Annals of Applied Probability

Queuing with future information

Joel Spencer, Madhu Sudan, and Kuang Xu

Full-text: Open access

Abstract

We study an admissions control problem, where a queue with service rate $1-p$ receives incoming jobs at rate $\lambda\in(1-p,1)$, and the decision maker is allowed to redirect away jobs up to a rate of $p$, with the objective of minimizing the time-average queue length.

We show that the amount of information about the future has a significant impact on system performance, in the heavy-traffic regime. When the future is unknown, the optimal average queue length diverges at rate $\sim\log_{1/(1-p)}\frac{1}{1-\lambda}$, as $\lambda\to1$. In sharp contrast, when all future arrival and service times are revealed beforehand, the optimal average queue length converges to a finite constant, $(1-p)/p$, as $\lambda\to1$. We further show that the finite limit of $(1-p)/p$ can be achieved using only a finite lookahead window starting from the current time frame, whose length scales as $\mathcal{O}(\log\frac{1}{1-\lambda})$, as $\lambda\to1$. This leads to the conjecture of an interesting duality between queuing delay and the amount of information about the future.

Article information

Source
Ann. Appl. Probab., Volume 24, Number 5 (2014), 2091-2142.

Dates
First available in Project Euclid: 26 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1403812371

Digital Object Identifier
doi:10.1214/13-AAP973

Mathematical Reviews number (MathSciNet)
MR3226173

Zentralblatt MATH identifier
1309.60090

Subjects
Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx] 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx] 90B36: Scheduling theory, stochastic [See also 68M20]

Keywords
Future information queuing theory admissions control resource pooling random walk online offline heavy-traffic asymptotics

Citation

Spencer, Joel; Sudan, Madhu; Xu, Kuang. Queuing with future information. Ann. Appl. Probab. 24 (2014), no. 5, 2091--2142. doi:10.1214/13-AAP973. https://projecteuclid.org/euclid.aoap/1403812371


Export citation

References

  • [1] Altman, E. and Shwartz, A. (1991). Markov decision problems and state-action frequencies. SIAM J. Control Optim. 29 786–809.
  • [2] Awerbuch, B., Azar, Y. and Plotkin, S. (1993). Throughput-competitive on-line routing. In Proceedings of Foundations of Computer Science (FOCS) 32–40. Palo Alto, CA.
  • [3] Azar, Y. (1998). On-line load balancing. In Online Algorithms (Schloss Dagstuhl, 1996). 178–195. Springer, Berlin.
  • [4] 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.
  • [5] Beutler, F. J. and Ross, K. W. (1986). Time-average optimal constrained semi-Markov decision processes. Adv. in Appl. Probab. 18 341–359.
  • [6] Borodin, A. and El-Yaniv, R. (2005). Online Computation and Competitive Analysis, Reissue ed. Cambridge Univ. Press, New York.
  • [7] Carr, M. and Hajek, B. (1993). Scheduling with asynchronous service opportunities with applications to multiple satellite systems. IEEE Trans. Automat. Control 38 1820–1833.
  • [8] Coffman, E. G. Jr., Jelenkovic, P. and Poonen, B. (1999). Reservation probabilities. Adv. Perf. Anal. 2 129–158.
  • [9] Fisher, M. and Raman, A. (1996). Reducing the cost of demand uncertainty through accurate response to early sales. Oper. Res. 44 87–99.
  • [10] Gallager, R. G. (1996). Discrete Stochastic Processes. Kluwer, Boston.
  • [11] Harrison, J. M. and López, M. J. (1999). Heavy traffic resource pooling in parallel-server systems. Queueing Systems Theory Appl. 33 339–368.
  • [12] Kim, S. C. and Horowitz, I. (2002). Scheduling hospital services: The efficacy of elective surgery quotas. Omega 30 335–346.
  • [13] Levi, R. and Shi, C. (2014). Revenue management of reusable resources with advanced reservations. Oper. Res. To appear.
  • [14] Lu, Y. and Radovanović, A. (2007). Asymptotic blocking probabilities in loss networks with subexponential demands. J. Appl. Probab. 44 1088–1102.
  • [15] Mandelbaum, A. and Reiman, M. I. (1998). On pooling in queueing networks. Management Science 44 971–981.
  • [16] 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. 52 836–855.
  • [17] Nawijn, W. M. (1990). Look-ahead policies for admission to a single server loss system. Oper. Res. 38 854–862.
  • [18] Smith, B. L., Williams, B. M. and Oswald, R. K. (2002). Comparison of parametric and nonparametric models for traffic flow forecasting. Cold Spring Harbor Symp. Quant. Biol. 10 303–321.
  • [19] Stidham, S. Jr. (1985). Optimal control of admission to a queueing system. IEEE Trans. Automat. Control 30 705–713.
  • [20] Tsitsiklis, J. N. and Xu, K. (2012). On the power of (even a little) resource pooling. Stoch. Syst. 2 66.