The Annals of Applied Probability

MaxWeight scheduling in a generalized switch: State space collapse and workload minimization in heavy traffic

Alexander L. Stolyar

Abstract

We consider a generalized switch model, which includes as special cases the model of multiuser data scheduling over a wireless medium, the input-queued cross-bar switch model and a discrete time version of a parallel server queueing system. Input flows $n=1,\ldots,N$ are served in discrete time by a switch. The switch state follows a finite state, discrete time Markov chain. In each state $m$, the switch chooses a scheduling decision $k$ from a finite set $K(m)$, which has the associated service rate vector $(\mu_1^m(k),\ldots,\mu_N^m(k))$.

We consider a heavy traffic regime, and assume a Resource Pooling (RP) condition. Associated with this condition is a notion of workload $X=\sum_n \zen Q_n$, where $\ze=(\ze_1,\ldots,\ze_N)$ is some fixed nonzero vector with nonnegative components, and $Q_1,\ldots,Q_N$ are the queue lengths. We study the MaxWeight discipline which always chooses a decision $k$ maximizing $\sum_n \gamma_n [Q_n]^{\beta} \mu_n^m(k)$, that is, $k \in \mathop{\arg\max}_{i} \sum_n \gamma_n [Q_n]^{\beta} \mu_n^m(i),$ where $\beta>0$, $\gamma_1>0,\ldots,\gamma_N>0$ are arbitrary parameters. We prove that under MaxWeight scheduling and the RP condition, in the heavy traffic limit, the queue length process has the following properties: (a) The vector $(\gamma_1 Q_1^{\beta},\ldots,\gamma_N Q_N^{\beta})$ is always proportional to $\ze$ (this is "State Space Collapse"), (b) the workload process converges to a Reflected Brownian Motion, (c) MaxWeight minimizes the workload among all disciplines. As a corollary of these properties, MaxWeight asymptotically minimizes the holding cost rate $\sum_n \gamma_n Q_{n}^{\beta+1}$ at all times, and cumulative cost (with this rate) over finite intervals.

Article information

Source
Ann. Appl. Probab. Volume 14, Number 1 (2004), 1-53.

Dates
First available in Project Euclid: 3 February 2004

https://projecteuclid.org/euclid.aoap/1075828046

Digital Object Identifier
doi:10.1214/aoap/1075828046

Mathematical Reviews number (MathSciNet)
MR2023015

Zentralblatt MATH identifier
1057.60092

Citation

Stolyar, Alexander L. MaxWeight scheduling in a generalized switch: State space collapse and workload minimization in heavy traffic. Ann. Appl. Probab. 14 (2004), no. 1, 1--53. doi:10.1214/aoap/1075828046. https://projecteuclid.org/euclid.aoap/1075828046

References

• Andrews, M., Kumaran, K., Ramanan, K., Stolyar, A. L., Vijayakumar, R. and Whiting, P. (2004). Scheduling in a queueing system with asynchronously varying service rates. Probab. Engrg. Inform. Sci. To appear.
• Armony, M. and Bambos, N. (1999). Queueing networks with interacting service resources. In Proc. 37th Annual Allerton Conference on Communication, Control, and Computing 42--51. Univ. of Illinois, Urbana-Champaign.
• Bell, S. L. and Williams, R. J. (2001). Dynamic scheduling of a system with two parallel servers in heavy traffic with complete resource pooling: Asymptotic optimality of a continuous review threshold policy. Ann. Appl. Probab. 11 608--649.
• Bramson, M. (1998). State space collapse with applications to heavy traffic limits for multiclass queueing networks. Queueing Systems Theory Appl. 30 89--148.
• Bramson, M. and Williams, R. (2000). On dynamic scheduling of stochastic networks in heavy traffic and some new results for the workload process. In Proc. 39th IEEE Conference on Decision and Control 516--521. IEEE, New York.
• Bramson, M. and Dai, J. G. (2001). Heavy traffic limits for some queueing networks. Ann. Appl. Probab. 11 49--88.
• Buche, R. and Kushner, H. J. (2002). Control of mobile communications with time-varying channels in heavy traffic. IEEE Trans. Automat. Control 47 992--1003.
• Chen, H. and Mandelbaum, A. (1991). Leontief systems, RBV's and RBM's. In Applied Stochastic Analysis (M. H. A. Davis and R. J. Elliott, eds.) 1--43. Gordon and Breach, New York.
• Dai, J. G. and Prabhakar, B. (2000). The throughput of data switches with and without speedup. In Proc. IEEE INFOCOM 2000 556--564.
• Ethier, S. N. and Kurtz, T. G. (1986). Markov Process: Characterization and Convergence. Wiley, New York.
• Gans, N. and van Ryzin, G. (1997). Optimal control of a multiclass, flexible queueing system. Oper. Res. 45 677--693.
• Gans, N. and van Ryzin, G. (1998). Optimal dynamic scheduling of a general class of parallel-processing queueing systems. Adv. in Appl. Probab. 30 1130--1156.
• Harrison, J. M. (1998). Heavy traffic analysis of a system with parallel servers: Asymptotic optimality of discrete review policies. Ann. Appl. Probab. 8 822--848.
• Harrison, J. M. (2000). Brownian models of open processing networks: Canonical representation of workload. Ann. Appl. Probab. 10 75--103.
• Harrison, J. M. and Van Mieghem, J. A. (1997). Dynamic control of Brownian networks: State space collapse and equivalent workload formulations. Ann. Appl. Probab. 7 747--771.
• Harrison, J. M. and Lopez, M. J. (1999). Heavy traffic resource pooling in parallel-server systems. Queueing Systems 33 339--368.
• Jalali, A., Padovani, R. and Pankaj, R. (2000). Data throughput of CDMA-HDR, a high efficiency-high data rate personal communication wireless system. In Proc. IEEE VTC 2000 Spring 1854--1858. IEEE, New York.
• Kahale, N. and Wright, P. E. (1997). Dynamic global packet routing in wireless networks. In Proc. IEEE INFOCOM'97 1414--1421.
• Karlin, S. (1992). Mathematical Methods in Games, Programming, and Economics. Dover, New York.
• Kelly, F. P. and Laws, C. N. (1993). Dynamic routing in open queueing networks: Brownian models, cut constraints and resource pooling. Queueing Systems 13 47--86.
• Laws, C. N. (1992). Resource pooling in queueing networks with dynamic routing. Adv. in Appl. Probab. 24 699--726.
• Mandelbaum, A. and Stolyar, A. L. (2004). Scheduling flexible servers with convex delay costs: Heavy-traffic optimality of the generalized $c\mu$-rule. Oper. Res. To appear.
• McKeown, N., Anantharam, V. and Walrand, J. (1996). Achieving 100% throughput in an input-queued switch. In Proc. IEEE INFOCOM'96 296--302.
• Mekkittikul, A. and McKeown, N. (1996). A starvation free algorithm for achieving 100% throughput in an input-queued switch. In Proc. IEEE ICCCN'96 226--231.
• Reiman, M. I. (1988). A multiclass feedback queue in heavy traffic. Adv. in Appl. Probab. 20 179--207.
• Reiman, M. I. (1984). Some diffusion approximations with state space collapse. Proc. International Seminar on Modeling and Performance Evaluation Methodology. Lecture Notes in Control and Inform. Sci. 209--240. Springer, New York.
• Shakkottai, S. and Stolyar, A. (2002). Scheduling for multiple flows sharing a time-varying channel: The exponential rule. In Analytic Methods in Applied Probability. In Memory of Fridrih Karpelevich (Yu. M. Suhov, ed.) 185--202. AMS, Providence, RI.
• Stolyar, A. L. (1995). On the stability of multiclass queueing networks: A relaxed sufficient condition via limiting fluid processes. Markov Process. Related Fields 1 491--512.
• Tassiulas, L. (1995). Adaptive back-pressure congestion control based on local information. IEEE Trans. Automat. Control 40 236--250.
• Tassiulas, L. and Ephremides, A. (1992). Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks. IEEE Trans. Automat. Control 37 1936--1948.
• Tassiulas, L. and Ephremides, A. (1993). Dynamic server allocation to parallel queues with randomly varying connectivity. IEEE Trans. Inform. Theory 39 466--478.
• Tassiulas, L. and Bhattacharya, P. P. (2000). Allocation of interdependent resources for maximal throughput. Comm. Statist. Stochastic Models 16 27--48.
• Whitt, W. (1971). Weak convergence theorems for priority queues: Preemptive resume discipline. J. Appl. Probab. 8 74--94.
• Williams, R. J. (1998). An invariance principle for semimartingale reflecting Brownian motions in an orthant. Queueing Systems 30 5--25.
• Williams, R. J. (1998). Diffusion approximations for open multiclass queueing networks: Sufficient conditions involving state space collapse. Queueing Systems 30 27--88.
• Williams, R. J. (2000). On dynamic scheduling of a parallel server system with complete resource pooling. Fields Institute Communications 28 49--71.