×

zbMATH — the first resource for mathematics

On the correlation structure of a Lévy-driven queue. (English) Zbl 1154.60348
Summary: We consider a single-server queue with Lévy input and, in particular, its workload process \((Q_t)_{t\geq 0}\), with a focus on the correlation structure. With the correlation function defined as \(r(t) := \)cov\((Q_{0}, Q_t) / var(Q_{0})\) (assuming that the workload process is in stationarity at time 0), we first determine its transform \(\int _{0}^{\infty }r(t)\)e\(^{-\vartheta t}\)d\(t\). This expression allows us to prove that \(r(\cdot )\) is positive, decreasing, and convex, relying on the machinery of completely monotone functions. We also show that \(r(\cdot )\) can be represented as the complementary distribution function of a specific random variable. These results are used to compute the asymptotics of \(r(t)\), for large \(t\), for the cases of light-tailed and heavy-tailed Lévy inputs.

MSC:
60K25 Queueing theory (aspects of probability theory)
60G51 Processes with independent increments; Lévy processes
90B05 Inventory, storage, reservoirs
PDF BibTeX Cite
Full Text: DOI
References:
[1] Abate, J. and Whitt, W. (1988). The correlation functions of RBM and M/M/1. Stoch. Models 4, 315–359. · Zbl 0712.60104
[2] Abate, J. and Whitt, W. (1994). Transient behavior of the M/G/1 workload process. Operat. Res. 42, 750–764. JSTOR: · Zbl 0833.90042
[3] Abate, J. and Whitt, W. (1997). Asymptotics for M/G/1 low-priority waiting-time tail probabilities. Queueing Systems 25, 173–233. · Zbl 0894.60088
[4] Asmussen, S. (2003). Applied Probability and Queues , 2nd edn. Springer, New York. · Zbl 1029.60001
[5] Asmussen, S. and Glynn, P. (2007). Stochastic Simulation: Algorithms and Analysis (Stoch. Modelling Appl. Prob. 57 ). Springer, New York.
[6] Beneš, V. (1957). On queues with Poisson arrivals. Ann. Math. Statist. 28 , 670–677. · Zbl 0085.34704
[7] Bernstein, S. N. (1929). Sur les fonctions absolument monotones. Acta Math. 52 , 1–66. · JFM 55.0142.07
[8] Bertoin, J. (1996). Lévy Processes (Camb. Tracts Math. 121 ). Cambridge University Press. · Zbl 0861.60003
[9] Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705–766. JSTOR: · Zbl 0322.60068
[10] Bingham, N. H. and Doney, R. A. (1974). Asymptotic properties of subcritical branching processes. I. The Galton–Watson process. Adv. Appl. Prob. 6, 711–731. JSTOR: · Zbl 0297.60044
[11] Bingham, N. H. and Pitts, S. M. (1999). Non-parametric estimation for the M/G/\(\infty\) queue. Ann. Inst. Statist. Math. 51, 71–97. · Zbl 0951.62024
[12] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation (Encyclopaedia Math. Appl. 27 ). Cambridge University Press. · Zbl 0617.26001
[13] Cox, D. R. and Smith, W. L. (1961). Queues . Methuen, London.
[14] Feller, W. (1971). An Introduction to Probability Theory and Its Applications , Vol. 2, 2nd edn. John Wiley, New York. · Zbl 0219.60003
[15] Hall, P. G. and Park, J. (2004). Nonparametric inference about service time distributions from indirect measurements. J. R. Statist. Soc. B 66, 861–875. JSTOR: · Zbl 1059.62029
[16] Kella, O., Boxma, O. J. and Mandjes, M. (2006). A Lévy process reflected at a Poisson age process. J. Appl. Prob. 43, 221–230. · Zbl 1104.60052
[17] Kyprianou, A. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications . Springer, Berlin. · Zbl 1104.60001
[18] Mandjes, M. (2007). Large Deviations for Gaussian Queues . John Wiley, Chichester. · Zbl 1125.60103
[19] Mandjes, M. and van de Meent, R. (2008). Resource dimensioning through buffer sampling. To appear in IEEE/ACM Trans. Networking .
[20] Mandjes, M. and Zwart, B. (2006). Large deviations for sojourn times in processor sharing queues. Queueing Systems 52, 237–250. · Zbl 1094.60062
[21] Morse, P. (1955). Stochastic properties of waiting lines. Operat. Res. 3, 255–261.
[22] Ott, T. (1977). The covariance function of the virtual waiting-time process in an M/G/1 queue. Adv. Appl. Prob. 9, 158–168. JSTOR: · Zbl 0382.60101
[23] Reynolds, J. F. (1975). The covariance structure of queues and related processes—a survey of recent work. Adv. Appl. Prob. 7, 383–415. JSTOR: · Zbl 0349.60096
[24] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions (Camb. Studies Adv. Math. 68 ). Cambridge University Press. · Zbl 0973.60001
[25] Zolotarev, V. (1964). The first passage time of a level and the behaviour at infinity for a class of processes with independent increments. Theory Prob. Appl. 9, 653–661. · Zbl 0149.12903
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.