Power series approximations for two-class generalized processor sharing systems

Walraevens, Joris; Leeuwaarden, Johan S. H. van; Boxma, Onno

doi:10.1007/s11134-010-9188-8

Cited by 34 publications

(41 citation statements)

References 20 publications

(40 reference statements)

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“…Note that if the moments of the continuous-time queue exist, there always exists a sufficiently small ∆ such that the discretized queue is stable and the moments of the discretized queue exist as well. Related approaches to obtain results for continuous-time systems from discrete-time results can be found, e.g., in [2][3][4][5].…”

Section: Discrete-time Modelingmentioning

confidence: 99%

Rejoinder on: Queueing models for the analysis of communication systems

Bruneel

Fiems

Walraevens

et al. 2014

TOP

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First of all, we want to express our gratitude to the former Editor-in-Chief of the journal TOP, our fondly-remembered colleague Professor Jesús Artalejo, who invited us about one year ago to write a review paper for TOP on the topic of queueing models for the analysis of communication systems. Also, we want to thank the three discussants for the effort of carefully reading our paper and adding interesting information. We mention in particular the insightful discussion of Onno Boxma on the role of transform methods in queueing analysis, their strengths and weaknesses. With respect to scheduling disciplines, Onno Boxma also adds the topic of dynamic priority scheduling schemes. As far as traffic modeling is concerned, Alexander Dudin not only refers to the well-known D-BMAP model as an alternative description of bursty, correlated traffic streams, but he also points to several related papers on continuous-time models with session-based arrivals. Harry Perros emphasizes the practical applications of discrete-time queueing models in the performance evaluation of various subsystems of modern telecommunication networks. In particular, he discusses the restriction introduced by the assumption in our paper that packets be of fixed length, which makes the results applicable in the context of ATM, in optical burst switching, and in HTTP adaptive video streaming, but not in a general IP network where packets typically are of variable length.A very striking observation is that all three discussants explicitly comment on the use of discrete-time queueing models in much of our work of the last thirty years (including the current paper), as opposed to the much more frequently encountered continuous-time models in literature. As these remarks are of a very generic nature, we isolate this topic from the more specific comments in this rejoinder. Therefore, our responses are structured around two main topics: (1) the aspect of modeling and queueing analysis in the discrete time domain and (2) possible extensions of and remarks on our presented analysis of the two-class priority queue with train arrivals. * SMACS: Stochastic Modeling and Analysis of Communication Systems

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Section: Discrete-time Modelingmentioning

confidence: 99%

Rejoinder on: Queueing models for the analysis of communication systems

Bruneel

Fiems

Walraevens

et al. 2014

TOP

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“…In our approach we generalize the method developed in [29] to develop power series approximations in a retrial system of weighted-fair orbit queues with structured batch arrivals. In particular, for the exponentially distributed service times, we first construct a power series expansion in ξ (rather than in load) for a bivariate pgf that corresponds to the set of states of an idle server, and then using the matrix functional equation we construct the power series expansion for the pgf that corresponds to the set of states of the busy server.…”

Section: Introductionmentioning

confidence: 99%

On the power series approximations of a structured batch arrival two-class retrial system with weighted fair orbit queues

Dimitriou

2019

Performance Evaluation

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We provide power series approximations for a structured batch arrival single server retrial system with two infinite capacity weighted fair orbit queues, i.e., the re-transmission rate of an orbit depends on the state of the other orbit queue. We consider both exponential and arbitrary distributed service times. In both cases we obtain power series expansions of the generating functions of the stationary joint orbit queue-length distributions, and provide a recursive approach to calculate their coefficients. We also show how to obtain the generating function of the stationary joint orbit queue-length distribution with the aid of a Riemann boundary value problem. Power series approximations are also provided for the model with two independent Poisson streams of jobs with single arrivals. Numerical illustrations are performed and show the accuracy of our approach.

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“…Priority queueing was, for instance, studied in [3,6,13,15,18,19]. Whereas, GPS was analyzed in [7,8,10,11,17,21].…”

Section: Introductionmentioning

confidence: 99%

On the Influence of High Priority Customers on a Generalized Processor Sharing Queue

Vanlerberghe

Walraevens

Maertens

et al. 2015

Analytical and Stochastic Modelling Techniques and Applications

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Abstract. In this paper, we study a hybrid scheduling mechanism in discrete-time. This mechanism combines the well-known Generalized Processor Sharing (GPS) scheduling with strict priority. We assume three customer classes with one class having strict priority over the other classes, whereby each customer requires a single slot of service. The latter share the remaining bandwith according to GPS. This kind of scheduling is used in practice for the scheduling of jobs on a processor and in Quality of Service modules of telecommunication network devices. First, we derive a functional equation of the joint probability generating function of the queue contents. To explicitly solve the functional equation, we introduce a power series in the weight parameter of GPS. Subsequently, an iterative procedure is presented to calculate consecutive coefficients of the power series. Lastly, the approximation resulting from a truncation of the power series is verified with simulation results. We also propose rational approximations. We argue that the approximation performs well and is extremely suited to study these systems and their sensitivity in their parameters (scheduling weights, arrival rates, loads ...). This method provides a fast way to observe the behaviour of such type of systems avoiding time-consuming simulations.

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Power series approximations for two-class generalized processor sharing systems

Cited by 34 publications

References 20 publications

Rejoinder on: Queueing models for the analysis of communication systems

Rejoinder on: Queueing models for the analysis of communication systems

On the power series approximations of a structured batch arrival two-class retrial system with weighted fair orbit queues

On the Influence of High Priority Customers on a Generalized Processor Sharing Queue

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