Long-range dependence of stationary processes in single-server queues. (English) Zbl 1178.60063

Summary: The stationary processes of waiting times \(\{W_{n}\}_{n = 1,2,\ldots}\) in a \(GI / G /1\) queue and queue sizes at successive departure epochs \(\{Q_{n}\}_{n = 1,2,\ldots}\) in an \(M / G /1\) queue are long-range dependent when \(3 < \kappa_{S} < 4\), where \(\kappa_{S}\) is the moment index of the independent identically distributed (i.i.d.) sequence of service times. When the tail of the service time is regularly varying at infinity the stationary long-range dependent process \(\{W_{n}\}\) has Hurst index \(\frac 12(5 - \kappa_{S})\), i.e. \[ \sup \left\{h : \limsup_{n\to\infty} \frac{\text{var}(W_1+\cdots+W_n)}{n^{2h}} = \infty \right\} = \frac{5-\kappa_S}{2}. \] If this assumption does not hold but the sequence of serial correlation coefficients \(\{\rho_{n}\}\) of the stationary process \(\{W_{n}\}\) behaves asymptotically as \(cn^{- \alpha}\) for some finite positive \(c\) and \(\alpha \in (0,1)\), where \(\alpha = \kappa_{S} - 3\), then \(\{W_{n}\}\) has Hurst index \((5 - \kappa_{S})\). If this condition also holds for the sequence of serial correlation coefficients \(\{r_{n}\}\) of the stationary process \(\{Q_{n}\}\) then it also has Hurst index \((5\kappa_{S})\)


60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research


Full Text: DOI


[1] A. Baltrūnas, D.J. Daley, and C. Klüppelberg, Tail behaviour of the busy period of a GI/G/1 queue with subexponential service times. Stochastic Processes and Their Applications 111 (2004) 237–258. · Zbl 1082.60080 · doi:10.1016/j.spa.2004.01.005
[2] J. Beran, Statistics for Long-Memory Processes. (Chapman & Hall, New York, 1994). · Zbl 0869.60045
[3] N.H. Bingham, C.M. Goldie, and J.L. Teugels, Regular Variation (Cambridge University Press, 1987). · Zbl 0617.26001
[4] K.J.E. Carpio, Long-Range Dependence of Markov Chains. Ph.D. Thesis, The Australian National University (2006).
[5] J.W. Cohen, Some results on regular variation for distributions in queueing and fluctuation theory. Journal of Applied Probability 10 (1973) 343–353. · Zbl 0258.60076 · doi:10.2307/3212351
[6] D.J. Daley, Stochastically monotone Markov chains. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 10 (1968) 305–317. · Zbl 0177.45604 · doi:10.1007/BF00531852
[7] D.J. Daley, The serial correlation coefficients of waiting times in a stationary single server queue. The Journal of the Australian Mathematical Society 8 (1968) 683–689. · Zbl 0164.47802 · doi:10.1017/S1446788700006509
[8] D.J. Daley, The Hurst index of long-range dependent renewal processes. Annals of Probability 27 (1999) 2035–2041. · Zbl 0961.60083 · doi:10.1214/aop/1022677560
[9] D.J. Daley and R. Vesilo, Long range dependence of point processes, with queueing examples. Stochastic Processes and Their Applications 70 (1997) 265–282. · Zbl 0911.60077 · doi:10.1016/S0304-4149(97)00045-8
[10] D.J. Daley and R. Vesilo, Long range dependence of inputs and outputs of classical queues. Fields Institute of Communications 28 (2000) 179–186. · Zbl 0983.60047
[11] J.C. Kiefer and J. Wolfowitz, On the theory of queues with many servers. Annals of Mathematical Statistics 27 (1956) 147–161. · Zbl 0070.36602 · doi:10.1214/aoms/1177728354
[12] A.W. Marshall, A one-sided analog of Kolmogorov’s inequality. The Annals of Mathematical Statistics 31 (1960) 483–487. · Zbl 0099.13105 · doi:10.1214/aoms/1177705912
[13] S.P. Meyn and R.L. Tweedie, Markov Chains and Stochastic Stability (Springer Verlag, London, 1993). · Zbl 0925.60001
[14] A.G. Pakes, The correlation coefficients of the queue lengths of some stationary single server queues. The Journal of the Australian Mathematical Society 12 (1971) 35–46. · Zbl 0209.19802 · doi:10.1017/S1446788700008272
[15] A.G. Pakes, On a class of Markov chains. The Journal of the Australian Mathematical Society 12 (1971) 91–97. · Zbl 0208.44201 · doi:10.1017/S1446788700008351
[16] N.U. Prabhu, Queues and Inventories: A Study of Their Basic Stochastic Processes (John Wiley & Sons, 1965). · Zbl 0131.16904
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.