## Stochastic Systems

### Resource sharing networks: Overview and an open problem

#### Abstract

This paper provides an overview of the resource sharing networks introduced by Massoulié and Roberts [20] to model the dynamic behavior of Internet flows. Striving to separate the model class from the applications that motivated its development, we assume no prior knowledge of communication networks. The paper also presents an open problem, along with simulation results, a formal analysis, and a selective literature review that provide context and motivation. The open problem is to devise a policy for dynamic resource allocation that achieves what we call hierarchical greedy ideal (HGI) performance in the heavy traffic limit. The existence of such a policy is suggested by formal analysis of an approximating Brownian control problem, assuming that there is “local traffic” on each processing resource.

#### Article information

Source
Stoch. Syst., Volume 4, Number 2 (2014), 524-555.

Dates
First available in Project Euclid: 27 March 2015

https://projecteuclid.org/euclid.ssy/1427462424

Digital Object Identifier
doi:10.1214/13-SSY130

Mathematical Reviews number (MathSciNet)
MR3353226

Zentralblatt MATH identifier
1314.68024

#### Citation

Harrison, J. Michael; Mandayam, Chinmoy; Shah, Devavrat; Yang, Yang. Resource sharing networks: Overview and an open problem. Stoch. Syst. 4 (2014), no. 2, 524--555. doi:10.1214/13-SSY130. https://projecteuclid.org/euclid.ssy/1427462424

#### References

• [1] Baskett, F., Chandy, K. M., Muntz, R. R. and Palacios, J. (1975). Open, closed, and mixed networks of queues with different classes of customers. J. of the ACM, 22 248–260.
• [2] Bell, S. L. and Williams, R. J. (2001). Dynamic scheduling of a system with two parallel servers in heavy traffic with resource pooling: Asymptotic optimality of a threshold policy. Ann. Appl. Probab., 11 608–649.
• [3] Bell, S. L. and Williams, R. J. (2005). Dynamic scheduling of a parallel server system in heavy traffic with complete resource pooling: Asymptotic optimality of a threshold policy. Elect. J. Probab., 10 1044–1115.
• [4] Bonald, T. and Proutiére, A. (2002). Insensitivity in processor-sharing networks. Performance Evaluation, 49 193–209.
• [5] Bonald, T. and Proutiére, A. (2003). Insensitive bandwidth sharing in data networks. Queueing Systems, 44 69–100.
• [6] Bramson, M. (2008). Stability of Queueing Networks. Springer, New York.
• [7] Chen, H. and Yao, D. D. (2001). Fundamentals of Queueing Networks. Springer, NewYork.
• [8] de Veciana, G., Lee, T.-J. and Konstantopoulos, T. (2001). Stability and performance analysis of networks supporting elastic services. IEEE/ACM Trans. on Networking, 9 2–14
• [9] Harrison, J. M. (1988). Brownian models of queueing networks with heterogeneous customer populations, in W. Fleming and P.-L. Lions (eds.), Stochastic Differential Systems, Stochastic Control Theory and Applications, IMA Volumes in Mathematics and Its Applications, 10 147–186. Springer-Verlag, New York.
• [10] Harrison, J. M. (1996). The BIGSTEP approach to flow management in stochastic processing networks, in F. P. Kelly, S. Zachary, and I. Ziedins (eds.), Stochastic Networks: Theory and Applications, 57–90. Oxford University Press.
• [11] Harrison, J. M. (2000). Brownian models of open processing networks: Canonical representation of workload. Ann. Appl. Probab., 10 75–103. Correction 13 (2003) 390–393.
• [12] Harrison, J. M. (2013). Brownian Models of Performance and Control. Cambridge University Press, New York.
• [13] Harrison, J. M. and J. A. Van Mieghem (1997). Dynamic control of Brownian networks: State space collapse and equivalent workload formulations. Ann. Appl. Probab., 7 747–771.
• [14] Kang, W. N., Kelly, F. P., Lee, N. H. and Williams, R. J. (2009). State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy. Ann. Appl. Probab., 19 1719–1780.
• [15] Kelly, F. P. (1979). Reversibility and Stochastic Networks. John Wiley and Sons, New York.
• [16] Kelly, F. P. (1997). Charging and rate control for elastic traffic. Eur. Trans. Telecom., 8 33–37.
• [17] Kelly, F. P., Maulloo, A. and Tan, D. (1998). Rate control for communication networks: Shadow price, proportional fairness and stability. J. Oper. Res. Soc., 49 237–252.
• [18] Kelly, F. P. and Williams, R. J. (2004). Fluid model for a network operating under a fair bandwidth-sharing policy. Ann. Appl. Probab., 14 1055–1083.
• [19] Kenyon, T. (2002). High-Performance Data Network Design. Digital Press. Boston.
• [20] Massoulié, L. and Roberts, J. (2000). Bandwidth sharing and admission control for elastic traffic. Telecommunication Systems, 15 185–201.
• [21] Mo, J. and Walrand, J. (2000). Fair end-to-end window-based congestion control. IEEE/ACM Trans. on Networking, 8 856–567.
• [22] Shah, D., Tsitsiklis, J. N. and Zhong, Y. (2013). Qualitative properties of $\alpha$-fair policies in bandwidth sharing networks. Ann. Appl. Probab., 24 76–113.
• [23] Verloop, I. M., Borst, S.C. and Núñez Queija, R. (2005). Stability of size-based scheduling disciplines in resource-sharing networks. Performance Evaluation, 62 247–262.
• [24] Verloop, I. M. and Núñez Queija, R. (2009). Assessing the efficiency of resource allocations in bandwidth-sharing networks. Performance Evaluation, 66 59–77.
• [25] Walkup, D. W. and Wets, R. J.-B. (1969). Lifting projections of convex polyhedra. Pacific J. of Math., 28 465–475.
• [26] Zachary, S. (2007). A note on insensitivity in stochastic networks. J. of Appl. Prob., 44 238–248.