Constructive Computation in Stochastic Models with Applications

Li, Quan‐Lin

doi:10.1007/978-3-642-11492-2

Cited by 68 publications

(116 citation statements)

References 198 publications

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“…We take some more efficient approaches to get these stationary results. On the one hand, based on the algorithmic method proposed by Li (2010), a simpler UL-type RG-factorization has been employed to find the distribution of the queue length at an arbitrary epoch. On the other hand, in order to develop an optimal management policy for the system, the expected length of the regeneration cycle is investigated by using an extended definition of the busy period that might be initiated with multiple customers.…”

Section: Discussionmentioning

confidence: 99%

“…However, the direct solving is time and resource consuming and hence can only be used for the low dimensional system. In order to efficiently solve the above system of linear algebraic equations, we use the UL-type RG-factorization, given in Li (2010), to find the solution of Eq. (1).…”

Section: Queueing Model Descriptionmentioning

confidence: 99%

“…Refer to the Subsection 2.6.3 of Li (2010), the stationary probability vectors of the DTMC are given by…”

Section: Queueing Model Descriptionmentioning

confidence: 99%

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Some analysis results associated with the optimization problem for a discrete-time finite-buffer NT-policy queue

Alfa

2015

Oper Res Int J

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The prime objective of this paperis to give some analysis results concerning the discrete-time finite-buffer NT-policy queue, which can be utilized to determine the optimal threshold values. By recording the waiting time of the leading customer in server's vacation period, the model is successfully described as a vectorvalued Markov chain. Meanwhile, depending on the special block structure of the one-step transition probability matrix, the equilibrium queue length distri-bution is calculated through a more effective UL-type RG-factorization. Due to the number of customers served in the busy period does not have the structure of a Galton-Watson branching process, analysis of the regeneration cycle is regarded as a difficult problem in establishing the cost structure of the queueing system. However, employing the concept of i-busy period and some difference equation solving skills, the explicit expression for the expected length of the regeneration cycle is easily derived, and the stochastic decomposition structure of the busy period is also demonstrated. Finally, numerical results are offered to illustrate how the direct search method can be implemented to obtain the optimal management policy.

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Section: Discussionmentioning

confidence: 99%

Section: Queueing Model Descriptionmentioning

confidence: 99%

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Some analysis results associated with the optimization problem for a discrete-time finite-buffer NT-policy queue

Alfa

2015

Oper Res Int J

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“…Comparing with the results given in Li and Lui [16] and Li [12], this paper provides more necessary phase-level probability analysis in setting up the infinitedimensional system of differential vector equations, which leads some new results and methodologies in the study of block-structured supermarket models. Note that the PH distributions constitute a versatile class of distributions that can approximate arbitrarily closely any probability distribution defined on the nonnegative real line, and the MAPs are a broad class of renewal or non-renewal point processes that can approximate arbitrarily closely any stochastic counting process (e.g., see Neuts [27,28] and Li [11] for more details), thus the results of this paper are a key advance of those given in Mitzenmacher [23] and Vvedenskaya et al [32] under the Poisson and exponential setting.…”

Section: Introductionmentioning

confidence: 99%

Block-structured supermarket models

Lui

2014

Discrete Event Dyn Syst

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Supermarket models are a class of parallel queueing networks with an adaptive control scheme that play a key role in the study of resource management of, such as, computer networks, manufacturing systems and transportation networks. When the arrival processes are non-Poisson and the service times are non-exponential, analysis of such a supermarket model is always limited, interesting, and challenging. This paper describes a supermarket model with non-Poisson inputs: Markovian Arrival Processes (MAPs) and with non-exponential service times: Phase-type (PH) distributions, and provides a generalized matrix-analytic method which is first combined with the operator semigroup and the mean-field limit. When discussing such a more general supermarket model, this paper makes some new results and advances as follows: (1) Providing a detailed probability analysis for setting up an infinitedimensional system of differential vector equations satisfied by the expected fraction vector, where the invariance of environment factors is given as an important result.(2) Introducing the phase-type structure to the operator semigroup and to the meanfield limit, and a Lipschitz condition can be obtained by means of a unified matrixdifferential algorithm. (3) The matrix-analytic method is used to compute the fixed point which leads to performance computation of this system. Finally, we use some * The main results of this paper will be published in "Discrete Event Dynamic Systems" 2014. On the other hand, the three appendices are the online supplementary material for this paper published in "Discrete Event Dynamic Systems" 2014 1 numerical examples to illustrate how the performance measures of this supermarket model depend on the non-Poisson inputs and on the non-exponential service times.Thus the results of this paper give new highlight on understanding influence of nonPoisson inputs and of non-exponential service times on performance measures of more general supermarket models.

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“…It is achieved by considering a Markov reward process (cf. Li, 2010) defined on an absorbing continuous time Markov chain (Horváth and Telek, 2007).…”

Section: Some Generalizationsmentioning

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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Constructive Computation in Stochastic Models with Applications

Cited by 68 publications

References 198 publications

Some analysis results associated with the optimization problem for a discrete-time finite-buffer NT-policy queue

Some analysis results associated with the optimization problem for a discrete-time finite-buffer NT-policy queue

Block-structured supermarket models

Modelling heavy-tails with phase-type scale mixture distributions

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