Regularly varying tail of the waiting time distribution in M/G/1 retrial queue

Kim, Jerim; Kim, Jeongsim; Kim, Bara

doi:10.1007/s11134-010-9180-3

Cited by 15 publications

(12 citation statements)

References 16 publications

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“…Multiplying (8) by z − 1, letting z tend to 1, and using (11) and (12), we have assertion (9). In a similar manner, we can obtain (10).…”

Section: Lemmamentioning

confidence: 74%

“…Kim et al [10] showed that the tail of the queue size distribution is asymptotically given by a geometric function multiplied by a power function. On the other hand, Shang et al [16] and Kim and Kim [9] studied heavy-tailed asymptotics in the M/G/1 retrial queue. Shang et al [16] showed that the stationary distribution of the queue size in the M/G/1 retrial queue is subexponential if the stationary distribution of the queue size in the corresponding ordinary M/G/1 queue is subexponential.…”

Section: Introductionmentioning

confidence: 99%

“…As a corollary of this property, they proved that the stationary distribution of the queue size has a regularly varying tail if the service time distribution has a regularly varying tail. Kim and Kim [9] showed that if the service time distribution has a regularly varying tail of index −α, 1 < α < 2, in the M/G/1 retrial queue, then the waiting time distribution has a regularly varying tail of index 1 − α.…”

Section: Introductionmentioning

confidence: 99%

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Tail asymptotics for the queue size distribution in the MAP/G/1 retrial queue

Kim

2010

Queueing Syst

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We consider a MAP/G/1 retrial queue where the service time distribution has a finite exponential moment. We derive matrix differential equations for the vector probability generating functions of the stationary queue size distributions. Using these equations, Perron-Frobenius theory, and the Karamata Tauberian theorem, we obtain the tail asymptotics of the queue size distribution. The main result on light-tailed asymptotics is an extension of the result in Kim et al. (J. Appl. Probab. 44:1111-1118, 2007 on the M/G/1 retrial queue.

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“…Multiplying (8) by z − 1, letting z tend to 1, and using (11) and (12), we have assertion (9). In a similar manner, we can obtain (10).…”

Section: Lemmamentioning

confidence: 74%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Tail asymptotics for the queue size distribution in the MAP/G/1 retrial queue

Kim

2010

Queueing Syst

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“…Recently, Kim et al [13] generalized the results in [10] to the MAP/G/1 retrial queue. Shang et al [16] and Kim and Kim [11] studied heavy-tailed asymptotics in the M/G/1 retrial queue. Shang et al [16] showed that the stationary distribution of the queue size in the M/G/1 retrial queue is subexponential if the stationary distribution of the queue size in the corresponding ordinary M/G/1 queue is subexponential.…”

Section: Introductionmentioning

confidence: 99%

Queue size distribution in a discrete-time D-BMAP/G/1 retrial queue

Kim

2010

Computers & Operations Research

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“…Heath et al, 1998;Mikosch et al, 2002),queueing theory (cf. Cohen, 1973;Frenk, 1982;Kim et al, 2010;Olvera-Cravioto et al, 2010) and approximation of the ruin probabilities (cf. Gaier and Grandits, 2002;Klüppelberg and Stadtmüller, 1998).…”

Section: Heavy-tailed Distributionsmentioning

confidence: 99%

Modelling heavy-tails with phase-type scale mixture distributions

Xie¹

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In this thesis, we present the new results found in exploring the asymptotic tail probability of the phase-type scale mixture distributions, and apply a subclass of the phase-type scale mixtures, the class of Erlangized scale mixture distributions, to the approximation of the probability of ruin.We consider the class of phase-type scale mixture distributions, which can be written as MellinStieltjes convolution Π G of a phase-type distribution G and a nonnegative but otherwise arbitrary distribution Π. Such a class can also be seen as the class of distributions of the product of two independent random variables: a phase-type random variable Y ∼ G and a nonnegative random variable S ∼ Π. We call S the scaling random variable and its corresponding distribution Π the scaling distribution. We investigate conditions for such a class of distributions to be either light-or heavy-tailed, we explore subexponentiality and determine their maximum domains of attraction.In this thesis, particular focus is on phase-type scale mixture distributions where the scaling distribution has discrete support -such a class has been recently used in risk applications to approximate heavy-tailed distributions. We extend an existing scheme for numerically calculating the probability of ruin of a classical Cramér-Lundberg reserve process having absolutely continuous but otherwise general claim size distributions. We employ a subclass of the phasetype scale mixtures, which is also a dense class of distributions that we denominate Erlangized scale mixtures (ESM). Such a class corresponds to nonnegative and absolutely continuous distributions which can be written as a Mellin-Stieltjes convolution Π G m of a discrete nonnegative distribution Π with an Erlang distribution G m . A distinctive feature of such a class is that it contains heavy-tailed distributions when the scaling distributions have unbounded support, and it provides sharp approximations to heavy-tailed distributions.We suggest a simple methodology for constructing a sequence of distributions having the form Π G m with the purpose of approximating the integrated tail distribution of the claim sizes.Then we adapt a recent result which delivers an explicit expression for the probability of ruin in the case that the claim size distribution is modelled as an Erlangized scale mixture. We provide simplified expressions for the approximation of the probability of ruin and construct explicit bounds for the error of approximation. We complement our results with a classical example where the claim sizes are heavy-tailed.ii

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Regularly varying tail of the waiting time distribution in M/G/1 retrial queue

Cited by 15 publications

References 16 publications

Tail asymptotics for the queue size distribution in the MAP/G/1 retrial queue

Tail asymptotics for the queue size distribution in the MAP/G/1 retrial queue

Queue size distribution in a discrete-time D-BMAP/G/1 retrial queue

Modelling heavy-tails with phase-type scale mixture distributions

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