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On Erlang Queue with Multiple Arrivals and its Time-Changed Variant

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  • Rohini Bhagwanrao Pote

    (Indian Institute of Technology Bhilai)

  • Kuldeep Kumar Kataria

    (Indian Institute of Technology Bhilai)

Abstract

We introduce and study a queue with the Erlang service system and whose arrivals are governed by a counting process in which there is a possibility of more than one arrival at any instant. We call it the Erlang queue with multiple arrivals. We derive a system of differential equations for its transient probabilities. Its probability generating function is obtained from which the explicit expressions of its transient probabilities are derived. Also, the probability of zero customers at any instant is obtained. Further, we define the queue length process for Erlang queue with multiple arrivals and obtain a system of differential equations for its mean queue length. We derive the explicit expression for the mean queue length and the second moment of the queue length process. Also, we study a time-changed variant of it by subordinating it with an independent inverse stable subordinator for which we obtain its state probabilities, mean queue length and distribution of busy period.

Suggested Citation

  • Rohini Bhagwanrao Pote & Kuldeep Kumar Kataria, 2025. "On Erlang Queue with Multiple Arrivals and its Time-Changed Variant," Methodology and Computing in Applied Probability, Springer, vol. 27(3), pages 1-36, September.
    • Handle: RePEc:spr:metcap:v:27:y:2025:i:3:d:10.1007_s11009-025-10200-7
    DOI: 10.1007/s11009-025-10200-7
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    References listed on IDEAS

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    1. H. J. Haubold & A. M. Mathai & R. K. Saxena, 2011. "Mittag‐Leffler Functions and Their Applications," Journal of Applied Mathematics, John Wiley & Sons, vol. 2011(1).
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    8. George Luchak, 1956. "The Solution of the Single-Channel Queuing Equations Characterized by a Time-Dependent Poisson-Distributed Arrival Rate and a General Class of Holding Times," Operations Research, INFORMS, vol. 4(6), pages 711-732, December.
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