Clearing Analysis on Phases: Exact Limiting Probabilities for Skip-Free, Unidirectional, Quasi-Birth-Death Processes

Published Online:https://doi.org/10.1287/15-SSY183

References

  • J. Abate and W. Whitt. Transient behavior of the M/M/1 queue via Laplace transforms. Advances in Applied Probability, pages 145–178, 1988. MR0932538Google Scholar
  • I. Adan and J. Resing. A class of Markov processes on a semi-infinite strip. Technical Report 99-03, Eindhoven University of Technology, Department of Mathematics and Computing Sciences, 1999.Google Scholar
  • S. Asmussen. Applied Probability and Queues, volume 51. Springer Science & Business Media, 2003. MR1978607Google Scholar
  • L. Bright and P. G. Taylor. Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes. Stochastic Models, 11(3):497–525, 1995. MR1340970Google Scholar
  • C. W. Chan, V. F. Farias, and G. J. Escobar. The impact of delays on service times in the intensive care unit. Management Science, 2016.Google Scholar
  • G. Ciardo, W. Mao, A. Riska, and E. Smirni. ETAQA-MG1: An efficient technique for the analysis of a class of M/G/1-type processes by aggregation. Performance Evaluation, 57(3):235–260, 2004.Google Scholar
  • G. Ciardo and E. Smirni. ETAQA: An efficient technique for the analysis of QBD-processes by aggregation. Performance Evaluation, 36:71–93, 1999.Google Scholar
  • M. Delasay, A. Ingolfsson, and B. Kolfal. Modeling load and overwork effects in queueing systems with adaptive service rates. Operations Research, 2016. MR3532859Google Scholar
  • S. Doroudi, B. Fralix, and M. Harchol-Balter. Clearing analysis on phases: Exact limiting probabilities for skip-free, unidirectional, quasi-birth-death processes. arXiv preprint arXiv:1503.05899v3, 2015.Google Scholar
  • A. Gandhi, S. Doroudi, M. Harchol-Balter, and A. Scheller-Wolf. Exact analysis of the M/M/k/setup class of Markov chains via recursive renewal reward. In Proceedings of the ACM SIGMETRICS/international conference on Measurement and modeling of computer systems, pages 153–166. ACM, 2013.Google Scholar
  • A. Gandhi, S. Doroudi, M. Harchol-Balter, and A. Scheller-Wolf. Exact analysis of the M/M/k/setup class of Markov chains via recursive renewal reward. Queueing Systems, 77(2):177–209, 2014. MR3206189Google Scholar
  • A. Gandhi, M. Harchol-Balter, R. Raghunathan, and M. A. Kozuch. Autoscale: Dynamic, robust capacity management for multi-tier data centers. ACM Transactions on Computer Systems (TOCS), 30(4):14, 2012.Google Scholar
  • M. Harchol-Balter. Performance Modeling and Design of Computer Systems: Queueing Theory in Action. Cambridge University Press, 2013.Google Scholar
  • Q.-M. He. Fundamentals of Matrix-Analytic Methods. Springer, 2014. MR3112230Google Scholar
  • R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, 2012.Google Scholar
  • S. Karlin and H. Taylor. A First Course in Stochastic Processes. Academic Press, New York, 1975.Google Scholar
  • G. Latouche and V. Ramaswami. A logarithmic reduction algorithm for quasi-birth-death processes. Journal of Applied Probability, pages 650–674, 1993.Google Scholar
  • G. Latouche and V. Ramaswami. Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM, Philadelphia, 1999.Google Scholar
  • Y. Levy and U. Yechiali. An M/M/s queue with servers’ vacations. INFOR, 14:153–163, 1976.Google Scholar
  • D. Liu and Y. Zhao. Determination of explicit solution for a general class of Markov processes. Matrix-Analytic Methods in Stochastic Models, page 343, 1996. MR1427280Google Scholar
  • I. Mitrani and R. Chakka. Spectral expansion solution for a class of Markov models: Application and comparison with the matrix-geometric method. Performance Evaluation, 23(3):241–260, 1995.Google Scholar
  • M. Neuts. Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. John Hopkins University Press, Baltimore, Maryland, 1981.Google Scholar
  • T. Phung-Duc. Exact solutions for M/M/c/setup queues. Telecommunication Systems, pages 1–16, 2014.Google Scholar
  • V. Ramaswami and G. Latouche. A general class of Markov processes with explicit matrix-geometric solutions. Operations-Research-Spektrum, 8(4):209–218, 1986.Google Scholar
  • A. Riska and E. Smirni. Exact aggregate solutions for M/G/1-type Markov processes. In ACM SIGMETRICS Performance Evaluation Review, volume 30, pages 86–96. ACM, 2002.Google Scholar
  • A. Riska and E. Smirni. ETAQA solutions for infinite Markov processes with repetitive structure. INFORMS Journal on Computing, 19(2):215–228, 2007.LinkGoogle Scholar
  • J. Selen, I. J. Adan, V. G. Kulkarni, and J. van Leeuwaarden. The snowball effect of customer slowdown in critical many-server systems. Stochastic Models, 32:366–391, 2016. MR3505449Google Scholar
  • J. Selen, I. J. Adan, and J. S. Van Leeuwaarden. Product-form solutions for a class of structured multidimensional Markov processes. SIAM Journal on Applied Mathematics, 74(3):844–863, 2014.Google Scholar
  • A. Sleptchenko, J. Selen, I. Adan, and G.-J. van Houtum. Joint queue length distribution of multi-class, single-server queues with preemptive priorities. Queueing Systems, 81(4):379–395, 2015.Google Scholar
  • A. Stathopoulos, A. Riska, Z. Hua, and E. Smirni. Bridging ETAQA and Ramaswami’s formula for the solution of M/G/1-type processes. Performance Evaluation, 62(1):331–348, 2005.Google Scholar
  • B. Van Houdt and J. van Leeuwaarden. Triangular M/G/1-Type and Tree-Like Quasi-Birth-Death Markov Chains. INFORMS Journal on Computing, 23(1):165–171, 2011.LinkGoogle Scholar
  • J. van Leeuwaarden, M. Squillante, and E. Winands. Quasi-birth-and-death processes, lattice path counting, and hypergeometric functions. Journal of Applied Probability, 46(2):507–520, 2009.Google Scholar
  • J. van Leeuwaarden and E. Winands. Quasi-birth-and-death processes with an explicit rate matrix. Stochastic Models, 22(1):77–98, 2006.Google Scholar
INFORMS site uses cookies to store information on your computer. Some are essential to make our site work; Others help us improve the user experience. By using this site, you consent to the placement of these cookies. Please read our Privacy Statement to learn more.