Electronic Communications in Probability

A note on non-existence of diffusion limits for serve-the-longest-queue when the buffers are equal in size

Rami Atar and Subhamay Saha

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Abstract

We consider the serve-the-longest-queue discipline for a multiclass queue with buffers of equal size, operating under (i) the conventional and (ii) the Halfin-Whitt heavy traffic regimes, and show that while the queue length process’ scaling limits are fully determined by the first and second order data in case (i), they depend on finer properties in case (ii). The proof of the latter relies on the construction of a deterministic arrival pattern.

Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 2, 10 pp.

Dates
Received: 16 June 2015
Accepted: 4 January 2016
First available in Project Euclid: 3 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1454514622

Digital Object Identifier
doi:10.1214/16-ECP4370

Mathematical Reviews number (MathSciNet)
MR3485371

Zentralblatt MATH identifier
1338.60096

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 60J60: Diffusion processes [See also 58J65] 60K25: Queueing theory [See also 68M20, 90B22]

Keywords
diffusion limits Halfin-Whitt regime serve-the-longest queue heavy-traffic

Rights
Creative Commons Attribution 4.0 International License.

Citation

Atar, Rami; Saha, Subhamay. A note on non-existence of diffusion limits for serve-the-longest-queue when the buffers are equal in size. Electron. Commun. Probab. 21 (2016), paper no. 2, 10 pp. doi:10.1214/16-ECP4370. https://projecteuclid.org/euclid.ecp/1454514622


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References

  • [1] Anderson, R. and Orey, S.: Small random perturbations of dynamical systems with reflecting boundary. Nagoya Math. J. 60, (1976), 189–216.
  • [2] Atar, R.: A diffusion regime with nondegenerate slowdown. Oper. Res. 60, (2012), 490–500.
  • [3] Atar, R. and saha, S.: An $\varepsilon$–nash equilibrium for strategic customers in heavy traffic, arXiv:1505.01328
  • [4] Kruk, L., Lehoczky, J., Ramanan, K. and Shreve, S.: An explicit formula for the Skorokhod map on $[0,a]$. Ann. Probab. 35, (2007), 1740–1768.
  • [5] van Mieghem, J. A.: Due-date scheduling: asymptotic optimality of generalized longest queue and generalized largest delay rules. Oper. Res. 51, (2003), 113–122.