Using Singularity Analysis to Approximate Transient Characteristics in Queueing Systems

Abstract

In this paper, we develop a simple method to approximate the transient behavior of queueing systems. In particular, it is shown how singularity analysis of a known generating function of a transient sequence of some performance measure leads to an approximation of this sequence. To illustrate our approach, several specific transient sequences are investigated in detail. By means of some numerical examples, we validate our approximations and demonstrate the usefulness of the technique.

On two generalisations of the final value theorem: scientific relevance, first applications, and physical foundations

“…The present work is motivated, on one side, by the appearance of the applications [1,2] and [3][4][5] of the generalization [6] of the Laplace transform final value theorem (FVT) [6][7][8][9], and, on the other side, by the more recent work [10] devoted to another generalization of the classical FVT. We discuss applications of both generalizations and explore the use of irrational functions in the last generalization.…”

Analysis of steady-state and transient delay in discrete-time single-arrival and batch-arrival systems

“…We will, for instance, calculate the transform function of the sequence of mean delays {d t } ∞ t=0 in section 5. We note that it is usually not straightforward to invert these generating functions, mainly because of the implicitly defined function Y (x), but that a couple of approaches have been suggested in literature, such as numerical inversion techniques [14,15], iterative techniques [13], as well as analytical asymptotical techniques [16]. We also refer to section 5 in our article [8] for some discussion on this.…”

“…Therefore, we calculate asymptotics from the generating functionsD(x) andD(x). These can be calculated by investigation of the transform functions (or related functions) in the neighbourhood of their dominant singularities (singularities with lowest norm), see [16] for a detailed procedure. We find…”

Combined Analysis of Transient Delay Characteristics and Delay Autocorrelation Function in theGeoX/G/1Queue

“…(40) tends to 0, no longer leading to a proper probability generating function. Note that V * (1) = 1 is then the analytic continuation of a solution V * (x) (|x| < 1) of z − A(xS(z)) = 0 outside the unit disk (see also [21]). …”

“…In [21], we showed how singularity analysis of a transform of a transient sequence leads to an approximation for (the tail of) the sequence. Given the expression of a transform F (x), the corresponding sequence {f k , k ≥ 1} is approximated according to the following procedure:…”