3528:
766:
1245:
578:
1500:
481:
874:
778:
402:
986:
516:
Stability conditions are known for RBMs in 1, 2, and 3 dimensions. "The problem of recurrence classification for SRBMs in four and higher dimensions remains open." In the special case where
1342:
362:
131:
263:
1040:
1280:
761:{\displaystyle \mathbb {P} (Z(t)\leq z)=\Phi \left({\frac {z-\mu t}{\sigma t^{1/2}}}\right)-e^{-2\mu z/\sigma ^{2}}\Phi \left({\frac {-z-\mu t}{\sigma t^{1/2}}}\right)}
1033:
60:
in a confined space and it is often called confined
Brownian motion. For example it can describe the motion of hard spheres in water confined between two walls.
1373:
562:(transient distribution) of a one-dimensional Brownian motion starting at 0 restricted to positive values (a single reflecting barrier at 0) with drift
3079:
3368:
3045:
2997:
1290:
1557:
for situations where the product form condition does not hold can be computed numerically as described below in the simulation section.
419:
2691:
2258:
796:
2220:
2806:
Steady-state analysis of reflected
Brownian motions: characterization, numerical methods and queueing applications (Ph. D. thesis)
3285:
3144:
1289:
The stationary distribution of a reflected
Brownian motion in multiple dimensions is tractable analytically when there is a
3356:
3072:
3437:
3326:
2150:
1362:
2546:
371:
3399:
901:
68:
2245:
2156:
3242:
3209:
2832:
3531:
3427:
3214:
3065:
2162:
786:
505:
3550:
3515:
3310:
3510:
3300:
3149:
2761:
1554:
333:
102:
3341:
3204:
2274:
Veestraeten, D. (2004). "The
Conditional Probability Density Function for a Reflected Brownian Motion".
559:
3331:
2798:
3252:
3171:
2752:(1991). "Steady-State Analysis of RBM in a Rectangle: Numerical Methods and A Queueing Application".
1301:
1240:{\displaystyle f(x,p_{b})={\frac {e^{-((x-u)/a)^{2}/2}+e^{-((x+u-2p_{b})/a)^{2}/2}}{a(2\pi )^{1/2}}}}
206:
3183:
3500:
3480:
3475:
3247:
2828:
2766:
2749:
2641:
2597:
2542:
2497:
2241:
993:
But be aware that the distributions of the processes as a whole are very different. In particular,
532:
3505:
3490:
3457:
3351:
2884:
2857:
2779:
2730:
2623:
2602:"Brownian Models of Feedforward Queueing Networks: Quasireversibility and Product Form Solutions"
2579:
2561:
2469:
2461:
2415:
2407:
2368:
2291:
1252:
96:
33:
3122:
3029:
2683:
3495:
3394:
3305:
3295:
3235:
3041:
2993:
2687:
2330:
2254:
2216:
2183:
2644:; Reiman, M. I. (1981). "On the Distribution of Multidimensional Reflected Brownian Motion".
3346:
3290:
3176:
3033:
3021:
2985:
2930:
2896:
2847:
2771:
2720:
2675:
2653:
2613:
2571:
2516:
2453:
2438:
2399:
2360:
2322:
2283:
2208:
3134:
2944:
1011:
3363:
3230:
3088:
2940:
1354:
187:
17:
3373:
3336:
2959:
2287:
1495:{\displaystyle p(z_{1},z_{2},\ldots ,z_{d})=\prod _{k=1}^{d}\eta _{k}e^{-\eta _{k}z_{k}}}
3432:
3013:
2914:
2501:
2364:
1575:
1571:
405:
64:
49:
2935:
2918:
2833:"Reflected Brownian Motion in an Orthant: Numerical Methods for Steady-State Analysis"
3544:
3485:
3470:
3447:
3259:
3022:
2676:
2473:
2419:
2295:
2310:
2212:
2167:
delayed reflection (the time spent on the boundary is positive with probability one)
3442:
3269:
3017:
2583:
2348:
2170:
partial reflection where the process is either immediately reflected or is absorbed
72:
2709:"Discretization Error in Simulation of One-Dimensional Reflecting Brownian Motion"
2502:"Brownian models of open queueing networks with homogeneous customer populations"
3465:
3422:
3389:
3166:
3161:
3156:
3139:
3129:
3117:
3112:
3107:
3102:
3037:
2989:
2887:(1962). "Stochastic Equations for Diffusion Processes in a Bounded Region. II".
2671:
2439:"Multiple Channel Queues in Heavy Traffic. II: Sequences, Networks, and Batches"
3188:
2852:
2775:
2725:
2708:
2618:
2601:
2520:
2434:
2387:
76:
2334:
2326:
3264:
2984:. Grundlehren der mathematischen Wissenschaften. Vol. 312. p. 31.
57:
2134:
521:
2861:
2783:
2734:
2627:
2547:"Positive recurrence of reflecting Brownian motion in three dimensions"
2465:
2411:
2372:
368:
The reflection matrix describes boundary behaviour. In the interior of
53:
2575:
1579:
2900:
2657:
2457:
2403:
2353:
Journal of the Royal
Statistical Society. Series B (Methodological)
3057:
2566:
476:{\displaystyle \scriptstyle \{z\in \mathbb {R} _{+}^{d}:z_{j}=0\}}
2205:
Wiley
Encyclopedia of Operations Research and Management Science
2126:
The error involved in discrete simulations has been quantified.
3061:
869:{\displaystyle \mathbb {P} (Z<z)=1-e^{-2\mu z/\sigma ^{2}}.}
2960:"Stochastic Differential Equations for Sticky Brownian Motion"
2145:
Feller described possible boundary condition for the process
508:
of the process on the corresponding section of the boundary.
779:
cumulative distribution function of the normal distribution
524:
then necessary and sufficient conditions for stability are
2980:
Chung, K. L.; Zhao, Z. (1995). "Killed
Brownian Motion".
2390:(1970). "Multiple Channel Queues in Heavy Traffic. I".
889:
coincides with the distribution of the running maximum
423:
375:
2309:
Faucheux, Luc P.; Libchaber, Albert J. (1994-06-01).
1376:
1304:
1255:
1043:
1014:
904:
799:
581:
422:
374:
336:
209:
105:
3456:
3415:
3382:
3319:
3278:
3223:
3197:
3095:
2808:(Thesis). Stanford University. Dept. of Mathematics
2203:Dieker, A. B. (2011). "Reflected Brownian Motion".
1494:
1336:
1274:
1239:
1027:
980:
868:
785: < 0) when taking t → ∞ an
760:
475:
396:
356:
257:
125:
2967:Probab. Statist. Group Manchester Research Report
2923:Transactions of the American Mathematical Society
1008:The heat kernel for reflected Brownian motion at
397:{\displaystyle \scriptstyle \mathbb {R} _{+}^{d}}
29:Wiener process with reflecting spatial boundaries
981:{\displaystyle Z(t)\sim M(t)=\sup _{s\in }X(s).}
936:
2236:
2234:
2232:
2155:instantaneous reflection, as described above a
52:in a space with reflecting boundaries. In the
2982:From Brownian Motion to Schrödinger's Equation
1570:In one dimension the simulated process is the
1293:, which occurs when the process is stable and
3073:
8:
2879:
2877:
2875:
2873:
2871:
2707:Asmussen, S.; Glynn, P.; Pitman, J. (1995).
2674:; Taimre, Thomas; Botev, Zdravko I. (2011).
469:
424:
2247:Brownian Motion and Stochastic Flow Systems
3080:
3066:
3058:
3024:Diffusion Processes and their Sample Paths
2889:Theory of Probability and Its Applications
2934:
2851:
2765:
2724:
2617:
2565:
2536:
2534:
2532:
2530:
1484:
1474:
1466:
1456:
1446:
1435:
1419:
1400:
1387:
1375:
1303:
1266:
1254:
1224:
1220:
1192:
1186:
1174:
1165:
1136:
1119:
1113:
1101:
1079:
1072:
1060:
1042:
1019:
1013:
939:
903:
855:
846:
833:
801:
800:
798:
741:
737:
710:
695:
686:
673:
649:
645:
621:
583:
582:
580:
457:
444:
439:
435:
434:
421:
387:
382:
378:
377:
373:
348:
343:
339:
338:
335:
208:
117:
112:
108:
107:
104:
2149:absorption or killed Brownian motion, a
2137:allows simulation of steady state RBMs.
2195:
91:–dimensional reflected Brownian motion
2919:"Diffusion processes in one dimension"
2492:
2490:
2351:(1962). "On Queues in Heavy Traffic".
289:is continuous and non–decreasing with
2958:Engelbert, H. J.; Peskir, G. (2012).
408:; on the boundary "roughly speaking,
7:
3020:(1996). "Time changes and killing".
1291:product form stationary distribution
549:Marginal and stationary distribution
357:{\displaystyle \mathbb {R} _{+}^{d}}
126:{\displaystyle \mathbb {R} _{+}^{d}}
2646:SIAM Journal on Applied Mathematics
56:literature, this process describes
2365:10.1111/j.2517-6161.1962.tb00465.x
2288:10.1023/B:CSEM.0000049491.13935.af
1308:
703:
614:
304:only increases at times for which
25:
2936:10.1090/S0002-9947-1954-0063607-6
2840:The Annals of Applied Probability
2754:The Annals of Applied Probability
2713:The Annals of Applied Probability
2682:. John Wiley & Sons. p.
2606:The Annals of Applied Probability
2554:The Annals of Applied Probability
63:RBMs have been shown to describe
3527:
3526:
2678:Handbook of Monte Carlo Methods
2446:Advances in Applied Probability
2392:Advances in Applied Probability
2213:10.1002/9780470400531.eorms0711
1582:program creates a sample path.
1337:{\displaystyle 2\Sigma =RD+DR'}
258:{\displaystyle Z(t)=X(t)+RY(t)}
155:non-singular covariance matrix
1425:
1380:
1217:
1207:
1183:
1171:
1143:
1140:
1110:
1098:
1086:
1083:
1066:
1047:
972:
966:
958:
946:
929:
923:
914:
908:
817:
805:
608:
599:
593:
587:
416:whenever the boundary surface
252:
246:
234:
228:
219:
213:
1:
3357:Flow-equivalent server method
2799:"Section A.5 (code for BNET)"
3438:Adversarial queueing network
3327:Continuous-time Markov chain
2797:Dai, Jiangang "Jim" (1990).
2151:Dirichlet boundary condition
1363:probability density function
1001:, which is not the case for
777: ≥ 0, (with Φ the
3400:Heavy traffic approximation
3145:Pollaczek–Khinchine formula
3038:10.1007/978-3-642-62025-6_6
2990:10.1007/978-3-642-57856-4_2
1275:{\displaystyle x\geq p_{b}}
404:the process behaves like a
75:and proven by Iglehart and
3567:
2600:; Williams, R. J. (1992).
2500:; Williams, R. J. (1987).
2311:"Confined Brownian motion"
2157:Neumann boundary condition
282:–dimensional vector where
141:–dimensional drift vector
18:Reflecting Brownian motion
3524:
3405:Reflected Brownian motion
3210:Markovian arrival process
2541:Bramson, M.; Dai, J. G.;
2521:10.1080/17442508708833469
2253:. John Wiley & Sons.
2141:Other boundary conditions
42:regulated Brownian motion
38:reflected Brownian motion
3428:Layered queueing network
3215:Rational arrival process
2327:10.1103/PhysRevE.49.5158
2163:Robin boundary condition
1584:
893:of the Brownian motion,
787:exponential distribution
491:th column of the matrix
44:, both with the acronym
3516:Teletraffic engineering
3311:Shortest remaining time
2853:10.1214/aoap/1177005771
2776:10.1214/aoap/1177005979
2726:10.1214/aoap/1177004597
2619:10.1214/aoap/1177005704
2276:Computational Economics
2173:sticky Brownian motion.
1555:Closed-form expressions
412:is pushed in direction
3511:Scheduling (computing)
3150:Matrix analytic method
2161:elastic reflection, a
1496:
1451:
1338:
1276:
1241:
1029:
982:
885:, the distribution of
870:
762:
477:
398:
358:
259:
186:) is an unconstrained
127:
3342:Product-form solution
3243:Gordon–Newell theorem
3205:Poisson point process
3096:Single queueing nodes
2433:Iglehart, Donald L.;
2386:Iglehart, Donald L.;
1497:
1431:
1339:
1277:
1242:
1030:
1028:{\displaystyle p_{b}}
983:
871:
763:
560:marginal distribution
478:
399:
359:
317: = 1,2,...,
260:
128:
71:as first proposed by
3369:Decomposition method
2750:Harrison, J. Michael
2242:Harrison, J. Michael
1374:
1361:). In this case the
1302:
1253:
1249:For the plane above
1041:
1012:
902:
797:
781:) which yields (for
579:
512:Stability conditions
420:
372:
334:
207:
133:uniquely defined by
103:
3501:Pipeline (software)
3481:Flow control (data)
3476:Erlang distribution
3458:Information systems
3248:Mean value analysis
2130:Multiple dimensions
1285:Multiple dimensions
533:non-singular matrix
449:
392:
353:
313: = 0 for
122:
3506:Quality of service
3491:Network congestion
3352:Quasireversibility
3332:Kendall's notation
1492:
1334:
1272:
1237:
1025:
978:
962:
866:
758:
544: < 0.
473:
472:
433:
394:
393:
376:
364:, t ≥ 0.
354:
337:
255:
170:reflection matrix
123:
106:
97:stochastic process
34:probability theory
3538:
3537:
3496:Network scheduler
3395:Mean-field theory
3306:Shortest job next
3296:Processor sharing
3253:Buzen's algorithm
3236:Traffic equations
3224:Queueing networks
3198:Arrival processes
3172:Kingman's formula
3047:978-3-540-60629-1
2999:978-3-642-63381-2
2576:10.1214/09-AAP631
2349:Kingman, J. F. C.
2315:Physical Review E
2184:Skorokhod problem
1235:
997:is increasing in
935:
752:
660:
293:(0) = 0
16:(Redirected from
3558:
3530:
3529:
3347:Balance equation
3279:Service policies
3177:Lindley equation
3082:
3075:
3068:
3059:
3052:
3051:
3027:
3010:
3004:
3003:
2977:
2971:
2970:
2964:
2955:
2949:
2948:
2938:
2911:
2905:
2904:
2885:Skorokhod, A. V.
2881:
2866:
2865:
2855:
2837:
2824:
2818:
2817:
2815:
2813:
2803:
2794:
2788:
2787:
2769:
2745:
2739:
2738:
2728:
2704:
2698:
2697:
2681:
2668:
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2638:
2632:
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2621:
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2587:
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2506:
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2482:
2480:
2443:
2430:
2424:
2423:
2383:
2377:
2376:
2345:
2339:
2338:
2321:(6): 5158–5163.
2306:
2300:
2299:
2271:
2265:
2264:
2252:
2238:
2227:
2226:
2200:
2122:
2119:
2116:
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2110:
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1620:
1617:
1613:
1610:
1607:
1604:
1601:
1597:
1594:
1591:
1588:
1578:. The following
1501:
1499:
1498:
1493:
1491:
1490:
1489:
1488:
1479:
1478:
1461:
1460:
1450:
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1423:
1405:
1404:
1392:
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1340:
1335:
1333:
1281:
1279:
1278:
1273:
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1246:
1244:
1243:
1238:
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1234:
1233:
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1228:
1202:
1201:
1200:
1196:
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1190:
1178:
1170:
1169:
1128:
1127:
1123:
1118:
1117:
1105:
1073:
1065:
1064:
1034:
1032:
1031:
1026:
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1023:
987:
985:
984:
979:
961:
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867:
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859:
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622:
586:
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264:
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111:
21:
3566:
3565:
3561:
3560:
3559:
3557:
3556:
3555:
3541:
3540:
3539:
3534:
3520:
3452:
3411:
3378:
3364:Arrival theorem
3315:
3274:
3231:Jackson network
3219:
3193:
3184:Fork–join queue
3123:Burke's theorem
3091:
3089:Queueing theory
3086:
3056:
3055:
3048:
3012:
3011:
3007:
3000:
2979:
2978:
2974:
2962:
2957:
2956:
2952:
2913:
2912:
2908:
2901:10.1137/1107002
2883:
2882:
2869:
2835:
2829:Harrison, J. M.
2826:
2825:
2821:
2811:
2809:
2801:
2796:
2795:
2791:
2747:
2746:
2742:
2706:
2705:
2701:
2694:
2672:Kroese, Dirk P.
2670:
2669:
2665:
2658:10.1137/0141030
2642:Harrison, J. M.
2640:
2639:
2635:
2598:Harrison, J. M.
2596:
2595:
2591:
2549:
2543:Harrison, J. M.
2540:
2539:
2528:
2504:
2498:Harrison, J. M.
2496:
2495:
2488:
2478:
2476:
2458:10.2307/1426324
2441:
2432:
2431:
2427:
2404:10.2307/3518347
2385:
2384:
2380:
2347:
2346:
2342:
2308:
2307:
2303:
2273:
2272:
2268:
2261:
2250:
2240:
2239:
2230:
2223:
2202:
2201:
2197:
2192:
2180:
2143:
2132:
2124:
2123:
2120:
2117:
2114:
2111:
2108:
2105:
2102:
2099:
2096:
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2087:
2084:
2081:
2078:
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2063:
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2039:
2036:
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2030:
2027:
2024:
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2018:
2015:
2012:
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2006:
2003:
2000:
1997:
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1991:
1988:
1985:
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1979:
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1928:
1925:
1922:
1919:
1916:
1913:
1910:
1907:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1840:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1765:
1762:
1759:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1681:
1678:
1675:
1672:
1669:
1666:
1663:
1660:
1657:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1611:
1608:
1605:
1602:
1599:
1595:
1592:
1589:
1586:
1568:
1563:
1541:
1532:
1524:
1515:
1480:
1470:
1462:
1452:
1415:
1396:
1383:
1372:
1371:
1326:
1300:
1299:
1287:
1262:
1251:
1250:
1216:
1203:
1182:
1161:
1132:
1109:
1075:
1074:
1056:
1039:
1038:
1015:
1010:
1009:
900:
899:
851:
829:
795:
794:
733:
729:
712:
706:
691:
669:
641:
637:
623:
617:
577:
576:
556:
551:
514:
503:
495:." The process
453:
418:
417:
370:
369:
332:
331:
312:
303:
205:
204:
188:Brownian motion
101:
100:
85:
65:queueing models
30:
23:
22:
15:
12:
11:
5:
3564:
3562:
3554:
3553:
3551:Wiener process
3543:
3542:
3536:
3535:
3525:
3522:
3521:
3519:
3518:
3513:
3508:
3503:
3498:
3493:
3488:
3483:
3478:
3473:
3468:
3462:
3460:
3454:
3453:
3451:
3450:
3445:
3440:
3435:
3433:Polling system
3430:
3425:
3419:
3417:
3413:
3412:
3410:
3409:
3408:
3407:
3397:
3392:
3386:
3384:
3383:Limit theorems
3380:
3379:
3377:
3376:
3371:
3366:
3361:
3360:
3359:
3354:
3349:
3339:
3334:
3329:
3323:
3321:
3317:
3316:
3314:
3313:
3308:
3303:
3298:
3293:
3288:
3282:
3280:
3276:
3275:
3273:
3272:
3267:
3262:
3257:
3256:
3255:
3250:
3240:
3239:
3238:
3227:
3225:
3221:
3220:
3218:
3217:
3212:
3207:
3201:
3199:
3195:
3194:
3192:
3191:
3186:
3181:
3180:
3179:
3174:
3164:
3159:
3154:
3153:
3152:
3147:
3137:
3132:
3127:
3126:
3125:
3115:
3110:
3105:
3099:
3097:
3093:
3092:
3087:
3085:
3084:
3077:
3070:
3062:
3054:
3053:
3046:
3005:
2998:
2972:
2950:
2906:
2867:
2819:
2789:
2767:10.1.1.44.5520
2740:
2699:
2693:978-1118014950
2692:
2663:
2652:(2): 345–361.
2633:
2589:
2526:
2486:
2452:(2): 355–369.
2425:
2398:(1): 150–177.
2378:
2359:(2): 383–392.
2340:
2301:
2282:(2): 185–207.
2266:
2260:978-0471819394
2259:
2228:
2221:
2194:
2193:
2191:
2188:
2187:
2186:
2179:
2176:
2175:
2174:
2171:
2168:
2165:
2159:
2153:
2142:
2139:
2131:
2128:
1585:
1576:Wiener process
1572:absolute value
1567:
1564:
1562:
1559:
1537:
1528:
1520:
1516: = 2
1511:
1505:
1504:
1503:
1502:
1487:
1483:
1477:
1473:
1469:
1465:
1459:
1455:
1449:
1444:
1441:
1438:
1434:
1430:
1427:
1422:
1418:
1414:
1411:
1408:
1403:
1399:
1395:
1390:
1386:
1382:
1379:
1347:
1346:
1345:
1344:
1332:
1329:
1325:
1322:
1319:
1316:
1313:
1310:
1307:
1286:
1283:
1269:
1265:
1261:
1258:
1231:
1227:
1223:
1219:
1215:
1212:
1209:
1206:
1199:
1195:
1189:
1185:
1181:
1177:
1173:
1168:
1164:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1135:
1131:
1126:
1122:
1116:
1112:
1108:
1104:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1078:
1071:
1068:
1063:
1059:
1055:
1052:
1049:
1046:
1022:
1018:
991:
990:
989:
988:
977:
974:
971:
968:
965:
960:
957:
954:
951:
948:
945:
942:
938:
934:
931:
928:
925:
922:
919:
916:
913:
910:
907:
879:
878:
877:
876:
865:
858:
854:
849:
845:
842:
839:
836:
832:
828:
825:
822:
819:
816:
813:
810:
807:
803:
771:
770:
769:
768:
756:
748:
744:
740:
736:
732:
727:
724:
721:
718:
715:
709:
705:
698:
694:
689:
685:
682:
679:
676:
672:
668:
664:
656:
652:
648:
644:
640:
635:
632:
629:
626:
620:
616:
613:
610:
607:
604:
601:
598:
595:
592:
589:
585:
555:
552:
550:
547:
546:
545:
536:
513:
510:
499:
483:is hit, where
471:
468:
465:
460:
456:
452:
447:
442:
437:
432:
429:
426:
406:Wiener process
390:
385:
380:
366:
365:
351:
346:
341:
330:) ∈
321:
308:
299:
294:
268:
267:
266:
265:
254:
251:
248:
245:
242:
239:
236:
233:
230:
227:
224:
221:
218:
215:
212:
176:
175:
160:
145:
120:
115:
110:
84:
81:
50:Wiener process
28:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3563:
3552:
3549:
3548:
3546:
3533:
3523:
3517:
3514:
3512:
3509:
3507:
3504:
3502:
3499:
3497:
3494:
3492:
3489:
3487:
3486:Message queue
3484:
3482:
3479:
3477:
3474:
3472:
3471:Erlang (unit)
3469:
3467:
3464:
3463:
3461:
3459:
3455:
3449:
3448:Retrial queue
3446:
3444:
3441:
3439:
3436:
3434:
3431:
3429:
3426:
3424:
3421:
3420:
3418:
3414:
3406:
3403:
3402:
3401:
3398:
3396:
3393:
3391:
3388:
3387:
3385:
3381:
3375:
3372:
3370:
3367:
3365:
3362:
3358:
3355:
3353:
3350:
3348:
3345:
3344:
3343:
3340:
3338:
3335:
3333:
3330:
3328:
3325:
3324:
3322:
3318:
3312:
3309:
3307:
3304:
3302:
3299:
3297:
3294:
3292:
3289:
3287:
3284:
3283:
3281:
3277:
3271:
3268:
3266:
3263:
3261:
3260:Kelly network
3258:
3254:
3251:
3249:
3246:
3245:
3244:
3241:
3237:
3234:
3233:
3232:
3229:
3228:
3226:
3222:
3216:
3213:
3211:
3208:
3206:
3203:
3202:
3200:
3196:
3190:
3187:
3185:
3182:
3178:
3175:
3173:
3170:
3169:
3168:
3165:
3163:
3160:
3158:
3155:
3151:
3148:
3146:
3143:
3142:
3141:
3138:
3136:
3133:
3131:
3128:
3124:
3121:
3120:
3119:
3116:
3114:
3111:
3109:
3106:
3104:
3101:
3100:
3098:
3094:
3090:
3083:
3078:
3076:
3071:
3069:
3064:
3063:
3060:
3049:
3043:
3039:
3035:
3031:
3026:
3025:
3019:
3018:McKean, H. P.
3015:
3009:
3006:
3001:
2995:
2991:
2987:
2983:
2976:
2973:
2968:
2961:
2954:
2951:
2946:
2942:
2937:
2932:
2928:
2924:
2920:
2916:
2910:
2907:
2902:
2898:
2894:
2890:
2886:
2880:
2878:
2876:
2874:
2872:
2868:
2863:
2859:
2854:
2849:
2845:
2841:
2834:
2830:
2823:
2820:
2807:
2800:
2793:
2790:
2785:
2781:
2777:
2773:
2768:
2763:
2759:
2755:
2751:
2748:Dai, Jim G.;
2744:
2741:
2736:
2732:
2727:
2722:
2718:
2714:
2710:
2703:
2700:
2695:
2689:
2685:
2680:
2679:
2673:
2667:
2664:
2659:
2655:
2651:
2647:
2643:
2637:
2634:
2629:
2625:
2620:
2615:
2611:
2607:
2603:
2599:
2593:
2590:
2585:
2581:
2577:
2573:
2568:
2563:
2559:
2555:
2548:
2544:
2537:
2535:
2533:
2531:
2527:
2522:
2518:
2514:
2510:
2503:
2499:
2493:
2491:
2487:
2475:
2471:
2467:
2463:
2459:
2455:
2451:
2447:
2440:
2436:
2429:
2426:
2421:
2417:
2413:
2409:
2405:
2401:
2397:
2393:
2389:
2382:
2379:
2374:
2370:
2366:
2362:
2358:
2354:
2350:
2344:
2341:
2336:
2332:
2328:
2324:
2320:
2316:
2312:
2305:
2302:
2297:
2293:
2289:
2285:
2281:
2277:
2270:
2267:
2262:
2256:
2249:
2248:
2243:
2237:
2235:
2233:
2229:
2224:
2222:9780470400531
2218:
2214:
2210:
2206:
2199:
2196:
2189:
2185:
2182:
2181:
2177:
2172:
2169:
2166:
2164:
2160:
2158:
2154:
2152:
2148:
2147:
2146:
2140:
2138:
2136:
2129:
2127:
1583:
1581:
1577:
1573:
1566:One dimension
1565:
1560:
1558:
1556:
1552:
1549:
1546: =
1545:
1540:
1536:
1531:
1527:
1523:
1519:
1514:
1510:
1485:
1481:
1475:
1471:
1467:
1463:
1457:
1453:
1447:
1442:
1439:
1436:
1432:
1428:
1420:
1416:
1412:
1409:
1406:
1401:
1397:
1393:
1388:
1384:
1377:
1370:
1369:
1368:
1367:
1366:
1364:
1360:
1356:
1353: =
1352:
1330:
1327:
1323:
1320:
1317:
1314:
1311:
1305:
1298:
1297:
1296:
1295:
1294:
1292:
1284:
1282:
1267:
1263:
1259:
1256:
1247:
1229:
1225:
1221:
1213:
1210:
1204:
1197:
1193:
1187:
1179:
1175:
1166:
1162:
1158:
1155:
1152:
1149:
1146:
1137:
1133:
1129:
1124:
1120:
1114:
1106:
1102:
1095:
1092:
1089:
1080:
1076:
1069:
1061:
1057:
1053:
1050:
1044:
1036:
1020:
1016:
1006:
1004:
1000:
996:
975:
969:
963:
955:
952:
949:
943:
940:
932:
926:
920:
917:
911:
905:
898:
897:
896:
895:
894:
892:
888:
884:
863:
856:
852:
847:
843:
840:
837:
834:
830:
826:
823:
820:
814:
811:
808:
793:
792:
791:
790:
789:
788:
784:
780:
776:
754:
746:
742:
738:
734:
730:
725:
722:
719:
716:
713:
707:
696:
692:
687:
683:
680:
677:
674:
670:
666:
662:
654:
650:
646:
642:
638:
633:
630:
627:
624:
618:
611:
605:
602:
596:
590:
575:
574:
573:
572:
571:
569:
566:and variance
565:
561:
554:One dimension
553:
548:
543:
540:
537:
534:
530:
527:
526:
525:
523:
519:
511:
509:
507:
502:
498:
494:
490:
486:
466:
463:
458:
454:
450:
445:
440:
430:
427:
415:
411:
407:
388:
383:
349:
344:
329:
325:
322:
320:
316:
311:
307:
302:
298:
295:
292:
288:
285:
284:
283:
281:
277:
273:
249:
243:
240:
237:
231:
225:
222:
216:
210:
203:
202:
201:
200:
199:
197:
194:and variance
193:
189:
185:
181:
173:
169:
165:
161:
158:
154:
150:
146:
144:
140:
136:
135:
134:
118:
113:
98:
94:
90:
82:
80:
78:
74:
70:
69:heavy traffic
67:experiencing
66:
61:
59:
55:
51:
47:
43:
39:
35:
27:
19:
3443:Loss network
3404:
3374:Beneš method
3337:Little's law
3320:Key concepts
3270:BCMP network
3023:
3008:
2981:
2975:
2966:
2953:
2926:
2922:
2909:
2892:
2888:
2846:(1): 65–86.
2843:
2839:
2827:Dai, J. G.;
2822:
2810:. Retrieved
2805:
2792:
2760:(1): 16–35.
2757:
2753:
2743:
2716:
2712:
2702:
2677:
2666:
2649:
2645:
2636:
2609:
2605:
2592:
2557:
2553:
2512:
2508:
2477:. Retrieved
2449:
2445:
2428:
2395:
2391:
2381:
2356:
2352:
2343:
2318:
2314:
2304:
2279:
2275:
2269:
2246:
2204:
2198:
2144:
2133:
2125:
2118:'k-'
2064:'k-'
1569:
1550:
1547:
1543:
1538:
1534:
1529:
1525:
1521:
1517:
1512:
1508:
1506:
1358:
1350:
1348:
1288:
1248:
1037:
1007:
1002:
998:
994:
992:
890:
886:
882:
880:
782:
774:
772:
567:
563:
557:
541:
538:
528:
517:
515:
500:
496:
492:
488:
484:
413:
409:
367:
327:
323:
318:
314:
309:
305:
300:
296:
290:
286:
279:
275:
271:
269:
195:
191:
183:
179:
177:
171:
167:
163:
156:
152:
148:
142:
138:
92:
88:
86:
62:
45:
41:
37:
31:
26:
3466:Data buffer
3423:Fluid queue
3390:Fluid limit
3301:Round-robin
3167:G/G/1 queue
3162:G/M/1 queue
3157:M/G/k queue
3140:M/G/1 queue
3135:M/M/∞ queue
3130:M/M/c queue
3118:M/M/1 queue
3113:M/D/c queue
3108:M/D/1 queue
3103:D/M/1 queue
3028:. pp.
2509:Stochastics
2435:Ward, Whitt
2388:Whitt, Ward
190:with drift
3416:Extensions
3189:Bulk queue
2915:Feller, W.
2812:5 December
2719:(4): 875.
2612:(2): 263.
2560:(2): 753.
2190:References
1561:Simulation
881:For fixed
506:local time
83:Definition
3265:G-network
2762:CiteSeerX
2567:1009.5746
2515:(2): 77.
2474:120281300
2420:202104090
2335:1063-651X
2296:121673717
1472:η
1468:−
1454:η
1433:∏
1410:…
1309:Σ
1260:≥
1214:π
1156:−
1138:−
1093:−
1081:−
944:∈
918:∼
853:σ
841:μ
835:−
827:−
731:σ
723:μ
720:−
714:−
704:Φ
693:σ
681:μ
675:−
667:−
639:σ
631:μ
628:−
615:Φ
603:≤
431:∈
58:diffusion
3545:Category
3532:Category
2929:: 1–31.
2917:(1954).
2895:: 3–23.
2831:(1992).
2545:(2010).
2437:(1970).
2244:(1985).
2178:See also
1331:′
773:for all
522:M-matrix
54:physical
3014:Itō, K.
2945:0063607
2862:2959654
2784:2959623
2735:2245096
2628:2959751
2584:2251853
2466:1426324
2412:3518347
2373:2984229
2070:subplot
2022:subplot
1587:% rbm.m
504:is the
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