Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-26T07:04:40.679Z Has data issue: false hasContentIssue false
  • English
  • Français

A fluid queue with a finite buffer and subexponential input

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  01 July 2016

A. P. Zwart
A. P. Zwart*
Affiliation:
Eindhoven University of Technology
*
∗ Postal address: Eindhoven University of Technology, Department of Mathematics and Computing Science, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: zwart@win.tue.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a fluid model similar to that of Kella and Whitt [32], but with a buffer having finite capacity K. The connections between the infinite buffer fluid model and the G/G/1 queue established by Kella and Whitt are extended to the finite buffer case: it is shown that the stationary distribution of the buffer content is related to the stationary distribution of the finite dam. We also derive a number of new results for the latter model. In particular, an asymptotic expansion for the loss fraction is given for the case of subexponential service times. The stationary buffer content distribution of the fluid model is also related to that of the corresponding model with infinite buffer size, by showing that the two corresponding probability measures are proportional on [0,K) if the silence periods are exponentially distributed. These results are applied to obtain large buffer asymptotics for the loss fraction and the mean buffer content when the fluid queue is fed by N On-Off sources with subexponential on-periods. The asymptotic results show a significant influence of heavy-tailed input characteristics on the performance of the fluid queue.

Keywords

On-Off processesfinite buffersfluid queuesloss fractionsproportionality of probability measuresregenerative processesregular variationsubexponentialityTauberian theorems

MSC classification

Primary: 60K25: Queueing theory 90B22: Queues and service
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 1 , March 2000 , pp. 221 - 243
Copyright
Copyright © Applied Probability Trust 2000 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Abate, J. and Whitt, W. (1998). Explicit M/G/1 waiting-time distributions for a class of long-tail service-time distributions. Preprint, AT&T Labs., Florham Park, NJ.Google Scholar
[2] Agrawal, R., Makowski. A. M and Nain, Ph. (1999). On a reduced load equivalence for fluid queues under subexponentiality. Queueing Systems, 33, 5–42.Google Scholar
[3] Asmussen, S. (1987). Applied Probability and Queues. Wiley, Chicester, UK.Google Scholar
[4] Asmussen, S. and Klüppelberg, C. (1996). Large deviations results for subexponential tails, with applications to insurance risk. Stoch. Proc. Appl. 64, 103–125.CrossRefGoogle Scholar
[5] Awater, G. A. (1994). Broadband Communication—Modeling, Analysis and Synthesis of an ATM Switching Element. , Delft University, The Netherlands.Google Scholar
[6] Beran, J., Sherman, R., Taqqu, M. S. and Willinger, W. (1995). Long-range dependence in variable-bit-rate video. IEEE Trans. on Communications 43, 1566–1579.CrossRefGoogle Scholar
[7] Bingham, N. H. and Doney, R. A. (1974). Asymptotic properties of supercritical branching processes I: The Galton–Watson process. Adv. Appl. Prob. 6, 711–731.CrossRefGoogle Scholar
[8] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
[9] Boots, N. K. and Tijms, H. C. (1999). A multi-server queueing system with impatient customers. Management Sci. 45, 444–448.CrossRefGoogle Scholar
[10] Boxma, O. J. and Cohen, J. W. (1998). The M/G/1 queue with heavy-tailed service time distribution. Queueing Systems, IEEE J. Sel. Area Commun. 16, 749–763.Google Scholar
[11] Boxma, O. J. and Dumas, V. (1998). Fluid queues with heavy-tailed activity periods. Computer Commun. 21, 1509–1529.CrossRefGoogle Scholar
[12] Boxma, O. J. and Dumas, V. (1998). The busy period in the fluid queue. Perform. Eval. Rev. 26, 100–110.CrossRefGoogle Scholar
[13] Cohen, J. W. (1974). Superimposed renewal processes and storage with gradual input. Stoch. Proc. Appl. 2, 31–58.CrossRefGoogle Scholar
[14] Cohen, J. W. (1976). Regenerative Processes in Queueing Theory. Springer, Berlin.CrossRefGoogle Scholar
[15] Cohen, J. W. (1977). On up- and downcrossings. J. Appl. Prob. 14, 405–410.CrossRefGoogle Scholar
[16] Cohen, J. W. (1978). On the maximal content of a dam and logarithmic concave renewal functions. Stoch. Proc. Appl. 6, 291–304.CrossRefGoogle Scholar
[17] Cohen, J. W. (1982). The Single Server Queue, 2nd edn. North Holland, Amsterdam.Google Scholar
[18] Cohen, J. W. (1994). On the effective bandwidth in buffer design for the multi-server channels. Rept BS-R9406, CWI, Amsterdam.Google Scholar
[19] Cohen, J. W. (1997). Heavy-traffic limit theorems for the heavy-tailed GI/G/1 queue. Rept PNA-R9719, CWI, Amsterdam.Google Scholar
[20] Daley, D. J. (1964). General customer impatience in the queue GI/G/1. J. Appl. Prob. 2, 186–205.CrossRefGoogle Scholar
[21] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.CrossRefGoogle Scholar
[22] Feller, W. (1971). An Introduction to Probability Theory and its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
[23] Gouweleeuw, F. (1996). A General Approach to Computing Loss Probabilities. , Free University, Amsterdam.Google Scholar
[24] Gouweleeuw, F. (1996). Calculating the loss probability in a BMAP/4G/1/N+1 queue. Stoch. Models 12, 473–492.CrossRefGoogle Scholar
[25] Green, L. (1982). A limit theorem on subintervals of interrenewal times. Operat. Res. 30, 210–216.CrossRefGoogle Scholar
[26] Heath, D. Resnick, S. and Samorodnitsky, G. (1997). Patterns of buffer overflow in a class of queues with long memory in the input stream. Ann. Appl. Prob. 7, 1021–1057.CrossRefGoogle Scholar
[27] Heath, D., Resnick, S. and Samorodnitsky, G. (1997). How system performance is affected by the interplay of averages in a fluid queue with long range dependence induced by heavy tails. Tech. rept, School of OR & IE, Cornell University, NY.Google Scholar
[28] Heath, D., Resnick, S. and Samorodnitsky, G. (1998). Heavy tails and long range dependence in on/off processes and associated fluid models. Math. Operat. Res. 23, 145–165.CrossRefGoogle Scholar
[29] Hooghiemstra, G. (1987). A path construction for the virtual waiting time of an M/G/1 queue. Statistica Neerlandica 41, 175–181.CrossRefGoogle Scholar
[30] Jelenković, P., (1999). Long-tailed loss rates in a GI/GI/1 queue with applications. Queueing Systems, 33, 91–125.CrossRefGoogle Scholar
[31] Jelenković, P. and Lazar, A. (1999). Asymptotic results for multiplexing on-off sources with subexponential on-periods. Adv. Appl. Prob. 31, 394–421.CrossRefGoogle Scholar
[32] Kennedy, D. (1973). Limit theorems for finite dams. Stoch. Proc. Appl. 1, 269–278.CrossRefGoogle Scholar
[33] Kella, O. and Whitt., W. (1992). A storage model with a two-state random environment. Operat. Res. 40, S257–S262.CrossRefGoogle Scholar
[34] Lindley, D. V. (1959). (Discussion of a paper by C. B. Winsten.) J. Roy. Statist. Soc. Ser. B, 21, 22–23.Google Scholar
[35] Loynes, R. M. (1965). On a property of the random walks describing simple queues and dams. J. Roy. Statist. Soc. Ser. B, 27, 125–129.Google Scholar
[36] Minh, Do Le. (1980). The GI/G/1 queue with uniformly limited virtual waiting times; the finite dam. Adv. Appl. Prob. 12, 501–516.CrossRefGoogle Scholar
[37] Pakes, A. G. (1975). On the tails of waiting-time distributions. J. Appl. Prob. 12, 555–564.CrossRefGoogle Scholar
[38] Parulekar, M. and Makowski, A. M. (1997). Tail probabilities for M/G/∞ input processes (I): Preliminary asymptotics. Queueing Systems 27, 271–296.CrossRefGoogle Scholar
[39] Paxson, V. and Floyd, S. (1995). Wide area traffic: the failure of Poisson modeling. IEEE/ACM Trans. on Networking 3, 226–244.CrossRefGoogle Scholar
[40] Resnick, S. and Samorodnitsky, G. (1997). Activity periods of an infinite server queue and performance of certain heavy tailed fluid queues. Tech. rept, School of OR & IE, Cornell University, NY.Google Scholar
[41] Stanford, R. E. (1979). Reneging phenomena in single channel queues. Math Operat. Res. 4, 162–178.CrossRefGoogle Scholar
[42] Takács, L., (1967). Combinatorial Methods in the Theory of Stochastic Processes. John Wiley, New York.Google Scholar
[43] Tijms, H. C. (1994). Stochastic Models—An Algorithmic Approach. John Wiley, Chicester.Google Scholar
[44] Veraverbeke, N. (1997). Asymptotic behaviour of Wiener–Hopf factors of a random walk. Stoch. Proc. Appl. 5, 27–37.CrossRefGoogle Scholar
[45] Whitt, W. (1991). A review of L=λ W and extensions. Queueing Systems 9, 235–268.CrossRefGoogle Scholar
[46] Willinger, W., Taqqu, M. S., Leland, W. E. and Wilson, D. V. (1995). Self-similarity in high-speed packet traffic: analysis and modeling of ethernet traffic measurements. Statist. Science 10, 67–85.CrossRefGoogle Scholar