## The Annals of Applied Probability

### Heavy traffic analysis of open processing networks with complete resource pooling: Asymptotic optimality of discrete review policies

#### Abstract

We consider a class of open stochastic processing networks, with feedback routing and overlapping server capabilities, in heavy traffic. The networks we consider satisfy the so-called complete resource pooling condition and therefore have one-dimensional approximating Brownian control problems. We propose a simple discrete review policy for controlling such networks. Assuming 2+ɛ moments on the interarrival times and processing times, we provide a conceptually simple proof of asymptotic optimality of the proposed policy.

#### Article information

Source
Ann. Appl. Probab. Volume 15, Number 1A (2005), 331-391.

Dates
First available in Project Euclid: 28 January 2005

http://projecteuclid.org/euclid.aoap/1106922331

Digital Object Identifier
doi:10.1214/105051604000000495

Mathematical Reviews number (MathSciNet)
MR2115046

Zentralblatt MATH identifier
1071.60081

#### Citation

Ata, Baris; Kumar, Sunil. Heavy traffic analysis of open processing networks with complete resource pooling: Asymptotic optimality of discrete review policies. Ann. Appl. Probab. 15 (2005), no. 1A, 331--391. doi:10.1214/105051604000000495. http://projecteuclid.org/euclid.aoap/1106922331.

#### References

• 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.
• Bertsekas, D. and Gallager, R. (1992). Data Networks. Prentice Hall, Englewood Cliffs, NJ.
• Bertsimas, D. and Tsitsiklis, J. N. (1997). Introduction to Linear Optimization. Athena Scientific, Belmont, MA.
• Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
• Bramson, M. (1998). State space collapse with applications to heavy-traffic limits of multiclass queueing networks. Queueing Systems Theory Appl. 30 89--148.
• Bramson, M. and Dai, J. G. (2001). Heavy traffic limits for some queueing networks. Ann. Appl. Probab. 11 49--90.
• Bramson, M. and Williams, R. J. (2000). On dynamic scheduling of stochastic networks in heavy traffic and some new results for the workload process. In Proceedings of the 39th IEEE Conference on Decision and Control 516--521. IEEE, New York.
• Bramson, M. and Williams, R. J. (2003). Two workload properties for Brownian networks. Queueing Systems Theory Appl. 45 191--221.
• Buzacott, J. A. and Shantikumar, J. G. (1993). Stochastic Analysis of Manufacturing Systems. Prentice Hall, Englewood Cliffs, NJ.
• Chevalier, P. B. and Wein, L. M. (1993). Scheduling networks of queues: Heavy traffic analysis of a multistation closed network. Oper. Res. 41 743--758.
• Ethier, E. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
• Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.
• Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems. Wiley, New York.
• Harrison, J. M. (1988). Brownian models of queueing networks with heterogenous customer populations. In Stochastic Differential Systems, Stochastic Control Theory and Their Applications (W. Fleming and P. L. Lions, eds.) 147--186. Springer, New York.
• Harrison, J. M. (1996). The bigstep approach to flow management in stochastic processing networks. In Stochastic Networks: Theory and Applications (F. P. Kelly, S. Zachary and I. Ziedins, eds.) 57--90. Oxford Univ. Press.
• 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. (2003). A broader view of Brownian networks. Ann. Appl. Probab. 13 1119--1150.
• Harrison, J. M. and Lopez, M. J. (1999). Heavy traffic resource pooling in parallel-server systems. Queueing Systems Theory Appl. 33 339--368.
• Harrison, J. M. and Van Mieghem, J. (1997). Dynamic control of Brownian networks: State space collapse and equivalent workload formulations. Ann. Appl. Probab. 7 747--771.
• Harrison, J. M. and Wein, L. (1989). Scheduling network of queues: Heavy traffic analysis of a simple open network. Queueing Systems Theory Appl. 5 265--280.
• Harrison, J. M. and Wein, L. (1990). Scheduling networks of queues: Heavy traffic analysis of a two-station closed network. Oper. Res. 38 1052--1064.
• Iglehart, D. L. and Whitt, W. (1971). The equivalence of functional central limit theorems for counting processes and associated partial sums. Ann. Math. Statist. 42 1372--1378.
• Kelly, F. P. and Laws, C. N. (1993). Dynamic routing in open queueing networks: Brownian models, cut constraints and resource pooling. Queueing Systems Theory Appl. 13 47--86.
• Kleinrock, L. (1976). Queueing Systems: Computer Applications II. Wiley, New York.
• Kumar, S. (1999). Scheduling open queueing networks with sufficiently flexible resources. In Proceedings of the 37th Allerton Conference. Univ. Illinois.
• Kumar, S. (2000). Two-server closed networks in heavy traffic: Diffusion limits and asymptotic optimality. Ann. Appl. Probab. 10 930--961.
• Kushner, H. J. and Chen, Y. N. (2000). Optimal control of assignment of jobs to processors under heavy traffic. Stochastics Stochastics Rep. 68 177--228.
• Kushner, H. J. and Martins, L. F. (1993). Limit theorems for pathwise average cost per unit time problems for controlled queues in heavy traffic. Stochastics Stochastics Rep. 42 25--51.
• Kushner, H. J. and Martins, L. F. (1996). Heavy traffic analysis of a controlled multi-class queueing network via weak convergence methods. SIAM J. Control Optim. 34 1781--1797.
• Kushner, H. J. and Ramachandran, K. M. (1989). Optimal and approximately optimal control policies for queues in heavy traffic. SIAM J. Control Optim. 27 1293--1318.
• Laws, C. N. (1992). Resource pooling in queueing networks with dynamic routing. Adv. in Appl. Probab. 24 699--726.
• Maglaras, C. (1999). Dynamic scheduling in multiclass queueing networks: Stability under discrete review policies. Queueing Systems Theory Appl. 31 171--206.
• Maglaras, C. (2000). Discrete-review policies for scheduling stochastic networks: Trajectory tracking and fluid-scale asymptotic optimality. Ann. Appl. Probab. 10 897--929.
• Maglaras, C. (2003). Continuous-review tracking policies for dynamic control of stochastic networks. Queueing Systems Theory Appl. 43 43--80.
• Mandelbaum, A. and Stolyar, A. (2004). Scheduling flexible servers with convex delay costs: Heavy traffic optimality of the generalized $c \mu$-rule. Oper. Res. To appear.
• Martins, L. F. and Kushner, H. J. (1990). Routing and singular control for queueing networks in heavy traffic. SIAM J. Control Optim. 28 1209--1233.
• Martins, L. F., Shreve, S. E. and Soner, H. M. (1996). Heavy traffic convergence of a controlled multiclass queueing system. SIAM J. Control Optim. 34 2133--2171.
• Meyn, S. P. (2003). Sequencing and routing in multiclass queueing networks. Part II: Workload relaxations. SIAM J. Control Optim. 42 178--217.
• Nahmias, S. (1997). Production and Operations Analysis, 3rd ed. McGraw-Hill, New York.
• Nguyen, V. and Williams, R. J. (1996). Reflecting Brownian motions and queueing networks. Unpublished manuscript.
• Plambeck, E., Kumar, S. and Harrison, J. M. (2001). A multiclass queue in heavy traffic with throughput time constraints: Asymptotically optimal dynamic controls. Queueing Systems Theory Appl. 39 23--54.
• Ross, M. R. (1996). Stochastic Processes. Wiley, New York.
• Stolyar, A. L. (2001). Maxweight scheduling in a generalized switch: State space collapse and equivalent workload minimization under complete resource pooling. Unpublished manuscript.
• Strang, G. (1988). Linear Algebra and Its Applications, 3rd ed. Saunders College Publishing, Philadelphia.
• Van Mieghem, J. (1995). Dynamic scheduling with convex delay costs: The generalized $c\mu$ rule. Ann. Appl. Probab. 5 809--833.
• Wein, L. (1990). Brownian networks with discretionary routing. Oper. Res. 38 1065--1078.
• Wein, L. (1991). Scheduling network of queues: Heavy traffic analysis of a two-station network with controllable inputs. Oper. Res. 39 322--340.
• Wein, L. (1992). Scheduling network of queues: Heavy traffic analysis of a multistation network with controllable inputs. Oper. Res. 40 312--334.
• Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York.
• Williams, R. J. (2000). On dynamic scheduling of stochastic networks in heavy traffic. Presentation to the Workshop on Stochastic Networks, Univ. Wisconsin, Madison.
• Williams, R. J. (1998). Diffusion approximations for open multiclass networks: Sufficient conditions for state space collapse. Queueing Systems Theory Appl. 30 27--88.
• Williams, R. J. (1998). An invariance principle for semimartingale reflecting Brownian motion. Queueing Systems Theory Appl. 30 5--25.
• Yao, D. D. (1994). Stochastic Modelling and Analysis of Manufacturing Systems. Springer, New York.