## The Annals of Applied Probability

### AIMD algorithms and exponential functionals

#### Abstract

The behavior of a connection transmitting packets into a network according to a general additive-increase multiplicative-decrease (AIMD) algorithm is investigated. It is assumed that loss of packets occurs in clumps. When a packet is lost, a certain number of subsequent packets are also lost (correlated losses). The stationary behavior of this algorithm is analyzed when the rate of occurrence of clumps becomes arbitrarily small. From a probabilistic point of view, it is shown that exponential functionals associated to compound Poisson processes play a key role. A formula for the fractional moments and some density functions are derived. Analytically, to get the explicit expression of the distributions involved, the natural framework of this study turns out to be the $q$-calculus. Different loss models are then compared using concave ordering. Quite surprisingly, it is shown that, for a fixed loss rate, the correlated loss model has a higher throughput than an uncorrelated loss model.

#### Article information

Source
Ann. Appl. Probab., Volume 14, Number 1 (2004), 90-117.

Dates
First available in Project Euclid: 3 February 2004

https://projecteuclid.org/euclid.aoap/1075828048

Digital Object Identifier
doi:10.1214/aoap/1075828048

Mathematical Reviews number (MathSciNet)
MR2023017

Zentralblatt MATH identifier
1041.60072

#### Citation

Guillemin, Fabrice; Robert, Philippe; Zwart, Bert. AIMD algorithms and exponential functionals. Ann. Appl. Probab. 14 (2004), no. 1, 90--117. doi:10.1214/aoap/1075828048. https://projecteuclid.org/euclid.aoap/1075828048

#### References

• Aldous, D. (1989). Probability Approximations via the Poisson Clumping Heuristic. Springer, New York.
• Altman, E., Avrachenko, K., Barakat, C. and Nuñez-Queija, R. (2001). State dependent $M/G/1$ type queueing analysis for congestion control in data networks. In Infocom 2001. IEEE, New York.
• Andrews, G. E., Askey, R. and Roy, R. (1999). Special Functions. Cambridge Univ. Press.
• Bertoin, J., Biane, P. and Yor, M. (2002). Poissonian exponential functionals, $q$-series, $q$-integrals and the moment problem for log-normal distributions. Technical Report PMA-705, Laboratoire de Probabilités, Univ. Paris 6.
• Bertoin, J. and Yor, M. (2001). On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes. Technical Report PMA-702, Laboratoire de Probabilités, Univ. Paris 6.
• Bolot, J. (1993). End-to-end packet delay and loss behavior in the Internet. In ACM SIGCOMM'93 289--298. ACM Press, New York.
• Carmona, P., Petit, F. and Yor, M. (1997). On the distribution and asymptotic results for exponential functionals of Lévy processes. In Exponential Functionals and Principal Values Related to Brownian Motion (M. Yor, ed.) 73--130. Rev. Mat. Iberoam., Madrid.
• Dumas, V., Guillemin, F. and Robert, P. (2002). A Markovian analysis of Additive-Increase Multiplicative-Decrease (AIMD) algorithms. Adv. in Appl. Probab. 34 85--111.
• Feller, W. (1971). An Introduction to Probability Theory and Its Applications, 2nd ed. Wiley, New York.
• Ferguson, T. S. (1972). Lose a dollar or double your fortune. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 3 657--666. Univ. California Press, Berkeley.
• Flajolet, P., Gourdon, X. and Dumas, P. (1995). Mellin transforms and asymptotics: Harmonic sums. Theoret. Comput. Sci. 144 3--58.
• Floyd, S. (1991). Connections with multiple congested gateways in packet-switched networks 1: One way traffic. Computer Comm. Rev. 21 30--47.
• Gasper, G. and Rahman, M. (1990). Basic Hypergeometric Series. Cambridge Univ. Press.
• Harrison, J. M. (1977). Ruin problems with compounding assets. Stochastic Process. Appl. 5 67--79.
• Jacobson, V. (1988). Congestion avoidance and control. In SIGCOMM'88 Symposium: Communications Architectures and Protocols. ACM, New York.
• Kingman, J. F. C. (1993). Poisson Processes. Clarendon Press, New York.
• Koornwinder, T. H. (1994). $q$-special functions, a tutorial. In Representation of Lie Groups and Quantum Groups (V. Baldoni and M. A. Picardello, eds.) 46--128. Longman, New York.
• Misra, V., Gong, W. and Towsley, D. (1999). Stochastic differential equation modeling and analysis of TCP windowsize behavior. In Performance'99.
• Neveu, J. (1977). Processus ponctuels. École d'Été de Probabilités de Saint-Flour VI. Lecture Notes in Math. 598 249--445. Springer, Berlin.
• Ott, T. J., Kemperman, J. H. B. and Mathis, M. (1996). The stationary behavior of ideal TCP congestion avoidance. Unpublished manuscript.
• Padhye, J., Firoiu, V., Towsley, D. and Kurose, J. (1998). Modeling TCP throughput: A simple model and its empirical validation. In ACM Sigcomm'98. ACM, New York.
• Paxson, V. (1999). End-to-end internet packet dynamics. IEEE/ACM Trans. Networking 7 277--292.
• Stevens, W. R. (1994). TCP-IP Illustrated 1: The Protocols. Addison--Wesley, Reading, MA.
• Stoyan, D. (1983). Comparison Methods for Queues and Other Stochastic Models. Wiley, New York.
• Yajnik, M., Kurose, J. and Towsley, D. (1995). Packet loss correlation in the MBone multicast network experimental measurements and Markov chain models. Technical Report UM-CS-1995-115, Univ. Massachusetts, Amherst.
• Yor, M. (1992). Some Aspects of Brownian Motion. Part I. Some Special Functionals. Birkhäuser, Basel.
• Yor, M. (1997). Some Aspects of Brownian motion. Part II. Some Recent Martingale Problems. Birkhäuser, Basel.
• Yor, M. (2001). Exponential Functionals of Brownian Motion and Related Processes. Springer, Berlin.
• Zhang, Y., Paxson, V. and Shenker, S. (2000). The stationarity of internet path properties: Routing, loss, and throughput. Technical report ACIRI.