### A scaling analysis of a transient stochastic network

#### Abstract

In this paper we use a simple transient Markov process with an absorbing point to investigate the qualitative behavior of a large-scale storage network of nonreliable file servers across which files can be duplicated. When the size of the system goes to ∞, we show that there is a critical value for the maximum number of files per server such that, below this quantity, most files have a maximum number of copies. Above this value, the network loses a significant number of files until some equilibrium is reached. When the network is stable, we show that, with convenient time scales, the evolution of the network towards the absorbing state can be described via a stochastic averaging principle.

#### Article information

Source
Adv. in Appl. Probab., Volume 46, Number 2 (2014), 516-535.

Dates
First available in Project Euclid: 29 May 2014

https://projecteuclid.org/euclid.aap/1401369705

Digital Object Identifier
doi:10.1239/aap/1401369705

Mathematical Reviews number (MathSciNet)
MR3215544

Zentralblatt MATH identifier
1303.60085

#### Citation

Feuillet, Mathieu; Robert, Philippe. A scaling analysis of a transient stochastic network. Adv. in Appl. Probab. 46 (2014), no. 2, 516--535. doi:10.1239/aap/1401369705. https://projecteuclid.org/euclid.aap/1401369705

#### References

• Anderson, R. F. and Orey, S. (1976). Small random perturbation of dynamical systems with reflecting boundary. Nagoya Math. J. 60, 189–216.
• Artalejo, J. R. and Gómez-Corral, A. (2008). Retrial Queueing Systems. A Computational Approach. Springer, Berlin.
• Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.
• Brown, T. (1978). A martingale approach to the Poisson convergence of simple point processes. Ann. Prob. 6, 615–628.
• Chun, B.-G. \et (2006). Efficient replica maintenance for distributed storage systems. In Proc. NSDI, IEEE, pp. 45–58.
• Darroch, J. N. and Seneta E. (1965). On quasi-stationary distributions in absorbing discrete-time finite Markov chains. J. Appl. Prob. 2, 88–100.
• Dawson, D. A. (1993). Measure-Valued Markov Processes. In École d'Été de Probabilités de Saint-Flour XXI–-1991 (Lecture Notes Math. 1541), Springer, Berlin, pp. 1–260.
• El Karoui, N. and Chaleyat-Maurel, M. (1978). Temps Locaux, vol. 52-53, ch. Un problème de réflexion et ses applications au temps local et aux équations différentielles stochastiques sur ${\mathbb R}$-cas continu. In Astérisque (Temps Locaux 52–53), Société Mathématique de France, pp. 117–144.
• Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York.
• Falin, G. I. and Templeton, J. G. C. (1997). Retrials Queues. Chapman & Hall, London.
• Ferrari, P. A., Kesten, H., Martinez, S. and Picco, P. (1995). Existence of quasi-stationary distributions. A renewal dynamical approach. Ann. Prob. 23, 501–521.
• Feuillet, M. (2012). On the flow-level stability of data networks without congestion control: the case of linear networks and upstream trees. Queueing Systems 70, 105–143.
• Feuillet, M. and Robert, P. (2012). On the transient behavior of Ehrenfest and Engset processes. Adv. Appl. Prob. 44, 562–582.
• Freidlin, M. I. and Wentzell, A. D. (1998). Random Perturbations of Dynamical Systems, 2nd edn. Springer, New York.
• Guckenheimer, J. and Holmes, P. (1990). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. (Appl. Math. Sci. 42). Springer, New York.
• Harrison, J. M. and Reiman, M. I. (1981). Reflected Brownian motion on an orthant. Ann. Prob. 9, 302–308.
• Has'minskiĭ, R. Z. (1980). Stochastic Stability of Differential Equations. Sijthoff & Noordhoff, Alphen aan den Rijn, Germantown, MD.
• Hunt, P. J. and Kurtz, T. G (1994). Large loss networks. Stoch. Process. Appl. 53, 363–378.
• Kasahara, Y. and Watanabe, S. (1986). Limit theorems for point processes and their functionals. J. Math. Soc. Japan 38, 543–574.
• Kelly, F. P. (1991). Loss networks. Ann. Appl. Prob. 1, 319–378.
• King, P. J. B. (1990). Computer and Communication Systems Performance Modelling. Prentice Hall, London.
• Kurtz, T. G. (1992). Averaging for Martingale Problems and Stochastic Approximation. In Applied Stochastic Analysis (Lecture Notes Control Inf. Sci. 177), Springer, Berlin, pp. 186–209.
• Legtchenko, S., Monnet, S., Sens, P. and Muller, G. (2011). RelaxDHT: a churn-resilient replication strategy for peer-to-peer distributed hash-tables. ACM Trans. Autonomous Adaptive Systems 7, 28.
• Papanicolaou, G. C., Stroock, D. and Varadhan, S. R. S. (1977). Martingale approach to some limit theorems. In Papers from the Duke Turbulence Conference (Duke University, Durham, NC, 1976), Paper number 6.
• Pavliotis, G. A. and Stuart, A. M. (2008). Multiscale Methods. Averaging and Homogenization. (Texts Appl. Math. 53). Springer, New York.
• Perry, O. and Whitt, W. (2011). An ODE for an overloaded X model involving a stochastic averaging principle. Stoch. Systems 1, 59–108.
• Picconi, F., Baynat, B. and Sens, P. (2007). An Analytical Estimation of Durability in DHTs. In Distributed Computing and Internet Technology (Lecture Notes Comput. Sci. 4882), Springer, Berlin, pp. 184–196.
• Ramabhadran, S. and Pasquale, J. (2006). Analysis of long-running replicated systems. In Proc. INFOCOM 2006, IEEE, pp. 1–9.
• Ramanan, K. (2006). Reflected diffusions defined via the extended Skorokhod map. Electron. J. Prob. 11, 934–992.
• Rhea, S. \et (2005). OpenDHT: a public DHT service and its uses. In Proc. SIGCOMM '05, ACM, New York, pp. 73–84.
• Robert, P. (2003). Stochastic Networks and Queues. (Appl. Math. (New York) 52). Springer, Berlin.
• Rowstron, A. and Druschel, P. (2001). Storage management and caching in PAST, a large-scale, persistent peer-to-peer storage utility. In Proc. SOSP'01, ACM, New York, pp. 188–201.
• Rudin, W. (1987). Real and Complex Analysis, 3rd edn. McGraw-Hill, New York.
• Skorokhod, A. V. (1962). Stochastic equations for diffusion processes in a bounded region. II. Theory Prob. Appl. 7, 3–23.
• Taylor, L. M. and Williams, R. J. (1993). Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant. Prob. Theory Relat. Fields 96, 283–317.