Systems of functional equations

Drmota, Michael

doi:10.1002/(sici)1098-2418(199701/03)10:1/2<103::aid-rsa5>3.0.co;2-z

Cited by 111 publications

(148 citation statements)

References 11 publications

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“…We thus want to find the dominant singularity of Y (z) and the behavior of Y (z) in the neighbourhood of this singularity. This has been studied in numerous papers in the field of combinatorics [6,11]. Under the condition that g(x, z) fulfills some mild requirements (see [6] for details),…”

Section: Implicitly-defined Functionsmentioning

confidence: 99%

Using Singularity Analysis to Approximate Transient Characteristics in Queueing Systems

Walraevens¹,

Fiems²,

Moeneclaey³

2009

Prob. Eng. Inf. Sci.

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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.

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Section: Implicitly-defined Functionsmentioning

confidence: 99%

Using Singularity Analysis to Approximate Transient Characteristics in Queueing Systems

Walraevens¹,

Fiems²,

Moeneclaey³

2009

Prob. Eng. Inf. Sci.

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“…Applying the results of [14], one can show that in the neighbourhood of s B , Y (z) is approximately given by…”

Section: Singularity S Bmentioning

confidence: 99%

Priority queueing systems: from probability generating functions to tail probabilities

2006

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Obtaining (tail) probabilities from a transform function is an important topic in queueing theory. To obtain these probabilities in discrete-time queueing systems, we have to invert probability generating functions, since most important distributions in discrete-time queueing systems can be determined in the form of probability generating functions. In this paper, we calculate the tail probabilities of two particular random variables in discrete-time priority queueing systems, by means of the dominant singularity approximation. We show that obtaining these tail probabilities can be a complex task, and that the obtained tail probabilities are not necessarily exponential (as in most 'traditional' queueing systems). Further, we show the impact and significance of the various system parameters on the type of tail behavior. Finally, we compare our approximation results with simulations.keywords priority queueing systems, tail probabilities, dominant pole, non-exponential behavior

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“…Here, x Y and x V are the dominant singularities on the real positive axis of Y (x) and V (x), and K Y (x Y ) and K V (x V ) are constants that can be calculated as function of the input functions A(x) and S(x) (but their precise form is not important for the following discussion). Since both Y (x) and V (x) are implicitly defined, the singularities are square-root branchpoints (see [19,20]) and we have…”

Section: Transient Delaysmentioning

confidence: 99%

“…The y i,j s are mutually independent, as are the v i,j s. From (19), it can be deduced that y is the number of slots that it takes to reduce the unfinished work at the beginning of a slot by one unit. Indeed, this equals one slot if there are no new arrivals in that slot.…”

Section: The Kernel Solutions Y (X) and V (X)mentioning

confidence: 99%

“…Indeed, this equals one slot if there are no new arrivals in that slot. If there are new arrivals, each new unit of work (there are a i=1 s i new units) takes a number of slots that is equally distributed as y, leading to (19). Equation (20) on the other hand demonstrates that v equals the total number of arrivals that is "generated" during a slot in the sense that not just the number of arrivals in the first slot are counted but also the arrivals in the slots when these customers are served, and the arrivals in the slots when these latter customers are served, etc.…”

Section: The Kernel Solutions Y (X) and V (X)mentioning

confidence: 99%

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

Walraevens

Claeys

Bruneel

2014

Applied Mathematics and Computation

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We perform an analysis of the transient delay in a discrete-time FIFO buffer with batch arrivals. As transient delay is an ambiguous concept, we first discuss different possible definitions of the term (delay of the k-th customer, delay of a customer arriving at time t, etc). In this paper, we focus on the analysis of the delay of a customer arriving in slot t, also sometimes called virtual delay in single-arrival systems. It turns out that the modeling in batch-arrival systems is more intricate. In analysis, we relate transient delay to transient unfinished work and characterize the latter. Some time-dependent as well as limiting steady-state delay measures are calculated. We also study a variation that is related to active probing measurements. A substantial part of the article finally focuses on some fundamental differences between alternative definitions of transient delay.

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Systems of functional equations

Cited by 111 publications

References 11 publications

Using Singularity Analysis to Approximate Transient Characteristics in Queueing Systems

Using Singularity Analysis to Approximate Transient Characteristics in Queueing Systems

Priority queueing systems: from probability generating functions to tail probabilities

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

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