The Annals of Applied Probability

Asymptotically optimal control for a multiclass queueing model in the moderate deviation heavy traffic regime

Abstract

A multi-class single-server queueing model with finite buffers, in which scheduling and admission of customers are subject to control, is studied in the moderate deviation heavy traffic regime. A risk-sensitive cost set over a finite time horizon $[0,T]$ is considered. The main result is the asymptotic optimality of a control policy derived via an underlying differential game. The result is the first to address a queueing control problem at the moderate deviation regime that goes beyond models having the so-called pathwise minimality property. Moreover, despite the well-known fact that an optimal control over a finite time interval is generically of a nonstationary feedback type, the proposed policy forms a stationary feedback, provided $T$ is sufficiently large.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 5 (2017), 2862-2906.

Dates
Revised: September 2016
First available in Project Euclid: 3 November 2017

https://projecteuclid.org/euclid.aoap/1509696036

Digital Object Identifier
doi:10.1214/16-AAP1269

Mathematical Reviews number (MathSciNet)
MR3719948

Zentralblatt MATH identifier
06822207

Citation

Atar, Rami; Cohen, Asaf. Asymptotically optimal control for a multiclass queueing model in the moderate deviation heavy traffic regime. Ann. Appl. Probab. 27 (2017), no. 5, 2862--2906. doi:10.1214/16-AAP1269. https://projecteuclid.org/euclid.aoap/1509696036

References

• [1] Atar, R. and Biswas, A. (2014). Control of the multiclass $G/G/1$ queue in the moderate deviation regime. Ann. Appl. Probab. 24 2033–2069.
• [2] Atar, R. and Cohen, A. (2016). A differential game for a multiclass queueing model in the moderate-deviation heavy-traffic regime. Math. Oper. Res. 41 1354–1380.
• [3] Atar, R., Dupuis, P. and Shwartz, A. (2003). An escape-time criterion for queueing networks: Asymptotic risk-sensitive control via differential games. Math. Oper. Res. 28 801–835.
• [4] Atar, R., Mandelbaum, A. and Reiman, M. I. (2004). Scheduling a multi class queue with many exponential servers: Asymptotic optimality in heavy traffic. Ann. Appl. Probab. 14 1084–1134.
• [5] Atar, R. and Mendelson, G. (2016). On the non-Markovian multiclass queue under risk-sensitive cost. Queueing Syst. 84 265–278.
• [6] Atar, R. and Shifrin, M. (2014). An asymptotic optimality result for the multiclass queue with finite buffers in heavy traffic. Stoch. Syst. 4 556–603.
• [7] Budhiraja, A. and Ghosh, A. P. (2012). Controlled stochastic networks in heavy traffic: Convergence of value functions. Ann. Appl. Probab. 22 734–791.
• [8] Chen, H. and Mandelbaum, A. (1991). Leontief systems, RBVs and RBMs. In Applied Stochastic Analysis (London, 1989). Stochastics Monogr. 5 1–43. Gordon and Breach, New York.
• [9] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Applications of Mathematics (New York) 38. Springer, New York.
• [10] Dupuis, P. and McEneaney, W. M. (1997). Risk-sensitive and robust escape criteria. SIAM J. Control Optim. 35 2021–2049.
• [11] Fleming, W. H. (1971). Stochastic control for small noise intensities. SIAM J. Control 9 473–517.
• [12] Fleming, W. H. (2006). Risk sensitive stochastic control and differential games. Commun. Inf. Syst. 6 161–177.
• [13] Fleming, W. H. and McEneaney, W. M. (1995). Risk-sensitive control on an infinite time horizon. SIAM J. Control Optim. 33 1881–1915.
• [14] Fleming, W. H. and Mete Soner, H. (2006). Controlled Markov Processes and Viscosity Solutions, 2nd ed. Stochastic Modelling and Applied Probability 25. Springer, New York.
• [15] Fleming, W. H. and Souganidis, P. E. (1986). PDE-viscosity solution approach to some problems of large deviations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 13 171–192.
• [16] Ganesh, A., O’Connell, N. and Wischik, D. (2004). Big Queues. Lecture Notes in Math. 1838. Springer, Berlin.
• [17] Harrison, J. M. and López, M. J. (1999). Heavy traffic resource pooling in parallel-server systems. Queueing Syst. Theory Appl. 33 339–368.
• [18] Jacobson, D. H. (1973). Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games. IEEE Trans. Automat. Control AC-18 124–131.
• [19] Kruk, L., Lehoczky, J., Ramanan, K. and Shreve, S. (2007). An explicit formula for the Skorokhod map on $[0,a]$. Ann. Probab. 35 1740–1768.
• [20] Majewski, K. (2006). Sample path large deviations for multiclass feedforward queueing networks in critical loading. Ann. Appl. Probab. 16 1893–1924.
• [21] Puhalskii, A. A. (1999). Moderate deviations for queues in critical loading. Queueing Syst. Theory Appl. 31 359–392.
• [22] Puhalskii, A. A. and Whitt, W. (1997). Functional large deviation principles for first-passage-time processes. Ann. Appl. Probab. 7 362–381.
• [23] Ward, A. R. and Kumar, S. (2008). Asymptotically optimal admission control of a queue with impatient customers. Math. Oper. Res. 33 167–202.
• [24] Whittle, P. (1990). A risk-sensitive maximum principle. Systems Control Lett. 15 183–192.