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On Poisson Approximations for Superposition Arrival Processes in Queues

Author

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  • S. L. Albin

    (Rutgers University)

Abstract

We report on simulations of \Sigma i GI i /M/1 queues; the arrival process is the superposition (sum) of up to 1024 i.i.d. renewal processes and there is a single exponential server. As one might anticipate, the simulation estimate of the expected number of customers in a \Sigma i GI i /M/1 queueing system approaches the expected number in an M/M/1 queueing system as the number of arrival processes, n, increases. However, for a given n, the difference between the expected numbers in the M/M/1 and \Sigma i GI i /M/1 queueing systems dramatically increases as the traffic intensity increases from \rho = 0.5 to \rho = 0.9. This difference is approximated by a formula which is a function of the traffic intensity, the number of component arrival processes and the squared coefficient of variation of the component interarrival times.

Suggested Citation

  • S. L. Albin, 1982. "On Poisson Approximations for Superposition Arrival Processes in Queues," Management Science, INFORMS, vol. 28(2), pages 126-137, February.
    • Handle: RePEc:inm:ormnsc:v:28:y:1982:i:2:p:126-137
    DOI: 10.1287/mnsc.28.2.126
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    Cited by:

    1. 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.
    2. 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.
    3. Jiangchuan Huang & Christoph M. Kirsch & Raja Sengupta, 2015. "Cloud Computing in Space," INFORMS Journal on Computing, INFORMS, vol. 27(4), pages 704-717, November.
    4. 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.
    5. 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.
    6. Seifbarghy, Mehdi & Jokar, Mohammad Reza Akbari, 2006. "Cost evaluation of a two-echelon inventory system with lost sales and approximately Poisson demand," International Journal of Production Economics, Elsevier, vol. 102(2), pages 244-254, August.
    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. Ananth. V. Iyer, 2002. "Inventory cost impact of order processing priorities based on demand uncertainty," Naval Research Logistics (NRL), John Wiley & Sons, vol. 49(4), pages 376-390, June.
    9. 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.
    10. 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.
    11. S. Rajagopalan, 2002. "Make to Order or Make to Stock: Model and Application," Management Science, INFORMS, vol. 48(2), pages 241-256, February.
    12. 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.
    13. 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.
    14. Van Nyen, Pieter L. M. & Van Ooijen, Henny P. G. & Bertrand, J.W.M.J. Will M., 2004. "Simulation results on the performance of Albin and Whitt's estimation method for waiting times in integrated production-inventory systems," International Journal of Production Economics, Elsevier, vol. 90(2), pages 237-249, July.
    15. Ward Whitt, 2004. "A Diffusion Approximation for the G/GI/n/m Queue," Operations Research, INFORMS, vol. 52(6), pages 922-941, December.
    16. 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.
    17. Tielemans, Peter F. J. & Kuik, Roelof, 1996. "An exploration of models that minimize leadtime through batching of arrived orders," European Journal of Operational Research, Elsevier, vol. 95(2), pages 374-389, December.

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