:
183:
178:
175:
172:
168:
164:
159:
155:
151:
146:
142:
102:
98:
55:
51:
34:
31:
13:
10:
9:
6:
4:
3:
2:
1052:
1041:
1038:
1037:
1035:
1022:
1012:
1006:
1003:
1001:
998:
996:
993:
991:
988:
986:
983:
981:
978:
976:
975:Message queue
973:
971:
968:
966:
963:
961:
960:Erlang (unit)
958:
956:
953:
952:
950:
948:
944:
938:
937:Retrial queue
935:
933:
930:
928:
925:
923:
920:
918:
915:
913:
910:
909:
907:
903:
895:
892:
891:
890:
887:
885:
882:
880:
877:
876:
874:
870:
864:
861:
859:
856:
854:
851:
847:
844:
842:
839:
837:
834:
833:
832:
829:
827:
824:
822:
819:
817:
814:
813:
811:
807:
801:
798:
796:
793:
791:
788:
786:
783:
781:
778:
776:
773:
772:
770:
766:
760:
757:
755:
752:
750:
749:Kelly network
747:
743:
740:
738:
735:
734:
733:
730:
726:
723:
722:
721:
718:
717:
715:
711:
705:
702:
700:
697:
695:
692:
691:
689:
685:
679:
676:
674:
671:
667:
664:
662:
659:
658:
657:
654:
652:
649:
647:
644:
640:
637:
635:
632:
631:
630:
627:
625:
622:
620:
617:
613:
610:
609:
608:
605:
603:
600:
598:
595:
593:
590:
589:
587:
583:
579:
572:
567:
565:
560:
558:
553:
552:
549:
538:
536:0-201-54419-9
532:
527:
526:
520:
514:
511:
507:
501:
498:
493:
491:9781139173087
487:
483:
479:
475:
468:
466:
462:
455:
437:
432:
428:
422:
419:
415:
409:
404:
401:
398:
394:
390:
385:
381:
373:
372:
371:
370:
369:
367:
363:
359:
351:
349:
347:
343:
340:. The matrix
339:
321:
317:
294:
290:
264:
260:
257:
251:
248:
245:
239:
232:
231:
230:
229:
228:
209:
204:
200:
194:
191:
187:
181:
176:
173:
170:
166:
162:
157:
153:
149:
144:
140:
132:
131:
130:
129:
128:
126:
122:
118:
100:
96:
87:
83:
79:
75:
71:
68:at each node
53:
49:
40:
32:
30:
27:
23:
19:
932:Loss network
863:Beneš method
826:Little's law
809:Key concepts
759:BCMP network
724:
524:
513:
500:
473:
365:
361:
355:
345:
341:
337:
281:
226:
124:
120:
116:
85:
81:
77:
73:
69:
36:
25:
15:
955:Data buffer
912:Fluid queue
879:Fluid limit
790:Round-robin
656:G/G/1 queue
651:G/M/1 queue
646:M/G/k queue
629:M/G/1 queue
624:M/M/∞ queue
619:M/M/c queue
607:M/M/1 queue
602:M/D/c queue
597:M/D/1 queue
592:D/M/1 queue
905:Extensions
678:Bulk queue
88:have rate
754:G-network
429:λ
395:∑
382:λ
318:γ
291:λ
261:γ
249:−
240:λ
201:λ
167:∑
154:γ
141:λ
97:γ
50:λ
1034:Category
1021:Category
82:internal
74:external
533:
488:
456:Notes
356:In a
37:In a
780:LIFO
775:FIFO
531:ISBN
486:ISBN
478:doi
16:In
1036::
484:.
464:^
368:)
127:)
24:,
570:e
563:t
556:v
539:.
494:.
480::
438:.
433:j
423:i
420:j
416:p
410:m
405:1
402:=
399:j
391:=
386:i
366:m
362:i
346:P
342:I
338:P
322:i
295:i
265:,
258:=
255:)
252:P
246:I
243:(
210:.
205:j
195:i
192:j
188:p
182:m
177:1
174:=
171:j
163:+
158:i
150:=
145:i
125:m
121:i
117:P
101:i
86:i
78:i
70:i
54:i
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