2983:
154:
additions. This dramatic improvement opened the door to applying the Gordon-Newell theorem to models of real world computer systems as well as flexible manufacturing systems and other cases where bottlenecks and queues can form within networks of inter-connected service facilities. The values of
2122:) is in principle a two dimensional matrix, it can be computed in a column by column fashion starting from the top of the leftmost column and running down each column to the bottom before proceeding to the next column on the right. The routine uses a single column vector
1561:
The Gordon-Newell theorem enables analysts to determine the stationary probability associated with each individual state of a closed queueing network. These individual probabilities must then be added together to evaluate other important probabilities. For example
699:
1252:, a surprising result emerges: the modified terms in the first group are identical to the terms used to compute the normalizing constant for the same network with one customer removed. Thus, the sum of the terms in the first group can be written as â
1885:
836:
2095:
401:
1969:
978:
280:
555:
2133:
The first loop in the algorithm below initializes the column vector C so that C = 1 and C(n) = 0 for nâ„1. Note that C remains equal to 1 throughout all subsequent iterations.
1142:
1093:
1045:
546:
497:
449:
908:
132:
1250:
1212:
1177:
1707:
717:
1281:
effectively disappears from all terms in this group (since it reduces in every case to a factor of 1). This leaves the total number of customers at the remaining
1982:
1518:-1). Buzenâs algorithm is simply the iterative application of this fundamental recurrence relation, along with the following boundary conditions.
310:
2534:
2823:
1891:
2367:
1144:. Note that this set of terms can be partitioned into two groups. The first group comprises all terms for which the exponent of
2740:
1617:
Many of these marginal probabilities can be computed with minimal additional effort. This is easy to see for the case of P(
2599:
3010:
913:
215:
694:{\displaystyle \mathbb {P} (n_{1},n_{2},\cdots ,n_{M})={\frac {1}{{\text{G}}(N)}}\prod _{i=1}^{M}\left(X_{i}\right)^{n_{i}}}
2811:
2527:
2431:
159:-1), which can be used to calculate other important quantities of interest, are computed as by-products of the algorithm.
59:
Performing a naĂŻve computation of the normalizing constant requires enumeration of all states. For a closed network with
2892:
2781:
2854:
1650:
can be factored out from each of these probabilities, leaving a set of modified probabilities whose sum is given by G(
2140:
as the algorithm proceeds down column m. This is achieved by setting each successive value of C(n) equal to:
2697:
302:
41:
2859:
2664:
180:
1098:
1049:
1001:
502:
453:
405:
3005:
2986:
2882:
2669:
2520:
2118:
have been computed by solving the relevant equations and are available as an input to our routine. Although g(
53:
2970:
2765:
850:
74:
2965:
2755:
2604:
33:
2796:
2659:
1695:
1220:
1182:
1147:
2786:
2136:
In the second loop, each successive value of C(n) for nâ„1 is set equal to the corresponding value of g(
2626:
1880:{\displaystyle \mathbb {P} (n_{i}=k)={\frac {X_{i}^{k}}{G(N)}}\quad {\text{ for }}k=0,1,\ldots ,N-1,}
2638:
2955:
2935:
2930:
2702:
48:
in his 1971 PhD dissertation and subsequently published in a refereed journal in 1973. Computing G(
831:{\displaystyle \mu _{j}X_{j}=\sum _{i=1}^{M}\mu _{i}X_{i}p_{ij}\quad {\text{ for }}j=1,\ldots ,M.}
2960:
2945:
2912:
2806:
2448:
2390:
21:
2577:
2507:
2950:
2849:
2760:
2750:
2690:
2801:
2745:
2631:
2484:
2440:
2426:
2382:
2363:
45:
2589:
980:
is equal to 1. Buzen's algorithm represents the first efficient procedure for computing G(
2818:
2685:
2543:
2398:
1265:-1)â. This insight provides the foundation for the development of the algorithm.
17:
2828:
2791:
2502:
1376:
Given this definition, the sum of the terms in the second group can now be written as g(
175:
circulating customers. Assume that the service time for a customer at service facility
2887:
1699:
2999:
2940:
2925:
2902:
2714:
1658:). This observation yields the following simple and highly efficient result:
2897:
2724:
2090:{\displaystyle \mathbb {E} (n_{i})=\sum _{k=1}^{N}X_{i}^{k}{\frac {G(N-k)}{G(N)}}.}
396:{\displaystyle \mathbb {P} (n_{1},n_{2},\cdots ,n_{M})={\frac {1}{{\text{G}}(N)}}}
2394:
282:
be the steady state probability that the number of customers at service facility
2920:
2877:
2844:
2621:
2616:
2611:
2594:
2584:
2572:
2567:
2562:
2557:
2368:"Computational algorithms for closed queueing networks with exponential servers"
2643:
1289:. The second group includes all possible arrangements of these N customers.
2719:
2340:
1435:) is equal to the combined sum of the terms in the first and second groups,
1476:
This same recurrence relation clearly exists for any intermediate value of
1277:
for every term in this group is zero. As a result, service facility
2444:
2386:
2472:
1635:
or higher in every state where the number of customers at service center
1214:
raised to the power 1 can be factored out of each of these terms.
1573:), the probability that the total number of customers at service center
2452:
1398:-1)â, the sum of the terms in the first group, can be re-written as â
2489:
2099:
These characterizations of quantities of interest in terms of the G(
2508:
Menasce: Convolution
Approach to Queueing Algorithms (presentation)
2512:
1964:{\displaystyle \mathbb {P} (n_{i}=N)={\frac {X_{i}^{N}}{G(N)}}.}
2516:
992:
The individual terms that must be added together to compute G(
2473:"Rethinking Randomness: An interview with Jeff Buzen, Part I"
2429:(1967). "Closed Queuing Systems with Exponential Servers".
2342:
DTIC AD0731575: Queueing
Network Models of Multiprogramming
2302:
At completion, the final values of C correspond to column
1420:) in the Gordon-Newell theorem can now be re-written as g(
847:) is a normalizing constant chosen so that the sum of all
1179:
is greater than or equal to 1. This implies that
973:{\displaystyle \mathbb {P} (n_{1},n_{2},\cdots ,n_{M})}
275:{\displaystyle \mathbb {P} (n_{1},n_{2},\cdots ,n_{M})}
192:
and that, after completing service at service facility
1268:
Next consider the second group. The exponent of
855:
79:
1985:
1973:
The expected number of customers at service facility
1894:
1710:
1223:
1185:
1150:
1101:
1052:
1004:
916:
853:
720:
558:
505:
456:
408:
313:
218:
77:
1557:
Marginal distributions, expected number of customers
2911:
2870:
2837:
2774:
2733:
2678:
2652:
2550:
196:, a customer will proceed next to service facility
2089:
1963:
1879:
1694:This relationship can then be used to compute the
1244:
1206:
1171:
1136:
1087:
1039:
972:
902:
830:
693:
540:
491:
443:
395:
274:
126:
2310:). Thus they represent the desired values G
1327:) as the normalizing constant for a network with
550:This result is usually written more compactly as
2503:Jain: The Convolution Algorithm (class handout)
1292:To express this concept precisely, assume that
134:individual terms, with each term consisting of
1702:number of customers at each service facility.
1610:customers can be distributed across the other
2528:
893:
858:
117:
82:
8:
2358:
2356:
2354:
2352:
1307:have been obtained for a given network with
2535:
2521:
2513:
1137:{\displaystyle \left(X_{M}\right)^{n_{M}}}
1088:{\displaystyle \left(X_{2}\right)^{n_{2}}}
1040:{\displaystyle \left(X_{1}\right)^{n_{1}}}
541:{\displaystyle \left(X_{M}\right)^{n_{M}}}
492:{\displaystyle \left(X_{2}\right)^{n_{2}}}
444:{\displaystyle \left(X_{1}\right)^{n_{1}}}
2488:
2046:
2040:
2035:
2025:
2014:
1998:
1987:
1986:
1984:
1936:
1931:
1925:
1907:
1896:
1895:
1893:
1836:
1802:
1752:
1747:
1741:
1723:
1712:
1711:
1709:
1232:
1222:
1194:
1184:
1159:
1149:
1126:
1121:
1111:
1100:
1077:
1072:
1062:
1051:
1029:
1024:
1014:
1003:
961:
942:
929:
918:
917:
915:
892:
857:
854:
852:
799:
789:
779:
769:
759:
748:
735:
725:
719:
683:
678:
668:
653:
642:
621:
615:
603:
584:
571:
560:
559:
557:
530:
525:
515:
504:
481:
476:
466:
455:
433:
428:
418:
407:
376:
370:
358:
339:
326:
315:
314:
312:
263:
244:
231:
220:
219:
217:
116:
81:
78:
76:
1416:In addition, the normalizing constant G(
167:Consider a closed queueing network with
52:) is required to compute the stationary
2331:
2165:) is the previous value of C(n), and g(
1599:, over all possible ways the remaining
20:, a discipline within the mathematical
903:{\displaystyle {\tbinom {N+M-1}{M-1}}}
138:factors raised to powers whose sum is
127:{\displaystyle {\tbinom {N+M-1}{M-1}}}
32:) is an algorithm for calculating the
7:
2471:Denning, Peter J. (24 August 2016).
2466:
2464:
2462:
2420:
2418:
1614:-1 service centers in the network.
1581:, must be summed over all values of
1484:, and for any intermediate value of
44:. This method was first proposed by
2126:to represent the current column of
2169:) is the current value of C(n-1)
1387:It also follows immediately that â
1245:{\displaystyle \left(X_{M}\right)}
1207:{\displaystyle \left(X_{M}\right)}
1172:{\displaystyle \left(X_{M}\right)}
862:
86:
14:
1358:members of the original sequence
996:) all have the following form:
2982:
2981:
142:. Buzen's algorithm computes G(
1835:
1631:must be raised to the power of
1285:-1 service facilities equal to
798:
183:random variable with parameter
2078:
2072:
2064:
2052:
2004:
1991:
1952:
1946:
1919:
1900:
1832:
1829:
1811:
1792:
1780:
1774:
1768:
1762:
1735:
1716:
967:
922:
632:
626:
609:
564:
387:
381:
364:
319:
269:
224:
56:of a closed queueing network.
1:
2812:Flow-equivalent server method
2893:Adversarial queueing network
2782:Continuous-time Markov chain
2111:It will be assumed that the
1639:is greater than or equal to
1592:and, for each such value of
1577:is greater than or equal to
1311:service facilities. For any
2855:Heavy traffic approximation
2600:PollaczekâKhinchine formula
1354: that match the first
1335:service facilities (1,2, âŠ
3027:
2339:Buzen, J.P. (1971-08-01).
712:are determined by solving
63:circulating customers and
2979:
2860:Reflected Brownian motion
2665:Markovian arrival process
2375:Communications of the ACM
2103:) are also due to Buzen.
181:exponentially distributed
2883:Layered queueing network
2670:Rational arrival process
2171:
54:probability distribution
2971:Teletraffic engineering
2766:Shortest remaining time
1339:), and values of
171:service facilities and
3011:Statistical algorithms
2966:Scheduling (computing)
2605:Matrix analytic method
2091:
2030:
1965:
1881:
1696:marginal distributions
1624:â„ k). Clearly,
1246:
1208:
1173:
1138:
1089:
1041:
974:
904:
832:
764:
695:
658:
542:
493:
445:
397:
276:
128:
67:service facilities, G(
34:normalization constant
2797:Product-form solution
2698:GordonâNewell theorem
2660:Poisson point process
2551:Single queueing nodes
2445:10.1287/opre.15.2.254
2387:10.1145/362342.362345
2092:
2010:
1966:
1882:
1247:
1209:
1174:
1139:
1090:
1042:
988:Algorithm description
975:
905:
833:
744:
696:
638:
543:
494:
446:
398:
303:GordonâNewell theorem
277:
129:
42:GordonâNewell theorem
30:convolution algorithm
22:theory of probability
2824:Decomposition method
2483:(August): 1:1â1:17.
1983:
1892:
1708:
1221:
1217:After factoring out
1183:
1148:
1099:
1050:
1002:
914:
851:
718:
556:
503:
454:
406:
311:
301:It follows from the
216:
150:multiplications and
75:
2956:Pipeline (software)
2936:Flow control (data)
2931:Erlang distribution
2913:Information systems
2703:Mean value analysis
2432:Operations Research
2045:
1941:
1757:
1539:,1) = (
2961:Quality of service
2946:Network congestion
2807:Quasireversibility
2787:Kendall's notation
2087:
2031:
1961:
1927:
1877:
1743:
1242:
1204:
1169:
1134:
1085:
1037:
970:
900:
898:
828:
691:
538:
489:
441:
393:
272:
124:
122:
2993:
2992:
2951:Network scheduler
2850:Mean-field theory
2761:Shortest job next
2751:Processor sharing
2708:Buzen's algorithm
2691:Traffic equations
2679:Queueing networks
2653:Arrival processes
2627:Kingman's formula
2082:
1956:
1839:
1772:
891:
802:
636:
624:
391:
379:
200:with probability
155:G(1), G(2) ... G(
115:
26:Buzen's algorithm
3018:
2985:
2984:
2802:Balance equation
2734:Service policies
2632:Lindley equation
2537:
2530:
2523:
2514:
2495:
2494:
2492:
2468:
2457:
2456:
2422:
2413:
2412:
2410:
2409:
2403:
2397:. Archived from
2372:
2360:
2347:
2346:
2336:
2306:in the matrix g(
2298:
2295:
2292:
2289:
2286:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2241:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2096:
2094:
2093:
2088:
2083:
2081:
2067:
2047:
2044:
2039:
2029:
2024:
2003:
2002:
1990:
1970:
1968:
1967:
1962:
1957:
1955:
1940:
1935:
1926:
1912:
1911:
1899:
1886:
1884:
1883:
1878:
1840:
1837:
1807:
1806:
1773:
1771:
1756:
1751:
1742:
1728:
1727:
1715:
1251:
1249:
1248:
1243:
1241:
1237:
1236:
1213:
1211:
1210:
1205:
1203:
1199:
1198:
1178:
1176:
1175:
1170:
1168:
1164:
1163:
1143:
1141:
1140:
1135:
1133:
1132:
1131:
1130:
1120:
1116:
1115:
1094:
1092:
1091:
1086:
1084:
1083:
1082:
1081:
1071:
1067:
1066:
1046:
1044:
1043:
1038:
1036:
1035:
1034:
1033:
1023:
1019:
1018:
979:
977:
976:
971:
966:
965:
947:
946:
934:
933:
921:
909:
907:
906:
901:
899:
897:
896:
890:
879:
861:
837:
835:
834:
829:
803:
800:
797:
796:
784:
783:
774:
773:
763:
758:
740:
739:
730:
729:
700:
698:
697:
692:
690:
689:
688:
687:
677:
673:
672:
657:
652:
637:
635:
625:
622:
616:
608:
607:
589:
588:
576:
575:
563:
547:
545:
544:
539:
537:
536:
535:
534:
524:
520:
519:
498:
496:
495:
490:
488:
487:
486:
485:
475:
471:
470:
450:
448:
447:
442:
440:
439:
438:
437:
427:
423:
422:
402:
400:
399:
394:
392:
390:
380:
377:
371:
363:
362:
344:
343:
331:
330:
318:
281:
279:
278:
273:
268:
267:
249:
248:
236:
235:
223:
133:
131:
130:
125:
123:
121:
120:
114:
103:
85:
71:) is the sum of
46:Jeffrey P. Buzen
3026:
3025:
3021:
3020:
3019:
3017:
3016:
3015:
3006:Queueing theory
2996:
2995:
2994:
2989:
2975:
2907:
2866:
2833:
2819:Arrival theorem
2770:
2729:
2686:Jackson network
2674:
2648:
2639:Forkâjoin queue
2578:Burke's theorem
2546:
2544:Queueing theory
2541:
2499:
2498:
2490:10.1145/2986329
2470:
2469:
2460:
2425:Gordon, W. J.;
2424:
2423:
2416:
2407:
2405:
2401:
2370:
2362:
2361:
2350:
2338:
2337:
2333:
2328:
2300:
2299:
2296:
2293:
2290:
2287:
2284:
2281:
2278:
2275:
2272:
2269:
2266:
2263:
2260:
2257:
2254:
2251:
2248:
2245:
2242:
2239:
2236:
2233:
2230:
2227:
2224:
2221:
2218:
2215:
2212:
2209:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2179:
2176:
2173:
2152:
2116:
2109:
2068:
2048:
1994:
1981:
1980:
1942:
1903:
1890:
1889:
1838: for
1798:
1758:
1719:
1706:
1705:
1677:
1666:
1648:
1629:
1622:
1608:
1597:
1586:
1567:
1559:
1545:
1504:
1495:This implies g(
1455:
1403:
1392:
1371:
1367:
1363:
1352:
1348:
1344:
1305:
1301:
1297:
1276:
1260:
1228:
1224:
1219:
1218:
1190:
1186:
1181:
1180:
1155:
1151:
1146:
1145:
1122:
1107:
1103:
1102:
1097:
1096:
1073:
1058:
1054:
1053:
1048:
1047:
1025:
1010:
1006:
1005:
1000:
999:
990:
957:
938:
925:
912:
911:
880:
863:
856:
849:
848:
801: for
785:
775:
765:
731:
721:
716:
715:
711:
679:
664:
660:
659:
620:
599:
580:
567:
554:
553:
526:
511:
507:
506:
501:
500:
477:
462:
458:
457:
452:
451:
429:
414:
410:
409:
404:
403:
375:
354:
335:
322:
309:
308:
291:
259:
240:
227:
214:
213:
208:
191:
179:is given by an
165:
104:
87:
80:
73:
72:
18:queueing theory
12:
11:
5:
3024:
3022:
3014:
3013:
3008:
2998:
2997:
2991:
2990:
2980:
2977:
2976:
2974:
2973:
2968:
2963:
2958:
2953:
2948:
2943:
2938:
2933:
2928:
2923:
2917:
2915:
2909:
2908:
2906:
2905:
2900:
2895:
2890:
2888:Polling system
2885:
2880:
2874:
2872:
2868:
2867:
2865:
2864:
2863:
2862:
2852:
2847:
2841:
2839:
2838:Limit theorems
2835:
2834:
2832:
2831:
2826:
2821:
2816:
2815:
2814:
2809:
2804:
2794:
2789:
2784:
2778:
2776:
2772:
2771:
2769:
2768:
2763:
2758:
2753:
2748:
2743:
2737:
2735:
2731:
2730:
2728:
2727:
2722:
2717:
2712:
2711:
2710:
2705:
2695:
2694:
2693:
2682:
2680:
2676:
2675:
2673:
2672:
2667:
2662:
2656:
2654:
2650:
2649:
2647:
2646:
2641:
2636:
2635:
2634:
2629:
2619:
2614:
2609:
2608:
2607:
2602:
2592:
2587:
2582:
2581:
2580:
2570:
2565:
2560:
2554:
2552:
2548:
2547:
2542:
2540:
2539:
2532:
2525:
2517:
2511:
2510:
2505:
2497:
2496:
2458:
2414:
2381:(9): 527â531.
2348:
2330:
2329:
2327:
2324:
2172:
2150:
2114:
2108:
2107:Implementation
2105:
2086:
2080:
2077:
2074:
2071:
2066:
2063:
2060:
2057:
2054:
2051:
2043:
2038:
2034:
2028:
2023:
2020:
2017:
2013:
2009:
2006:
2001:
1997:
1993:
1989:
1960:
1954:
1951:
1948:
1945:
1939:
1934:
1930:
1924:
1921:
1918:
1915:
1910:
1906:
1902:
1898:
1876:
1873:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1805:
1801:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1770:
1767:
1764:
1761:
1755:
1750:
1746:
1740:
1737:
1734:
1731:
1726:
1722:
1718:
1714:
1675:
1664:
1646:
1627:
1620:
1606:
1595:
1584:
1565:
1558:
1555:
1543:
1502:
1453:
1401:
1390:
1369:
1365:
1361:
1350:
1346:
1342:
1303:
1299:
1295:
1272:
1256:
1240:
1235:
1231:
1227:
1202:
1197:
1193:
1189:
1167:
1162:
1158:
1154:
1129:
1125:
1119:
1114:
1110:
1106:
1080:
1076:
1070:
1065:
1061:
1057:
1032:
1028:
1022:
1017:
1013:
1009:
989:
986:
969:
964:
960:
956:
953:
950:
945:
941:
937:
932:
928:
924:
920:
895:
889:
886:
883:
878:
875:
872:
869:
866:
860:
827:
824:
821:
818:
815:
812:
809:
806:
795:
792:
788:
782:
778:
772:
768:
762:
757:
754:
751:
747:
743:
738:
734:
728:
724:
707:
703:The values of
686:
682:
676:
671:
667:
663:
656:
651:
648:
645:
641:
634:
631:
628:
619:
614:
611:
606:
602:
598:
595:
592:
587:
583:
579:
574:
570:
566:
562:
533:
529:
523:
518:
514:
510:
484:
480:
474:
469:
465:
461:
436:
432:
426:
421:
417:
413:
389:
386:
383:
374:
369:
366:
361:
357:
353:
350:
347:
342:
338:
334:
329:
325:
321:
317:
297:= 1, 2, ... ,
289:
271:
266:
262:
258:
255:
252:
247:
243:
239:
234:
230:
226:
222:
204:
187:
164:
161:
119:
113:
110:
107:
102:
99:
96:
93:
90:
84:
13:
10:
9:
6:
4:
3:
2:
3023:
3012:
3009:
3007:
3004:
3003:
3001:
2988:
2978:
2972:
2969:
2967:
2964:
2962:
2959:
2957:
2954:
2952:
2949:
2947:
2944:
2942:
2941:Message queue
2939:
2937:
2934:
2932:
2929:
2927:
2926:Erlang (unit)
2924:
2922:
2919:
2918:
2916:
2914:
2910:
2904:
2903:Retrial queue
2901:
2899:
2896:
2894:
2891:
2889:
2886:
2884:
2881:
2879:
2876:
2875:
2873:
2869:
2861:
2858:
2857:
2856:
2853:
2851:
2848:
2846:
2843:
2842:
2840:
2836:
2830:
2827:
2825:
2822:
2820:
2817:
2813:
2810:
2808:
2805:
2803:
2800:
2799:
2798:
2795:
2793:
2790:
2788:
2785:
2783:
2780:
2779:
2777:
2773:
2767:
2764:
2762:
2759:
2757:
2754:
2752:
2749:
2747:
2744:
2742:
2739:
2738:
2736:
2732:
2726:
2723:
2721:
2718:
2716:
2715:Kelly network
2713:
2709:
2706:
2704:
2701:
2700:
2699:
2696:
2692:
2689:
2688:
2687:
2684:
2683:
2681:
2677:
2671:
2668:
2666:
2663:
2661:
2658:
2657:
2655:
2651:
2645:
2642:
2640:
2637:
2633:
2630:
2628:
2625:
2624:
2623:
2620:
2618:
2615:
2613:
2610:
2606:
2603:
2601:
2598:
2597:
2596:
2593:
2591:
2588:
2586:
2583:
2579:
2576:
2575:
2574:
2571:
2569:
2566:
2564:
2561:
2559:
2556:
2555:
2553:
2549:
2545:
2538:
2533:
2531:
2526:
2524:
2519:
2518:
2515:
2509:
2506:
2504:
2501:
2500:
2491:
2486:
2482:
2478:
2474:
2467:
2465:
2463:
2459:
2454:
2450:
2446:
2442:
2438:
2434:
2433:
2428:
2427:Newell, G. F.
2421:
2419:
2415:
2404:on 2016-05-13
2400:
2396:
2392:
2388:
2384:
2380:
2376:
2369:
2365:
2359:
2357:
2355:
2353:
2349:
2344:
2343:
2335:
2332:
2325:
2323:
2321:
2317:
2313:
2309:
2305:
2170:
2168:
2164:
2159:
2157:
2153:
2146:
2141:
2139:
2134:
2131:
2129:
2125:
2121:
2117:
2106:
2104:
2102:
2097:
2084:
2075:
2069:
2061:
2058:
2055:
2049:
2041:
2036:
2032:
2026:
2021:
2018:
2015:
2011:
2007:
1999:
1995:
1978:
1976:
1971:
1958:
1949:
1943:
1937:
1932:
1928:
1922:
1916:
1913:
1908:
1904:
1887:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1841:
1826:
1823:
1820:
1817:
1814:
1808:
1803:
1799:
1795:
1789:
1786:
1783:
1777:
1765:
1759:
1753:
1748:
1744:
1738:
1732:
1729:
1724:
1720:
1703:
1701:
1697:
1692:
1690:
1686:
1682:
1678:
1671:
1667:
1659:
1657:
1653:
1649:
1642:
1638:
1634:
1630:
1623:
1615:
1613:
1609:
1602:
1598:
1591:
1587:
1580:
1576:
1572:
1568:
1556:
1554:
1553:
1549:
1542:
1538:
1533:
1532:
1528:
1524:
1519:
1517:
1513:
1509:
1505:
1498:
1493:
1491:
1487:
1483:
1479:
1474:
1472:
1468:
1464:
1460:
1456:
1449:
1445:
1441:
1436:
1434:
1429:
1427:
1423:
1419:
1414:
1412:
1408:
1404:
1397:
1393:
1385:
1383:
1379:
1374:
1372:
1357:
1353:
1338:
1334:
1330:
1326:
1322:
1318:
1314:
1310:
1306:
1290:
1288:
1284:
1280:
1275:
1271:
1266:
1264:
1259:
1255:
1238:
1233:
1229:
1225:
1215:
1200:
1195:
1191:
1187:
1165:
1160:
1156:
1152:
1127:
1123:
1117:
1112:
1108:
1104:
1078:
1074:
1068:
1063:
1059:
1055:
1030:
1026:
1020:
1015:
1011:
1007:
997:
995:
987:
985:
983:
962:
958:
954:
951:
948:
943:
939:
935:
930:
926:
887:
884:
881:
876:
873:
870:
867:
864:
846:
842:
838:
825:
822:
819:
816:
813:
810:
807:
804:
793:
790:
786:
780:
776:
770:
766:
760:
755:
752:
749:
745:
741:
736:
732:
726:
722:
713:
710:
706:
701:
684:
680:
674:
669:
665:
661:
654:
649:
646:
643:
639:
629:
617:
612:
604:
600:
596:
593:
590:
585:
581:
577:
572:
568:
551:
548:
531:
527:
521:
516:
512:
508:
482:
478:
472:
467:
463:
459:
434:
430:
424:
419:
415:
411:
384:
372:
367:
359:
355:
351:
348:
345:
340:
336:
332:
327:
323:
306:
304:
300:
296:
292:
285:
264:
260:
256:
253:
250:
245:
241:
237:
232:
228:
210:
207:
203:
199:
195:
190:
186:
182:
178:
174:
170:
163:Problem setup
162:
160:
158:
153:
149:
146:) using only
145:
141:
137:
111:
108:
105:
100:
97:
94:
91:
88:
70:
66:
62:
57:
55:
51:
47:
43:
39:
35:
31:
27:
23:
19:
2898:Loss network
2829:BeneĆĄ method
2792:Little's law
2775:Key concepts
2725:BCMP network
2707:
2480:
2476:
2436:
2430:
2406:. Retrieved
2399:the original
2378:
2374:
2364:Buzen, J. P.
2341:
2334:
2319:
2315:
2311:
2307:
2303:
2301:
2166:
2162:
2161:Note that g(
2160:
2155:
2148:
2144:
2142:
2137:
2135:
2132:
2127:
2123:
2119:
2112:
2110:
2100:
2098:
1979:
1977:is given by
1974:
1972:
1888:
1704:
1693:
1688:
1684:
1680:
1673:
1669:
1662:
1660:
1655:
1651:
1644:
1640:
1636:
1632:
1625:
1618:
1616:
1611:
1604:
1600:
1593:
1589:
1582:
1578:
1574:
1570:
1563:
1560:
1551:
1547:
1540:
1536:
1534:
1530:
1526:
1522:
1520:
1515:
1511:
1507:
1500:
1496:
1494:
1489:
1485:
1481:
1477:
1475:
1470:
1466:
1462:
1458:
1451:
1447:
1443:
1439:
1437:
1432:
1430:
1425:
1421:
1417:
1415:
1410:
1406:
1399:
1395:
1388:
1386:
1381:
1377:
1375:
1359:
1355:
1340:
1336:
1332:
1328:
1324:
1320:
1316:
1312:
1308:
1293:
1291:
1286:
1282:
1278:
1273:
1269:
1267:
1262:
1257:
1253:
1216:
998:
993:
991:
981:
844:
840:
839:
714:
708:
704:
702:
552:
549:
307:
298:
294:
287:
286:is equal to
283:
211:
205:
201:
197:
193:
188:
184:
176:
172:
168:
166:
156:
151:
147:
143:
139:
135:
68:
64:
60:
58:
49:
37:
29:
25:
15:
2921:Data buffer
2878:Fluid queue
2845:Fluid limit
2756:Round-robin
2622:G/G/1 queue
2617:G/M/1 queue
2612:M/G/k queue
2595:M/G/1 queue
2590:M/M/â queue
2585:M/M/c queue
2573:M/M/1 queue
2568:M/D/c queue
2563:M/D/1 queue
2558:D/M/1 queue
2158:).
1413:)â.
1331:customers,
3000:Categories
2871:Extensions
2644:Bulk queue
2439:(2): 254.
2408:2006-04-15
2326:References
2316:(1), ... ,
1550:= 0, 1, âŠ
1525:) = 1 for
1488:from 1 to
1480:from 1 to
910:values of
2720:G-network
2059:−
2012:∑
1869:−
1860:…
1824:−
1818:−
1796:−
1787:−
1529:= 1, 2, âŠ
1492:.
1323:define g(
952:⋯
885:−
874:−
817:…
767:μ
746:∑
723:μ
640:∏
594:⋯
349:⋯
254:⋯
109:−
98:−
40:) in the
2987:Category
2477:Ubiquity
2366:(1973).
2154:times g(
1700:expected
1431:Since G(
1405:times g(
1394:times G(
1319:and m â€
1261:times G(
2147:) plus
1643:. Thus
2453:168557
2451:
2393:
1654:-k)/G(
1546:) for
1514:) + g(
1465:) + g(
1442:) = g(
305:that
2449:JSTOR
2402:(PDF)
2395:10702
2391:S2CID
2371:(PDF)
2267:until
2240:until
2201:until
2167:n-1,m
2163:n,m-1
2156:n-1,m
2145:n,m-1
1672:) = (
1428:).
1384:-1).
1368:, ⊠X
1349:, ⊠X
1302:, ⊠X
1095:....
499:....
28:(or
2746:LIFO
2741:FIFO
2481:2016
2312:(0),
2261:step
2234:step
2195:step
2138:n,m)
1698:and
1687:)/G(
1679:) G(
1521:g(0,
1499:) =
1473:-1)
1450:) =
293:for
212:Let
2485:doi
2441:doi
2383:doi
2322:.
2320:(N)
2308:n,m
2249:for
2222:for
2183:for
2130:.
2120:n,m
1516:n,m
1510:-1,
1497:n,m
1461:-1,
1409:-1,
1373:.
1364:, X
1345:, X
1325:n,m
1298:, X
984:).
299:M .
16:In
3002::
2479:.
2475:.
2461:^
2447:.
2437:15
2435:.
2417:^
2389:.
2379:16
2377:.
2373:.
2351:^
2279::=
2273:do
2255::=
2246:do
2228::=
2213::=
2207:do
2189::=
2177::=
2143:g(
1691:)
1668:â„
1661:P(
1603:â
1588:â„
1569:â„
1562:P(
1535:g(
1506:g(
1457:g(
1446:,
1438:G(
1380:,
1321:M,
1315:â€
209:.
206:ij
152:NM
148:NM
36:G(
24:,
2536:e
2529:t
2522:v
2493:.
2487::
2455:.
2443::
2411:.
2385::
2345:.
2318:G
2314:G
2304:M
2297:;
2294:C
2291:*
2288:X
2285:+
2282:C
2276:C
2270:N
2264:1
2258:1
2252:n
2243:M
2237:1
2231:1
2225:m
2219:;
2216:0
2210:C
2204:N
2198:1
2192:1
2186:n
2180:1
2174:C
2151:m
2149:X
2128:g
2124:C
2115:m
2113:X
2101:n
2085:.
2079:)
2076:N
2073:(
2070:G
2065:)
2062:k
2056:N
2053:(
2050:G
2042:k
2037:i
2033:X
2027:N
2022:1
2019:=
2016:k
2008:=
2005:)
2000:i
1996:n
1992:(
1988:E
1975:i
1959:.
1953:)
1950:N
1947:(
1944:G
1938:N
1933:i
1929:X
1923:=
1920:)
1917:N
1914:=
1909:i
1905:n
1901:(
1897:P
1875:,
1872:1
1866:N
1863:,
1857:,
1854:1
1851:,
1848:0
1845:=
1842:k
1833:]
1830:)
1827:1
1821:k
1815:N
1812:(
1809:G
1804:i
1800:X
1793:)
1790:k
1784:N
1781:(
1778:G
1775:[
1769:)
1766:N
1763:(
1760:G
1754:k
1749:i
1745:X
1739:=
1736:)
1733:k
1730:=
1725:i
1721:n
1717:(
1713:P
1689:N
1685:k
1683:-
1681:N
1676:i
1674:X
1670:k
1665:i
1663:n
1656:N
1652:N
1647:i
1645:X
1641:k
1637:i
1633:k
1628:i
1626:X
1621:i
1619:n
1612:M
1607:i
1605:n
1601:N
1596:i
1594:n
1590:k
1585:i
1583:n
1579:k
1575:i
1571:k
1566:i
1564:n
1552:N
1548:n
1544:i
1541:X
1537:n
1531:M
1527:m
1523:m
1512:m
1508:n
1503:m
1501:X
1490:M
1486:m
1482:N
1478:n
1471:M
1469:,
1467:N
1463:M
1459:N
1454:M
1452:X
1448:M
1444:N
1440:N
1433:N
1426:M
1424:,
1422:N
1418:N
1411:M
1407:N
1402:M
1400:X
1396:N
1391:M
1389:X
1382:M
1378:N
1370:M
1366:2
1362:1
1360:X
1356:m
1351:m
1347:2
1343:1
1341:X
1337:m
1333:m
1329:n
1317:N
1313:n
1309:M
1304:M
1300:2
1296:1
1294:X
1287:N
1283:M
1279:M
1274:M
1270:X
1263:N
1258:M
1254:X
1239:)
1234:M
1230:X
1226:(
1201:)
1196:M
1192:X
1188:(
1166:)
1161:M
1157:X
1153:(
1128:M
1124:n
1118:)
1113:M
1109:X
1105:(
1079:2
1075:n
1069:)
1064:2
1060:X
1056:(
1031:1
1027:n
1021:)
1016:1
1012:X
1008:(
994:N
982:N
968:)
963:M
959:n
955:,
949:,
944:2
940:n
936:,
931:1
927:n
923:(
919:P
894:)
888:1
882:M
877:1
871:M
868:+
865:N
859:(
845:N
843:(
841:G
826:.
823:M
820:,
814:,
811:1
808:=
805:j
794:j
791:i
787:p
781:i
777:X
771:i
761:M
756:1
753:=
750:i
742:=
737:j
733:X
727:j
709:i
705:X
685:i
681:n
675:)
670:i
666:X
662:(
655:M
650:1
647:=
644:i
633:)
630:N
627:(
623:G
618:1
613:=
610:)
605:M
601:n
597:,
591:,
586:2
582:n
578:,
573:1
569:n
565:(
561:P
532:M
528:n
522:)
517:M
513:X
509:(
483:2
479:n
473:)
468:2
464:X
460:(
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431:n
425:)
420:1
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412:(
388:)
385:N
382:(
378:G
373:1
368:=
365:)
360:M
356:n
352:,
346:,
341:2
337:n
333:,
328:1
324:n
320:(
316:P
295:i
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288:n
284:i
270:)
265:M
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251:,
246:2
242:n
238:,
233:1
229:n
225:(
221:P
202:p
198:j
194:i
189:i
185:Ό
177:i
173:N
169:M
157:N
144:N
140:N
136:M
118:)
112:1
106:M
101:1
95:M
92:+
89:N
83:(
69:N
65:M
61:N
50:N
38:N
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