2380:
1286:, which requires log-quadratic time. The method of moments can solve in practice models with up to 10 classes of customers or sometimes larger, which are typically inaccessible by means of exact MVA. This technique however does not use the arrival theorem and relies on solving systems of linear equations involving the
71:
customers circulating in the system. Suppose that the customers are indistinguishable from each other, so that the network has a single class of customers. To compute the mean queue length and waiting time at each of the nodes and throughput of the system we use an iterative algorithm starting with a
1278:
For networks with a single customer class the MVA algorithm is very fast and time taken grows linearly with the number of customers and number of queues. However, in multiclass models the number of multiplications and additions and the storage requirements for MVA grow exponentially with the number
36:
queue lengths, waiting time at queueing nodes and throughput in equilibrium for a closed separable system of queues. The first approximate techniques were published independently by
Schweitzer and Bard, followed later by an exact version by Lavenberg and Reiser published in 1980.
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of customer classes. Practically, the algorithm works well for 3-4 customer classes, although this generally depends on the implementation and the structure of the model. For example, the Tree-MVA method can scale to larger models if the routing matrix is sparse.
814:
1116:
933:
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Approximate MVA (AMVA) algorithms, such as the Bard-Schweitzer method, offer instead an alternative solution technique that provides low complexity also on multiclass networks and typically deliver highly accurate results.
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which can be solved numerically. This iterative approach often goes under the name of approximate MVA (AMVA) and it is typically faster than the recursive approach of MVA.
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The algorithm starts with an empty network (zero customers), then increases the number of customers by 1 at each iteration until there are the required number (
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48:-customer closed system arrives at a service facility he/she observes the rest of the system to be in the equilibrium state for a system with
1866:
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1454:
1421:
1414:
Proceedings of the Third
International Symposium on Modelling and Performance Evaluation of Computer Systems: Performance of Computer Systems
1396:
352:
245:
1744:
Zahorjan, John; Eager, Derek L.; Sweillam, Hisham M. (1988). "Accuracy, speed, and convergence of approximate mean value analysis".
2137:
1996:
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1327:
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Schweitzer, P. J.; Serazzi, G.; Broglia, M. (1993). "A survey of bottleneck analysis in closed networks of queues".
967:, although certain restrictions exist in the case of first-come first-served stations due to the assumptions of the
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809:{\displaystyle \lambda _{m}={\frac {m}{\sum _{k=1}^{K}{\frac {{\frac {m-1}{m}}L_{k}(m)+1}{\mu _{k}}}v_{k}}}}
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2001:
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1441:. International Series in Operations Research & Management Science. Vol. 154. pp. 561–586.
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21:
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1111:{\displaystyle W_{k,r}(\mathbb {m} )={\frac {1+L_{k}\left(\mathbb {m} -1_{r}\right)}{\mu _{k,r}}}.}
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1705:"CoMoM: A Class-Oriented Algorithm for Probabilistic Evaluation of Multiclass Queueing Networks"
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215:. (This sets the average queue length in a system with no customers to zero at all nodes.)
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17:
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2188:
1573:
Schweitzer, Paul (1979). "Approximate analysis of multiclass closed networks of queues".
447:
3. Finally, use Little's law applied to each queue to compute the mean queue lengths for
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928:{\displaystyle L_{k}(m)=v_{k}\lambda _{m}{\frac {{\frac {m-1}{m}}L_{k}(m)+1}{\mu _{k}}}}
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Exact values for mean performance metrics can be obtained in large models using the
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which is a linear interpolation. From the above formulas, this approximation yields
2294:
2121:
1876:
1835:
1499:
968:
1412:
Bard, Yonathan (1979). "Some
Extensions to Multiclass Queueing Network Analysis".
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2018:
2013:
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Thomas, N.; Zhao, Y. (2010). "Mean value analysis for a class of PEPA models".
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The Bard–Schweitzer approximation estimates the average number of jobs at node
145:
customers in the system (this includes the job currently being served at queue
109:
for service. To use the algorithm, we first compute the visit ratio row vector
2040:
1896:. Handout from Helsinki Tech gives good overview of Jackson's Theorem and MVA.
1575:
Proceedings of
International Conference on Stochastic Control and Optimization
1337:
1522:
2116:
1827:
1785:
1588:
Tay, Y. C. (2010). "Analytical
Performance Modeling for Computer Systems".
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1471:
1853:. Lecture Notes in Computer Science. Vol. 8657. pp. 174–177.
1517:. Lecture Notes in Computer Science. Vol. 1769. pp. 491–504.
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1323:
434:{\displaystyle \lambda _{m}={\frac {m}{\sum _{k=1}^{K}W_{k}(m)v_{k}}}.}
1346:, a MATLAB toolbox that includes exact and approximate MVA algorithms.
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328:{\displaystyle W_{k}(m)={\frac {1+L_{k}\left(m-1\right)}{\mu _{k}}}.}
1909:
101:
denotes the probability that a customer finishing service at node
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Hoyme, K. P.; Bruell, S. C.; Afshari, P. V.; Kain, R. Y. (1986).
1383:. Lecture Notes in Computer Science. Vol. 729. p. 491.
1302:
The mean value analysis algorithm has been applied to a class of
238:
compute the waiting time at each node using the arrival theorem:
1303:
170:
customers in the system. Denote the throughput of a system with
1913:
1628:"Exact analysis of performance models by the Method of Moments"
1513:
Reiser, M. (2000). "Mean Value
Analysis: A Personal Account".
1381:
Performance
Evaluation of Computer and Communication Systems
991:
can still be related to the total mean queue-length at node
1472:"Mean-Value Analysis of Closed Multichain Queuing Networks"
636:{\displaystyle L_{k}(m-1)\approx {\frac {m-1}{m}}L_{k}(m)}
1809:"JMT: performance engineering tools for system modeling"
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567:
464:
355:
248:
1849:
Marzolla, M. (2014). "The Octave
Queueing Package".
1437:
Adan, I.; Wal, J. (2011). "Mean Values
Techniques".
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Simon Lam: A simple derivation of the MVA algorithm
532:{\displaystyle L_{k}(m)=v_{k}\lambda _{m}W_{k}(m).}
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1178:{\displaystyle \mathbb {m} =(m_{1},\ldots ,m_{R})}
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1672:"A tree-structured mean value analysis algorithm"
1290:of state probabilities for the queueing network.
162:) for the mean time spent by a customer in queue
1416:. North-Holland Publishing Co. pp. 51–62.
995:using a generalization of the arrival theorem:
1515:Performance Evaluation: Origins and Directions
92:for the customer routing matrix where element
1925:
1807:Bertoli, M.; Casale, G.; Serazzi, G. (2009).
8:
1816:ACM SIGMETRICS Performance Evaluation Review
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341:2. Then compute the system throughput using
137:) for the mean number of customers at queue
44:, which states that when one customer in an
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1709:IEEE Transactions on Software Engineering
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946:In the case of multiclass networks with
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20:, a discipline within the mathematical
1590:Synthesis Lectures on Computer Science
60:Consider a closed queueing network of
1470:Reiser, M.; Lavenberg, S. S. (1980).
7:
1676:ACM Transactions on Computer Systems
954:can feature different service rates
1549:An introduction to queueing systems
1851:Quantitative Evaluation of Systems
14:
1602:10.2200/S00282ED1V01Y201005CSL002
950:classes of customers, each queue
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2209:Flow-equivalent server method
1902:. Shows relationship between
84:for the service rate at node
2290:Adversarial queueing network
2179:Continuous-time Markov chain
1894:J. Virtamo: Queuing networks
1859:10.1007/978-3-319-10696-0_14
1758:10.1016/0166-5316(88)90028-4
1447:10.1007/978-1-4419-6472-4_13
1235:{\displaystyle \mathbb {m} }
2252:Heavy traffic approximation
1997:Pollaczek–Khinchine formula
1268:{\displaystyle m_{r}\geq 1}
2419:
1731:10.1016/j.peva.2010.12.009
1657:10.1016/j.peva.2010.12.009
682:repeat until convergence:
72:network with 0 customers.
2376:
2257:Reflected Brownian motion
2062:Markovian arrival process
1552:. Springer. p. 174.
1310:and the performance of a
648:fixed-point relationships
166:when there is a total of
141:when there is a total of
2280:Layered queueing network
2067:Rational arrival process
1546:Bose, Sanjay K. (2001).
1523:10.1007/3-540-46506-5_22
971:in the multiclass case.
2368:Teletraffic engineering
2163:Shortest remaining time
1828:10.1145/1530873.1530877
1312:key distribution center
1216:subtracts one from the
2363:Scheduling (computing)
2002:Matrix analytic method
1746:Performance Evaluation
1635:Performance Evaluation
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550:Bard–Schweitzer method
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207:(0) = 0 for
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2095:Gordon–Newell theorem
2057:Poisson point process
1948:Single queueing nodes
1786:10.1093/comjnl/bxq064
1689:10.1145/214419.214423
1491:10.1145/322186.322195
1330:which implements MVA.
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1209:{\displaystyle 1_{r}}
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451: = 1, ...,
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22:theory of probability
2221:Decomposition method
1326:, a tool written in
1288:normalizing constant
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2328:Erlang distribution
2310:Information systems
2100:Mean value analysis
1703:Casale, G. (2008).
1626:Casale, G. (2011).
1340:which includes MVA.
963:for each job class
942:Multiclass networks
222: = 1,...,
211: = 1,...,
198:To initialise, set
40:It is based on the
26:mean value analysis
2358:Quality of service
2343:Network congestion
2204:Quasireversibility
2184:Kendall's notation
1477:Journal of the ACM
1389:10.1007/BFb0013865
1306:models describing
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2158:Shortest job next
2148:Processor sharing
2105:Buzen's algorithm
2088:Traffic equations
2076:Queueing networks
2050:Arrival processes
2024:Kingman's formula
1904:Buzen's algorithm
1868:978-3-319-10695-3
1559:978-0-306-46734-9
1532:978-3-540-67193-0
1456:978-1-4419-6471-7
1439:Queueing Networks
1423:978-0-444-85332-5
1398:978-3-540-57297-8
1308:queueing networks
1284:method of moments
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974:The waiting time
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443:
442:
441:
430:
422:
418:
414:
411:
408:
403:
399:
393:
388:
385:
382:
378:
373:
368:
363:
359:
339:
338:
337:
336:
335:
324:
317:
313:
307:
303:
300:
297:
293:
287:
283:
279:
276:
270:
267:
264:
261:
256:
252:
202:
188:
185:
178:
153:
128:
105:moves to node
96:
79:
57:
54:
13:
10:
9:
6:
4:
3:
2:
2415:
2404:
2401:
2400:
2398:
2385:
2375:
2369:
2366:
2364:
2361:
2359:
2356:
2354:
2351:
2349:
2346:
2344:
2341:
2339:
2338:Message queue
2336:
2334:
2331:
2329:
2326:
2324:
2323:Erlang (unit)
2321:
2319:
2316:
2315:
2313:
2311:
2307:
2301:
2300:Retrial queue
2298:
2296:
2293:
2291:
2288:
2286:
2283:
2281:
2278:
2276:
2273:
2272:
2270:
2266:
2258:
2255:
2254:
2253:
2250:
2248:
2245:
2243:
2240:
2239:
2237:
2233:
2227:
2224:
2222:
2219:
2217:
2214:
2210:
2207:
2205:
2202:
2200:
2197:
2196:
2195:
2192:
2190:
2187:
2185:
2182:
2180:
2177:
2176:
2174:
2170:
2164:
2161:
2159:
2156:
2154:
2151:
2149:
2146:
2144:
2141:
2139:
2136:
2135:
2133:
2129:
2123:
2120:
2118:
2115:
2113:
2112:Kelly network
2110:
2106:
2103:
2101:
2098:
2097:
2096:
2093:
2089:
2086:
2085:
2084:
2081:
2080:
2078:
2074:
2068:
2065:
2063:
2060:
2058:
2055:
2054:
2052:
2048:
2042:
2039:
2037:
2034:
2030:
2027:
2025:
2022:
2021:
2020:
2017:
2015:
2012:
2010:
2007:
2003:
2000:
1998:
1995:
1994:
1993:
1990:
1988:
1985:
1983:
1980:
1976:
1973:
1972:
1971:
1968:
1966:
1963:
1961:
1958:
1956:
1953:
1952:
1950:
1946:
1942:
1935:
1930:
1928:
1923:
1921:
1916:
1915:
1912:
1905:
1901:
1898:
1895:
1892:
1891:
1887:
1878:
1874:
1870:
1864:
1860:
1856:
1852:
1845:
1842:
1837:
1833:
1829:
1825:
1821:
1817:
1810:
1803:
1800:
1795:
1791:
1787:
1783:
1779:
1776:
1775:
1767:
1764:
1759:
1755:
1751:
1747:
1740:
1737:
1732:
1728:
1723:
1718:
1714:
1710:
1706:
1699:
1696:
1690:
1685:
1681:
1677:
1673:
1666:
1663:
1658:
1654:
1649:
1644:
1640:
1636:
1629:
1622:
1620:
1616:
1611:
1607:
1603:
1599:
1595:
1591:
1584:
1581:
1576:
1569:
1566:
1561:
1555:
1551:
1550:
1542:
1539:
1534:
1528:
1524:
1520:
1516:
1509:
1506:
1501:
1497:
1492:
1487:
1483:
1479:
1478:
1473:
1466:
1463:
1458:
1452:
1448:
1444:
1440:
1433:
1430:
1425:
1419:
1415:
1408:
1405:
1400:
1394:
1390:
1386:
1382:
1375:
1373:
1369:
1362:
1358:
1355:
1354:
1350:
1345:
1342:
1339:
1335:
1332:
1329:
1325:
1322:
1321:
1317:
1315:
1313:
1309:
1305:
1297:
1295:
1291:
1289:
1285:
1280:
1276:
1262:
1259:
1254:
1250:
1219:
1201:
1197:
1188:
1167:
1163:
1159:
1156:
1153:
1148:
1144:
1137:
1105:
1098:
1095:
1092:
1088:
1082:
1076:
1072:
1068:
1059:
1053:
1049:
1045:
1042:
1036:
1020:
1017:
1014:
1010:
1002:
1001:
1000:
999:
998:
997:
996:
994:
990:
986:
981:
977:
972:
970:
966:
961:
957:
953:
949:
941:
918:
914:
909:
906:
900:
892:
888:
882:
878:
875:
872:
861:
857:
851:
847:
843:
837:
829:
825:
817:
798:
794:
786:
782:
777:
774:
768:
760:
756:
750:
746:
743:
740:
729:
724:
721:
718:
714:
709:
704:
699:
695:
687:
686:
685:
684:
683:
680:
679:
675:
671:
666:
662:
653:
651:
649:
627:
619:
615:
609:
605:
602:
599:
593:
587:
584:
581:
573:
569:
561:
560:
559:
549:
547:
526:
520:
512:
508:
502:
498:
492:
488:
484:
478:
470:
466:
458:
457:
456:
455:
454:
450:
446:
428:
420:
416:
409:
401:
397:
391:
386:
383:
380:
376:
371:
366:
361:
357:
349:
348:
347:
346:
344:
340:
322:
315:
311:
305:
301:
298:
295:
291:
285:
281:
277:
274:
268:
262:
254:
250:
242:
241:
240:
239:
237:
233:
229:
228:
227:
225:
221:
216:
214:
210:
205:
201:
196:
194:
186:
184:
181:
177:
174:customers by
173:
169:
165:
161:
156:
152:
148:
144:
140:
136:
131:
127:
122:
120:
117: =
116:
112:
108:
104:
99:
95:
91:
87:
82:
78:
73:
70:
66:
63:
56:Problem setup
55:
53:
51:
47:
43:
38:
35:
31:
27:
23:
19:
2295:Loss network
2226:Beneš method
2189:Little's law
2172:Key concepts
2122:BCMP network
2099:
1850:
1844:
1819:
1815:
1802:
1777:
1772:
1766:
1749:
1745:
1739:
1712:
1708:
1698:
1679:
1675:
1665:
1638:
1634:
1593:
1589:
1583:
1574:
1568:
1548:
1541:
1514:
1508:
1481:
1475:
1465:
1438:
1432:
1413:
1407:
1380:
1301:
1292:
1283:
1281:
1277:
1217:
1189:classes and
1186:
1124:
992:
988:
984:
979:
975:
973:
969:BCMP theorem
964:
959:
955:
951:
947:
945:
681:
677:
673:
669:
664:
660:
658:
645:
553:
546:End repeat.
545:
452:
448:
343:Little's law
235:
231:
223:
219:
217:
212:
208:
203:
199:
197:
192:
190:
179:
175:
171:
167:
163:
159:
154:
150:
146:
142:
138:
134:
129:
125:
123:
118:
114:
110:
106:
102:
97:
93:
89:
85:
80:
76:
74:
68:
65:M/M/1 queues
61:
59:
49:
45:
39:
29:
25:
15:
2318:Data buffer
2275:Fluid queue
2242:Fluid limit
2153:Round-robin
2019:G/G/1 queue
2014:G/M/1 queue
2009:M/G/k queue
1992:M/G/1 queue
1987:M/M/∞ queue
1982:M/M/c queue
1970:M/M/1 queue
1965:M/D/c queue
1960:M/D/1 queue
1955:D/M/1 queue
218:Repeat for
2268:Extensions
2041:Bulk queue
1774:Comput. J.
1484:(2): 313.
1363:References
1338:GNU Octave
1298:Extensions
654:Pseudocode
124:Now write
2117:G-network
1822:(4): 10.
1717:CiteSeerX
1643:CiteSeerX
1610:207318911
1596:: 1–116.
1260:≥
1157:…
1089:μ
1069:−
965:r=1,...,R
915:μ
876:−
858:λ
783:μ
744:−
715:∑
696:λ
603:−
594:≈
585:−
499:λ
377:∑
358:λ
312:μ
299:−
187:Algorithm
121: P.
2397:Category
2384:Category
1906:and MVA.
1794:12824669
1351:See also
1334:queueing
1318:Software
34:expected
1877:4978676
1836:6920559
1500:8694947
558:to be:
230:1. For
67:, with
1875:
1865:
1834:
1792:
1719:
1645:
1608:
1556:
1529:
1498:
1453:
1420:
1395:
1125:where
149:) and
75:Write
1873:S2CID
1832:S2CID
1812:(PDF)
1790:S2CID
1631:(PDF)
1606:S2CID
1496:S2CID
2143:LIFO
2138:FIFO
1863:ISBN
1554:ISBN
1527:ISBN
1451:ISBN
1418:ISBN
1393:ISBN
1344:Line
1328:Java
1324:JMVA
1304:PEPA
672:) =
659:set
88:and
1855:doi
1824:doi
1782:doi
1754:doi
1727:doi
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1653:doi
1598:doi
1519:doi
1486:doi
1443:doi
1385:doi
980:k,r
960:k,r
30:MVA
16:In
2399::
1871:.
1861:.
1830:.
1820:36
1818:.
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1788:.
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1725:.
1713:35
1711:.
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1637:.
1633:.
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1600::
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