# 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)
Balancing queues by mean field interaction. (English) Zbl 1080.90026
Summary: Consider a queueing network with $N$ nodes in which queue lengths are balanced through mean-field interaction. When $N$ is large, we study the performance of such a network in terms of limiting results as $N$ goes to infinity.
##### MSC:
 90B22 Queues and service (optimization) 60K25 Queueing theory
##### References:
 [1] C. Adjih, P. Jacquet and N. Vvedenskaya, Performance evalution of a single queue under multi-user TCP/IP connection, INRIA Research report, #4141 (2001). [2] G.L. Arsenišvili, Single-channel queuing systems with varying intensities, Trudy Tbiliss. Univ. 189, (1977) 65–79. [3] F. Baccelli, R.D. McDonald, and J. Reynier, A mean field model for multiple TCP connections through a buffer implementing RED. INRIA Research report #4449 (2002). [4] M.F. Chen, Existence theorems for interacting particle systems with non-compact state space. Sci Sinica. A 30 (1987) 148–156. [5] D.A. Dawson, Critical dynamics and fluctuations for a mean-field model of cooperative behavior. Journal of Statistical Physics, 31(1) (1983) 29–85. · doi:10.1007/BF01010922 [6] D.A. Dawson and J. Gärtner, Long-time behavior of interaction diffusions. Stochastic calculus in application, in Proceedings of the Cambridge Symposium, eds. J.R. Norris, Pitman Research Notes in Mathematics Series 197, (John Wiley & Sons, New York 1988). [7] D.A. Dawson and X. Zheng, Law of large numbers and a central limit theorem for unbounded jump mean field models, Adv. Appl. Math. 12(3) (1991) 293–326. · Zbl 0751.60080 · doi:10.1016/0196-8858(91)90015-B [8] N. Duffield, Local mean-field Markov processes: An application to message-switching networks. Probab. Theory Related Fields 93(4) (1992) 485–505. · Zbl 0767.60096 · doi:10.1007/BF01192718 [9] W. Feller, On the integro-differential equations of purely discontinuous Markoff processes, Trans. Amer. Math. Soc. 48 (1940) 488–515. · doi:10.1090/S0002-9947-1940-0002697-3 [10] S. Feng and X. Zheng, Solutions of a class of nonlinear master equations, Stochastic Processes and their Applications 43 (1992) 65–84. · Zbl 0762.60075 · doi:10.1016/0304-4149(92)90076-3 [11] S. Feng, Large deviations for unbounded jump type Markov processes with mean field interaction, Technical Report Series of the Laboratory for Research in Statistics and Probability, Carleton University. No. 182, (1991). [12] S. Feng, Nonlinear master equation of multitype particle systems, Stochastic Processes and their Applications 57 (1995) 247–271. · Zbl 0821.60098 · doi:10.1016/0304-4149(94)00055-X [13] R.D. Foley and D.R. McDonald, Join the shortest queue: Stability and exact asymptotics, Ann. Appl. Probab. 11(3) (2001) 569–607. [14] T. Fujisawa, On a queuing process with queue-length dependent service, Yokohama Math. J. 10 (1962) 53–72. [15] T. Funaki, A certain class of diffusion processes associated with nonlinear parabolic equations, Zeitschrift fur Wahr. 67 (1984) 331–348. · Zbl 0546.60081 · doi:10.1007/BF00535008 [16] H.C. Gromoll, A.L. Puha and R.J. Williams. The fluid limit of a heavily loaded processor sharing queue. Preprint (2001). [17] N. Hadidi, A queueing model with variable arrival rates, Period. Math. Hungar. 6 (1975) 39–47. · Zbl 0333.60095 · doi:10.1007/BF02018394 [18] F.A. Haight, Two queues in parallel, Biometrika 45 (1958) 401–410. [19] K. Hepp and E.H. Lieb, On the superradiant phase transition for molecules in a quantized radiation field: The Dicke Maser model, Ann. Physics 76 (1973) 360–404. · doi:10.1016/0003-4916(73)90039-0 [20] L.L. Hoffman, Characterizing linear birth and death processes, J. Amer. Statist. Assoc. 87 (420) (1992) 1183–1187. · Zbl 0770.60078 · doi:10.2307/2290658 [21] C. Knessl, B.J. Matkowsky, Z. Schuss and C. Tier, Distribution of the maximum buffer content during a busy period for state-dependent M/G/l queues, Comm. Statist. Stochastic Models 3(2) (1987) 191–226. · Zbl 0618.60093 · doi:10.1080/15326348708807053 [22] A. Mandelbaum and G. Pats, State-dependent queues: Approximations and applications. Stochastic Networks, IMA Vol. Math. Appl. 71, 239–282, (Springer, New York) (1995). [23] J.B. Martin and Y.M. Suhov, Fast Jackson networks, Annals of Applied Probability 9(3) (1999) 854–870. · Zbl 0972.90008 · doi:10.1214/aoap/1029962816 [24] S.P. Meyn, Sequencing and routing in multiclass queueing networks, I. Feedback regulation, SIAM J. Control Optim. 40(3) (2001) 741–776 (electronic). · Zbl 1060.90043 · doi:10.1137/S0363012999362724 [25] M. Mitzenmather, The power of two choices in randomized load balancing. Ph.D. thesis, University of Carlifornia, Berkeley (1996). [26] M. Mitzenmacher and B. Voecking, Selecting the shortest of two, improved. Analytic Methods in applied probability, Amer. Math. Soc. Transl. Ser. 2 207, (2002). 165–176, (Amer. Math. Soc., Providence, R1). [27] J.A. Morrison, Diffusion approximation for head-of-the-line processor sharing for two parallel queues. SIAM J. Appl. Math. 53(2) (1993) 471–490. · Zbl 0768.60093 · doi:10.1137/0153028 [28] B. Natvig, On a queuing model where potential customers are discouraged by queue length, Scand. J. Statist. 2(1) (1975) 34–42. [29] G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium System (John Wiley & Sons, New York, 1977). [30] V.I. Oseledets and D.V. Khmelev, Global stability of infinite systems of nonlinear differential equations, and nonhomogeneous countable Markov chains, Problemy Peredachi Informatsii 36(1) (2000) 60–76; translation in Probl. Inf. Transm. 36(1) 54–70. [31] V.I. Oseledets and D.V. Khmelev, Stochastic Transportation Networks and Stability of Dynamical Systems, Theory of Probability and Its Applications 46(1) (2002) 154–161. · Zbl 1002.60090 · doi:10.1137/S0040585X97978786 [32] P. Tinnakornsrisuphap and A. M. Makowski, TCP traffic modeling via limit theorems. Technical research Report, http://www.isr.umd.edu/TechReports/ISR/2002/TR_2002–23/TR_2002-23.phtml (2002). [33] N.D. Vvedenskaya, R.L. Dobrushin and F.I. Karpelevich, Queueing system with selection of the shortest of two queues: An asymptotic approach, Problems of Information Transmission 32(1) (1996) 15–27. [34] N. D. Vvedenskaya and Yu. M. Suhov, Dobrushin’s mean-field approximation for a queue with dynamic routing. Markov Processes and Related Fields 3 (1997) 493–526. [35] N.D. Vvedenskaya and M.Y. Suhov, Fast Jackson networks with dynamic routing, Problems of Information Transmission 38(2) (2002) 136–153. · Zbl 1021.60072 · doi:10.1023/A:1020010710507 [36] W. Winston, Optimality of the shortest line discipline, J. Appl. Probability 14(1) (1977) 181–189. · Zbl 0357.60023 · doi:10.2307/3213271 [37] S. Yan and Z. Li, The stochastic models for non-equilibrium system and formulation of master equations, Acta Phys. Sinica 29 (1980).