IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v335y2024i1d10.1007_s10479-024-05851-7.html
   My bibliography  Save this article

The superposition of Markovian arrival processes: moments and the minimal Laplace transform

Author

Listed:
  • Sunkyo Kim

    (Ajou University)

Abstract

The superposition of two independent Markovian arrival processes (MAPs) is also a Markovian arrival process of which the Markovian representation is given as the Kronecker sum of the transition rate matrices of the component processes. The moments of stationary intervals of the superposition can be obtained by differentiating the Laplace transform (LT) given in terms of the transition rate matrices. In this paper, we propose a streamlined procedure to determine the minimal LT of the merged process in terms of the minimal LT coefficients of the component processes. Combined with the closed-form transformation between moments and LT coefficients, our result enables us to determine the moments of the superposed process based on the moments of the component processes. The main contribution is that the whole procedure can be implemented without explicit Markovian representations. In order to transform the minimal LT coefficients of the component processes into the minimal LT representation of the merged process, we also introduce another minimal representation. A numerical example is provided to illustrate the procedure.

Suggested Citation

  • Sunkyo Kim, 2024. "The superposition of Markovian arrival processes: moments and the minimal Laplace transform," Annals of Operations Research, Springer, vol. 335(1), pages 237-259, April.
    • Handle: RePEc:spr:annopr:v:335:y:2024:i:1:d:10.1007_s10479-024-05851-7
    DOI: 10.1007/s10479-024-05851-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10479-024-05851-7
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10479-024-05851-7?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Bodrog, L. & Heindl, A. & Horváth, G. & Telek, M., 2008. "A Markovian canonical form of second-order matrix-exponential processes," European Journal of Operational Research, Elsevier, vol. 190(2), pages 459-477, October.
    2. Levente Bodrog & András Horváth & Miklós Telek, 2008. "Moment characterization of matrix exponential and Markovian arrival processes," Annals of Operations Research, Springer, vol. 160(1), pages 51-68, April.
    3. S. L. Albin, 1982. "On Poisson Approximations for Superposition Arrival Processes in Queues," Management Science, INFORMS, vol. 28(2), pages 126-137, February.
    4. Whitt, Ward, 1985. "Queues with superposition arrival processes in heavy traffic," Stochastic Processes and their Applications, Elsevier, vol. 21(1), pages 81-91, December.
    5. Kim, Sunkyo, 2004. "The heavy-traffic bottleneck phenomenon under splitting and superposition," European Journal of Operational Research, Elsevier, vol. 157(3), pages 736-745, September.
    6. Sunkyo Kim, 2011. "Modeling Cross Correlation in Three-Moment Four-Parameter Decomposition Approximation of Queueing Networks," Operations Research, INFORMS, vol. 59(2), pages 480-497, April.
    7. S. Suresh & W. Whitt, 1990. "Arranging Queues in Series: A Simulation Experiment," Management Science, INFORMS, vol. 36(9), pages 1080-1091, September.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Irina Peshkova & Evsey Morozov & Michele Pagano, 2024. "Regenerative Analysis and Approximation of Queueing Systems with Superposed Input Processes," Mathematics, MDPI, vol. 12(14), pages 1-22, July.
    2. Rabta, Boualem, 2013. "A hybrid method for performance analysis of G/G/m queueing networks," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 89(C), pages 38-49.
    3. Sunkyo Kim, 2016. "Minimal LST representations of MAP(n)s: Moment fittings and queueing approximations," Naval Research Logistics (NRL), John Wiley & Sons, vol. 63(7), pages 549-561, October.
    4. Hill, R.M. & Seifbarghy, M. & Smith, D.K., 2007. "A two-echelon inventory model with lost sales," European Journal of Operational Research, Elsevier, vol. 181(2), pages 753-766, September.
    5. Rodríguez, Joanna & Lillo, Rosa E. & Ramírez-Cobo, Pepa, 2015. "Failure modeling of an electrical N-component framework by the non-stationary Markovian arrival process," Reliability Engineering and System Safety, Elsevier, vol. 134(C), pages 126-133.
    6. S. Rajagopalan, 2002. "Make to Order or Make to Stock: Model and Application," Management Science, INFORMS, vol. 48(2), pages 241-256, February.
    7. Morabito, Reinaldo & de Souza, Mauricio C. & Vazquez, Mariana, 2014. "Approximate decomposition methods for the analysis of multicommodity flow routing in generalized queuing networks," European Journal of Operational Research, Elsevier, vol. 232(3), pages 618-629.
    8. M. Rosário Moreira & Rui Alves, 2006. "Does Order Negotiation Improve The Job-Shop Workload Control?," FEP Working Papers 213, Universidade do Porto, Faculdade de Economia do Porto.
    9. Thangam, A. & Uthayakumar, R., 2008. "A two-level supply chain with partial backordering and approximated Poisson demand," European Journal of Operational Research, Elsevier, vol. 187(1), pages 228-242, May.
    10. Hau L. Lee & Kamran Moinzadeh, 1987. "Operating characteristics of a two‐echelon inventory system for repairable and consumable items under batch ordering and shipment policy," Naval Research Logistics (NRL), John Wiley & Sons, vol. 34(3), pages 365-380, June.
    11. Pepa Ramírez-Cobo & Rosa E. Lillo, 2012. "New Results About Weakly Equivalent MAP 2 and MAP 3 Processes," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 421-444, September.
    12. Vinarskiy, Miron, 2017. "A method of approximate analysis of an open exponential queuing network with losses due to finite shared buffers in multi-queue nodes," European Journal of Operational Research, Elsevier, vol. 258(1), pages 207-215.
    13. Sunkyo Kim, 2011. "Modeling Cross Correlation in Three-Moment Four-Parameter Decomposition Approximation of Queueing Networks," Operations Research, INFORMS, vol. 59(2), pages 480-497, April.
    14. Sarang Deo & Milind Sohoni, 2015. "Optimal Decentralization of Early Infant Diagnosis of HIV in Resource-Limited Settings," Manufacturing & Service Operations Management, INFORMS, vol. 17(2), pages 191-207, May.
    15. A. Korhan Aras & Xinyun Chen & Yunan Liu, 2018. "Many-server Gaussian limits for overloaded non-Markovian queues with customer abandonment," Queueing Systems: Theory and Applications, Springer, vol. 89(1), pages 81-125, June.
    16. Olsson, Ralph J. & Hill, Roger M., 2007. "A two-echelon base-stock inventory model with Poisson demand and the sequential processing of orders at the upper echelon," European Journal of Operational Research, Elsevier, vol. 177(1), pages 310-324, February.
    17. Rodríguez César, Joanna Virginia & Ramírez-Cobo, Pepa, 2012. "On the identifiability of the two-state BMAP," DES - Working Papers. Statistics and Econometrics. WS ws120201, Universidad Carlos III de Madrid. Departamento de Estadística.
    18. Casale, Giuliano & Sansottera, Andrea & Cremonesi, Paolo, 2016. "Compact Markov-modulated models for multiclass trace fitting," European Journal of Operational Research, Elsevier, vol. 255(3), pages 822-833.
    19. Sarat Babu Moka & Yoni Nazarathy & Werner Scheinhardt, 2023. "Diffusion parameters of flows in stable multi-class queueing networks," Queueing Systems: Theory and Applications, Springer, vol. 103(3), pages 313-346, April.
    20. Bitran, Gabriel R. & Morabito, Reinaldo., 1994. "Open queueing networks : optimization and performance evaluation models for discrete manufacturing systems," Working papers 3743-94., Massachusetts Institute of Technology (MIT), Sloan School of Management.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:annopr:v:335:y:2024:i:1:d:10.1007_s10479-024-05851-7. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.