The Korean Journal of Applied Statistics (응용통계연구)

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

Stochastic Upper Bound for the Stationary Queue Lengths of GPS Servers

  • Kim, Sung-Gon (Department of Information Statistics, Gyeongsang National University)
  • Published : 2009.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

Generalized processor sharing(GPS) service policy is a scheduling algorithm to allocate the bandwidth of a queueing system with multi-class input traffic. In a queueing system with single-class traffic, the stationary queue length becomes larger stochastically when the bandwidth (i.e. the service rate) of the system decreases. For a given GPS server, we consider the similar problem to this. We define the monotonicity for the head of the line processor sharing(HLPS) servers in which the units in the heads of the queues are served simultaneously and the bandwidth allocated to each queue are determined by the numbers of units in the queues. GPS is a type of monotonic HLPS. We obtain the HLPS server whose queue length of a class stochastically bounds upper that of corresponding class in the given monotonic HLPS server for all classes. The queue lengths process of all classes in the obtained HLPS server has the stationary distribution of product form. When the given monotonic HLPS server is GPS server, we obtain the explicit form of the stationary queue lengths distribution of the bounding HLPS server. Numerical result shows how tight the stochastic bound is.

Keywords

References

  1. Adan, I. J. B. F., Boxma, O. J. and Resing, J. A. C. (2001). Queueing models with multiple waiting lines, Queueing Systems, 37, 65-98 https://doi.org/10.1023/A:1011040100856
  2. Bertsimas, D., Paschalidis, I. C. and Tsitsiklis, J. N. (1999). Large deviations analysis of the generalized processor sharing policy, Queueing Systems, 32, 319-349 https://doi.org/10.1023/A:1019151423773
  3. Bonald, T. and Proutiere, A. (2004). On stochastic bounds for monotonic processor sharing networks, Queueing Systems, 47, 81-106 https://doi.org/10.1023/B:QUES.0000032802.41986.c6
  4. Borst,S., Mandjes, M. and van Uitert, M. (2003). Generalized processor sharing queues with heterogeneous traffic classes, Advances in Applied Probability, 35, 806-845 https://doi.org/10.1239/aap/1059486830
  5. Borst, S. and Zwart, B. (2003). A reduced-peak equivalence for queues with a mixture of light-tailed and heavy-tailed input flows, Advances in Applied Probability, 35, 793-805 https://doi.org/10.1239/aap/1059486829
  6. Brandt, A. and Bradnt, M. (1998). On the sojourn times for many-queue head-of-the-line processor-sharing systems with permanent customers, Mathematical Methods of Operations Research, 47, 181-220 https://doi.org/10.1007/BF01194397
  7. Cohen, J. W. (1988). Boundary value problems in queueing theory, Queueing Systems, 3, 97-128 https://doi.org/10.1007/BF01189045
  8. de Veciana, G. and Kesidis, G. (1996). Bandwidth allocation for multiple qualities of service using generalized processor sharing, IEEE Transactions on Information Theory, 42, 268-272 https://doi.org/10.1109/18.481801
  9. Dupuis, P. and Ramanan, K. (1998). A Skorokhod problem formulation and large deviation analysis of a processor sharing model, Queueing Systems, 28, 109-124 https://doi.org/10.1023/A:1019186720196
  10. Fayolle, G. and lasnogorodski, R. (1979). Two coupled processors: The reduction to a Riemann-Hilbert problem, Probability Theorp and Related Fields, 47, 325-351 https://doi.org/10.1007/BF00535168
  11. Parekh, A. K. and Gallager, R. G. (1993). A generalized processor sharing approach to flow control in integrated services networks: The single node case, IEEE/ACM Transactions on Networking, 1, 344-357 https://doi.org/10.1109/90.234856
  12. Serfozo, R. (1999). Introduction to Stochastic Networks, Springer, New York
  13. Zhang, Z. L. (1998). Large deviations and the generalized processor sharing scheduling for a multiple-queue system, Queueing Systems, 28, 349-376 https://doi.org/10.1023/A:1019163509718