Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-15T18:55:53.028Z Has data issue: false hasContentIssue false
  • English
  • Français

PERFORMANCE MEASURES FOR THE TWO-NODE QUEUE WITH FINITE BUFFERS

Published online by Cambridge University Press:  26 June 2019

Yanting Chen ,
Xinwei Bai ,
Richard J. Boucherie  and
Jasper Goseling
Yanting Chen
Affiliation:
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan410082, P. R. China E-mail: yantingchen@hnu.edu.cn
Xinwei Bai
Affiliation:
Stochastic Operations Research, University of Twente, P.O. Box 217, 7500AEEnschede, The Netherlands E-mail: xinwei.bai@ibeo-as.com; r.j.boucherie@utwente.nl; j.goseling@utwente.nl
Richard J. Boucherie
Affiliation:
Stochastic Operations Research, University of Twente, P.O. Box 217, 7500AEEnschede, The Netherlands E-mail: xinwei.bai@ibeo-as.com; r.j.boucherie@utwente.nl; j.goseling@utwente.nl
Jasper Goseling
Affiliation:
Stochastic Operations Research, University of Twente, P.O. Box 217, 7500AEEnschede, The Netherlands E-mail: xinwei.bai@ibeo-as.com; r.j.boucherie@utwente.nl; j.goseling@utwente.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a two-node queue modeled as a two-dimensional random walk. In particular, we consider the case that one or both queues have finite buffers. We develop an approximation scheme based on the Markov reward approach to error bounds in order to bound performance measures of such random walks. The approximation scheme is developed in terms of a perturbed random walk in which the transitions along the boundaries are different from those in the original model and the invariant measure of the perturbed random walk is of product-form. We then apply this approximation scheme to a tandem queue and some variants of this model, for the case that both buffers are finite. The modified approximation scheme and the corresponding applications for a two-node queueing system in which only one of the buffers has finite capacity have also been discussed.

Keywords

error boundsfinite state spaceperformance measureproduct-formrandom walktwo-node queue
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 34 , Issue 4 , October 2020 , pp. 522 - 549
Copyright
Copyright © Cambridge University Press 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

Current address: Ibeo Automotive Eindhoven, High Tech Campus 69, 5656 AE Eindhoven, The Netherlands

References

1.Asadathorn, N. & Chao., X. (1999). A decomposition approximation for assembly-disassembly queueing networks with finite buffer and blocking. Annals of Operations Research 87: 247–261.CrossRefGoogle Scholar
2.Balsamo, S. (2011) Queueing networks with blocking: Analysis, solution algorithms and properties. In Network Performance Engineering. Kouvatsos, D.D (ed). Berlin, Germany: Springer, pp. 233–257.CrossRefGoogle Scholar
3.Balsamo, S. & De Nitto-Personé., V. (1991). Closed queueing networks with finite capacities: blocking types, product-form solution and performance indices. Performance Evaluation 12(2): 85–102.CrossRefGoogle Scholar
4.Balsamo, S. & De Nitto-Personé., V. (1994). A survey of product form queueing networks with blocking and their equivalences. Annals of Operations Research 48(1-4): 31–61.CrossRefGoogle Scholar
5.Berezner, S.A., Krzesinski, A.E., & Taylor., P.G. (1997). A product-form “loss network” with a form of queueing. Journal of Applied Probability 34(4): 1075–1078.CrossRefGoogle Scholar
6.Bini, D.A., Latouche, G., & Meini., B. (2002). Solving matrix polynomial equations arising in queueing problems. Linear Algebra and its Applications 340: 225–244.CrossRefGoogle Scholar
7.Bini, D.A., Latouche, G., & Meini., B. (2005). Numerical methods for structured Markov chains. Oxford, England: Oxford University Press.CrossRefGoogle Scholar
8.Bini, D.A., Meini, B., Steffe, S., & Van Houdt., B. (2006) Structured Markov chains solver: software tools. Proceedings from the 2006 Workshop on Tools for Solving Structured Markov Chains.CrossRefGoogle Scholar
9.Bini, D.A., Meini, B., Steffe, S., & Van Houdt., B. (2009) Structured Markov chains solver: tool extension. Proceedings of the Fourth International ICST Conference on Performance Evaluation Methodologies and Tools.CrossRefGoogle Scholar
10.Bini, D.A., Meini, B., Steffe, S., Perez, J.F., & Van Houdt., B. (2012). SMCsolver and Q-MAM: tools for matrix-analytic methods. ACM SIGMETRICS Performance Evaluation Review 39(4): 46–46.CrossRefGoogle Scholar
11.Boucherie, R.J. & van Dijk., N.M. (2009). Monotonicity and error bounds for networks of Erlang loss queues. Queueing Systems 62(1–2): 159–193.CrossRefGoogle Scholar
12.Chen, Y., Boucherie, R.J., & Goseling., J. (2016). Invariant measures and error bounds for random walks in the quarter-plane based on sums of geometric terms. Queueing Systems 84(1–2): 21–48.CrossRefGoogle Scholar
13.de Nitto Persone, V. & Grassi., V. (1996). Solution of finite QBD processes. Journal of Applied Probability 33(4): 1003–1010.CrossRefGoogle Scholar
14.Economou, A. & Fakinos., D. (1998). Product form stationary distributions for queueing networks with blocking the rerouting. Queueing Systems 30: 251–260.CrossRefGoogle Scholar
15.Elhafsi, E.H. & Molle, M. (2007). On the solution to QBD processes with finite state space. Stochastic Analysis and Applications 25: 763–779.CrossRefGoogle Scholar
16.Fayolle, G. & Iasnogorodski., R. (1979). Two coupled processors: the reduction to a riemann-hilbert problem. Probability Theory and Related Fields 47(3): 325–351.Google Scholar
17.Gershwin., S.B. (1987). An efficient decomposition method for the approximate evaluation of tandem queues with finite storage space and blocking. Operations Research 35(2): 291–305.CrossRefGoogle Scholar
18.Goseling, J., Boucherie, R.J., & van Ommeren., J.C.W. (2013). Energy–delay tradeoff in a two-way relay with network coding. Performance Evaluation 70(11): 981–994.CrossRefGoogle Scholar
19.Goseling, J., Boucherie, R.J., & van Ommeren., J.C.W (2016). A linear programming approach to error bounds for random walks in the quarter-plane. Kybernetika (Prague) 52(5): 757–784.Google Scholar
20.Grassmann, W.K. & Tavakoli., J. (2005). Two-stations queueing networks with moving servers, blocking, and customer loss. Electronic Journal of Linear Algebra 13: 72–89.CrossRefGoogle Scholar
21.Gun, L. & Makowski, A.M. (1988). Matrix-geometric solution for finite capacity queues with phase-type distributions. In Courtouis, P.J. & Latouche, G., (eds.), Performance '87, Brussels, Belgium, 269–282.Google Scholar
22.Hajek., B.E. (1982). Birth-and-death processes on the integers with phases and general boundaries. Journal of Applied Probability 19(3): 488–499.CrossRefGoogle Scholar
23.He, C., Meini, B., Rhee, N.H., & Sohraby., K. (2004). A quadratically convergent Bernoulli-like algorithm for solving matrix polynomial equations in Markov chains. Electronic Transactions on Numerical Analysis 17: 151–167.Google Scholar
24.Hillier, F.S. & So., K.C. (1995). On the optimal design of tandem queueing systems with finite buffers. Queueing Systems 21(3–4): 245–266.CrossRefGoogle Scholar
25.Knessl, C. & Morrison., J.A. (2012). Asymptotic analysis of two coupled queues with vastly different arrival rates and finite customer capacities. Studies in Applied Mathematics 128(2): 107–143.CrossRefGoogle Scholar
26.Kroese, D.P., Scheinhardt, W.R.W., & Taylor., P.G. (2004). Spectral properties of the Tandem Jackson network seen as a quasi-birth-and-death process. The Annals of Applied Probability 14(4): 2057–2089.CrossRefGoogle Scholar
27.Latouche, G. & Ramaswami, V. (1999). Introduction to Matrix-Analytic Methods in Stochastic Modeling. Philadelphia: ASA-SIAM.CrossRefGoogle Scholar
28.Latouche, G., Nguyen, G.T., & Taylor., P.G. (2011). Queues with boundary assistance and the many effects of truncations. Queueing Systems 69(2): 175–197.CrossRefGoogle Scholar
29.Le Boudec, J.Y. (1991). An efficient solution method for Markov models of ATM links with loss priorities. IEEE Journal on Selected Areas in Communications 9(3): 408–417.CrossRefGoogle Scholar
30.Li., S.Q. (1989). Overload control in a finite message storage buffer. IEEE Transactions on Communications 37(12): 1330–1338.CrossRefGoogle Scholar
31.Miretskiy, D.I., Scheinhardt, W.R.W., & Mandjes., M.R.H. (2011). State-dependent importance sampling for a slowdown tandem queue. Annals of Operations Research 189(1): 299–329.CrossRefGoogle Scholar
32.Miyazawa., M. (2009). Tail decay rates in double QBD processes and related reflected random walks. Mathematics of Operations Research 34(3): 547–575.CrossRefGoogle Scholar
33.Perez, J.F. & Van Houdt., B. (2011). Quasi-birth-and-death processes with restricted transitions and its applications. Performance Evaluation (Special issue QEST 2009) 68(2): 126–141.Google Scholar
34.Perros., H.G. (1994). Queueing networks with blocking. Oxford, England: Oxford University Press, Inc.Google Scholar
35.Poloni., F. (2010) Algorithms for quadratic matrix and vector equations. PhD thesis, University of Pisa.CrossRefGoogle Scholar
36.Shanthikumar, J.G. & Jafari., M.A. (1994). Bounding the performance of tandem queues with finite buffer spaces. Annals of Operations Research 48(2): 185–195.CrossRefGoogle Scholar
37.van Dijk., N.M. (1987). A formal proof for the insensitivity of simple bounds for finite multi-server non-exponential tandem queues based on monotonicity results. Stochastic Processes and Their Applications 27: 261–277.CrossRefGoogle Scholar
38.van Dijk., N.M. (1988). Simple bounds for queueing systems with breakdowns. Performance Evaluation 8(2): 117–128.CrossRefGoogle Scholar
39.van Dijk., N.M. (1998). Bounds and error bounds for queueing networks. Annals of Operations Research 79: 295–319.CrossRefGoogle Scholar
40.van Dijk, N.M. (2011). Error bounds and comparison results: The Markov reward approach for queueing networks. In Boucherie, R.J. & Van Dijk, N.M., (eds.), Queueing Networks: A Fundamental Approach, volume 154 of International Series in Operations Research & Management Science. Berlin, Germany: Springer.Google Scholar
41.van Dijk, N.M. & Lamond., B.F. (1988). Simple bounds for finite single-server exponential tandem queues. Operations Research 36(3): 470–477.CrossRefGoogle Scholar
42.van Dijk, N.M. & Miyazawa., M. (2004). Error bounds for perturbing nonexponential queues. Mathematics of Operations Research 29(3): 525–558.CrossRefGoogle Scholar
43.van Dijk, N.M. & Puterman., M.L. (1988). Perturbation theory for Markov reward processes with applications to queueing systems. Advances in Applied Probability 20(1): 79–98.CrossRefGoogle Scholar
44.van Dijk, N.M. & van der Wal., J. (1989). Simple bounds and monotonicity results for finite multi-server exponential tandem queues. Queueing Systems 4(1): 1–15.CrossRefGoogle Scholar
45.van Foreest, N.D., van Ommeren, J.C.W., Mandjes, M.R.H., & Scheinhardt., W.R.W. (2005). A tandem queue with server slow-down and blocking. Stochastic Models 21(2-3): 695–724.CrossRefGoogle Scholar
46.van Vuuren, M., Adan, I.J.B.F., & Resing-Sassen, S.A.E. (2006). Performance analysis of multi-server tandem queues with finite buffers and blocking. In Liberopoulos, G., Papadopoulos, C.T., Tan, B., Smith, J.M., Gershwin, S.B. (eds.), Stochastic Modeling of Manufacturing Systems. Berlin, Germany: Springer, 169–192.CrossRefGoogle Scholar