The Annals of Applied Probability

Exponential penalty function control of loss networks

Garud Iyengar and Karl Sigman

Full-text: Open access


We introduce penalty-function-based admission control policies to approximately maximize the expected reward rate in a loss network. These control policies are easy to implement and perform well both in the transient period as well as in steady state. A major advantage of the penalty approach is that it avoids solving the associated dynamic program. However, a disadvantage of this approach is that it requires the capacity requested by individual requests to be sufficiently small compared to total available capacity. We first solve a related deterministic linear program (LP) and then translate an optimal solution of the LP into an admission control policy for the loss network via an exponential penalty function. We show that the penalty policy is a target-tracking policy—it performs well because the optimal solution of the LP is a good target. We demonstrate that the penalty approach can be extended to track arbitrarily defined target sets. Results from preliminary simulation studies are included.

Article information

Ann. Appl. Probab., Volume 14, Number 4 (2004), 1698-1740.

First available in Project Euclid: 5 November 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 93E03: Stochastic systems, general 93E35: Stochastic learning and adaptive control 90C59: Approximation methods and heuristics

Exponential penalty loss networks mathematical programming bounds stochastic control


Iyengar, Garud; Sigman, Karl. Exponential penalty function control of loss networks. Ann. Appl. Probab. 14 (2004), no. 4, 1698--1740. doi:10.1214/105051604000000936.

Export citation


  • Aspnes, J., Azar, Y., Fiat, A., Plotkin, S. and Waarts, O. (1997). On-line routing of virtual circuits with applications to load balancing and machine scheduling. J. Assoc. Comput. Mach. 44 486–504.
  • Azar, Y., Kalyanasundaram, B., Plotkin, S., Pruhs, K. R. and Waarts, O. (1997). On-line load balancing of temporary tasks. J. Algorithms 22 93–110.
  • Bean, N., Gibbens, R. and Zachary, S. (1995). Asymptotic analysis of single resource loss systems in heavy traffic with applications to integrated networks. Adv. in Appl. Probab. 27 273–292.
  • Bertsimas, D. and Chryssikou, T. (1999). Bounds and policies for dynamic routing in loss networks. Oper. Res. 47 379–394.
  • Bertsimas, D. and Niño Mora, J. (1999a). Optimization of multiclass queueing networks with changeover times via the achievable region approach. I. The single-station case. Math. Oper. Res. 24 306–330.
  • Bertsimas, D. and Niño Mora, J. (1999b). Optimization of multiclass queueing networks with changeover times via the achievable region approach. II. The multi-station case. Math. Oper. Res. 24 331–361.
  • Bertsimas, D., Paschalidis, I. C. and Tsitsiklis, J. N. (1994). Optimization of multiclass queueing networks: Polyhedral and nonlinear characterizations of achievable performance. Ann. Appl. Probab. 4 43–75.
  • Bertsimas, D. and Sethuraman, J. (2002). From fluid relaxations to practical algorithms for job-shop scheduling: The makespan objective. Math. Program. 92 61–102.
  • Bertsimas, D., Sethuraman, J. and Gamarnik, D. (2003). From fluid relaxations to practical algorithms for job-shop scheduling: The holding cost objective. Oper. Res. 51 798–813.
  • Bienstock, D. (2002). Potential Function Methods for Approximately Solving Linear Programs: Theory and Practice. Kluwer, Boston.
  • Blondel, V. D. and Tsitsiklis, J. N. (2000). A survey of computational complexity results in systems and control. Automatica 36 1249–1274.
  • Cosyn, J. (2003). Exponential penalty function control of queues with applications to bandwidth allocation. Ph.D. dissertation, IEOR Dept., Columbia Univ.
  • Cosyn, J. and Sigman, K. (2004). Stochastic networks: Admission and routing using penalty functions. Unpublished manuscript.
  • Foschini, G. J. and Gopinath, B. (1983). Sharing memory optimally. IEEE Trans. Comm. 31 352–360.
  • Gavois, A. and Rosberg, Z. (1994). A restricted complete sharing policy for a stochastic knapsack problem in a B-ISDN. IEEE Trans. Comm. 42 2375–2379.
  • Gibbens, R. J. and Kelly, F. P. (1995). Network programming methods for loss networks. IEEE Journal on Selected Areas in Communications 13 1189–1198.
  • Halfin, S. and Whitt, W. (1981). Heavy-traffic limits for queues with many exponential servers. Oper. Res. 29 567–588.
  • Harrison, J. M. (2003). A broader view of Brownian networks. Ann. Appl. Probab. 13 1119–1150.
  • Hochbaum, D., ed. (1996). Approximation Algorithms for NP-Hard Problems. Brooks/Cole, Monterey, CA.
  • Hui, J. Y. (1990). Switching and Traffic Theory for Integrated Broadband Networks. Kluwer, Boston.
  • Hunt, P. J. and Kurtz, T. G. (1994). Large loss networks. Stochastic Process. Appl. 53 363–378.
  • Hunt, P. J. and Laws, C. N. (1993). Asymptotically optimal loss network control. Math. Oper. Res. 18 880–900.
  • Hunt, P. J. and Laws, C. N. (1997). Optimization via trunk reservation in single resource loss systems in heavy traffic. Ann. Appl. Probab. 7 1058–1079.
  • Jordan, A. and Varaiya, P. P. (1994). Control of multiple service, multiple resource communication networks. IEEE Trans. Comm. 42 2979–2988.
  • Kamath, A., Palmon, O. and Plotkin, S. (1998). Routing and admission control in general topology networks with Poisson arrivals. J. Algorithms 27 236–258.
  • Kelly, F. P. (1985). Stochastic models for computer communication systems. J. R. Stat. Soc. Ser. B Stat. Methodol. 47 379–395.
  • Kelly, F. P. (1991). Loss networks. Ann. Appl. Probab. 1 319–378.
  • Key, P. B. (1990). Optimal control and trunk reservation in loss networks. Probab. Engrg. Inform. Sci. 4 203–242.
  • Key, P. B. (1994). Some control issues in telecommunications. In Probability, Statistics, and Optimization (F. P. Kelly, ed.) 383–395. Wiley, New York.
  • Ku, C.-Y. and Jordan, S. (1997). Access control to two multi-server loss queues in series. IEEE Trans. Automat. Control 42 1017–1023.
  • Lagarias, J. C., Odlyzko, A. M. and Zagier, D. B. (1985). Realizable traffic patterns and capacity of disjointly shared networks. Computer Networks 10 275–285.
  • Lippman, S. A. and Ross, S. M. (1971). The streetwalker's dilemma: A job shop model. SIAM J. Appl. Math. 20 336–342.
  • Luenberger, D. G. (1984). Linear and Nonlinear Programming. Addison–Wesley, Reading, MA.
  • Maglaras, C. (2000). Discrete-review policies for scheduling stochastic networks: Trajectory tracking and fluid-scale asymptotic optimality. Ann. Appl. Probab. 10 897–929.
  • McGill, J. I. and van Ryzin, G. J. (1999). Revenue management: Research overview and prospects. Transportation Sci. 33 233–256.
  • Miller, B. L. (1969). A queueing reward system with several customer classes. Management Science 16 234–245.
  • Mitra, D., Morrison, J. A. and Ramakrishnan, K. G. (1996). ATM network design: A multirate loss network framework. IEEE/ACM Transactions on Networking 4 531–543.
  • Mitra, D. and Weinberger, P. J. (1987). Probabilistic models for database locking: Solutions, computational algorithms and asymptotics. J. Assoc. Comput. Mach. 31 855–878.
  • Ott, T. J. and Krishnan, K. R. (1992). Separable routing: A scheme for state dependent routing of circuit switched telephone networks. Ann. Oper. Res. 35 43–68.
  • Papadimitriou, C. and Tsitsiklis, J. T. (1999). The complexity of optimal queueing network controls. Math. Oper. Res. 24 293–305.
  • Plotkin, S. A., Shmoys, D. B. and Tardos, É. (1991). Fast approximation algorithms for fractional packing and covering problems. In 32nd FOCS 495–504.
  • Reiman, M. I. and Schwartz, A. (2001). Call admission: A new approach to quality of service. Queueing Systems Theory Appl. 38 125–148.
  • Ross, K. W. (1995). Multiservice Loss Models for Broadband Telecommunication Networks. Springer, New York.
  • Ross, K. W. and Tsang, D. H. K. (1989a). Optimal circuit access policies in an ISDN environment: A Markov decision approach. IEEE Trans. Comm. 37 934–939.
  • Ross, K. W. and Tsang, D. H. K. (1989b). The stochastic knapsack problem. IEEE Trans. Comm. 37 740–747.
  • Ross, K. W. and Yao, D. D. (1990). Monotonicity properties for the stochastic knapsack. IEEE Trans. Inform. Theory 36 1173–1179.
  • Savin, S. V., Cohen, M. A., Gans, N. and Katalan, Z. (2000). Capacity management in rental businesses with heterogeneous customer bases. Technical report, Business School, Columbia Univ.
  • Shahrokhi, F. and Matula, D. W. (1990). The maximum concurrent flow problem. J. Assoc. Comput. Mach. 37 318–334.
  • Wolff, R. W. (1989). Stochastic Modeling and the Theory of Queues. Prentice-Hall, Englewood Cliffs, NJ.
  • Young, N. (1995). Randomized rounding without solving the linear program. In Proceedings of the 6th ACM–SIAM Symposium on Discrete Algorithms 170–178. SIAM, Philadelphia.