Exact waiting time and queue size distributions for equilibrium M/G/1 queues with Pareto service

Ramsay, Colin M.

doi:10.1007/s11134-007-9052-7

Cited by 20 publications

(23 citation statements)

References 21 publications

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“…Pareto claims, ρ = 0.80 φ = 1.5 example because exact calculations of the ruin probabilities are available in this case, due to [17,18]. We apply the calibration procedure descrubed in Section 4 and approximate the distribution of X as a suitable mixture of scaled versions of Y .…”

Section: Note Thatmentioning

confidence: 99%

“…In Tables 1 -4 we compare the obtained approximate ruin probability with the exact (for Pareto claims) numbers of [17,18]. The comparison is performed for a risk reserve process with unit premium rate in terms of the expected net claim amount per unit time, ρ, and the parameter φ of the Pareto distribution.…”

Section: Note Thatmentioning

confidence: 99%

“…This has turned out to be difficult. Practical computation algorithms have been developed for claim sizes with different versions of the pure Pareto distribution; see [17,18] and [1]. Our approach for calculating the ruin probability will apply to a wide range of levels of u and to a wide variety of claim size distributions.…”

Section: Introductionmentioning

confidence: 99%

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Calculation of ruin probabilities for a dense class of heavy tailed distributions

Bladt¹,

Nielsen

Samorodnitsky

2014

Scandinavian Actuarial Journal

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Abstract. In this paper we propose a class of infinite-dimensional phase-type distributions with finitely many parameters as models for heavy tailed distributions. The class of finite-dimensional distributions is dense in the class of distributions on the positive reals and may hence approximate any such distribution. We prove that formulas from renewal theory, and with a particular attention to ruin probabilities, which are true for common phase-type distributions also hold true for the infinite-dimensional case. We provide algorithms for calculating functionals of interest such as the renewal density and the ruin probability. It might be of interest to approximate a given heavy-tailed distribution of some other type by a distribution from the class of infinite-dimensional phase-type distributions and to this end we provide a calibration procedure which works for the approximation of distributions with a slowly varying tail. An example from risk theory, comparing ruin probabilities for a classical risk process with Pareto distributed claim sizes, is presented and exact known ruin probabilities for the Pareto case are compared to the ones obtained by approximating by an infinite-dimensional hyper-exponential distribution.

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Section: Note Thatmentioning

confidence: 99%

Section: Note Thatmentioning

confidence: 99%

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Calculation of ruin probabilities for a dense class of heavy tailed distributions

Bladt¹,

Nielsen

Samorodnitsky

2014

Scandinavian Actuarial Journal

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“…Then, we refer here to the case 1 < γ 2, where the Pareto distribution has finite mean and infinite variance. This entails a sort of paradox: the M/P/1 queue is stable {there exists the state probability distribution as well as the distribution of the delay [6]}, but its mean delay is infinite (actually, it does not exists); this is a very special case where the infinite mean delay does not imply the queue instability. Note that in these queuing systems with heavy-tailed distributions of the service time, the interest is not on the study of the mean delay, but rather on the queue overflow probability (case with finite rooms in the queue) on the basis of There are also problems in simulating M/P/1 queues as γ approaches 2 from above; the simulation can be extremely slow to approach the regime condition.…”

Section: 332mentioning

confidence: 99%

M/G/1 Queuing Theory and Applications

Giambene

2014

Queuing Theory and Telecommunications

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The M/G/1 theory is a powerful tool, generalizing the solution of Markovian queues to the case of general service time distributions. There are many applications of the M/G/1 theory in the field of telecommunications; for instance, it can be used to study the queuing of fixed-size packets to be transmitted on a given link (i.e., M/D/1 case). Moreover, this theory yields results which are compatible with the M/M/1 theory, based on birth-death Markov chains.In the M/G/1 theory, the arrival process is Poisson with mean arrival rate λ, but, in general, the service time is not exponentially distributed. Hence, the service process has a certain memory: if there is a request in service at a given instant, its residual service time has a distribution depending on the time elapsed since the beginning of its service. Let us refer to a generic instant t. The system is described by a two-dimensional state S(t), characterized as follows:• Number of requests in the system at instant t, n(t).• Elapsed time from the beginning of the service of the currently served request, τ(t). Note that in the Markovian M/M/1 case, the pdf of the residual service time does not depend on τ(t) because of the memoryless property of the exponential distribution.Hence, S(t) ¼ {n(t), τ(t)}. In order to characterize these queues, we study their behaviors at specific time instants ζ i where we obtain a mono-dimensional simplification of state S(ζ i ). The M/G/1 queue is studied at specific imbedding instants, where we obtain again a Markovian system; this is a so-called imbedded Markov chain [1,2]. Different alternatives are available to select instants ζ i . It is not requested that instants ζ i be equally spaced in time. Typical choices for ζ i instants are:The online version of this chapter (

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“…In [15] this integral representation was generalized to arbritrary (non-integer) m > 0, in [19] to a renewal risk model with Erlang interclaim times and in [16] to the case of compound shifted Pareto sums. In all these cases the used method is akin to the one used in [7] and [8] to determine the density function of the finite sum of certain Pareto variables.…”

Section: Introductionmentioning

confidence: 99%

On ruin probability and aggregate claim representations for Pareto claim size distributions

Albrecher

Kortschak

2009

Insurance: Mathematics and Economics

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We generalize an integral representation for the ruin probability in a Crámer-Lundberg risk model with shifted (or also called US-)Pareto claim sizes, obtained by Ramsay [14], to classical Pareto(a) claim size distributions with arbitrary real values a > 1 and derive its asymptotic expansion. Furthermore an integral representation for the tail of compound sums of Pareto-distributed claims is obtained and numerical illustrations of its performance in comparison to other aggregate claim approximations are provided.

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Exact waiting time and queue size distributions for equilibrium M/G/1 queues with Pareto service

Cited by 20 publications

References 21 publications

Calculation of ruin probabilities for a dense class of heavy tailed distributions

Calculation of ruin probabilities for a dense class of heavy tailed distributions

M/G/1 Queuing Theory and Applications

On ruin probability and aggregate claim representations for Pareto claim size distributions

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