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A New Method for Finding the Characteristic Roots of E n /E m /1 Queues

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  • Winfried K. Grassmann

    (University of Saskatchewan)

Abstract

In 1953, Smith (Proc Camb Philos Soc 49:449–461, 1953), and, following him, Syski (1960) suggested a method to find the waiting time distribution for one server queues with Erlang-n arrivals and Erlang-m service times by using characteristic roots. Syski shows that these roots can be determined from a very simple equation, but an equation of degree n + m. Syski also shows that almost all of the characteristic roots are complex. In this paper, we derive a set of equations, one for each complex root, which can be solved by Newton’s method using real arithmetic. This method simplifies the programming logic because it avoids deflation and the subsequent polishing of the roots. Using the waiting time distribution, Syski then derived the distribution of the number in the system after a departure. E n /E m /1 queues can also formulated as quasi birth-death (QBD) processes, and in this case, the characteristic roots discussed by Syski are closely related to the eigenvalues of the QBD process. The QBD process provides information about the number in system at random times, but they are much more difficult to formulate and solve.

Suggested Citation

  • Winfried K. Grassmann, 2011. "A New Method for Finding the Characteristic Roots of E n /E m /1 Queues," Methodology and Computing in Applied Probability, Springer, vol. 13(4), pages 873-886, December.
    • Handle: RePEc:spr:metcap:v:13:y:2011:i:4:d:10.1007_s11009-010-9199-2
    DOI: 10.1007/s11009-010-9199-2
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    Cited by:

    1. Xiaoyuan Liu & Brian Fralix, 2019. "On Lattice Path Counting and the Random Product Representation, with Applications to the Er/M/1 Queue and the M/Er/1 Queue," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1119-1149, December.
    2. B. H. Margolius, 2023. "The periodic steady-state solution for queues with Erlang arrivals and service and time-varying periodic transition rates," Queueing Systems: Theory and Applications, Springer, vol. 103(1), pages 45-94, February.

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