237:. By introducing "negative customers" which can destroy or eliminate other customers, he generalised the family of product form networks. Then this was further extended in several steps, first by Gelenbe's "triggers" which are customers which have the power of moving other customers from some queue to another. Another new form of customer that also led to product form was Gelenbe's "batch removal". This was further extended by Erol Gelenbe and Jean-Michel Fourneau with customer types called "resets" which can model the repair of failures: when a queue hits the empty state, representing (for instance) a failure, the queue length can jump back or be "reset" to its steady-state distribution by an arriving reset customer, representing a repair. All these previous types of customers in G-Networks can exist in the same network, including with multiple classes, and they all together still result in the product form solution, taking us far beyond the reversible networks that had been considered before.
2050:
313:. Walrand and Varaiya suggest that non-overtaking (where customers cannot overtake other customers by taking a different route through the network) may be a necessary condition for the result to hold. Mitrani offers exact solutions to some simple networks with overtaking, showing that none of these exhibit product-form sojourn time distributions.
365:
254:
note that "virtually all of the models that have been successfully analyzed in classical queueing network theory are models having a so-called product-form stationary distribution" More recently, product-form solutions have been published for Markov process algebras (e.g.
169:
232:
model was the first to show that this is not the case. Motivated by the need to model biological neurons which have a point-process like spiking behaviour, he introduced the precursor of G-Networks, calling it the
332:
Semi-product-form solutions are solutions where a distribution can be written as a product where terms have a limited functional dependency on the global state space, which can be approximated.
29:
is a particularly efficient form of solution for determining some metric of a system with distinct sub-components, where the metric for the collection of components can be written as a
329:
Approximate product-form solutions are computed assuming independent marginal distributions, which can give a good approximation to the stationary distribution under some conditions.
214:
gives the joint equilibrium distribution of an open queueing network as the product of the equilibrium distributions of the individual queues. After numerous extensions, chiefly the
43:
316:
For closed networks, Chow showed a result to hold for two service nodes, which was later generalised to a cycle of queues and to overtake–free paths in
1601:
1890:
211:
240:
Product-form solutions are sometimes described as "stations are independent in equilibrium". Product form solutions also exist in networks of
1568:
1488:
909:
925:
Anderson, D. F.; Craciun, G.; Kurtz, T. G. (2010). "Product-Form
Stationary Distributions for Deficiency Zero Chemical Reaction Networks".
338:
solutions which are not the product of marginal densities, but the marginal densities describe the distribution in a product-type manner or
1506:
203:
751:
1807:
178:
is some constant. Solutions of this form are of interest as they are computationally inexpensive to evaluate for large values of
1666:
1413:
Baynat, B.; Dallery, Y. (1993). "A unified view of product-form approximation techniques for general closed queueing networks".
671:; Williams, R. J. (1992). "Brownian models of feedforward queueing networks: quasireversibility and product-form solutions".
1878:
1594:
1551:
Angius, A.; Horváth, A. S.; Wolf, V. (2013). "Approximate
Transient Analysis of Queuing Networks by Quasi Product Forms".
673:
297:
has also been used to refer to the sojourn time distribution in a cyclic queueing system, where the time spent by jobs at
1959:
1848:
1921:
1375:
1068:
1764:
710:
Henderson, W.; Taylor, P. G. (1990). "Product form in networks of queues with batch arrivals and batch services".
317:
1926:
1731:
275:
206:
sub-components exhibit a product-form solution by the definition of independence. Initially the term was used in
2072:
2053:
1949:
1736:
1587:
195:
393:
285:, and furthermore that such networks can be used to approximate bounded and continuous real-valued functions.
779:
2037:
1832:
798:
341:
approximate form for transient probability distributions which allows transient moments to be approximated.
2032:
1822:
1671:
1028:
887:
682:
1726:
282:
234:
30:
1853:
1774:
1693:
1504:
Dębicki, K.; Dieker, A. B.; Rolski, T. (2007). "Quasi-Product Forms for Levy-Driven Fluid
Networks".
1705:
892:
2022:
2002:
1997:
1769:
1033:
687:
668:
247:
34:
301:
nodes is given as the product of time spent at each node. In 1957 Reich showed the result for two
2027:
2012:
1979:
1873:
1533:
1515:
1395:
1357:
1322:
1281:
1262:
1238:
1203:
1096:
1077:
998:
960:
934:
729:
587:
549:
507:
469:
450:
423:
183:
22:
1644:
1440:
Dallery, Y.; Cao, X. R. (1992). "Operational analysis of stochastic closed queueing networks".
490:(1989). "Random Neural Networks with Negative and Positive Signals and Product Form Solution".
2017:
1916:
1827:
1817:
1757:
1564:
1484:
1468:
1046:
952:
905:
842:
794:
164:{\displaystyle {\text{P}}(x_{1},x_{2},x_{3},\ldots ,x_{n})=B\prod _{i=1}^{n}{\text{P}}(x_{i})}
1019:; Mao, Zhi-Hong; Li, Yan-Da (1991). "Function approximation with the random neural network".
1868:
1812:
1698:
1556:
1525:
1476:
1449:
1422:
1387:
1349:
1312:
1271:
1230:
1195:
1160:
1127:
1086:
1038:
990:
944:
897:
862:
854:
810:
774:
766:
721:
712:
692:
647:
617:
579:
541:
499:
459:
413:
405:
374:
360:
251:
1656:
1186:; Varaiya, P. (1980). "Sojourn Times and the Overtaking Condition in Jacksonian Networks".
1885:
1752:
1610:
310:
271:
207:
1895:
1858:
281:
The work by
Gelenbe also shows that product form G-Networks can be used to model spiking
1954:
1234:
441:
1340:
Daduna, H. (1982). "Passage Times for
Overtake-Free Paths in Gordon-Newell Networks".
815:
770:
651:
2066:
2007:
1992:
1969:
1781:
1453:
1426:
1064:
747:
511:
219:
1537:
1326:
1285:
1002:
733:
529:
427:
1964:
1791:
1183:
1100:
1016:
978:
964:
882:
Mairesse, J.; Nguyen, H. T. (2009). "Deficiency Zero Petri Nets and
Product Form".
635:
605:
567:
525:
487:
473:
225:
215:
199:
1480:
1987:
1944:
1911:
1688:
1683:
1678:
1661:
1651:
1639:
1634:
1629:
1624:
1560:
1471:(2010). "State-Dependent Rates and Semi-Product-Form via the Reversed Process".
901:
309:
M/M/1 queues in tandem and it has been shown to apply to overtake–free paths in
302:
858:
1710:
1165:
1148:
1132:
1115:
1073:"The Product Form for Sojourn Time Distributions in Cyclic Exponential Queues"
994:
948:
696:
621:
503:
267:
264:
241:
1786:
229:
1529:
1050:
956:
867:
1317:
1300:
1276:
1257:
464:
445:
378:
394:"Local balance in queueing networks with positive and negative customers"
1221:
Mitrani, I. (1985). "Response Time
Problems in Communication Networks".
1091:
1072:
1399:
1361:
1242:
1207:
725:
591:
553:
409:
1378:; Pollett, P. K. (1983). "Sojourn Times in Closed Queueing Networks".
1042:
418:
1520:
530:"Product-form queueing networks with negative and positive customers"
1391:
1353:
1199:
583:
545:
1223:
Journal of the Royal
Statistical Society. Series B (Methodological)
1579:
1475:. Lecture Notes in Computer Science. Vol. 6342. p. 207.
939:
886:. Lecture Notes in Computer Science. Vol. 5606. p. 103.
210:
where the sub-components would be individual queues. For example,
1555:. Lecture Notes in Computer Science. Vol. 7984. p. 22.
820:
260:
256:
182:. Such solutions in queueing networks are important for finding
1583:
186:
in models of multiprogrammed and time-shared computer systems.
1553:
Analytical and
Stochastic Modeling Techniques and Applications
1301:"The Time for a Round Trip in a Cycle of Exponential Queues"
274:'s deficiency zero theorem gives a sufficient condition for
981:(1993). "Learning in the recurrent random neural network".
1258:"The Cycle Time Distribution of Exponential Cyclic Queues"
845:(2012). "Analysis of stochastic Petri nets with signals".
638:; Fourneau, Jean-Michel (2002). "G-Networks with resets".
610:
Probability in the
Engineering and Informational Sciences
608:(1993). "G-Networks with triggered customer movement".
570:(1993). "G-networks with triggered customer movement".
446:"Product form and local balance in queueing networks"
46:
33:
of the metric across the different components. Using
1978:
1937:
1904:
1841:
1800:
1745:
1719:
1617:
663:
661:
278:to exhibit a product-form stationary distribution.
752:"Product form solution for a class of PEPA models"
163:
780:20.500.11820/13c57018-5854-4f34-a4c9-833262a71b7c
799:"Turning back time in Markovian process algebra"
194:The first product-form solutions were found for
392:Boucherie, Richard J.; van Dijk, N. M. (1994).
222:was a requirement for a product-form solution.
1595:
8:
202:. Trivially, models composed of two or more
444:; Howard, J. H. Jr; Towsley, D. F. (1977).
37:a product-form solution has algebraic form
1602:
1588:
1580:
1178:
1176:
1519:
1316:
1275:
1164:
1131:
1116:"Waiting Times when Queues are in Tandem"
1090:
1032:
938:
891:
866:
814:
778:
686:
463:
417:
363:(1963). "Jobshop-like queueing systems".
152:
140:
134:
123:
104:
85:
72:
59:
47:
45:
335:Quasi-product-form solutions are either
352:
18:Form of solution in probability theory
1299:Schassberger, R.; Daduna, H. (1983).
1153:The Annals of Mathematical Statistics
1120:The Annals of Mathematical Statistics
884:Applications and Theory of Petri Nets
7:
1021:IEEE Transactions on Neural Networks
305:in tandem, later extending this to
1507:Mathematics of Operations Research
1235:10.1111/j.2517-6161.1985.tb01368.x
14:
1071:; Konheim, A. G. (January 1984).
2049:
2048:
1473:Computer Performance Engineering
927:Bulletin of Mathematical Biology
1380:Advances in Applied Probability
1342:Advances in Applied Probability
1188:Advances in Applied Probability
572:Journal of Applied Probability
534:Journal of Applied Probability
158:
145:
110:
52:
1:
1879:Flow-equivalent server method
816:10.1016/S0304-3975(02)00375-4
771:10.1016/S0166-5316(99)00005-X
674:Annals of Applied Probability
652:10.1016/S0166-5316(02)00127-X
398:Annals of Operations Research
1960:Adversarial queueing network
1849:Continuous-time Markov chain
1481:10.1007/978-3-642-15784-4_14
1454:10.1016/0166-5316(92)90019-D
1427:10.1016/0166-5316(93)90017-O
803:Theoretical Computer Science
1922:Heavy traffic approximation
1667:Pollaczek–Khinchine formula
1561:10.1007/978-3-642-39408-9_3
1256:Chow, We-Min (April 1980).
902:10.1007/978-3-642-02424-5_8
2089:
1149:"Note on Queues in Tandem"
859:10.1016/j.peva.2012.06.003
289:Sojourn time distributions
276:chemical reaction networks
2046:
1927:Reflected Brownian motion
1732:Markovian arrival process
995:10.1162/neco.1993.5.1.154
949:10.1007/s11538-010-9517-4
622:10.1017/S0269964800002953
504:10.1162/neco.1989.1.4.502
196:equilibrium distributions
190:Equilibrium distributions
1950:Layered queueing network
1737:Rational arrival process
841:Marin, A.; Balsamo, S.;
2038:Teletraffic engineering
1833:Shortest remaining time
1166:10.1214/aoms/1177704275
1133:10.1214/aoms/1177706889
697:10.1214/aoap/1177005704
2033:Scheduling (computing)
1672:Matrix analytic method
1530:10.1287/moor.1070.0259
1442:Performance Evaluation
1415:Performance Evaluation
847:Performance Evaluation
759:Performance Evaluation
640:Performance Evaluation
318:Gordon–Newell networks
283:random neural networks
165:
139:
1864:Product-form solution
1765:Gordon–Newell theorem
1727:Poisson point process
1618:Single queueing nodes
1318:10.1145/322358.322369
1277:10.1145/322186.322193
1114:Reich, Edgar (1957).
750:; Thomas, N. (1999).
465:10.1145/322003.322009
379:10.1287/mnsc.10.1.131
235:random neural network
166:
119:
27:product-form solution
1891:Decomposition method
44:
2023:Pipeline (software)
2003:Flow control (data)
1998:Erlang distribution
1980:Information systems
1770:Mean value analysis
1092:10.1145/2422.322419
184:performance metrics
35:capital Pi notation
2028:Quality of service
2013:Network congestion
1874:Quasireversibility
1854:Kendall's notation
1469:Harrison, Peter G.
1305:Journal of the ACM
1263:Journal of the ACM
1147:Reich, E. (1963).
1078:Journal of the ACM
983:Neural Computation
726:10.1007/BF02411466
492:Neural Computation
451:Journal of the ACM
410:10.1007/BF02033315
366:Management Science
161:
23:probability theory
2060:
2059:
2018:Network scheduler
1917:Mean-field theory
1828:Shortest job next
1818:Processor sharing
1775:Buzen's algorithm
1758:Traffic equations
1746:Queueing networks
1720:Arrival processes
1694:Kingman's formula
1570:978-3-642-39407-2
1490:978-3-642-15783-7
1043:10.1109/72.737488
911:978-3-642-02423-8
361:Jackson, James R.
212:Jackson's theorem
208:queueing networks
143:
50:
2080:
2052:
2051:
1869:Balance equation
1801:Service policies
1699:Lindley equation
1604:
1597:
1590:
1581:
1575:
1574:
1548:
1542:
1541:
1523:
1501:
1495:
1494:
1464:
1458:
1457:
1437:
1431:
1430:
1410:
1404:
1403:
1372:
1366:
1365:
1337:
1331:
1330:
1320:
1296:
1290:
1289:
1279:
1253:
1247:
1246:
1218:
1212:
1211:
1194:(4): 1000–1018.
1180:
1171:
1170:
1168:
1144:
1138:
1137:
1135:
1111:
1105:
1104:
1094:
1061:
1055:
1054:
1036:
1013:
1007:
1006:
975:
969:
968:
942:
933:(8): 1947–1970.
922:
916:
915:
895:
879:
873:
872:
870:
838:
832:
831:
829:
828:
819:. Archived from
818:
809:(3): 1947–2013.
791:
785:
784:
782:
765:(3–4): 171–192.
756:
744:
738:
737:
713:Queueing Systems
707:
701:
700:
690:
665:
656:
655:
632:
626:
625:
602:
596:
595:
564:
558:
557:
522:
516:
515:
484:
478:
477:
467:
438:
432:
431:
421:
389:
383:
382:
357:
311:Jackson networks
170:
168:
167:
162:
157:
156:
144:
141:
138:
133:
109:
108:
90:
89:
77:
76:
64:
63:
51:
48:
2088:
2087:
2083:
2082:
2081:
2079:
2078:
2077:
2073:Queueing theory
2063:
2062:
2061:
2056:
2042:
1974:
1933:
1900:
1886:Arrival theorem
1837:
1796:
1753:Jackson network
1741:
1715:
1706:Fork–join queue
1645:Burke's theorem
1613:
1611:Queueing theory
1608:
1578:
1571:
1550:
1549:
1545:
1503:
1502:
1498:
1491:
1467:Thomas, Nigel;
1466:
1465:
1461:
1439:
1438:
1434:
1412:
1411:
1407:
1392:10.2307/1426623
1374:
1373:
1369:
1354:10.2307/1426680
1339:
1338:
1334:
1298:
1297:
1293:
1255:
1254:
1250:
1220:
1219:
1215:
1200:10.2307/1426753
1182:
1181:
1174:
1146:
1145:
1141:
1113:
1112:
1108:
1063:
1062:
1058:
1015:
1014:
1010:
977:
976:
972:
924:
923:
919:
912:
893:10.1.1.745.1585
881:
880:
876:
853:(11): 551–572.
843:Harrison, P. G.
840:
839:
835:
826:
824:
795:Harrison, P. G.
793:
792:
788:
754:
746:
745:
741:
709:
708:
704:
669:Harrison, J. M.
667:
666:
659:
634:
633:
629:
604:
603:
599:
584:10.2307/3214781
566:
565:
561:
546:10.2307/3214499
524:
523:
519:
486:
485:
481:
442:Chandy, K. Mani
440:
439:
435:
391:
390:
386:
359:
358:
354:
350:
326:
291:
272:Martin Feinberg
218:it was thought
192:
148:
100:
81:
68:
55:
42:
41:
19:
12:
11:
5:
2086:
2084:
2076:
2075:
2065:
2064:
2058:
2057:
2047:
2044:
2043:
2041:
2040:
2035:
2030:
2025:
2020:
2015:
2010:
2005:
2000:
1995:
1990:
1984:
1982:
1976:
1975:
1973:
1972:
1967:
1962:
1957:
1955:Polling system
1952:
1947:
1941:
1939:
1935:
1934:
1932:
1931:
1930:
1929:
1919:
1914:
1908:
1906:
1905:Limit theorems
1902:
1901:
1899:
1898:
1893:
1888:
1883:
1882:
1881:
1876:
1871:
1861:
1856:
1851:
1845:
1843:
1839:
1838:
1836:
1835:
1830:
1825:
1820:
1815:
1810:
1804:
1802:
1798:
1797:
1795:
1794:
1789:
1784:
1779:
1778:
1777:
1772:
1762:
1761:
1760:
1749:
1747:
1743:
1742:
1740:
1739:
1734:
1729:
1723:
1721:
1717:
1716:
1714:
1713:
1708:
1703:
1702:
1701:
1696:
1686:
1681:
1676:
1675:
1674:
1669:
1659:
1654:
1649:
1648:
1647:
1637:
1632:
1627:
1621:
1619:
1615:
1614:
1609:
1607:
1606:
1599:
1592:
1584:
1577:
1576:
1569:
1543:
1514:(3): 629–647.
1496:
1489:
1459:
1432:
1421:(3): 205–224.
1405:
1386:(3): 638–656.
1367:
1348:(3): 672–686.
1332:
1291:
1270:(2): 281–286.
1248:
1229:(3): 396–406.
1213:
1172:
1139:
1126:(3): 768–773.
1106:
1085:(1): 128–133.
1056:
1034:10.1.1.46.7710
1008:
989:(1): 154–164.
970:
917:
910:
874:
833:
786:
739:
702:
688:10.1.1.56.1572
681:(2): 263–293.
657:
646:(1): 179–191.
627:
616:(3): 335–342.
597:
578:(3): 742–748.
559:
540:(3): 656–663.
517:
498:(4): 502–510.
479:
458:(2): 250–263.
433:
404:(5): 463–492.
384:
373:(1): 131–142.
351:
349:
346:
345:
344:
343:
342:
339:
333:
330:
325:
322:
290:
287:
191:
188:
172:
171:
160:
155:
151:
147:
137:
132:
129:
126:
122:
118:
115:
112:
107:
103:
99:
96:
93:
88:
84:
80:
75:
71:
67:
62:
58:
54:
17:
13:
10:
9:
6:
4:
3:
2:
2085:
2074:
2071:
2070:
2068:
2055:
2045:
2039:
2036:
2034:
2031:
2029:
2026:
2024:
2021:
2019:
2016:
2014:
2011:
2009:
2008:Message queue
2006:
2004:
2001:
1999:
1996:
1994:
1993:Erlang (unit)
1991:
1989:
1986:
1985:
1983:
1981:
1977:
1971:
1970:Retrial queue
1968:
1966:
1963:
1961:
1958:
1956:
1953:
1951:
1948:
1946:
1943:
1942:
1940:
1936:
1928:
1925:
1924:
1923:
1920:
1918:
1915:
1913:
1910:
1909:
1907:
1903:
1897:
1894:
1892:
1889:
1887:
1884:
1880:
1877:
1875:
1872:
1870:
1867:
1866:
1865:
1862:
1860:
1857:
1855:
1852:
1850:
1847:
1846:
1844:
1840:
1834:
1831:
1829:
1826:
1824:
1821:
1819:
1816:
1814:
1811:
1809:
1806:
1805:
1803:
1799:
1793:
1790:
1788:
1785:
1783:
1782:Kelly network
1780:
1776:
1773:
1771:
1768:
1767:
1766:
1763:
1759:
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1311:: 146–150.
1184:Walrand, J.
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242:bulk queues
204:independent
1938:Extensions
1711:Bulk queue
1027:(1): 3–9.
827:2015-08-29
419:1871/12327
348:References
324:Extensions
268:petri nets
265:stochastic
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