Abstract
Motivated by the desire to appropriately account for complex features of network traffic revealed in traffic measurements, such as heavy-tail probability distributions, long-range dependence, self similarity and nonstationarity, we propose a nonstationary offered-load model. Connections of multiple types arrive according to independent nonhomogeneous Poisson processes, and general bandwidth stochastic processes (not necessarily Markovian) describe the individual user bandwidth requirements at multiple links of a communication network during their connections. We obtain expressions for the moment generating function, mean and variance of the total required bandwidth of all customers on each link at any designated time. We justify Gaussian approximations by establishing a central limit theorem for the offered-load process. We also obtain a Gaussian approximation for the time-dependent buffer-content distribution in an infinite-capacity buffer with constant processing rate. The offered-load model can be used for predicting future bandwidth requirements; we then advocate exploiting information about the history of connections in progress.
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References
J. Abate and W. Whitt, Numerical inversion of probability generating functions, Operations Research Letters 12 (1992) 245-251.
J. Abate and W. Whitt, Numerical inversion of Laplace transforms of probability distributions, ORSA Journal on Computing 7 (1995) 36-43.
R.G. Addie and M. Zukerman, An approximation for performance evaluation of stationary single server queues, IEEE Transactions on Communications 42 (1994) 3150-3160.
R.G. Addie, M. Zukerman and T. Neame, Fractal traffic-measurements, modeling and performance evaluation, in: Proc. of IEEE Infocom '95 (1995) pp. 977-984.
H. Ahmadi, P.F. Chimento, R.A. Guerin, L. Gün, B. Lin, R.O. Onvural and T.E. Tedijanto, NBBS traffic management overview, IBM Systems Journal 34 (1995) 604-628.
J.P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1968).
D.D. Botvich and N.G. Duffield, Large deviations, the shape of the loss curve, and economies of scale in large multiplexes, Queueing Systems 20 (1995) 293-320.
R. Cáceres, P.G. Danzig, S. Jamin and D.J. Mitzel, Characteristics of wide-area TCP/IP conversations, Computer Communications Review 21 (1991) 101-112.
C.S. Chang, Stability, queue length, and delay of deterministic and stochastic queueing networks, IEEE Transactions on Automatic Control 39 (1994) 913-931.
J. Choe and N.B. Shroff, A central limit theorem based approach for analyzing queue behavior in high-speed networks, in: Teletraffic Contributions for the Information Age, Proc. of ITC 15, eds. V. Ramaswami and P.E. Wirth (Elsevier, Amsterdam, 1997) pp. 1129-1138.
K.L. Chung, A Course in Probability Theory, 2nd ed. (Academic Press, New York, 1974).
E. Çinlar, Superposition of point processes, in: Stochastic Point Processes: Statistical Analysis, Theory and Applications, ed. P.A.W. Lewis (Wiley, New York, 1972) pp. 549-606.
P.J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2nd ed. (Academic Press, New York, 1984).
N.G. Duffield and N. O'Connell, Large deviations and overflow probabilities for the general singleserver queue, with applications, Mathematical Proceedings of the Cambridge Philosophical Society 118 (1995) 363-374.
N.G. Duffield and W. Whitt, Control and recovery from rare congestion events in a large multi-server system, Queueing Systems 26 (1997) 69-104.
N.G. Duffield and W. Whitt, A source traffic model and its transient analysis for network control, Stochastic Models 14 (1998) 51-78.
N.G. Duffield and W. Whitt, Network design and control using on—off and multi-level source traffic models with heavy-tailed distributions, in: Self-Similar Network Traffic and Performance Evaluation, eds. K. Park and W. Willinger (Wiley, New York, 2000) chapter 17, pp. 421-445.
S.G. Eick, W.A. Massey and W. Whitt, The physics of the M t/G/∞ queue, Operations Research 41 (1993) 731-742.
A. Feldmann, A.C. Gilbert, W. Willinger and T.G. Kurtz, The changing nature of network traffic: scaling phenomena, Computer Communications Review 28 (1998) 5-29.
C.M. Fortuin, P.W. Kastelyn and J. Ginibre, Correlation inequalities on some partially ordered sets, Communications in Mathematical Physics 22 (1971) 89-103.
A.G. Greenberg, R. Srikant and W. Whitt, Resource sharing for book-ahead and instantaneous-request calls, in: Teletraffic Contributions for the Information Age, Proc. of ITC 15, eds. V. Ramaswami and P.E. Wirth (Elsevier, Amsterdam, 1997) pp. 539-548. Longer version to appear in IEEE/ACM Transaction Networking.
R. Guerin, H. Ahmadi and M. Naghshineh, Equivalent capacity and its application to bandwidth allocation in high-speed networks, IEEE Journal on Selected Areas in Communications 9 (1991) 968-991.
J. Keilson and L. Servi, Networks of non-homogeneous M/G/∞ systems, Journal of Applied Probabibility 31A (1994) 157-168.
F.P. Kelly, Notes on effective bandwidths, in: Stochastic Networks, Theory and Applications, eds. F.P. Kelly, S. Zachary and I. Ziedins (Clarendon Press, Oxford, 1996) pp. 141-168.
T.G. Kurtz, Limit theorems for workload input models, in: Stochastic Networks, Theory and Applications, eds. F.P. Kelly, S. Zachary and I. Ziedins (Clarendon Press, Oxford, 1996) pp. 119-139.
C. Klüppelberg and T. Mikosch, Explosive Poisson shot noise processes with application to risk reserves, Bernoulli 1 (1995) 125-147.
W.E. Leland, M.S. Taqqu, W. Willinger and D.V. Wilson, On the self-similar nature of Ethernet traffic, IEEE/ACM Transactions on Networking 2 (1994) 1-15.
K.K. Leung, W.A. Massey and W. Whitt, Traffic models for wireless communication networks, IEEE Journal on Selected Areas in Communications 12 (1994) 1353-1364.
A. Mandelbaum, W.A. Massey and M.I. Reiman, Strong approximations for markovian service networks, Queueing Systems 30 (1998) 149-201.
W.A. Massey and W. Whitt, Networks of infinite-server queues with nonstationary Poisson input, Queueing Systems 13 (1993) 183-250.
W.A. Massey and W. Whitt, A stochastic model to capture space and time dynamics in wireless communication systems, Probability in the Engineering and Informational Sciences 8 (1994) 541-569.
W.A. Massey and W. Whitt, Peak congestion in multi-server service systems with slowly varying arrival rates, Queueing Systems 25 (1997) 157-172.
W.A. Massey and W. Whitt, A probabilistic generalization of Taylor's theorem, Statistics and Probability Letters 16 (1993) 51-54.
I. Norros, A storage model with self-similar input, Queueing Systems 16 (1994) 387-396.
I. Norros, On the use of fractal Brownian motion in the theory of connectionless networks, IEEE Journal on Selected Areas in Communications 13 (1995) 953-962.
K.R. Parthasarathy, Probability Measures on Metric Spaces (Academic Press, New York, 1967).
P.F. Pawlita, Traffic measurements in data networks, recent measurement results, and some implications, IEEE Transactions on Communications 29 (1981) 525-535.
V. Paxson and S. Floyd, Wide-area traffic: the failure of Poisson modeling, IEEE/ACM Transactions on Networking 3 (1995) 226-244.
J. Rice, On generalized shot noise, Advances in Applied Probability 9 (1977) 553-565.
W. Whitt, Dynamic staffing in a telephone call center aiming to immediately answer all calls, Operations Research Letters 24 (1999) 205-212.
W. Willinger, M.S. Taqqu and A. Erramilli, A bibliographical guide to self-similar traffic and performance modeling for modern high-speed networks, in: Stochastic Networks, eds. F.P. Kelly, S. Zachary and I. Ziedins (Clarendon Press, Oxford, 1996) pp. 339-366.
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Duffield, N., Massey, W. & Whitt, W. A Nonstationary Offered-Load Model for Packet Networks. Telecommunication Systems 16, 271–296 (2001). https://doi.org/10.1023/A:1016654625257
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DOI: https://doi.org/10.1023/A:1016654625257