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14
Optimal control of a large dam
 J. Appl. Prob
, 2007
"... Abstract. A large dam model is an object of study of this paper. The parameters L lower and L upper are its lower and upper levels, L = L upper − L lower is large, and if a current level of water is between these bounds, then the dam is assumed to be in normal state. Passage one or other bound leads ..."
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Abstract. A large dam model is an object of study of this paper. The parameters L lower and L upper are its lower and upper levels, L = L upper − L lower is large, and if a current level of water is between these bounds, then the dam is assumed to be in normal state. Passage one or other bound leads to damage. It is assumed that input stream of water is described by a Poisson process, while the output stream is statedependent (the exact formulation of the problem is given in the paper). Let Lt denote the dam level at time t, and let p1 = limt→ ∞ P{Lt = L lower}, p2 = limt→ ∞ P{Lt> L upper} exist. Then the expected longrun damage J = p1J1+p2J2 for the long time interval T proportional to L (J1 and J2 are the corresponding damage costs per time T associated with passage the bounds) is a performance measure, and the aim of the paper is to choose the parameter of output stream (exactly specified in the paper) minimizing J. 1.
Asymptotic analysis of loss probabilities in GI/M/m/n queueing systems as n increases to infinity
 Qual. Technol. Quantit. Manag
, 2007
"... Abstract. The paper studies asymptotic behavior of the loss probability for the GI/M/m/n queueing system as n increases to infinity. The approach of the paper is based on applications of classic results of Takács (1967) and the Tauberian theorem with remainder of Postnikov (19791980) associated wit ..."
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Abstract. The paper studies asymptotic behavior of the loss probability for the GI/M/m/n queueing system as n increases to infinity. The approach of the paper is based on applications of classic results of Takács (1967) and the Tauberian theorem with remainder of Postnikov (19791980) associated with the recurrence relation of convolution type. The main result of the paper is associated with asymptotic behavior of the loss probability. Specifically it is shown that in some cases (precisely described in the paper) where the load of the system approaches 1 from the left and n increases to infinity, the loss probability of the GI/M/m/n queue becomes asymptotically independent of the parameter m.
On The Effects Of The Packet Size Distribution On The Packet Loss Process
, 2006
"... Realtime multimedia applications have to use forward error correction (FEC) and error concealment techniques to cope with losses in today's besteffort Internet. The efficiency ..."
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Cited by 5 (0 self)
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Realtime multimedia applications have to use forward error correction (FEC) and error concealment techniques to cope with losses in today's besteffort Internet. The efficiency
Classes of probability distributions and their applications. arXiv/0804.2310
"... Abstract. The aim of this paper is a nontrivial application of certain classes of probability distribution functions with further establishing the bounds for the least root of the functional equation x = b G(µ − µx), where b G(s) is the LaplaceStieltjes transform of an unknown probability distribut ..."
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Abstract. The aim of this paper is a nontrivial application of certain classes of probability distribution functions with further establishing the bounds for the least root of the functional equation x = b G(µ − µx), where b G(s) is the LaplaceStieltjes transform of an unknown probability distribution function G(x) of a positive random variable having the first two moments g1 and g2, and µ is a positive parameter satisfying the condition µg1> 1. The additional information characterizing G(x) is that it belongs to the special class of distributions such that the difference between two elements of that class in the Kolmogorov (uniform) metric is not greater than κ. The obtained result is then used to establish the lower and upper bounds for loss probabilities in certain loss queueing systems with large buffers as well as continuity theorems in large M/M/1/n queueing systems. 1.
BOUNDS FOR THE LOSS PROBABILITIES OF LARGE LOSS QUEUEING SYSTEMS
, 804
"... Abstract. The aim of this paper is to establish the bounds for the least root of the functional equation x = b G(µ − µx), where b G(s) is the LaplaceStieltjes transform of an unknown probability distribution function G(x) of a positive random variable having the first two moments g1 and g2, and µ i ..."
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Abstract. The aim of this paper is to establish the bounds for the least root of the functional equation x = b G(µ − µx), where b G(s) is the LaplaceStieltjes transform of an unknown probability distribution function G(x) of a positive random variable having the first two moments g1 and g2, and µ is a positive parameter satisfying the condition µg1> 1. The additional information characterizing G(x) is an empirical probability distribution function Gemp(x), and it is assumed that the distance in the uniform (Kolmogorov) metric between G(x) and Gemp(x) is not greater than κ. The obtained bounds for the positive least root of the functional equation x = b G(µ − µx) are then used to find the asymptotic bounds for the loss probabilities in certain queueing systems with a large number of waiting places, when only an empirical probability distribution function of an interarrival or service time is known, as well as to study the continuity of the loss probabilities in M/M/1/n queueing systems when n is large. 1.
Optimal control of a large dam, taking into account the water costs. arXiv:math
, 2007
"... Abstract. Consider a dam model, Lupper and Llower are upper and, respectively, lower levels, L = Lupper − Llower is large and if the level of water is between these bounds, then the dam is said to be in a normal state. Passage across lower or upper levels leads to damage. Let J1 = j1L and J2 = j2L d ..."
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Abstract. Consider a dam model, Lupper and Llower are upper and, respectively, lower levels, L = Lupper − Llower is large and if the level of water is between these bounds, then the dam is said to be in a normal state. Passage across lower or upper levels leads to damage. Let J1 = j1L and J2 = j2L denote the damage costs per time unit of crossing the lower and, correspondingly, upper level where j1 and j2 are given constants. It is assumed that input stream of water is described by a Poisson process, while the output stream is state dependent. Let Lt denote the level of water in time t, and cLt denote the water cost at level Lt (Llower < Lt ≤ Lupper). Assuming that p1 = limt→ ∞ P{Lt = Llower}, p2 = limt→ ∞ P{Lt> Lupper} and qi = limt→ ∞ P{Lt = i} (Llower < i ≤ Lupper) exist, the aim of the paper is to choose the parameters of an output stream (specifically defined in the paper) minimizing the longrun expenses J = p1J1 + p2J2 + L upper X i=L lower +1 qici. The more particular problem, not taking into account the water costs, has been recently studied in [Abramov, J. Appl. Prob., 44 (2007), 249258]. The present paper partially answers the question: How does the structure of water costs affect the optimal solution? 1.
CERTAIN CLASSES OF PROBABILITY DISTRIBUTIONS AND THEIR APPLICATIONS TO QUEUEING PROBLEMS
, 804
"... Abstract. The aim of this paper is a nontrivial application of certain classes of probability distribution functions with further establishing the bounds for the least root of the functional equation x = b G(µ − µx) (or similar functional equations appearing in queueing problems), where bG(s) is the ..."
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Abstract. The aim of this paper is a nontrivial application of certain classes of probability distribution functions with further establishing the bounds for the least root of the functional equation x = b G(µ − µx) (or similar functional equations appearing in queueing problems), where bG(s) is the LaplaceStieltjes transform of an unknown probability distribution function G(x) of a positive random variable having the first two moments g1 and g2, and µ is a positive parameter satisfying the condition µg1> 1. The additional information characterizing G(x) is that it belongs to the special class of distributions such that the difference between two elements of that class in Kolmogorov’s metric is not greater than κ. The obtained result is then used to establish the lower and upper bounds for loss probabilities and continuity theorems in certain loss queueing systems with large buffers. 1.
Continuity theorems for M/M/1/n queueing systems
 Queueing Syst
, 2008
"... Abstract. In this paper continuity theorems are established for the number of losses during a busy period of the M/M/1/n queue, when the service time probability distribution, slightly different in certain sense from the exponential distribution, is approximated by that exponential distribution. Con ..."
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Abstract. In this paper continuity theorems are established for the number of losses during a busy period of the M/M/1/n queue, when the service time probability distribution, slightly different in certain sense from the exponential distribution, is approximated by that exponential distribution. Continuity theorems are obtained in the form of one or twoside stochastic inequalities. The paper shows how the bounds of these inequalities are changed if one or other assumption, associated with specific properties of the service time distribution (precisely described in the paper), is done. Specifically, some parametric families of service time distributions are discussed, and the paper establishes uniform estimations (given for all possible values of the parameter) and local estimations (where the parameter is fixed and takes only the given value). The analysis of the paper is based on the level crossing approach and some characterization properties of exponential distribution. 1.
DOI 10.1007/s1113400890767 Continuity theorems for the M/M/1/n queueing system
"... this paper continuity theorems are established for the number of losses during a busy period of the M/M/1/n queue. We consider an M/GI/1/n queueing system where the service time probability distribution, slightly different in a certain sense from the exponential distribution, is approximated by that ..."
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this paper continuity theorems are established for the number of losses during a busy period of the M/M/1/n queue. We consider an M/GI/1/n queueing system where the service time probability distribution, slightly different in a certain sense from the exponential distribution, is approximated by that exponential distribution. Continuity theorems are obtained in the form of one or twosided stochastic inequalities. The paper shows how the bounds of these inequalities are changed if further assumptions, associated with specific properties of the service time distribution (precisely described in the paper), are made. Specifically, some parametric families of service time distributions are discussed, and the paper establishes uniform estimates (given for all possible values of the parameter) and local estimates (where the parameter is fixed and takes only the given value). The analysis of the paper is based on the level crossing approach and some characterization properties of the exponential distribution.
© Applied Probability Trust 2007
"... A large dam model is the object of study of this paper. The parameters Llower and Lupper define its lower and upper levels, L = Lupper − Llower is large, and if the current level of water is between these bounds, the dam is assumed to be in a normal state. Passage across one or other of the levels l ..."
Abstract
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A large dam model is the object of study of this paper. The parameters Llower and Lupper define its lower and upper levels, L = Lupper − Llower is large, and if the current level of water is between these bounds, the dam is assumed to be in a normal state. Passage across one or other of the levels leads to damage. Let J1 and J2 denote the damage costs of crossing the lower and, respectively, the upper levels. It is assumed that the input stream of water is described by a Poisson process, while the output stream is state dependent. Let Lt denote the dam level at time t, and let p1 = limt→ ∞ P{Lt = Llower} and p2 = limt→ ∞ P{Lt>Lupper} exist. The longrun average cost, J = p1J1 + p2J2, is a performance measure. The aim of the paper is to choose the parameter controlling the output stream so as to minimize J.