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Brownian models of open queueing networks with homogeneous customer populations. (English) Zbl 0632.60095
This paper considers a family of multidimensional diffusion processes that arise as heavy traffic approximations for open queueing networks. More precisely, the diffusion processes considered arise as approximate models of open queueing networks with homogeneous customer populations which means that customers occupying any given node or station of the network are essentially indistinguishable from one another. The Jackson network fits this description but multiclass network models do not.
The objectives of the paper are (a) to explain in concrete terms how one approximates a conventional queueing model or a real physical system by a corresponding Brownian model, and (b) to state and prove some new results regarding stationary distributions of such Brownian models.
The part of the paper aimed at objective (a) is largely a recapitulation of previous work on weak convergence theorems, with the emphasis placed on modelling intuition. With respect to objective (b) some foundational issues are resolved. Also, it is shown that the stationary distribution of the Brownian model has a separable (product form) density if and only if its data satisfy a certain condition, in which case the stationary density is exponential, and all relevant performance measures can be written out in explicit formulas.
Reviewer: F.P.Kelly

60K25 Queueing theory (aspects of probability theory)
60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.)
60J65 Brownian motion
Full Text: DOI
[1] DOI: 10.1007/BF00531519 · Zbl 0178.20302 · doi:10.1007/BF00531519
[2] DOI: 10.1145/321879.321887 · Zbl 0313.68055 · doi:10.1145/321879.321887
[3] Billingsley P., Convergence of Probability Measures (1968) · Zbl 0172.21201
[4] Chung K. L., Introduction to Stochastic Integration (1983) · Zbl 0527.60058 · doi:10.1007/978-1-4757-9174-7
[5] DOI: 10.1007/BF00531446 · Zbl 0443.60074 · doi:10.1007/BF00531446
[6] Goffman E. G., Mathematical Computer Performance and Reliability pp 33– (1984)
[7] Meyer P. A, Probabilites et Potentiel (1980)
[8] Ethier S. N., Markov Processes, Characterization and Convergence (1986) · Zbl 0592.60049 · doi:10.1002/9780470316658
[9] Flores C., Computer Communications (1985) · Zbl 0581.90027
[10] Harrison J. M., Brownian Motion and Stochastic Flow Systems (1985) · Zbl 0659.60112
[11] Harrison J. M., IMA Workshop on Stochastic Differential Systems (1987)
[12] DOI: 10.1214/aop/1176994471 · Zbl 0462.60073 · doi:10.1214/aop/1176994471
[13] DOI: 10.1137/0141030 · Zbl 0464.60081 · doi:10.1137/0141030
[14] DOI: 10.1214/aop/1176992259 · Zbl 0615.60072 · doi:10.1214/aop/1176992259
[15] Iglehart D. L., Adv. Appl. Prob 2 pp 150,177– (1970) · doi:10.1017/S0001867800037241
[16] DOI: 10.1287/opre.5.4.518 · doi:10.1287/opre.5.4.518
[17] DOI: 10.1287/mnsc.10.1.131 · doi:10.1287/mnsc.10.1.131
[18] Johnson D. P., Diffusion approximations for optimal filtering of jump processes and for queueing networks (1983)
[19] Kelly F. P., Reversibility and Stochastic Networks (1979)
[20] DOI: 10.1287/mnsc.24.11.1175 · Zbl 0396.60088 · doi:10.1287/mnsc.24.11.1175
[21] McKean H. P., Stochastic Integrals (1968)
[22] Peterson W. P., Diffusion approximations for networks of queues with multiple customer types (1985)
[23] DOI: 10.1287/moor.9.3.441 · Zbl 0549.90043 · doi:10.1287/moor.9.3.441
[24] Reiman M. I., Williams, A boundary property of semimartingale reflecting Brownian motions, preprint (1986) · Zbl 0617.60081
[25] Tanaka H., Hiroshima Math. J 9 pp 163– (1979)
[26] DOI: 10.1002/cpa.3160380405 · Zbl 0579.60082 · doi:10.1002/cpa.3160380405
[27] Whitt W., Mathematical Models in Queueing Theory (1974)
[28] Whitt W., Bell Sys. Tech. J 62 pp 2779– (1983) · doi:10.1002/j.1538-7305.1983.tb03204.x
[29] Whitt W., Bell Sys. Tech. J 62 pp 2817– (1983) · doi:10.1002/j.1538-7305.1983.tb03205.x
[30] Williams R. J., to appear in Probability Theory and Related Fields
[31] Yosida K., Functional Analysis (1978) · Zbl 0365.46001 · doi:10.1007/978-3-642-96439-8
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