On the application of Rouché's theorem in queueing theory

Adan, I.J.B.F.; Leeuwaarden, Johan S. H. van; Winands, E.M.M.

doi:10.1016/j.orl.2005.05.012

Cited by 59 publications

(114 citation statements)

References 13 publications

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“…In this case, it can be proved my means of Rouché's theorem (or by the generalized version proved in [4]) that (10) has exactly one solution inside and on the unit disk.…”

Section: The Probability Of An Empty Buffermentioning

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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“…In this case, it can be proved my means of Rouché's theorem (or by the generalized version proved in [4]) that (10) has exactly one solution inside and on the unit disk.…”

Section: The Probability Of An Empty Buffermentioning

confidence: 99%

Using Singularity Analysis to Approximate Transient Characteristics in Queueing Systems

Walraevens¹,

Fiems²,

Moeneclaey³

2009

Prob. Eng. Inf. Sci.

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“…First, the function p → p N − φ * (p)β * (p) N has N zeroes in B(0, 1), one of which equals 1 [1]. Second, on account of the analyticity property, the right-hand side of (28) vanishes at those zeroes.…”

Section: N Constantmentioning

confidence: 99%

A queueing system with vacations afterNservices

Boxma

Claeys

Gulikers

et al. 2015

Naval Research Logistics

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This paper is devoted to the study of an M/G/1 queue with a particular vacation discipline. The server is due to take a vacation as soon as it has served exactly N customers since the end of the previous vacation. N may be either a constant or a random variable. If the system becomes empty before the server has served N customers, then it stays idle until the next customer arrival. Such a vacation discipline arises, e.g., in production systems and in order picking in warehouses.We determine the joint transform of the length of a visit period and the number of customers in the system at the end of that period. We also derive the generating function of the number of customers at a random instant, and the Laplace-Stieltjes transform of the delay of a customer.

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“…By using a generalization of Rouché's theorem [1], we can prove that z 1 − A(z 1 , z 2 , z 3 ) has one zero in the unit disk of z 1 for an arbitrary z 2 and z 3 in the unit disk. We denote this zero by Y 2,3 (z 2 , z 3 ) and it is thus implicitly defined as…”

Section: Eliminatingmentioning

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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On the application of Rouché's theorem in queueing theory

Cited by 59 publications

References 13 publications

Using Singularity Analysis to Approximate Transient Characteristics in Queueing Systems

Using Singularity Analysis to Approximate Transient Characteristics in Queueing Systems

A queueing system with vacations afterNservices

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

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