Optimal heavy-traffic queue length scaling in an incompletely saturated switch

We consider an input queued switch operating under the MaxWeight scheduling algorithm. This system is interesting to study because it is a model for Internet routers and data center networks. Recently, it was shown that the MaxWeight algorithm has optimal heavy-traffic queue length scaling when all ports are uniformly saturated. Here we consider the case where a fraction of the ports are saturated and others are not (which we call the incompletely saturated case), and also the case where the rates at which the ports are saturated can be different. We use a recently developed drift technique to show that the heavy-traffic queue length under the MaxWeight scheduling algorithm has optimal scaling with respect to the switch size even in these cases.

[1]  Devavrat Shah,et al.  Switched networks with maximum weight policies: Fluid approximation and multiplicative state space collapse , 2010, ArXiv.

[2]  B. Hajek.  Hitting-time and occupation-time bounds implied by drift analysis with applications , 1982, Advances in Applied Probability.

[3]  Leandros Tassiulas,et al.  Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks , 1992 .

[4]  Bruce E. Hajek,et al.  Scheduling nonuniform traffic in a packet switching system with small propagation delay , 1994, Proceedings of INFOCOM '94 Conference on Computer Communications.

[5]  A. Stolyar.  MaxWeight scheduling in a generalized switch: State space collapse and workload minimization in heavy traffic , 2004 .

[6]  John N. Tsitsiklis,et al.  Optimal scaling of average queue sizes in an input-queued switch: an open problem , 2011, Queueing Syst. Theory Appl..

[7]  R. Srikant,et al.  Optimal heavy-traffic queue length scaling in an incompletely saturated switch , 2018, Queueing Syst. Theory Appl..

[8]  Nick McKeown,et al.  pFabric: minimal near-optimal datacenter transport , 2013, SIGCOMM.

[9]  G. Ziegler.  Lectures on Polytopes , 1994 .

[10]  Ruth J. Williams,et al.  Diffusion approximation for an input-queued switch operating under a maximum weight matching policy , 2012 .

[11]  John N. Tsitsiklis,et al.  On Queue-Size Scaling for Input-Queued Switches , 2014, ArXiv.

[12]  R. Srikant,et al.  Asymptotically tight steady-state queue length bounds implied by drift conditions , 2011, Queueing Syst. Theory Appl..

[13]  Devavrat Shah,et al.  Optimal queue-size scaling in switched networks , 2011, SIGMETRICS '12.

[14]  Alexandre Proutière,et al.  Insensitivity in processor-sharing networks , 2002, Perform. Evaluation.

[15]  Jean C. Walrand,et al.  Achieving 100% throughput in an input-queued switch , 1996, Proceedings of IEEE INFOCOM '96. Conference on Computer Communications.

[16]  Devavrat Shah,et al.  Fastpass , 2014, SIGCOMM.

[17]  J. Tsitsiklis,et al.  Performance of Multiclass Markovian Queueing Networks Via Piecewise Linear Lyapunov Functions , 2001 .

[18]  Eytan Modiano,et al.  Logarithmic delay for N × N packet switches under the crossbar constraint , 2007, TNET.

[19]  Kyomin Jung,et al.  Stability of the max-weight routing and scheduling protocol in dynamic networks and at critical loads , 2007, STOC '07.

[20]  Alexander L. Stolyar,et al.  MaxWeight Scheduling: Asymptotic Behavior of Unscaled Queue-Differentials in Heavy Traffic , 2015, SIGMETRICS 2015.

[21]  Bruce Hajek,et al.  Scheduling nonuniform traffic in a packet-switching system with small propagation delay , 1997, TNET.

[22]  Lei Ying,et al.  Communication Networks - An Optimization, Control, and Stochastic Networks Perspective , 2014 .

[23]  R. Srikant,et al.  Heavy-Traffic Behavior of the MaxWeight Algorithm in a Switch with Uniform Traffic , 2015, PERV.

[24]  R. Srikant,et al.  Queue Length Behavior in a Switch under the MaxWeight Algorithm , 2015 .