ASMTA 2015: Analytical and Stochastic Modelling Techniques and Applications pp 203-216 | Cite as
On the Influence of High Priority Customers on a Generalized Processor Sharing Queue
Abstract
In this paper, we study a hybrid scheduling mechanism in discrete-time. This mechanism combines the well-known Generalized Processor Sharing (GPS) scheduling with strict priority. We assume three customer classes with one class having strict priority over the other classes, whereby each customer requires a single slot of service. The latter share the remaining bandwith according to GPS. This kind of scheduling is used in practice for the scheduling of jobs on a processor and in Quality of Service modules of telecommunication network devices. First, we derive a functional equation of the joint probability generating function of the queue contents. To explicitly solve the functional equation, we introduce a power series in the weight parameter of GPS. Subsequently, an iterative procedure is presented to calculate consecutive coefficients of the power series. Lastly, the approximation resulting from a truncation of the power series is verified with simulation results. We also propose rational approximations. We argue that the approximation performs well and is extremely suited to study these systems and their sensitivity in their parameters (scheduling weights, arrival rates, loads ...). This method provides a fast way to observe the behaviour of such type of systems avoiding time-consuming simulations.
Keywords
Generalized Processor Sharing (GPS) Priority Queueing Scheduling Power seriesPreview
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References
- 1.Adan, I.J., Van Leeuwaarden, J., Winands, E.M.: On the application of Rouché’s theorem in queueing theory. Operations Research Letters 34(3), 355–360 (2006)CrossRefMATHMathSciNetGoogle Scholar
- 2.Asmussen, S., Glynn, P.W.: Stochastic Simulation: Algorithms and Analysis: Algorithms and Analysis, vol. 57. Springer (2007)Google Scholar
- 3.Choi, B., Choi, D., Lee, Y., Sung, D.: Priority queueing system with fixed-length packet-train arrivals. IEE Proceedings-Communications 145(5), 331–336 (1998)CrossRefGoogle Scholar
- 4.Jin, X., Min, G.: Analytical modelling of hybrid PQ-GPS scheduling systems under long-range dependent traffic. In: 21st International Conference on Advanced Information Networking and Applications, 2007, AINA 2007. pp. 1006–1013. IEEE (2007)Google Scholar
- 5.Jin, X., Min, G.: Performance modelling of hybrid PQ-GPS systems under long-range dependent network traffic. IEEE Communications Letters 11(5), 446–448 (2007)CrossRefGoogle Scholar
- 6.Kim, K., Chae, K.C.: Discrete-time queues with discretionary priorities. European Journal of Operational Research 200(2), 473–485 (2010)CrossRefMATHGoogle Scholar
- 7.Lee, J.Y., Kim, S., Kim, D., Sung, D.K.: Bandwidth optimization for internet traffic in generalized processor sharing servers. IEEE Transactions on Parallel and Distributed Systems 16(4), 324–334 (2005)CrossRefGoogle Scholar
- 8.Lieshout, P., Mandjes, M.: Generalized processor sharing: Characterization of the admissible region and selection of optimal weights. Computers & Operations Research 35(8), 2497–2519 (2008)CrossRefMATHGoogle Scholar
- 9.Nichols, K., Blake, S., Baker, F., Black, D.: Definition of the differentiated services field (DS field) in the IPv4 and IPv6 headers. RFC 2474 (Proposed Standard) (dec 1998). http://www.ietf.org/rfc/rfc2474.txt, updated by RFCs 3168, 3260
- 10.Parekh, A.K., Gallager, R.G.: A generalized processor sharing approach to flow control in integrated services networks: the single-node case. IEEE/ACM Transactions on Networking (TON) 1(3), 344–357 (1993)CrossRefGoogle Scholar
- 11.Parekh, A.K., Gallagher, R.G.: A generalized processor sharing approach to flow control in integrated services networks: the multiple node case. IEEE/ACM Transactions on Networking (TON) 2(2), 137–150 (1994)CrossRefGoogle Scholar
- 12.Parveen, A.S.: A survey of an integrated scheduling scheme with long-range and short-range dependent traffic. International Journal of Engineering Sciences & Research Technology 3(1), 430–439 (2014)Google Scholar
- 13.Smith, P.J., Firag, A., Dmochowski, P.A., Shafi, M.: Analysis of the M/M/N/N queue with two types of arrival process: Applications to future mobile radio systems. Journal of Applied Mathematics 2012 (2012)Google Scholar
- 14.Spall, J.C.: Introduction to stochastic search and optimization: estimation, simulation, and control, vol. 65. John Wiley & Sons (2005)Google Scholar
- 15.Takine, T., Sengupta, B., Hasegawa, T.: An analysis of a discrete-time queue for broadband ISDN with priorities among traffic classes. IEEE Transactions on Communications 42(234), 1837–1845 (1994)CrossRefGoogle Scholar
- 16.Vanlerberghe, J., Walraevens, J., Maertens, T., Bruneel, H.: Approximating the optimal weights for discrete-time generalized processor sharing. In: Networking Conference, 2014 IFIP, pp. 1–9. IEEE (2014)Google Scholar
- 17.Walraevens, J., van Leeuwaarden, J., Boxma, O.: Power series approximations for two-class generalized processor sharing systems. Queueing systems 66(2), 107–130 (2010)CrossRefMATHMathSciNetGoogle Scholar
- 18.Walraevens, J., Steyaert, B., Bruneel, H.: Delay characteristics in discrete-time GI-G-1 queues with non-preemptive priority queueing discipline. Performance Evaluation 50(1), 53–75 (2002)CrossRefGoogle Scholar
- 19.Walraevens, J., Steyaert, B., Bruneel, H.: Performance analysis of a single-server ATM queue with a priority scheduling. Computers & Operations Research 30(12), 1807–1829 (2003)CrossRefMATHGoogle Scholar
- 20.Wang, L., Min, G., Kouvatsos, D.D., Jin, X.: Analytical modeling of an integrated priority and WFQ scheduling scheme in multi-service networks. Computer Communications 33, S93–S101 (2010)CrossRefGoogle Scholar
- 21.Zhang, Z.L., Towsley, D., Kurose, J.: Statistical analysis of the generalized processor sharing scheduling discipline. IEEE Journal on Selected Areas in Communications 13(6), 1071–1080 (1995)CrossRefGoogle Scholar