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

Winfried K. Grassmann ()
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Winfried K. Grassmann: University of Saskatchewan

Methodology and Computing in Applied Probability, 2011, vol. 13, issue 4, 873-886

Abstract: 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.

Keywords: Erlang queues; Characteristic equations; Waiting time; Matrix analytic methods; Solutions of polynomials; 60K25; 65H04 (search for similar items in EconPapers)
Date: 2011
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s11009-010-9199-2

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