Renewal theory. Queueing theory

Some reflections on the Renewal-theory paradox in queueing theory

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s104895339800029x ◽

1998 ◽

Vol 11 (3) ◽

pp. 355-368 ◽

Cited By ~ 3

Author(s):

Robert B. Cooper ◽

Shun-Chen Niu ◽

Mandyam M. Srinivasan

Keyword(s):

Waiting Time ◽

Queueing Theory ◽

Waiting Times ◽

Renewal Theory ◽

Vacation Models ◽

Polling Models ◽

Mean Waiting Time ◽

The Mean ◽

Expository Paper ◽

Set Up

The classical renewal-theory (waiting time, or inspection) paradox states that the length of the renewal interval that covers a randomly-selected time epoch tends to be longer than an ordinary renewal interval. This paradox manifests itself in numerous interesting ways in queueing theory, a prime example being the celebrated Pollaczek-Khintchine formula for the mean waiting time in the M/G/1 queue. In this expository paper, we give intuitive arguments that explain why the renewal-theory paradox is ubiquitous in queueing theory, and why it sometimes produces anomalous results. In particular, we use these intuitive arguments to explain decomposition in vacation models, and to derive formulas that describe some recently-discovered counterintuitive results for polling models, such as the reduction of waiting times as a consequence of forcing the server to set up even when no work is waiting.

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On the non-closure under convolution of the subexponential family

Journal of Applied Probability ◽

10.1017/s0021900200041796 ◽

1989 ◽

Vol 26 (01) ◽

pp. 58-66 ◽

Cited By ~ 5

Author(s):

J. R. Leslie

Keyword(s):

Queueing Theory ◽

Branching Processes ◽

Renewal Theory ◽

Random Variable ◽

Infinite Divisibility ◽

Family Of Distributions

A distribution F of a non-negative random variable belongs to the subexponential family of distributions S if 1 – F (2)(x) ~ 2(1 – F(x)) as x →∞. This family is of considerable interest in branching processes, queueing theory, transient renewal theory and infinite divisibility theory. Much is known about the kind of distributions that belong to S but the question of whether S is closed under convolution has remained unresolved for some time. This paper contains an example which demonstrates that S is not in fact closed.

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Recursive estimation of distributional fix-points

Journal of Applied Probability ◽

10.1017/s0021900200015266 ◽

2000 ◽

Vol 37 (01) ◽

pp. 73-87

Author(s):

Paul Embrechts ◽

Harro Walk

Keyword(s):

Stochastic Process ◽

Distribution Function ◽

Stochastic Approximation ◽

Stochastic Models ◽

Queueing Theory ◽

Renewal Theory ◽

Random Variable ◽

Recursive Estimation ◽

Real Random Variable ◽

Index Space

In various stochastic models the random equation of implicit renewal theory appears where the real random variable S and the stochastic process Ψ with index space and state space R are independent. By use of stochastic approximation the distribution function of S is recursively estimated on the basis of independent or ergodic copies of Ψ. Under integrability assumptions almost sure L 1-convergence is proved. The choice of gains in the recursion is discussed. Applications are given to insurance mathematics (perpetuities) and queueing theory (stationary waiting and queueing times).

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Recursive estimation of distributional fix-points

Journal of Applied Probability ◽

10.1239/jap/1014842269 ◽

2000 ◽

Vol 37 (1) ◽

pp. 73-87

Author(s):

Paul Embrechts ◽

Harro Walk

Keyword(s):

Stochastic Process ◽

Distribution Function ◽

Stochastic Approximation ◽

Stochastic Models ◽

Queueing Theory ◽

Renewal Theory ◽

Random Variable ◽

Recursive Estimation ◽

Real Random Variable ◽

Index Space

In various stochastic models the random equation of implicit renewal theory appears where the real random variable S and the stochastic process Ψ with index space and state space R are independent. By use of stochastic approximation the distribution function of S is recursively estimated on the basis of independent or ergodic copies of Ψ. Under integrability assumptions almost sure L1-convergence is proved. The choice of gains in the recursion is discussed. Applications are given to insurance mathematics (perpetuities) and queueing theory (stationary waiting and queueing times).

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On the non-closure under convolution of the subexponential family

Journal of Applied Probability ◽

10.2307/3214316 ◽

1989 ◽

Vol 26 (1) ◽

pp. 58-66 ◽

Cited By ~ 41

Author(s):

J. R. Leslie

Keyword(s):

Queueing Theory ◽

Branching Processes ◽

Renewal Theory ◽

Random Variable ◽

Infinite Divisibility ◽

Family Of Distributions

A distribution F of a non-negative random variable belongs to the subexponential family of distributions S if 1 – F(2)(x) ~ 2(1 – F(x)) as x →∞. This family is of considerable interest in branching processes, queueing theory, transient renewal theory and infinite divisibility theory. Much is known about the kind of distributions that belong to S but the question of whether S is closed under convolution has remained unresolved for some time. This paper contains an example which demonstrates that S is not in fact closed.

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Queueing theory

AccessScience ◽

10.1036/1097-8542.564700 ◽

2015 ◽

Keyword(s):

Queueing Theory

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The Approximation of Sum of Independent Distributions by the Erlang Distribution Function

Nonlinear Analysis Modelling and Control ◽

10.15388/na.2002.7.1.15202 ◽

2002 ◽

Vol 7 (1) ◽

pp. 55-60 ◽

Cited By ~ 1

Author(s):

Antanas Karoblis

Keyword(s):

Distribution Function ◽

Exponential Distribution ◽

Queueing Theory ◽

Erlang Distribution ◽

Estimation Of Error

The exponential distribution and the Erlang distribution function are been used in numerous areas of mathematics, and specifically in the queueing theory. Such and similar applications emphasize the importance of estimation of error of approximation by the Erlang distribution function. The article gives an analysis and technique of error’s estimation of an accuracy of such approximation, especially in some specific cases.

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