Institute of Mathematical Statistics Collections

Double Skorokhod Map and Reneging Real-Time Queues

Łukasz Kruk, John Lehoczky, Kavita Ramanan, and Steven Shreve

Full-text: Open access

Abstract

An explicit formula for the Skorokhod map Γ0,a on [0,a] for a>0 is provided and related to similar formulas in the literature. Specically, it is shown that on the space $\mathcal{D}$[0,) of right-continuous functions with left limits taking values in ℝ,

Γ0,a(ψ)(t)=ψ(t)[(ψ(0)a)+infu[0,t]ψ(u)]sups[0,t][(ψ(s)a)infu[s,t]ψ(u)]

is the unique function taking values in [0,a] that is obtained from by minimal “pushing” at the endpoints 0 and a. An application of this result to real-time queues with reneging is outlined.

Chapter information

Source
Stewart N. Ethier, Jin Feng and Richard H. Stockbridge, eds., Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2008), 169-193

Dates
First available in Project Euclid: 28 January 2009

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1233152942

Digital Object Identifier
doi:10.1214/074921708000000372

Mathematical Reviews number (MathSciNet)
MR2574231

Zentralblatt MATH identifier
1170.60314

Rights
Copyright © 2008, Institute of Mathematical Statistics

Citation

Kruk, Łukasz; Lehoczky, John; Ramanan, Kavita; Shreve, Steven. Double Skorokhod Map and Reneging Real-Time Queues. Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz, 169--193, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2008. doi:10.1214/074921708000000372. https://projecteuclid.org/euclid.imsc/1233152942


Export citation

References

  • [1] Anderson, R. and Orey, S. (1976). Small random perturbations of dynamical systems with reflecting boundary. Nagoya Math. J. 60 189–216.
  • [2] Anulova, S. V. and Liptser, R. Sh. (1990). Diffusion approximation for processes with normal reflection. Theory Probab. Appl. 35 (3) 411–423.
  • [3] Burdzy, K., Kang, W., and Ramanan, K. (2008). The Skorokhod problem in a time-dependent interval. Stoch. Proc. Appl., to appear.
  • [4] Chitashvili, R. J. and Lazrieva, N. L. (1981). Strong solutions of stochastic differential equations with boundary conditions. Stochastics 5 255–309.
  • [5] Cooper, W., Schmidt, V. and Serfozo, R. (2001). Skorohod–Loynes characterizations of queueing, fluid, and inventory processes. Queueing Systems 37 233–257.
  • [6] Doytchinov, B., Lehoczky, J., and Shreve, S. (2001). Real-time queues in heavy trafic with earliest-deadline-first queue discipline. Annals of Applied Probability 11 332–378.
  • [7] Ganesh, A., O’Connell, N. and Wischik, D. (2004). Big Queues. Lecture Notes in Mathematics 1838. Springer, New York.
  • [8] Iglehart, D. and Whitt, W. (1970). Multiple channel queues in heavy trafic I. Adv. Appl. Probab. 2 150–177.
  • [9] Kingman, J. F. C. (1961). A single server queue in heavy trafic. Proc. Cambridge Phil. Soc. 48 277–289.
  • [10] 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.
  • [11] Kruk, L., Lehoczky, J., Ramanan, K. and Shreve, S. (2007). Heavy trafic analysis for EDF queues with reneging. Preprint.
  • [12] Skorokhod, A. V. (1961). Stochastic equations for di usions in a bounded region. Theor. of Prob. and Its Appl. 6 264–274.
  • [13] Tanaka, H. (1979). Stochastic differential equations with reflecting boundary conditions in convex regions. Hiroshima Math. J. 9 163–177.
  • [14] Toomey, T. (1998). Bursty trafic and finite capacity queues. Ann. Oper. Research 79 45–62.
  • [15] Whitt, W. (2002). Stochastic-Process Limits. Springer.