×

zbMATH — the first resource for mathematics

On the rate of convergence for infinite server Erlang-Sevastyanov’s problem. (English) Zbl 1307.60138
Summary: Polynomial convergence rates in total variation are established in Erlang-Sevastyanov type problems with an infinite number of servers and a general distribution of service under assumptions on the intensity of service.

MSC:
60K25 Queueing theory (aspects of probability theory)
60J27 Continuous-time Markov processes on discrete state spaces
68M20 Performance evaluation, queueing, and scheduling in the context of computer systems
90B22 Queues and service in operations research
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Adan, I., Weiss, G.: A loss system with skill based servers under assign to longest idle server policy. Probab. Eng. Inf. Sci. 26(03), 307-321 (2012) · Zbl 1277.60149
[2] Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, New York (2003) · Zbl 1029.60001
[3] Bambos, N; Walrand, J, On stability of state-dependent queues and acyclic queueing networks, Adv. Appl. Probab., 21, 681-701, (1989) · Zbl 0674.60087
[4] Baxendale, P, T. E. harris’s contribution to recurrent Markov processes and stochastic flows, Ann. Probab., 39, 417428, (2011) · Zbl 1213.60116
[5] Bonald, T; Proutiére, A, Insensitive bandwidth sharing in data networks, Queueing Syst., 44, 69-100, (2003) · Zbl 1023.90003
[6] Borovkov, A.A.: Asymptotic Methods in Queuing Theory. Wiley, New York (1984) · Zbl 0544.60085
[7] Borovkov, A.A.: Ergodicity and Stability of Stochastic Processes. Wiley, New York (1998) · Zbl 0917.60005
[8] Browne, S; Sigman, K, Work-modulated queues with applications to storage processes, J. Appl. Probab., 29, 699-712, (1992) · Zbl 0755.60077
[9] Cheong, CK; Heathcote, CR, On the rate of convergence of waiting times, J. Aust. Math. Soc., 5, 365-373, (1965) · Zbl 0143.20403
[10] Davis, MHA, Piecewise-deterministic Markov processes: a general class of non-diffusion stochastic models, J. R. Stat. Soc. Ser. B, 46, 353-388, (1984) · Zbl 0565.60070
[11] Doeblin, W, Sur deux problèmes de M. kolmogoroff concernant LES chaînes dénombrables, Bull. Soc. Math. Fr., 66, 210-220, (1938) · Zbl 0020.14604
[12] Doob, J.L.: Stochastic Processes. Wiley, New York (1953)
[13] Dynkin, E.B.: Markov Processes, vol. I, II. Springer, Berlin (1965) · Zbl 0132.37901
[14] Erlang, A. K.: Solution of some problems in the theory of probabilities of significance in automatic telephone exchanges. Elektroteknikeren 13, 1917, 5-13 (1917) (in Danish); Engl. transl. P.O. Elect. Eng. J. 10, 189-197 (1918); Reprinted as: WEB-based edition by permission from the Danish Acad. Techn. Sci. at http://oldwww.com.dtu.dk/teletraffic/Erlang.html · Zbl 0363.60094
[15] Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, New York (1986) · Zbl 0592.60049
[16] Falin, GI, Ergodicity, stability, and insensitivity for one class of circuit switching networks, Probl. Inf. Transm., 24, 56-67, (1988) · Zbl 0667.60097
[17] Falin, GI, Ergodicity and stability of systems with repeated calls, Ukr. Math. J., 41, 559-562, (1989) · Zbl 0687.60080
[18] Fortet, R.: Calcul des probabilités. CNRS, Paris (1950) · Zbl 0040.21802
[19] Foss, S; Konstantopoulos, T, An overview of some stochastic stability methods, J. Oper. Res. Soc. Jpn., 47, 275-303, (2004) · Zbl 1134.93412
[20] Gnedenko, B.V., Kovalenko, I.N.: Introduction to queueing theory. 2nd edn., rev. and suppl. Birkhäuser, Boston, MA (1991) · Zbl 0744.60111
[21] Harris, T. E.: The Existence of Stationary Measures for Certain Markov Processes. In: Proceedings Third Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, pp. 113-124. University of California Press, Berkeley, CA (1956)
[22] Ivnickiǐ, VA, On the condition for the invariance of the stationary probabilities for queueing networks, Theory Probab. Appl., 27, 196-201, (1982) · Zbl 0503.60096
[23] Kalashnikov, VV, The property of \(γ \)-reflexivity for Markov sequences, Sov. Math. Dokl., 14, 1869-1873, (1973) · Zbl 0325.60058
[24] Kalashnikov, VV, Some properties of piecewise linear Markov processes, Theory Probab. Appl., 20, 560-571, (1975) · Zbl 0361.60034
[25] Kalashnikov, V.V.: Topics on regenerative processes. CRC Press, Boca Raton (1993) · Zbl 0872.60063
[26] Kelbert, M; Veretennikov, A, On the estimation of mixing coefficients for a multiphase service system, Queueing Syst., 25, 325-337, (1997) · Zbl 0878.90043
[27] Khasminskii, R.: Stochastic Stability of Differential Equations, 2nd edn. Springer, Berlin (2010) · Zbl 1259.60058
[28] Khinchin, A.Ya.: Mathematical Methods in the Theory of Queueing, Trudy Mat. Inst. Steklov. 49, 3-122 (1955); Engl. transl. Griffin, London (1960).
[29] Khinchin, A.Ya.: Studies in Mathematical Theory of Queueing. State Publishing House for Physics-Mathematics Literature, Moscow (1963) (in Russian).
[30] Khinchin, A.Ya.: Erlang’s formulas in the theory of mass service. Theory Probab. Appl. 7, 320-325 (1963) · Zbl 0124.34302
[31] Koenig, D; Jansen, U, Eine invarianzeigenschaftzufaelliger bedienungsprozesse mit positiven geschwindigkeiten, Math. Nachr., 70, 321-364, (1975) · Zbl 0324.60076
[32] Kolmogorov, AN, A local limit theorem for classical Markov chains, Izv. Akad. Nauk SSSR Ser. Mat., 13, 281-300, (1949)
[33] Kovalenko, IN, On the conditions of independence of the stationary distribution laws of servicing time, Probl. Inf. Transm., 2, 147-152, (1962)
[34] Kovalenko, IN, Sur la condition pour que, en régime stationnaire, la distribution soit independante des lois des durées de conversation, Ann. Télecomm., 17, 190-191, (1962)
[35] Liptser, R.Sh., Shiryaev, A.N.: Stochastic calculus on filtered probability spaces. In: Prokhorov, YuV, Shiryaev, A.N. (eds.) Probability Theory III. Stochastic Calculus, pp. 111-157. Encyclopaedia of mathematical sciences. Springer, Berlin (1998)
[36] Massoulié, L, Structural properties of proportional fairness: stability and insensitivity, Ann. Appl. Probab., 17, 809-839, (2007) · Zbl 1125.60104
[37] Matthes, K.: Zür Theorie der Bedienungsprozesse. In: Trans. 3rd Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, pp. 513-528. Publishing House of the Czechoslovak Academy of Science, Liblice, Czechoslovakia (1962).
[38] Mejzler, D, A note on erlang’s formulas, Isr. J. Math., 3, 157-162, (1965) · Zbl 0151.22503
[39] Miller, HD, Geometric ergodicity in a class of denumerable Markov chains, Z. Wahrsch., 4, 354-373, (1965) · Zbl 0138.11603
[40] Neuts, MF; Teugels, JL, Exponential ergodicity of the \(M/G/1\) queue, SIAM J. Appl. Math., 17, 921-929, (1969) · Zbl 0183.49303
[41] Pechinkin, AV; Solovyev, AD; Yashkov, SF, A system with servicing discipline whereby the order of minimum remaining length is serviced first, Eng. Cybern., 17, 38-45, (1979) · Zbl 0458.60086
[42] Pechinkin, AV, On an invariant queueing system, Math. Operationsforsch. Statist. Ser. Optim., 14, 433-444, (1983) · Zbl 0519.60094
[43] Schassberger, R, Insensitivity of steady-state distributions of generalized semi-Markov processes. I., Ann. Probab., 5, 87-99, (1977) · Zbl 0363.60094
[44] Schassberger, R, Insensitivity of steady-state distributions of generalized semi-Markov processes. II., Ann. Probab., 6, 85-93, (1978) · Zbl 0396.60086
[45] Sevastyanov, B. A.: Erlang formulae in telephone systems with arbitrary distribution function of call, Proceedings third All-Union Mathematical Congress, Moscow, June-July 1956, vol. 4, Publishing House of the Academy of Sciences of the USSR, 121-135 (1959) (in Russian).
[46] Sevastyanov, BA, An ergodic theorem for Markov processes and its application to telephone systems with refusals, Theory Probab. Appl., 2, 104-112, (1957)
[47] Thorisson, H, The queue GI/G/1: finite moments of the cycle variables and uniform rates of convergence, Stoch. Proc. Appl., 19, 85-99, (1985) · Zbl 0554.60088
[48] Thorisson, H.: Coupling, Stationarity, and Regeneration. Springer, New York (2000) · Zbl 0949.60007
[49] Tuominen, P; Tweedie, RL, Exponential decay and ergodicity of general Markov processes, J. Appl. Probab., 16, 867-880, (1979) · Zbl 0422.60074
[50] Tweedie, R. L.: Criteria for Rates of Convergence of Markov Chains, with Application to Queueing and Storage Theory. In: Probability Statistics Analysis, London Mathematical Society, Lecture Note Series, 79, pp. 260-276, Cambridge University Press, Cambridge (1983). · Zbl 0501.60072
[51] Doorn, EA; Zeifman, AI, On the speed of convergence to stationarity of the Erlang loss system, Queueing Syst., 63, 241-252, (2009) · Zbl 1209.90122
[52] Veretennikov, A.Yu., Zverkina, G.A. Simple proof of Dynkin’s formula for single-server systems and polynomial convergence rates, Markov Proc. Rel. Fileds (in press); arXiv:1306.2359 [math.PR] (2013).
[53] Veretennikov, A.Yu.: The ergodicity of service systems with an infinite number of servomechanisms. Math. Notes 22(4), 804-808 (1977) · Zbl 0419.60093
[54] Veretennikov, A.Yu.: Ergodicity and invariance principle for multiphase queuing systems. Autom. Remote Control 42(7), 906-909 (1981) · Zbl 0489.60094
[55] Veretennikov, A.Yu.: On polynomial mixing and convergence rate for stochastic difference and differential equations. Theory Probab. Appl. 44(2), 361-374 (2000) · Zbl 0969.60070
[56] Veretennikov, A.Yu.: On the mixing rate and convergence to the stationary distribution in the discrete Erlang problem. Autom. Remote Control 70, 1992-2002 (2009) · Zbl 1213.60116
[57] Veretennikov, A.Yu.: On the rate of mixing and convergence to a stationary distribution in Erlang-type systems in continuous time. Probl Inf. Transm. 46(4), 382-389 (2010) · Zbl 1235.60098
[58] Veretennikov, A.Yu.: On the rate of convergence to the stationary distribution in the single-server queuing systems. Avtomatika i Telemekhanika, 74(10), 23-35 (2013) (in Russian); Engl. transl. Autom. Remote Control 74(10), 1620-1629 (2013) · Zbl 1297.90021
[59] Walton, NS, Insensitive, maximum stable allocations converge to proportional fairness, Queueing Syst., 68, 51-60, (2011) · Zbl 1214.90030
[60] Whittle, P, Partial balance and insensitivity, J. Appl. Probab., 2, 104-112, (1985) · Zbl 0561.60095
[61] Zachary, S, A note on insensitivity in stochastic networks, J. Appl. Probab., 44, 238-248, (2007) · Zbl 1133.60356
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.