3500:
1145:
720:
118:
A fluid queue can be viewed as a large tank, typically assumed to be of infinite capacity, connected to a series of pipes that pour fluid in to the tank and a series of pumps which remove fluid from the tank. An operator controls the pipes and pumps controlling the rate at which fluid pours in to the
302:
1308:
A feedback fluid queue is a model where the model parameters (transition rate matrix and drift vector) are allowed to some extent to depend on the buffer content. Typically the buffer content is partitioned and the parameters depend on which partition the buffer content process is in. The ordered
568:
963:
841:
576:
944:
439:
1228:
150:
450:
1140:{\displaystyle W^{\ast }(s)={\frac {\beta \lambda +s\lambda -\beta \mu +\alpha \mu -{\sqrt {4\beta \alpha \mu (\mu -\lambda )+(s\lambda +\beta (\lambda -\mu )+\alpha \mu )^{2}}}}{2\beta (\lambda -\mu )}}}
731:
715:{\displaystyle F(x,2)={\frac {\alpha }{\alpha +\beta }}-{\frac {\beta \left(\lambda -\mu \right)}{\alpha +\beta }}e^{\left({\frac {\beta }{\mu }}-{\frac {\alpha }{\lambda -\mu }}\right)x}}
3579:
4114:
1509:
Stanford, David A.; Latouche, Guy; Woolford, Douglas G.; Boychuk, Dennis; Hunchak, Alek (2005). "Erlangized Fluid Queues with
Application to Uncontrolled Fire Perimeter".
3938:
1888:
1859:
1284:
The term mountain process has been coined to describe the maximum buffer content process value achieved during a busy period and can be computed using results from a
2170:
4541:
3534:
4071:
4051:
4455:
884:
379:
1160:
297:{\displaystyle {\frac {\mathrm {d} X(t)}{\mathrm {d} t}}={\begin{cases}r_{i}&{\text{ if }}X(t)>0\\\max(r_{i},0)&{\text{ if }}X(t)=0.\end{cases}}}
2657:
1915:
Gaeta, R.; Gribaudo, M.; Manini, D.; Sereno, M. (2006). "Analysis of resource transfers in peer-to-peer file sharing applications using fluid models".
1276:
In a finite buffer the rate at which fluid is lost (rejected from the system due to a full buffer) can be computed using
Laplace-Stieltjes transforms.
4372:
3051:
4382:
4056:
369:
For a simple system where service has a constant rate Ό and arrival fluctuate between rates λ and 0 (in states 1 and 2 respectively) according to a
4066:
40:) is a mathematical model used to describe the fluid level in a reservoir subject to randomly determined periods of filling and emptying. The term
4424:
1240:
There are two main approaches to solving for the busy period in general, using either spectral decomposition or an iterative recurrent method. A
4139:
1954:
Yazici, M. A.; Akar, N. (2013). "Analysis of continuous feedback Markov fluid queues and its applications to modeling
Optical Burst Switching".
4321:
3340:
878:
For an infinite buffer with constant service rate Ό and arrivals at rates λ and 0, modulated by a continuous time Markov chain with parameters
563:{\displaystyle F(x,1)={\frac {\beta }{\alpha +\beta }}\left(1-e^{\left({\frac {\beta }{\mu }}-{\frac {\alpha }{\lambda -\mu }}\right)x}\right)}
331:
1325:
with parameters controlled by a Markov process. Two different types of boundary conditions are commonly considered: absorbing and reflecting.
4611:
4601:
4124:
2273:
1981:
1604:
63:
in 1954 where a discrete-time model was considered. Fluid queues allow arrivals to be continuous rather than discrete, as in models like the
4511:
4475:
1297:
4428:
1857:
Norros, I.; Roberts, J. W.; Simonian, A.; Virtamo, J. T. (1991). "The superposition of variable bit rate sources in an ATM multiplexer".
4779:
4516:
1886:
Rasmussen, C.; Sorensen, J. H.; Kvols, K. S.; Jacobsen, S. B. (1991). "Source-independent call acceptance procedures in ATM networks".
3626:
3527:
2787:
1955:
1800:
1472:(1991). "Analysis and design of rate-based congestion control of high speed networks, I: Stochastic fluid models, access regulation".
4581:
4159:
4129:
1775:
Proceedings of the joint international conference on
Measurement and modeling of computer systems - SIGMETRICS 2004/PERFORMANCE 2004
4432:
4416:
4626:
4331:
3551:
3257:
1321:
Second order fluid queues (sometimes called Markov modulated diffusion processes or fluid queues with
Brownian noise) consider a
3116:
2957:
Karandikar, R. L.; Kulkarni, V. G. (1995). "Second-Order Fluid Flow Models: Reflected
Brownian Motion in a Random Environment".
868:
4531:
4496:
4465:
4460:
4099:
3896:
3813:
4470:
3798:
4094:
3901:
44:
was used in earlier literature for these models. The model has been used to approximate discrete models, model the spread of
3820:
107:
4556:
4436:
836:{\displaystyle {\frac {(\lambda -\mu )\beta }{(\mu (\alpha +\beta )-\beta \lambda )(\alpha +\beta )}}(\mu ,\lambda -\mu ).}
4784:
4561:
4397:
4296:
4281:
3693:
3609:
3520:
3328:
3044:
2988:
2959:
1342:
4571:
4207:
4566:
3409:
3298:
2092:
370:
308:
4169:
2986:
Gribaudo, M.; Manini, D.; Sericola, B.; Telek, M. (2007). "Second order fluid models with general boundary behaviour".
3753:
3698:
3614:
3371:
357:
The additive decomposition method is numerically stable and separates the eigenvalues necessary for computation using
4501:
4491:
4134:
4104:
4506:
3671:
3569:
87:
4217:
3793:
3574:
3214:
4586:
4387:
4301:
4286:
3676:
3376:
3181:
1322:
95:
4420:
4306:
3728:
4805:
3808:
3783:
3503:
3399:
3186:
3037:
1620:
1241:
60:
4526:
4109:
3644:
132:
for the net fluid arrival rate in this state (input less output). When the buffer contains fluid, if we write
2055:
Govorun, M.; Latouche, G.; Remiche, M. A. (2013). "Stability for Fluid Queues: Characteristic
Inequalities".
4721:
4711:
4402:
4184:
3923:
3788:
3599:
3487:
3282:
1732:
1234:
347:
99:
4006:
2042:
Teletraffic
Engineering in a Competitive World (Proceedings of the 16th International Teletraffic Congress)
1773:
Hohn, N.; Veitch, D.; Papagiannaki, K.; Diot, C. (2004). "Bridging router performance and queuing theory".
4663:
4591:
3850:
3482:
3272:
3121:
2997:
2626:
2302:
1926:
1917:
1778:
867:) returns to zero). In earlier literature it is sometimes referred to as the wet period (of the dam). The
351:
53:
2168:
Asmussen, SĂžren (1995). "Stationary distributions for fluid flow models with or without brownian noise".
4686:
4668:
4648:
4643:
4362:
4194:
4174:
4021:
3964:
3803:
3713:
3313:
3176:
851:
The busy period is the period of time measured from the instant that fluid first arrives in the buffer (
4154:
3303:
4761:
4716:
4706:
4447:
4392:
4367:
4336:
4316:
4076:
4061:
3928:
3224:
3143:
2785:
DÄbicki, K.; Dieker, A. B.; Rolski, T. (2007). "Quasi-Product Forms for Levy-Driven Fluid
Networks".
1690:
3155:
3002:
1931:
1296:
The stationary distribution of two tandem fluid queues has been computed and shown not to exhibit a
195:
4756:
4596:
4521:
4326:
4086:
3996:
3886:
3472:
3452:
3447:
3219:
2758:
Kella, O. (2000). "Non-product form of two-dimensional fluid networks with dependent LĂ©vy inputs".
2555:
RACSAM - Revista de la Real
Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas
2198:
1310:
358:
335:
1783:
4726:
4691:
4606:
4576:
4346:
4341:
4164:
4001:
3666:
3604:
3543:
3477:
3462:
3429:
3323:
3015:
2939:
2921:
2894:
2814:
2796:
2707:
2570:
2494:
2231:
2112:
2074:
2022:
1987:
1806:
1755:
1706:
1565:
1528:
1491:
1450:
1403:
79:
21:
4407:
3094:
1237:
function which means that the longer a busy period has lasted the longer it is likely to last.
4746:
4551:
4202:
3959:
3876:
3845:
3738:
3718:
3708:
3564:
3559:
3467:
3366:
3277:
3267:
3207:
2867:
2589:
2396:
2293:
2269:
2057:
2040:
Ramaswami, V. Smith, D.; Hey, P (eds.). "Matrix analytic methods for stochastic fluid flows".
1977:
1796:
1600:
1548:
1511:
1433:
4412:
4149:
2436:
1386:(1988). "Stochastic Theory of a Fluid Model of Producers and Consumers Coupled by a Buffer".
4766:
4653:
4536:
3906:
3881:
3830:
3758:
3681:
3634:
3318:
3262:
3148:
3007:
2968:
2931:
2884:
2876:
2845:
2806:
2767:
2738:
2699:
2669:
2635:
2604:
2562:
2528:
2486:
2451:
2412:
2377:
2345:
2311:
2261:
2221:
2213:
2179:
2147:
2104:
2066:
2014:
1969:
1961:
1936:
1897:
1868:
1837:
1788:
1747:
1698:
1681:
1655:
1557:
1520:
1483:
1474:
1442:
1395:
3106:
1425:
4731:
4631:
4616:
4377:
4311:
3989:
3933:
3916:
3661:
3335:
3202:
3060:
871:
of the busy period distribution is known for the fluid queue with infinite buffer and the
17:
4546:
3778:
3345:
3308:
119:
buffer and the rate at which fluid leaves. When the operator puts the system in to state
1694:
939:{\displaystyle Q={\begin{pmatrix}-\alpha &\alpha \\\beta &-\beta \end{pmatrix}}}
434:{\displaystyle Q={\begin{pmatrix}-\alpha &\alpha \\\beta &-\beta \end{pmatrix}}}
4736:
4701:
4621:
4227:
3974:
3891:
3860:
3855:
3835:
3825:
3768:
3763:
3743:
3723:
3688:
3656:
3639:
3404:
2131:
1751:
1233:
In this case, of a single on/off source, the busy period distribution is known to be a
1151:
872:
75:
2350:
1546:
Remiche, M. A. (2005). "Compliance of the Token-Bucket Model with Markovian Traffic".
1223:{\displaystyle \mathbb {E} (W)={\frac {\lambda }{\alpha \mu +\beta (\lambda -\mu )}}.}
4799:
4638:
4179:
4016:
4011:
3969:
3911:
3733:
3649:
3589:
3457:
3442:
3419:
3231:
2912:
Ivanovs, J. (2010). "Markov-modulated Brownian motion with two reflecting barriers".
2365:
2250:
2078:
1728:
1569:
1532:
1469:
1383:
2943:
2898:
2818:
2624:
O'Reilly, M. G. M.; Palmowski, Z. (2013). "Loss rates for stochastic fluid models".
2574:
2477:; Scheinhardt, W. R. W. (2001). "Joint Distributions for Interacting Fluid Queues".
1810:
1759:
1710:
1495:
1244:
algorithm for computing points of the transform was published by Ahn and Ramaswami.
4696:
4658:
4212:
4144:
4033:
4028:
3840:
3773:
3748:
3584:
3414:
3241:
3019:
2711:
2498:
1676:
1454:
1268: = 3 then the fluid queue has busy period with mean 1 and variance 5/3.
4276:
2333:
2260:. Communications in Computer and Information Science. Vol. 356. p. 167.
1991:
327:
represents the level of fluid in the buffer it can only take non-negative values.
2513:
2265:
2070:
1352:
4741:
4260:
4255:
4250:
4043:
3984:
3979:
3943:
3703:
3594:
3437:
3361:
3138:
3133:
3128:
3111:
3101:
3089:
3084:
3079:
3074:
2474:
1842:
1825:
1589:
1346:
1313:
can be used to efficiently compute the stationary distribution of such a model.
1285:
875:
busy period in the case of a finite buffer and arrivals as instantaneous jumps.
83:
68:
64:
49:
2865:
Kankaya, H. E.; Akar, N. (2008). "Solving Multi-Regime Feedback Fluid Queues".
2639:
2437:"Busy period analysis, rare events and transient behavior in fluid flow models"
2315:
2298:"An approximate compositional approach to the analysis of fluid queue networks"
2297:
1940:
4751:
4291:
4235:
4119:
3160:
3011:
2880:
2850:
2743:
2726:
2703:
2687:
2673:
2653:
2532:
2490:
2456:
2183:
2151:
1660:
1643:
1362:
2935:
2771:
2727:"Stability and nonproduct form of stochastic fluid networks with LĂ©vy inputs"
2609:
2590:"Efficient algorithms for transient analysis of stochastic fluid flow models"
2548:"Applications of fluid flow matrix analytic methods in ruin theory âa review"
2417:
2400:
2217:
1965:
4245:
3236:
2832:
Malhotra, R.; Mandjes, M. R. H.; Scheinhardt, W. R. W.; Berg, J. L. (2008).
1792:
957:) for the LaplaceâStieltjes transform of the busy period distribution, then
2810:
2547:
2690:; Perry, D. (2009). "On the Cycle Maximum of Mountains, Dams and Queues".
2381:
1561:
1524:
1446:
2972:
1597:
Frontiers in Queueing: Models and Applications in Science and Engineering
45:
1334:
4072:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
3512:
2889:
2566:
2258:
Modern Probabilistic Methods for Analysis of Telecommunication Networks
2235:
2226:
2116:
2026:
1973:
1487:
1407:
444:
the stationary distribution can be computed explicitly and is given by
2801:
2199:"Infinite- and finite-buffer Markov fluid queues: A unified analysis"
1957:
Proceedings of the 2013 25th International Teletraffic Congress (ITC)
1901:
1872:
1702:
1356:
1338:
91:
2251:"Analysis of Fluid Queues in Saturation with Additive Decomposition"
2108:
2018:
1399:
2005:
Gani, J. (1969). "Recent Advances in Storage and Flooding Theory".
1733:"Stochastic Theory of a Data-Handling System with Multiple Sources"
3029:
2926:
2401:"Busy periods in fluid queues with multiple emptying input states"
2095:; Shi, Z. (1994). "Computing the Invariant Law of a Fluid Model".
1826:"A fluid queue modulated by two independent birthâdeath processes"
1426:"Fluid Flow Models and QueuesâA Connection by Stochastic Coupling"
3516:
3033:
2834:"A feedback fluid queue with two congestion control thresholds"
2833:
2512:
Gautam, N.; Kulkarni, V. G.; Palmowski, Z.; Rolski, T. (1999).
2334:"The expected wet period of finite dam with exponential inputs"
2135:
103:
1644:"Application of Methods in Sequential Analysis to Dam Theory"
52:
and to model high speed data networks. The model applies the
1623:(1954). "A probability theory of dams and storage systems".
290:
102:, and has applications in civil engineering when designing
4052:
Autoregressive conditional heteroskedasticity (ARCH) model
2368:; Dumas, V. (1998). "The busy period in the fluid queue".
1679:(1958). "Continuous Time Treatment of a Storage Problem".
1256: = 2 is fed by an on/off source with parameters
74:
Fluid queues have been used to model the performance of a
2662:
Probability in the Engineering and Informational Sciences
2521:
Probability in the Engineering and Informational Sciences
3580:
Independent and identically distributed random variables
4057:
Autoregressive integrated moving average (ARIMA) model
2514:"Bounds for Fluid Models Driven by Semi-Markov Inputs"
2444:
Journal of Applied Mathematics and Stochastic Analysis
899:
394:
1163:
966:
887:
859:) becomes non-zero) until the buffer is again empty (
734:
579:
453:
382:
350:
as first shown by Asmussen and can be computed using
153:
4679:
4484:
4446:
4355:
4269:
4226:
4193:
4085:
4042:
3952:
3869:
3625:
3550:
3428:
3387:
3354:
3291:
3250:
3195:
3169:
3067:
1824:Arunachalam, V.; Gupta, V.; Dharmaraja, S. (2010).
2656:; Perry, D.; Van Der Duyn Schouten, F. A. (1999).
1222:
1139:
938:
835:
714:
562:
433:
296:
110:, for which efficient solution methods are known.
2692:Communications in Statistics - Theory and Methods
2332:Lee, Eui Yong; Kinateder, Kimberly K. J. (2000).
3939:Stochastic chains with memory of variable length
1889:IEEE Journal on Selected Areas in Communications
1860:IEEE Journal on Selected Areas in Communications
1252:For example, if a fluid queue with service rate
237:
2171:Communications in Statistics. Stochastic Models
2163:
2161:
2546:Badescu, Andrei L.; Landriault, David (2009).
2287:
2285:
3528:
3045:
1830:Computers & Mathematics with Applications
8:
2370:ACM SIGMETRICS Performance Evaluation Review
1349:implementation of some of the above results.
2838:Mathematical Methods of Operations Research
2430:
2428:
2338:Stochastic Processes and Their Applications
4067:Autoregressiveâmoving-average (ARMA) model
3535:
3521:
3513:
3052:
3038:
3030:
1583:
1581:
1579:
3001:
2925:
2888:
2849:
2800:
2742:
2608:
2469:
2467:
2455:
2416:
2349:
2225:
1930:
1841:
1782:
1659:
1181:
1165:
1164:
1162:
1103:
1028:
989:
971:
965:
894:
886:
735:
733:
684:
671:
665:
622:
601:
578:
527:
514:
508:
475:
452:
389:
381:
264:
247:
210:
202:
190:
176:
157:
154:
152:
1722:
1720:
1590:"Fluid models for single buffer systems"
1375:
20:, a discipline within the mathematical
4373:Doob's martingale convergence theorems
2327:
2325:
332:piecewise deterministic Markov process
106:. The process is closely connected to
4125:Constant elasticity of variance (CEV)
4115:ChanâKarolyiâLongstaffâSanders (CKLS)
2658:"Fluid Queues and Mountain Processes"
1648:The Annals of Mathematical Statistics
1419:
1417:
7:
2249:Telek, M. S.; VĂ©csei, M. S. (2013).
1298:product form stationary distribution
4612:Skorokhod's representation theorem
4393:Law of large numbers (weak/strong)
2788:Mathematics of Operations Research
2136:"Continuous feedback fluid queues"
2130:Scheinhardt, W.; Van Foreest, N.;
1752:10.1002/j.1538-7305.1982.tb03089.x
330:The model is a particular type of
177:
158:
59:The model was first introduced by
14:
4582:Martingale representation theorem
2731:The Annals of Applied Probability
1740:The Bell System Technical Journal
346:The stationary distribution is a
4627:Stochastic differential equation
4517:Doob's optional stopping theorem
4512:DoobâMeyer decomposition theorem
3499:
3498:
4497:Convergence of random variables
4383:FisherâTippettâGnedenko theorem
2588:Ahn, S.; Ramaswami, V. (2005).
2007:Advances in Applied Probability
1588:Kulkarni, Vidyadhar G. (1997).
1424:Ahn, S.; Ramaswami, V. (2003).
1388:Advances in Applied Probability
4095:Binomial options pricing model
2914:Journal of Applied Probability
2760:Journal of Applied Probability
2597:Journal of Applied Probability
2405:Journal of Applied Probability
2206:Journal of Applied Probability
2197:Akar, N.; Sohraby, K. (2004).
2097:Journal of Applied Probability
1211:
1199:
1175:
1169:
1131:
1119:
1100:
1087:
1075:
1060:
1054:
1042:
983:
977:
827:
809:
803:
791:
788:
776:
764:
758:
750:
738:
595:
583:
469:
457:
278:
272:
259:
240:
224:
218:
171:
165:
140:) for the fluid level at time
1:
4562:Kolmogorov continuity theorem
4398:Law of the iterated logarithm
3329:Flow-equivalent server method
2989:Annals of Operations Research
2351:10.1016/S0304-4149(00)00034-X
90:(the intended technology for
4567:Kolmogorov extension theorem
4246:Generalized queueing network
3754:Interacting particle systems
3410:Adversarial queueing network
3299:Continuous-time Markov chain
2266:10.1007/978-3-642-35980-4_19
2071:10.1080/15326349.2013.750533
1359:code for multi-regime models
371:continuous time Markov chain
334:and can also be viewed as a
309:continuous time Markov chain
3699:Continuous-time random walk
3372:Heavy traffic approximation
3117:PollaczekâKhinchine formula
2140:Operations Research Letters
1843:10.1016/j.camwa.2010.08.039
869:LaplaceâStieltjes transform
108:quasi-birthâdeath processes
4822:
4707:Extreme value theory (EVT)
4507:Doob decomposition theorem
3799:OrnsteinâUhlenbeck process
3570:Chinese restaurant process
2640:10.1016/j.peva.2013.05.005
2316:10.1016/j.peva.2007.06.025
1941:10.1016/j.peva.2005.01.001
1363:Fluid flow models tutorial
338:with boundary conditions.
311:and is usually called the
88:Asynchronous Transfer Mode
4775:
4587:Optional stopping theorem
4388:Large deviation principle
4140:HeathâJarrowâMorton (HJM)
4077:Moving-average (MA) model
4062:Autoregressive (AR) model
3887:Hidden Markov model (HMM)
3821:SchrammâLoewner evolution
3496:
3377:Reflected Brownian motion
3182:Markovian arrival process
3012:10.1007/s10479-007-0297-7
2881:10.1080/15326340802232285
2851:10.1007/s00186-008-0235-8
2704:10.1080/03610910902936232
2674:10.1017/S0269964899134028
2533:10.1017/S026996489913403X
2457:10.1155/S1048953394000262
2184:10.1080/15326349508807330
2152:10.1016/j.orl.2004.11.008
1642:Phatarfod, R. M. (1963).
1323:reflected Brownian motion
1317:Second order fluid queues
96:peer-to-peer file sharing
4502:Doléans-Dade exponential
4332:Progressively measurable
4130:CoxâIngersollâRoss (CIR)
3400:Layered queueing network
3187:Rational arrival process
2435:Asmussen, S. R. (1994).
1966:10.1109/ITC.2013.6662952
1731:; Sondhi, M. M. (1982).
1292:Networks of fluid queues
1242:quadratically convergent
725:and average fluid level
56:to a stochastic source.
4722:Mathematical statistics
4712:Large deviations theory
4542:Infinitesimal generator
4403:Maximal ergodic theorem
4322:Piecewise-deterministic
3924:Random dynamical system
3789:Markov additive process
3488:Teletraffic engineering
3283:Shortest remaining time
2744:10.1214/aoap/1034968070
2491:10.1023/A:1011044217695
2044:. Elsevier Science B.V.
1793:10.1145/1005686.1005728
1661:10.1214/aoms/1177703892
1365:by V. Ramaswami at MAM8
1235:decreasing failure rate
352:matrix-analytic methods
348:phase-type distribution
342:Stationary distribution
100:optical burst switching
4557:KarhunenâLoĂšve theorem
4492:CameronâMartin formula
4456:BurkholderâDavisâGundy
3851:Variance gamma process
3483:Scheduling (computing)
3122:Matrix analytic method
2936:10.1239/jap/1294170517
2811:10.1287/moor.1070.0259
2772:10.1239/jap/1014843090
2627:Performance Evaluation
2610:10.1239/jap/1118777186
2418:10.1239/jap/1276784904
2303:Performance Evaluation
2218:10.1239/jap/1082999086
1918:Performance Evaluation
1224:
1141:
940:
837:
716:
564:
435:
373:with generator matrix
298:
54:leaky bucket algorithm
38:stochastic fluid model
4687:Actuarial mathematics
4649:Uniform integrability
4644:Stratonovich integral
4572:LĂ©vyâProkhorov metric
4476:MarcinkiewiczâZygmund
4363:Central limit theorem
3965:Gaussian random field
3794:McKeanâVlasov process
3714:Dyson Brownian motion
3575:GaltonâWatson process
3314:Product-form solution
3215:GordonâNewell theorem
3177:Poisson point process
3068:Single queueing nodes
2382:10.1145/277858.277881
1562:10.1081/STM-200057884
1525:10.1081/STM-200056242
1447:10.1081/STM-120023564
1304:Feedback fluid queues
1300:in nontrivial cases.
1225:
1142:
941:
838:
717:
565:
436:
299:
22:theory of probability
4762:Time series analysis
4717:Mathematical finance
4602:Reflection principle
3929:Regenerative process
3729:FlemingâViot process
3544:Stochastic processes
3341:Decomposition method
2973:10.1287/opre.43.1.77
1599:. pp. 321â338.
1161:
964:
885:
732:
577:
451:
380:
151:
4757:Stochastic analysis
4597:Quadratic variation
4592:Prokhorov's theorem
4527:FeynmanâKac formula
3997:Markov random field
3645:Birthâdeath process
3473:Pipeline (software)
3453:Flow control (data)
3448:Erlang distribution
3430:Information systems
3220:Mean value analysis
2960:Operations Research
1695:1958Natur.182...39G
1311:Schur factorization
1264: = 1 and
359:Schur decomposition
336:Markov reward model
313:environment process
4727:Probability theory
4607:Skorokhod integral
4577:Malliavin calculus
4160:Korn-Kreer-Lenssen
4044:Time series models
4007:PitmanâYor process
3478:Quality of service
3463:Network congestion
3324:Quasireversibility
3304:Kendall's notation
2725:Kella, O. (1996).
2567:10.1007/BF03191912
1625:Aust. J. Appl. Sci
1488:10.1007/BF01158791
1220:
1137:
936:
930:
833:
712:
560:
431:
425:
317:background process
307:The operator is a
294:
289:
4793:
4792:
4747:Signal processing
4466:Doob's upcrossing
4461:Doob's martingale
4425:EngelbertâSchmidt
4368:Donsker's theorem
4302:Feller-continuous
4170:RendlemanâBartter
3960:Dirichlet process
3877:Branching process
3846:Telegraph process
3739:Geometric process
3719:Empirical process
3709:Diffusion process
3565:Branching process
3560:Bernoulli process
3510:
3509:
3468:Network scheduler
3367:Mean-field theory
3278:Shortest job next
3268:Processor sharing
3225:Buzen's algorithm
3208:Traffic equations
3196:Queueing networks
3170:Arrival processes
3144:Kingman's formula
2868:Stochastic Models
2275:978-3-642-35979-8
2058:Stochastic Models
1983:978-0-9836283-7-8
1606:978-0-8493-8076-1
1549:Stochastic Models
1512:Stochastic Models
1434:Stochastic Models
1215:
1135:
1109:
807:
700:
679:
659:
617:
543:
522:
491:
323:. As the process
267:
213:
185:
114:Model description
4813:
4767:Machine learning
4654:Usual hypotheses
4537:Girsanov theorem
4522:Dynkin's formula
4287:Continuous paths
4195:Actuarial models
4135:GarmanâKohlhagen
4105:BlackâKarasinski
4100:BlackâDermanâToy
4087:Financial models
3953:Fields and other
3882:Gaussian process
3831:Sigma-martingale
3635:Additive process
3537:
3530:
3523:
3514:
3502:
3501:
3319:Balance equation
3251:Service policies
3149:Lindley equation
3054:
3047:
3040:
3031:
3024:
3023:
3005:
2983:
2977:
2976:
2954:
2948:
2947:
2929:
2920:(4): 1034â1047.
2909:
2903:
2902:
2892:
2862:
2856:
2855:
2853:
2829:
2823:
2822:
2804:
2782:
2776:
2775:
2766:(4): 1117â1122.
2755:
2749:
2748:
2746:
2722:
2716:
2715:
2684:
2678:
2677:
2650:
2644:
2643:
2621:
2615:
2614:
2612:
2594:
2585:
2579:
2578:
2552:
2543:
2537:
2536:
2518:
2509:
2503:
2502:
2479:Queueing Systems
2471:
2462:
2461:
2459:
2441:
2432:
2423:
2422:
2420:
2392:
2386:
2385:
2362:
2356:
2355:
2353:
2329:
2320:
2319:
2289:
2280:
2279:
2255:
2246:
2240:
2239:
2229:
2203:
2194:
2188:
2187:
2165:
2156:
2155:
2127:
2121:
2120:
2093:Rogers, L. C. G.
2089:
2083:
2082:
2052:
2046:
2045:
2037:
2031:
2030:
2002:
1996:
1995:
1960:. pp. 1â8.
1951:
1945:
1944:
1934:
1912:
1906:
1905:
1902:10.1109/49.76633
1883:
1877:
1876:
1873:10.1109/49.76636
1854:
1848:
1847:
1845:
1836:(8): 2433â2444.
1821:
1815:
1814:
1786:
1770:
1764:
1763:
1746:(8): 1871â1894.
1737:
1724:
1715:
1714:
1703:10.1038/182039a0
1672:
1666:
1665:
1663:
1654:(4): 1588â1592.
1639:
1633:
1632:
1617:
1611:
1610:
1594:
1585:
1574:
1573:
1556:(2â3): 615â630.
1543:
1537:
1536:
1506:
1500:
1499:
1475:Queueing Systems
1468:Elwalid, A. I.;
1465:
1459:
1458:
1430:
1421:
1412:
1411:
1380:
1280:Mountain process
1260: = 2,
1229:
1227:
1226:
1221:
1216:
1214:
1182:
1168:
1150:which gives the
1146:
1144:
1143:
1138:
1136:
1134:
1111:
1110:
1108:
1107:
1029:
990:
976:
975:
945:
943:
942:
937:
935:
934:
842:
840:
839:
834:
808:
806:
756:
736:
721:
719:
718:
713:
711:
710:
706:
702:
701:
699:
685:
680:
672:
660:
658:
647:
646:
642:
623:
618:
616:
602:
569:
567:
566:
561:
559:
555:
554:
553:
549:
545:
544:
542:
528:
523:
515:
492:
490:
476:
440:
438:
437:
432:
430:
429:
303:
301:
300:
295:
293:
292:
268:
265:
252:
251:
214:
211:
207:
206:
186:
184:
180:
174:
161:
155:
34:fluid flow model
4821:
4820:
4816:
4815:
4814:
4812:
4811:
4810:
4806:Queueing theory
4796:
4795:
4794:
4789:
4771:
4732:Queueing theory
4675:
4617:Skorokhod space
4480:
4471:KunitaâWatanabe
4442:
4408:Sanov's theorem
4378:Ergodic theorem
4351:
4347:Time-reversible
4265:
4228:Queueing models
4222:
4218:SparreâAnderson
4208:CramĂ©râLundberg
4189:
4175:SABR volatility
4081:
4038:
3990:Boolean network
3948:
3934:Renewal process
3865:
3814:Non-homogeneous
3804:Poisson process
3694:Contact process
3657:Brownian motion
3627:Continuous time
3621:
3615:Maximal entropy
3546:
3541:
3511:
3506:
3492:
3424:
3383:
3350:
3336:Arrival theorem
3287:
3246:
3203:Jackson network
3191:
3165:
3156:Forkâjoin queue
3095:Burke's theorem
3063:
3061:Queueing theory
3058:
3028:
3027:
3003:10.1.1.484.6192
2985:
2984:
2980:
2956:
2955:
2951:
2911:
2910:
2906:
2864:
2863:
2859:
2831:
2830:
2826:
2784:
2783:
2779:
2757:
2756:
2752:
2724:
2723:
2719:
2698:(16â17): 2706.
2686:
2685:
2681:
2652:
2651:
2647:
2623:
2622:
2618:
2592:
2587:
2586:
2582:
2550:
2545:
2544:
2540:
2516:
2511:
2510:
2506:
2473:
2472:
2465:
2439:
2434:
2433:
2426:
2397:Harrison, P. G.
2394:
2393:
2389:
2364:
2363:
2359:
2331:
2330:
2323:
2291:
2290:
2283:
2276:
2253:
2248:
2247:
2243:
2201:
2196:
2195:
2191:
2167:
2166:
2159:
2129:
2128:
2124:
2109:10.2307/3215314
2091:
2090:
2086:
2054:
2053:
2049:
2039:
2038:
2034:
2019:10.2307/1426410
2004:
2003:
1999:
1984:
1953:
1952:
1948:
1932:10.1.1.102.3905
1914:
1913:
1909:
1885:
1884:
1880:
1856:
1855:
1851:
1823:
1822:
1818:
1803:
1777:. p. 355.
1772:
1771:
1767:
1735:
1726:
1725:
1718:
1674:
1673:
1669:
1641:
1640:
1636:
1621:Moran, P. A. P.
1619:
1618:
1614:
1607:
1592:
1587:
1586:
1577:
1545:
1544:
1540:
1508:
1507:
1503:
1467:
1466:
1462:
1428:
1423:
1422:
1415:
1400:10.2307/1427040
1382:
1381:
1377:
1372:
1331:
1319:
1306:
1294:
1282:
1274:
1250:
1186:
1159:
1158:
1112:
1099:
991:
967:
962:
961:
929:
928:
920:
914:
913:
908:
895:
883:
882:
849:
757:
737:
730:
729:
689:
670:
666:
661:
648:
632:
628:
624:
606:
575:
574:
532:
513:
509:
504:
497:
493:
480:
449:
448:
424:
423:
415:
409:
408:
403:
390:
378:
377:
367:
344:
321:driving process
288:
287:
262:
243:
234:
233:
208:
198:
191:
175:
156:
149:
148:
131:
116:
18:queueing theory
12:
11:
5:
4819:
4817:
4809:
4808:
4798:
4797:
4791:
4790:
4788:
4787:
4782:
4780:List of topics
4776:
4773:
4772:
4770:
4769:
4764:
4759:
4754:
4749:
4744:
4739:
4737:Renewal theory
4734:
4729:
4724:
4719:
4714:
4709:
4704:
4702:Ergodic theory
4699:
4694:
4692:Control theory
4689:
4683:
4681:
4677:
4676:
4674:
4673:
4672:
4671:
4666:
4656:
4651:
4646:
4641:
4636:
4635:
4634:
4624:
4622:Snell envelope
4619:
4614:
4609:
4604:
4599:
4594:
4589:
4584:
4579:
4574:
4569:
4564:
4559:
4554:
4549:
4544:
4539:
4534:
4529:
4524:
4519:
4514:
4509:
4504:
4499:
4494:
4488:
4486:
4482:
4481:
4479:
4478:
4473:
4468:
4463:
4458:
4452:
4450:
4444:
4443:
4441:
4440:
4421:BorelâCantelli
4410:
4405:
4400:
4395:
4390:
4385:
4380:
4375:
4370:
4365:
4359:
4357:
4356:Limit theorems
4353:
4352:
4350:
4349:
4344:
4339:
4334:
4329:
4324:
4319:
4314:
4309:
4304:
4299:
4294:
4289:
4284:
4279:
4273:
4271:
4267:
4266:
4264:
4263:
4258:
4253:
4248:
4243:
4238:
4232:
4230:
4224:
4223:
4221:
4220:
4215:
4210:
4205:
4199:
4197:
4191:
4190:
4188:
4187:
4182:
4177:
4172:
4167:
4162:
4157:
4152:
4147:
4142:
4137:
4132:
4127:
4122:
4117:
4112:
4107:
4102:
4097:
4091:
4089:
4083:
4082:
4080:
4079:
4074:
4069:
4064:
4059:
4054:
4048:
4046:
4040:
4039:
4037:
4036:
4031:
4026:
4025:
4024:
4019:
4009:
4004:
3999:
3994:
3993:
3992:
3987:
3977:
3975:Hopfield model
3972:
3967:
3962:
3956:
3954:
3950:
3949:
3947:
3946:
3941:
3936:
3931:
3926:
3921:
3920:
3919:
3914:
3909:
3904:
3894:
3892:Markov process
3889:
3884:
3879:
3873:
3871:
3867:
3866:
3864:
3863:
3861:Wiener sausage
3858:
3856:Wiener process
3853:
3848:
3843:
3838:
3836:Stable process
3833:
3828:
3826:Semimartingale
3823:
3818:
3817:
3816:
3811:
3801:
3796:
3791:
3786:
3781:
3776:
3771:
3769:Jump diffusion
3766:
3761:
3756:
3751:
3746:
3744:Hawkes process
3741:
3736:
3731:
3726:
3724:Feller process
3721:
3716:
3711:
3706:
3701:
3696:
3691:
3689:Cauchy process
3686:
3685:
3684:
3679:
3674:
3669:
3664:
3654:
3653:
3652:
3642:
3640:Bessel process
3637:
3631:
3629:
3623:
3622:
3620:
3619:
3618:
3617:
3612:
3607:
3602:
3592:
3587:
3582:
3577:
3572:
3567:
3562:
3556:
3554:
3548:
3547:
3542:
3540:
3539:
3532:
3525:
3517:
3508:
3507:
3497:
3494:
3493:
3491:
3490:
3485:
3480:
3475:
3470:
3465:
3460:
3455:
3450:
3445:
3440:
3434:
3432:
3426:
3425:
3423:
3422:
3417:
3412:
3407:
3405:Polling system
3402:
3397:
3391:
3389:
3385:
3384:
3382:
3381:
3380:
3379:
3369:
3364:
3358:
3356:
3355:Limit theorems
3352:
3351:
3349:
3348:
3343:
3338:
3333:
3332:
3331:
3326:
3321:
3311:
3306:
3301:
3295:
3293:
3289:
3288:
3286:
3285:
3280:
3275:
3270:
3265:
3260:
3254:
3252:
3248:
3247:
3245:
3244:
3239:
3234:
3229:
3228:
3227:
3222:
3212:
3211:
3210:
3199:
3197:
3193:
3192:
3190:
3189:
3184:
3179:
3173:
3171:
3167:
3166:
3164:
3163:
3158:
3153:
3152:
3151:
3146:
3136:
3131:
3126:
3125:
3124:
3119:
3109:
3104:
3099:
3098:
3097:
3087:
3082:
3077:
3071:
3069:
3065:
3064:
3059:
3057:
3056:
3049:
3042:
3034:
3026:
3025:
2978:
2949:
2904:
2857:
2824:
2777:
2750:
2717:
2679:
2668:(4): 407â427.
2645:
2616:
2580:
2561:(2): 353â372.
2538:
2504:
2463:
2450:(3): 269â299.
2424:
2395:Field, A. J.;
2387:
2357:
2321:
2310:(9â12): 1137.
2281:
2274:
2241:
2189:
2157:
2122:
2103:(4): 885â896.
2084:
2047:
2032:
1997:
1982:
1946:
1907:
1878:
1849:
1816:
1802:978-1581138733
1801:
1765:
1716:
1667:
1634:
1612:
1605:
1575:
1538:
1501:
1482:(1â2): 29â63.
1460:
1413:
1394:(3): 646â676.
1374:
1373:
1371:
1368:
1367:
1366:
1360:
1350:
1330:
1329:External links
1327:
1318:
1315:
1305:
1302:
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1273:
1270:
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1246:
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627:
621:
615:
612:
609:
605:
600:
597:
594:
591:
588:
585:
582:
571:
570:
558:
552:
548:
541:
538:
535:
531:
526:
521:
518:
512:
507:
503:
500:
496:
489:
486:
483:
479:
474:
471:
468:
465:
462:
459:
456:
442:
441:
428:
422:
419:
416:
414:
411:
410:
407:
404:
402:
399:
396:
395:
393:
388:
385:
366:
363:
343:
340:
305:
304:
291:
286:
283:
280:
277:
274:
271:
266: if
263:
261:
258:
255:
250:
246:
242:
239:
236:
235:
232:
229:
226:
223:
220:
217:
212: if
209:
205:
201:
197:
196:
194:
189:
183:
179:
173:
170:
167:
164:
160:
127:
115:
112:
76:network switch
13:
10:
9:
6:
4:
3:
2:
4818:
4807:
4804:
4803:
4801:
4786:
4783:
4781:
4778:
4777:
4774:
4768:
4765:
4763:
4760:
4758:
4755:
4753:
4750:
4748:
4745:
4743:
4740:
4738:
4735:
4733:
4730:
4728:
4725:
4723:
4720:
4718:
4715:
4713:
4710:
4708:
4705:
4703:
4700:
4698:
4695:
4693:
4690:
4688:
4685:
4684:
4682:
4678:
4670:
4667:
4665:
4662:
4661:
4660:
4657:
4655:
4652:
4650:
4647:
4645:
4642:
4640:
4639:Stopping time
4637:
4633:
4630:
4629:
4628:
4625:
4623:
4620:
4618:
4615:
4613:
4610:
4608:
4605:
4603:
4600:
4598:
4595:
4593:
4590:
4588:
4585:
4583:
4580:
4578:
4575:
4573:
4570:
4568:
4565:
4563:
4560:
4558:
4555:
4553:
4550:
4548:
4545:
4543:
4540:
4538:
4535:
4533:
4530:
4528:
4525:
4523:
4520:
4518:
4515:
4513:
4510:
4508:
4505:
4503:
4500:
4498:
4495:
4493:
4490:
4489:
4487:
4483:
4477:
4474:
4472:
4469:
4467:
4464:
4462:
4459:
4457:
4454:
4453:
4451:
4449:
4445:
4438:
4434:
4430:
4429:HewittâSavage
4426:
4422:
4418:
4414:
4413:Zeroâone laws
4411:
4409:
4406:
4404:
4401:
4399:
4396:
4394:
4391:
4389:
4386:
4384:
4381:
4379:
4376:
4374:
4371:
4369:
4366:
4364:
4361:
4360:
4358:
4354:
4348:
4345:
4343:
4340:
4338:
4335:
4333:
4330:
4328:
4325:
4323:
4320:
4318:
4315:
4313:
4310:
4308:
4305:
4303:
4300:
4298:
4295:
4293:
4290:
4288:
4285:
4283:
4280:
4278:
4275:
4274:
4272:
4268:
4262:
4259:
4257:
4254:
4252:
4249:
4247:
4244:
4242:
4239:
4237:
4234:
4233:
4231:
4229:
4225:
4219:
4216:
4214:
4211:
4209:
4206:
4204:
4201:
4200:
4198:
4196:
4192:
4186:
4183:
4181:
4178:
4176:
4173:
4171:
4168:
4166:
4163:
4161:
4158:
4156:
4153:
4151:
4148:
4146:
4143:
4141:
4138:
4136:
4133:
4131:
4128:
4126:
4123:
4121:
4118:
4116:
4113:
4111:
4110:BlackâScholes
4108:
4106:
4103:
4101:
4098:
4096:
4093:
4092:
4090:
4088:
4084:
4078:
4075:
4073:
4070:
4068:
4065:
4063:
4060:
4058:
4055:
4053:
4050:
4049:
4047:
4045:
4041:
4035:
4032:
4030:
4027:
4023:
4020:
4018:
4015:
4014:
4013:
4012:Point process
4010:
4008:
4005:
4003:
4000:
3998:
3995:
3991:
3988:
3986:
3983:
3982:
3981:
3978:
3976:
3973:
3971:
3970:Gibbs measure
3968:
3966:
3963:
3961:
3958:
3957:
3955:
3951:
3945:
3942:
3940:
3937:
3935:
3932:
3930:
3927:
3925:
3922:
3918:
3915:
3913:
3910:
3908:
3905:
3903:
3900:
3899:
3898:
3895:
3893:
3890:
3888:
3885:
3883:
3880:
3878:
3875:
3874:
3872:
3868:
3862:
3859:
3857:
3854:
3852:
3849:
3847:
3844:
3842:
3839:
3837:
3834:
3832:
3829:
3827:
3824:
3822:
3819:
3815:
3812:
3810:
3807:
3806:
3805:
3802:
3800:
3797:
3795:
3792:
3790:
3787:
3785:
3782:
3780:
3777:
3775:
3772:
3770:
3767:
3765:
3762:
3760:
3759:ItĂŽ diffusion
3757:
3755:
3752:
3750:
3747:
3745:
3742:
3740:
3737:
3735:
3734:Gamma process
3732:
3730:
3727:
3725:
3722:
3720:
3717:
3715:
3712:
3710:
3707:
3705:
3702:
3700:
3697:
3695:
3692:
3690:
3687:
3683:
3680:
3678:
3675:
3673:
3670:
3668:
3665:
3663:
3660:
3659:
3658:
3655:
3651:
3648:
3647:
3646:
3643:
3641:
3638:
3636:
3633:
3632:
3630:
3628:
3624:
3616:
3613:
3611:
3608:
3606:
3605:Self-avoiding
3603:
3601:
3598:
3597:
3596:
3593:
3591:
3590:Moran process
3588:
3586:
3583:
3581:
3578:
3576:
3573:
3571:
3568:
3566:
3563:
3561:
3558:
3557:
3555:
3553:
3552:Discrete time
3549:
3545:
3538:
3533:
3531:
3526:
3524:
3519:
3518:
3515:
3505:
3495:
3489:
3486:
3484:
3481:
3479:
3476:
3474:
3471:
3469:
3466:
3464:
3461:
3459:
3458:Message queue
3456:
3454:
3451:
3449:
3446:
3444:
3443:Erlang (unit)
3441:
3439:
3436:
3435:
3433:
3431:
3427:
3421:
3420:Retrial queue
3418:
3416:
3413:
3411:
3408:
3406:
3403:
3401:
3398:
3396:
3393:
3392:
3390:
3386:
3378:
3375:
3374:
3373:
3370:
3368:
3365:
3363:
3360:
3359:
3357:
3353:
3347:
3344:
3342:
3339:
3337:
3334:
3330:
3327:
3325:
3322:
3320:
3317:
3316:
3315:
3312:
3310:
3307:
3305:
3302:
3300:
3297:
3296:
3294:
3290:
3284:
3281:
3279:
3276:
3274:
3271:
3269:
3266:
3264:
3261:
3259:
3256:
3255:
3253:
3249:
3243:
3240:
3238:
3235:
3233:
3232:Kelly network
3230:
3226:
3223:
3221:
3218:
3217:
3216:
3213:
3209:
3206:
3205:
3204:
3201:
3200:
3198:
3194:
3188:
3185:
3183:
3180:
3178:
3175:
3174:
3172:
3168:
3162:
3159:
3157:
3154:
3150:
3147:
3145:
3142:
3141:
3140:
3137:
3135:
3132:
3130:
3127:
3123:
3120:
3118:
3115:
3114:
3113:
3110:
3108:
3105:
3103:
3100:
3096:
3093:
3092:
3091:
3088:
3086:
3083:
3081:
3078:
3076:
3073:
3072:
3070:
3066:
3062:
3055:
3050:
3048:
3043:
3041:
3036:
3035:
3032:
3021:
3017:
3013:
3009:
3004:
2999:
2995:
2991:
2990:
2982:
2979:
2974:
2970:
2966:
2962:
2961:
2953:
2950:
2945:
2941:
2937:
2933:
2928:
2923:
2919:
2915:
2908:
2905:
2900:
2896:
2891:
2886:
2882:
2878:
2874:
2870:
2869:
2861:
2858:
2852:
2847:
2843:
2839:
2835:
2828:
2825:
2820:
2816:
2812:
2808:
2803:
2798:
2794:
2790:
2789:
2781:
2778:
2773:
2769:
2765:
2761:
2754:
2751:
2745:
2740:
2736:
2732:
2728:
2721:
2718:
2713:
2709:
2705:
2701:
2697:
2693:
2689:
2683:
2680:
2675:
2671:
2667:
2663:
2659:
2655:
2649:
2646:
2641:
2637:
2633:
2629:
2628:
2620:
2617:
2611:
2606:
2602:
2598:
2591:
2584:
2581:
2576:
2572:
2568:
2564:
2560:
2556:
2549:
2542:
2539:
2534:
2530:
2526:
2522:
2515:
2508:
2505:
2500:
2496:
2492:
2488:
2484:
2480:
2476:
2475:Kroese, D. P.
2470:
2468:
2464:
2458:
2453:
2449:
2445:
2438:
2431:
2429:
2425:
2419:
2414:
2410:
2406:
2402:
2398:
2391:
2388:
2383:
2379:
2375:
2371:
2367:
2361:
2358:
2352:
2347:
2343:
2339:
2335:
2328:
2326:
2322:
2317:
2313:
2309:
2305:
2304:
2299:
2295:
2288:
2286:
2282:
2277:
2271:
2267:
2263:
2259:
2252:
2245:
2242:
2237:
2233:
2228:
2223:
2219:
2215:
2211:
2207:
2200:
2193:
2190:
2185:
2181:
2177:
2173:
2172:
2164:
2162:
2158:
2153:
2149:
2145:
2141:
2137:
2133:
2126:
2123:
2118:
2114:
2110:
2106:
2102:
2098:
2094:
2088:
2085:
2080:
2076:
2072:
2068:
2064:
2060:
2059:
2051:
2048:
2043:
2036:
2033:
2028:
2024:
2020:
2016:
2013:(1): 90â110.
2012:
2008:
2001:
1998:
1993:
1989:
1985:
1979:
1975:
1971:
1967:
1963:
1959:
1958:
1950:
1947:
1942:
1938:
1933:
1928:
1924:
1920:
1919:
1911:
1908:
1903:
1899:
1895:
1891:
1890:
1882:
1879:
1874:
1870:
1866:
1862:
1861:
1853:
1850:
1844:
1839:
1835:
1831:
1827:
1820:
1817:
1812:
1808:
1804:
1798:
1794:
1790:
1785:
1784:10.1.1.1.3208
1780:
1776:
1769:
1766:
1761:
1757:
1753:
1749:
1745:
1741:
1734:
1730:
1723:
1721:
1717:
1712:
1708:
1704:
1700:
1696:
1692:
1688:
1684:
1683:
1678:
1677:Prabhu, N. U.
1671:
1668:
1662:
1657:
1653:
1649:
1645:
1638:
1635:
1630:
1626:
1622:
1616:
1613:
1608:
1602:
1598:
1591:
1584:
1582:
1580:
1576:
1571:
1567:
1563:
1559:
1555:
1551:
1550:
1542:
1539:
1534:
1530:
1526:
1522:
1518:
1514:
1513:
1505:
1502:
1497:
1493:
1489:
1485:
1481:
1477:
1476:
1471:
1464:
1461:
1456:
1452:
1448:
1444:
1440:
1436:
1435:
1427:
1420:
1418:
1414:
1409:
1405:
1401:
1397:
1393:
1389:
1385:
1379:
1376:
1369:
1364:
1361:
1358:
1354:
1351:
1348:
1344:
1340:
1336:
1333:
1332:
1328:
1326:
1324:
1316:
1314:
1312:
1303:
1301:
1299:
1291:
1289:
1287:
1279:
1277:
1271:
1269:
1267:
1263:
1259:
1255:
1247:
1245:
1243:
1238:
1236:
1217:
1208:
1205:
1202:
1196:
1193:
1190:
1187:
1183:
1178:
1172:
1157:
1156:
1155:
1153:
1128:
1125:
1122:
1116:
1113:
1104:
1096:
1093:
1090:
1084:
1081:
1078:
1072:
1069:
1066:
1063:
1057:
1051:
1048:
1045:
1039:
1036:
1033:
1030:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
986:
980:
972:
968:
960:
959:
958:
956:
952:
931:
925:
922:
917:
910:
905:
902:
896:
891:
888:
881:
880:
879:
876:
874:
870:
866:
862:
858:
854:
846:
830:
824:
821:
818:
815:
812:
800:
797:
794:
785:
782:
779:
773:
770:
767:
761:
753:
747:
744:
741:
728:
727:
726:
707:
703:
696:
693:
690:
686:
681:
676:
673:
667:
662:
655:
652:
649:
643:
639:
636:
633:
629:
625:
619:
613:
610:
607:
603:
598:
592:
589:
586:
580:
573:
572:
556:
550:
546:
539:
536:
533:
529:
524:
519:
516:
510:
505:
501:
498:
494:
487:
484:
481:
477:
472:
466:
463:
460:
454:
447:
446:
445:
426:
420:
417:
412:
405:
400:
397:
391:
386:
383:
376:
375:
374:
372:
364:
362:
360:
355:
353:
349:
341:
339:
337:
333:
328:
326:
322:
318:
314:
310:
284:
281:
275:
269:
256:
253:
248:
244:
230:
227:
221:
215:
203:
199:
192:
187:
181:
168:
162:
147:
146:
145:
143:
139:
135:
130:
126:
122:
113:
111:
109:
105:
101:
97:
93:
89:
85:
81:
77:
72:
70:
66:
62:
57:
55:
51:
47:
43:
39:
35:
31:
27:
23:
19:
4697:Econometrics
4659:Wiener space
4547:ItĂŽ integral
4448:Inequalities
4337:Self-similar
4307:GaussâMarkov
4297:Exchangeable
4277:CĂ dlĂ g paths
4240:
4213:Risk process
4165:LIBOR market
4034:Random graph
4029:Random field
3841:Superprocess
3779:LĂ©vy process
3774:Jump process
3749:Hunt process
3585:Markov chain
3415:Loss network
3394:
3346:BeneĆĄ method
3309:Little's law
3292:Key concepts
3242:BCMP network
2993:
2987:
2981:
2964:
2958:
2952:
2917:
2913:
2907:
2872:
2866:
2860:
2841:
2837:
2827:
2802:math/0512119
2792:
2786:
2780:
2763:
2759:
2753:
2734:
2730:
2720:
2695:
2691:
2688:Boxma, O. J.
2682:
2665:
2661:
2654:Boxma, O. J.
2648:
2631:
2625:
2619:
2600:
2596:
2583:
2558:
2554:
2541:
2524:
2520:
2507:
2482:
2478:
2447:
2443:
2408:
2404:
2390:
2373:
2369:
2366:Boxma, O. J.
2360:
2341:
2337:
2307:
2301:
2294:Harrison, P.
2257:
2244:
2209:
2205:
2192:
2175:
2169:
2143:
2139:
2125:
2100:
2096:
2087:
2062:
2056:
2050:
2041:
2035:
2010:
2006:
2000:
1956:
1949:
1922:
1916:
1910:
1893:
1887:
1881:
1864:
1858:
1852:
1833:
1829:
1819:
1774:
1768:
1743:
1739:
1689:(4627): 39.
1686:
1680:
1670:
1651:
1647:
1637:
1628:
1624:
1615:
1596:
1553:
1547:
1541:
1519:(2â3): 631.
1516:
1510:
1504:
1479:
1473:
1463:
1438:
1432:
1391:
1387:
1378:
1320:
1307:
1295:
1283:
1275:
1265:
1261:
1257:
1253:
1251:
1239:
1232:
1154:busy period
1149:
954:
950:
948:
877:
864:
860:
856:
852:
850:
724:
443:
368:
365:On/off model
356:
345:
329:
324:
320:
316:
312:
306:
141:
137:
133:
128:
124:
120:
117:
73:
69:M/G/1 queues
58:
41:
37:
33:
29:
25:
15:
4742:Ruin theory
4680:Disciplines
4552:ItĂŽ's lemma
4327:Predictable
4002:Percolation
3985:Potts model
3980:Ising model
3944:White noise
3902:Differences
3764:ItĂŽ process
3704:Cox process
3600:Loop-erased
3595:Random walk
3438:Data buffer
3395:Fluid queue
3362:Fluid limit
3273:Round-robin
3139:G/G/1 queue
3134:G/M/1 queue
3129:M/G/k queue
3112:M/G/1 queue
3107:M/M/â queue
3102:M/M/c queue
3090:M/M/1 queue
3085:M/D/c queue
3080:M/D/1 queue
3075:D/M/1 queue
2890:11693/23071
2844:: 149â169.
2737:: 186â199.
2376:: 100â110.
2344:: 175â180.
2292:Field, A.;
2227:11693/24279
2132:Mandjes, M.
1974:11693/28055
1727:Anick, D.;
1347:Mathematica
1286:G/M/1 queue
847:Busy period
84:IEEE 802.11
50:ruin theory
30:fluid model
26:fluid queue
4752:Statistics
4532:Filtration
4433:Kolmogorov
4417:Blumenthal
4342:Stationary
4282:Continuous
4270:Properties
4155:HullâWhite
3897:Martingale
3784:Local time
3672:Fractional
3650:pure birth
3388:Extensions
3161:Bulk queue
2875:(3): 425.
2795:(3): 629.
2634:(9): 593.
2603:(2): 531.
2527:(4): 429.
2485:: 99â139.
2411:(2): 474.
2212:(2): 557.
2146:(6): 551.
1925:(3): 149.
1896:(3): 351.
1867:(3): 378.
1675:Gani, J.;
1631:: 116â124.
1441:(3): 325.
1370:References
86:protocol,
42:dam theory
4664:Classical
3677:Geometric
3667:Excursion
3237:G-network
2998:CiteSeerX
2996:: 69â82.
2967:: 77â88.
2927:1003.4107
2178:: 21â49.
2079:120102947
2065:: 64â88.
1927:CiteSeerX
1779:CiteSeerX
1729:Mitra, D.
1570:121190780
1533:123591340
1470:Mitra, D.
1384:Mitra, D.
1353:PevaTools
1272:Loss rate
1209:μ
1206:−
1203:λ
1197:β
1191:μ
1188:α
1184:λ
1129:μ
1126:−
1123:λ
1117:β
1097:μ
1094:α
1085:μ
1082:−
1079:λ
1073:β
1067:λ
1052:λ
1049:−
1046:μ
1040:μ
1037:α
1034:β
1026:−
1023:μ
1020:α
1014:μ
1011:β
1008:−
1005:λ
996:λ
993:β
973:∗
926:β
923:−
918:β
911:α
906:α
903:−
825:μ
822:−
819:λ
813:μ
801:β
795:α
786:λ
783:β
780:−
774:β
768:α
762:μ
754:β
748:μ
745:−
742:λ
697:μ
694:−
691:λ
687:α
682:−
677:μ
674:β
656:β
650:α
640:μ
637:−
634:λ
626:β
620:−
614:β
608:α
604:α
540:μ
537:−
534:λ
530:α
525:−
520:μ
517:β
502:−
488:β
482:α
478:β
421:β
418:−
413:β
406:α
401:α
398:−
123:we write
61:Pat Moran
46:wildfires
4800:Category
4785:Category
4669:Abstract
4203:BĂŒhlmann
3809:Compound
3504:Category
2944:19329962
2899:53363967
2819:16150704
2575:53498442
2399:(2010).
2296:(2007).
2134:(2005).
1811:14416842
1760:16836549
1711:42193342
1496:19379411
873:expected
4292:Ergodic
4180:VaĆĄĂÄek
4022:Poisson
3682:Meander
3020:1735120
2712:9973624
2499:3482641
2236:3216036
2117:3215314
2027:1426410
1691:Bibcode
1455:6733796
1408:1427040
1335:BuTools
1248:Example
4632:Tanaka
4317:Mixing
4312:Markov
4185:Wilkie
4150:HoâLee
4145:Heston
3917:Super-
3662:Bridge
3610:Biased
3018:
3000:
2942:
2897:
2817:
2710:
2573:
2497:
2272:
2234:
2115:
2077:
2025:
1992:863180
1990:
1980:
1929:
1809:
1799:
1781:
1758:
1709:
1682:Nature
1603:
1568:
1531:
1494:
1453:
1406:
1357:MATLAB
1343:Python
1339:MATLAB
949:write
92:B-ISDN
82:, the
80:router
4485:Tools
4261:M/M/c
4256:M/M/1
4251:M/G/1
4241:Fluid
3907:Local
3016:S2CID
2940:S2CID
2922:arXiv
2895:S2CID
2815:S2CID
2797:arXiv
2708:S2CID
2593:(PDF)
2571:S2CID
2551:(PDF)
2517:(PDF)
2495:S2CID
2440:(PDF)
2254:(PDF)
2232:JSTOR
2202:(PDF)
2113:JSTOR
2075:S2CID
2023:JSTOR
1988:S2CID
1807:S2CID
1756:S2CID
1736:(PDF)
1707:S2CID
1593:(PDF)
1566:S2CID
1529:S2CID
1492:S2CID
1451:S2CID
1429:(PDF)
1404:JSTOR
65:M/M/1
48:, in
4437:LĂ©vy
4236:Bulk
4120:Chen
3912:Sub-
3870:Both
3263:LIFO
3258:FIFO
2270:ISBN
1978:ISBN
1797:ISBN
1601:ISBN
1345:and
1337:, a
1152:mean
228:>
104:dams
78:, a
67:and
24:, a
4017:Cox
3008:doi
2994:160
2969:doi
2932:doi
2885:hdl
2877:doi
2846:doi
2807:doi
2768:doi
2739:doi
2700:doi
2670:doi
2636:doi
2605:doi
2563:doi
2559:103
2529:doi
2487:doi
2452:doi
2413:doi
2378:doi
2346:doi
2312:doi
2262:doi
2222:hdl
2214:doi
2180:doi
2148:doi
2105:doi
2067:doi
2015:doi
1970:hdl
1962:doi
1937:doi
1898:doi
1869:doi
1838:doi
1789:doi
1748:doi
1699:doi
1687:182
1656:doi
1558:doi
1521:doi
1484:doi
1443:doi
1396:doi
319:or
238:max
94:),
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