On the dynamics of a finite buffer queue conditioned on the amount of loss

Zhang, Xiaowei

doi:10.1007/s11134-010-9204-z

Cited by 7 publications

(17 citation statements)

References 20 publications

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“…On the other hand, since V is regenerative and [M, M](t) = a 2 t , we may conclude directly from results for regenerative processes (this is precisely what the authors do in [22]) that …”

Section: But J~t' (V (S) Dm(s) Is a Local Martingale With Quadratic Vsupporting

confidence: 59%

“…Zhang and Glynn [22] contains the corresponding investigation for reflected Brownian motion. We provide a law of large numbers and a central limit theorem for U.…”

Section: A Few Asymptotic Results For a Continuous Inputmentioning

confidence: 99%

“…The idea of investigating the differential operator £, comes from [22]. Example 1 below contains a short discussion about the differences between the techniques used in [22] and this paper.…”

Section: S»d[m M](s) =It !(V(s)) Dm(s) + !(O)l(t)f'(b)u(t) + It{~fmentioning

confidence: 99%

“…This example was treated in [22]. Suppose that we take M(t) = aB(t), so that X(t) = aB(t) + ut , i.e.…”

Section: But J~t' (V (S) Dm(s) Is a Local Martingale With Quadratic Vmentioning

confidence: 99%

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Local martingales with two reflecting barriers

Pihlsgård

2015

J. Appl. Probab.

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We give an account of the characteristics that result from reflecting a drifting local martingale (Le. the sum of a local martingale and a multiple of its quadratic variation process) in 0 and b > O. We present conditions which guarantee the existence of finite moments of what is required to keep the reflected process within its boundaries. Also, we derive an associated law of large numbers and a central limit theorem which apply when the input is continuous. Similar results for integrals of the paths of the reflected process are also presented. These results are in close agreement to what has previously been shown for Brownian motion.

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“…On the other hand, since V is regenerative and [M, M](t) = a 2 t , we may conclude directly from results for regenerative processes (this is precisely what the authors do in [22]) that …”

Section: But J~t' (V (S) Dm(s) Is a Local Martingale With Quadratic Vsupporting

confidence: 59%

“…Zhang and Glynn [22] contains the corresponding investigation for reflected Brownian motion. We provide a law of large numbers and a central limit theorem for U.…”

Section: A Few Asymptotic Results For a Continuous Inputmentioning

confidence: 99%

Section: S»d[m M](s) =It !(V(s)) Dm(s) + !(O)l(t)f'(b)u(t) + It{~fmentioning

confidence: 99%

“…This example was treated in [22]. Suppose that we take M(t) = aB(t), so that X(t) = aB(t) + ut , i.e.…”

Section: But J~t' (V (S) Dm(s) Is a Local Martingale With Quadratic Vmentioning

confidence: 99%

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Local martingales with two reflecting barriers

Pihlsgård

2015

J. Appl. Probab.

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“…The second part of the paper focuses on DROU, with emphasis on properties of the loss process U t (and also the idleness process L t ), for t large. Zhang and Glynn's martingale approach, as developed in Zhang and Glynn (2011), is employed to tackle a problem of this type. With h(Á) being a twice continuously differentiable real function, we apply Itô's formula on h(Z t ) and require h(Á) to satisfy certain ordinary differential equations (ODEs) and specific initial and boundary conditions in order to construct martingales related to U t and L t .…”

Section: Introductionmentioning

confidence: 99%

Limit theorems for reflected Ornstein–Uhlenbeck processes

Huang

Mandjes

Spreij

2014

Statistica Neerlandica

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This paper studies one-dimensional Ornstein-Uhlenbeck processes, with the distinguishing feature that they are reflected on a single boundary (put at level 0) or two boundaries (put at levels 0 and d>0). In the literature they are referred to as reflected OU (ROU) and doubly-reflected OU (DROU) respectively. For both cases, we explicitly determine the decay rates of the (transient) probability to reach a given extreme level. The methodology relies on sample-path large deviations, so that we also identify the associated most likely paths. For DROU, we also consider the `idleness process' $L_t$ and the `loss process' $U_t$, which are the minimal nondecreasing processes which make the OU process remain $\geqslant 0$ and $\leqslant d$, respectively. We derive central limit theorems for $U_t$ and $L_t$, using techniques from stochastic integration and the martingale central limit theorem

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Central Limit Theorems and Large Deviations for Additive Functionals of Reflecting Diffusion Processes

Wang

2015

Asymptotic Laws and Methods in Stochastics

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This paper develops central limit theorems (CLT's) and large deviations results for additive functionals associated with reflecting diffusions in which the functional may include a term associated with the cumulative amount of boundary reflection that has occurred. Extending the known central limit and large deviations theory for Markov processes to include additive functionals that incorporate boundary reflection is important in many applications settings in which reflecting diffusions arise, including queueing theory and economics. In particular, the paper establishes the partial differential equations that must be solved in order to explicitly compute the mean and variance for the CLT, as well as the associated rate function for the large deviations principle.

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On the dynamics of a finite buffer queue conditioned on the amount of loss

Cited by 7 publications

References 20 publications

Local martingales with two reflecting barriers

Local martingales with two reflecting barriers

Limit theorems for reflected Ornstein–Uhlenbeck processes

Central Limit Theorems and Large Deviations for Additive Functionals of Reflecting Diffusion Processes

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