## Stochastic Systems

### Heavy traffic queue length behavior in a switch under the MaxWeight algorithm

#### Abstract

We consider a switch operating under the MaxWeight scheduling algorithm, under any traffic pattern such that all the ports are loaded. This system is interesting to study since the queue lengths exhibit a multi-dimensional state-space collapse in the heavy-traffic regime. We use a Lyapunov-type drift technique to characterize the heavy-traffic behavior of the expectation of the sum queue lengths in steady-state, under the assumption that all ports are saturated and all queues receive non-zero traffic. Under these conditions, we show that the heavy-traffic scaled queue length is given by $(1-\frac{1}{2n})||\sigma||^{2}$, where $\sigma$ is the vector of the standard deviations of arrivals to each port in the heavy-traffic limit. In the special case of uniform Bernoulli arrivals, the corresponding formula is given by $(n-\frac{3}{2}+\frac{1}{2n})$. The result shows that the heavy-traffic scaled queue length has optimal scaling with respect to $n,$ thus settling one version of an open conjecture; in fact, it is shown that the heavy-traffic queue length is at most within a factor of two from the optimal. We then consider certain asymptotic regimes where the load of the system scales simultaneously with the number of ports. We show that the MaxWeight algorithm has optimal queue length scaling behavior provided that the arrival rate approaches capacity sufficiently fast.

#### Article information

Source
Stoch. Syst., Volume 6, Number 1 (2016), 211-250.

Dates
First available in Project Euclid: 16 November 2016

https://projecteuclid.org/euclid.ssy/1479287408

Digital Object Identifier
doi:10.1214/15-SSY193

Mathematical Reviews number (MathSciNet)
MR3581000

Zentralblatt MATH identifier
1356.60146

#### Citation

Maguluri, Siva Theja; Srikant, R. Heavy traffic queue length behavior in a switch under the MaxWeight algorithm. Stoch. Syst. 6 (2016), no. 1, 211--250. doi:10.1214/15-SSY193. https://projecteuclid.org/euclid.ssy/1479287408

#### References

• [1] L. Tassiulas and A. Ephremides, “Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks,” IEEE Transactions on Automatic Control, vol. 37, no. 12, pp. 1936–1948, 1992.
• [2] N. McKeown, V. Anantharam, and J. Walrand, “Achieving 100% throughput in an input queued switch,” in Proceedings of IEEE INFOCOM, 1996, pp. 296–302.
• [3] A. L. Stolyar, “Maxweight scheduling in a generalized switch: State space collapse and workload minimization in heavy traffic,” Annals of Applied Probability, pp. 1–53, 2004.
• [4] M. Andrews, K. Jung, and A. Stolyar, “Stability of the max-weight routing and scheduling protocol in dynamic networks and at critical loads,” in Proceedings of the Thirty-ninth Annual ACM Symposium on Theory of Computing, ser. STOC ’07, 2007, pp. 145–154.
• [5] D. Shah and D. Wischik, “Switched networks with maximum weight policies: Fluid approximation and multiplicative state space collapse,” The Annals of Applied Probability, vol. 22, no. 1, pp. 70–127, 2012.
• [6] W. N. Kang and R. J. Williams, “Diffusion approximation for an input-queued packet switch operating under a maximum weight algorithm,” Stochastic Systems, 2012.
• [7] A. Eryilmaz and R. Srikant, “Asymptotically tight steady-state queue length bounds implied by drift conditions,” Queueing Systems, vol. 72, no. 3-4, pp. 311–359, 2012.
• [8] J. M. Harrison and R. J. Williams, “Brownian models of open queueing networks with homogeneous customer populations,” Stochastics, vol. 22, no. 2, pp. 77–115, 1987.
• [9] W. N. Kang, F. P. Kelly, N. H. Lee, and R. J. Williams, “State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy,” The Annals of Applied Probability, pp. 1719–1780, 2009.
• [10] T. Ji, E. Athanasopoulou, and R. Srikant, “On optimal scheduling algorithms for small generalized switches,” IEEE/ACM Transactions on Networking, vol. 18, no. 5, pp. 1585–1598, 2010.
• [11] D. Shah, J. Tsitsiklis, and Y. Zhong, “Optimal scaling of average queue sizes in an input-queued switch: an open problem,” Queueing Systems, vol. 68, no. 3-4, pp. 375–384, 2011.
• [12] M. J. Neely, E. Modiano, and Y.-S. Cheng, “Logarithmic delay for n$\times$ n packet switches under the crossbar constraint,” IEEE/ACM Transactions on Networking, vol. 15, no. 3, pp. 657–668, 2007.
• [13] D. Shah, N. S. Walton, and Y. Zhong, “Optimal queue-size scaling in switched networks,” Ann. Appl. Probab., vol. 24, no. 6, pp. 2207–2245, 12 2014.
• [14] D. Shah, J. N. Tsitsiklis, and Y. Zhong, “On queue-size scaling for input-queued switches,” 2014, arxiv.
• [15] R. Srikant and L. Ying, Communication Networks: An Optimization, Control and Stochastic Networks Perspective. Cambridge University Press, 2014.