# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
A reflected fBm limit for fluid models with ON/OFF sources under heavy traffic. (English) Zbl 1108.60074
Summary: We consider a family of non-deterministic fluid models that can be approximated under heavy traffic conditions by a multidimensional reflected fractional Brownian motion (rfBm). Specifically, we prove a heavy traffic limit theorem for multi-station fluid models with feedback and non-deterministic arrival process generated by a large enough number of heavy tailed ON/OFF sources, say $N$. Scaling in time by a factor $r$ and in state space conveniently, and letting $N$ and $r$ approach infinity (in this order) we prove that the scaled immediate workload process converges in some sense to an rfBm.

##### MSC:
 60K25 Queueing theory 60G15 Gaussian processes 90B22 Queues and service (optimization)
Full Text:
##### References:
 [1] Bernard, A.; El Kharroubi, A.: Régulations déterministes et stochastiques dans le premier orthant de rn. Stochastics and stochastics reports 34, 149-167 (1991) · Zbl 0727.60109 [2] Billingsley, P.: Convergence of probability measures. (1968) · Zbl 0172.21201 [3] Debicki, K.; Mandjes, M.: Traffic with an fBm limit: convergence of the stationary workload process. Queueing systems 46, 113-127 (2004) · Zbl 1061.90019 [4] El Karoui, N.; Chaleyat-Maurel, M.: Un problème de réflexion et ses aplications au temps local et aux équations différentielles stochastiques sur R. Cas continu. Société mathématique de France, astérisque 52--53, 117-144 (1978) [5] Harrison, J. M.: Balanced fluid models of multiclass queueing networks: A heavy traffic conjecture. IMA volumes in mathematics and its applications 71, 1-20 (1995) · Zbl 0838.90045 [6] Mandelbrot, B. B.; Van Ness, J. W.: Fractional Brownian motions, fractional noises and applications. SIAM review 10, No. 4, 422-437 (1968) · Zbl 0179.47801 [7] Taqqu, M. S.; Willinger, W.; Sherman, R.: Proof of a fundamental result in self-similar traffic modeling. Computer communication review 27, 5-23 (1997) [8] Williams, R. J.: An invariance principle for semimartingale reflecting Brownian motions in an orthant. Queueing systems 30, 5-25 (1998) · Zbl 0911.90170