138 results
Search Results
2. Comparison results for M/G/1 queues with waiting and sojourn time deadlines
- Author
-
Yoshiaki Inoue
- Subjects
Statistics and Probability ,Waiting time ,Discrete mathematics ,021103 operations research ,Service time ,General Mathematics ,0211 other engineering and technologies ,Comparison results ,02 engineering and technology ,01 natural sciences ,010104 statistics & probability ,M/G/1 queue ,0101 mathematics ,Statistics, Probability and Uncertainty ,Queue ,Mathematics - Abstract
This paper considers two variants of M/G/1 queues with impatient customers, which are denoted by M/G/1+Gw and M/G/1+Gs. In the M/G/1+Gw queue customers have deadlines for their waiting times, and they leave the system immediately if their services do not start before the expiration of their deadlines. On the other hand, in the M/G/1+Gs queue customers have deadlines for their sojourn times, where customers in service also immediately leave the system when their deadlines expire. In this paper we derive comparison results for performance measures of these models. In particular, we show that if the service time distribution is new better than used in expectation, then the loss probability in the M/G/1+Gs queue is greater than that in the M/G/1+Gw queue.
- Published
- 2019
3. Quasistochastic matrices and Markov renewal theory
- Author
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Gerold Alsmeyer
- Subjects
Statistics and Probability ,Markov kernel ,General Mathematics ,perpetuity ,01 natural sciences ,age-dependent multitype branching process ,010104 statistics & probability ,Matrix (mathematics) ,random difference equation ,60K05 ,Markov renewal process ,Quasistochastic matrix ,60J45 ,Nonnegative matrix ,Renewal theory ,Markov renewal equation ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematics ,Discrete mathematics ,Markov chain ,010102 general mathematics ,Stochastic matrix ,Stone-type decomposition ,60K15 ,Markov renewal theorem ,spread out ,60J10 ,Statistics, Probability and Uncertainty ,Markov random walk - Abstract
Let 𝓈 be a finite or countable set. Given a matrix F = (F ij ) i,j∈𝓈 of distribution functions on R and a quasistochastic matrix Q = (q ij ) i,j∈𝓈 , i.e. an irreducible nonnegative matrix with maximal eigenvalue 1 and associated unique (modulo scaling) positive left and right eigenvectors u and v, the matrix renewal measure ∑ n≥0 Q n ⊗ F *n associated with Q ⊗ F := (q ij F ij ) i,j∈𝓈 (see below for precise definitions) and a related Markov renewal equation are studied. This was done earlier by de Saporta (2003) and Sgibnev (2006, 2010) by drawing on potential theory, matrix-analytic methods, and Wiener-Hopf techniques. In this paper we describe a probabilistic approach which is quite different and starts from the observation that Q ⊗ F becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate Q ⊗ F to a Markov random walk {(M n , S n )} n≥0 with discrete recurrent driving chain {M n } n≥0. It is then shown that renewal theorems including a Choquet-Deny-type lemma may be easily established by resorting to standard renewal theory for ordinary random walks. The paper concludes with two typical examples.
- Published
- 2014
4. Asymptotic Bounds for the Distribution of the Sum of Dependent Random Variables
- Author
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Ruodu Wang
- Subjects
Statistics and Probability ,General Mathematics ,Structure (category theory) ,Value (computer science) ,91E30 ,01 natural sciences ,value at risk ,Combinatorics ,010104 statistics & probability ,0502 economics and business ,60E05 ,Limit (mathematics) ,0101 mathematics ,Mathematics ,Discrete mathematics ,050208 finance ,05 social sciences ,Expected shortfall ,Distribution (mathematics) ,Dependence bound ,complete mixability ,modeling uncertainty ,60E15 ,Marginal distribution ,Statistics, Probability and Uncertainty ,Random variable ,Value at risk - Abstract
Suppose that X 1, …, X n are random variables with the same known marginal distribution F but unknown dependence structure. In this paper we study the smallest possible value of P(X 1 + · · · + X n < s) over all possible dependence structures, denoted by m n,F (s). We show that m n,F (ns) → 0 for s no more than the mean of F under weak assumptions. We also derive a limit of m n,F (ns) for any s ∈ R with an error of at most n -1/6 for general continuous distributions. An application of our result to risk management confirms that the worst-case value at risk is asymptotically equivalent to the worst-case expected shortfall for risk aggregation with dependence uncertainty. In the last part of this paper we present a dual presentation of the theory of complete mixability and give dual proofs of theorems in the literature on this concept.
- Published
- 2014
5. Hazard rate ordering of order statistics and systems
- Author
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Moshe Shaked and Jorge Navarro
- Subjects
Statistics and Probability ,Discrete mathematics ,Reliability theory ,Series (mathematics) ,Multivariate random variable ,General Mathematics ,010102 general mathematics ,Order statistic ,Function (mathematics) ,01 natural sciences ,Stochastic ordering ,010104 statistics & probability ,Probability theory ,Statistics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Rate function ,Mathematics - Abstract
Let X = (X1, X2, …, Xn) be an exchangeable random vector, and write X(1:i) = min{X1, X2, …, Xi}, 1 ≤ i ≤ n. In this paper we obtain conditions under which X(1:i) decreases in i in the hazard rate order. A result involving more general (that is, not necessarily exchangeable) random vectors is also derived. These results are applied to obtain the limiting behaviour of the hazard rate function of the lifetimes of various coherent systems in reliability theory. The notions of the Samaniego signatures and the minimal signatures of such systems are extensively used in the paper. An interesting relationship between these two signatures is obtained. The results are illustrated in a series of examples.
- Published
- 2006
6. The classification of matrix GI/M/1-type Markov chains with a tree structure and its applications to queueing
- Author
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Qi-Ming He
- Subjects
Statistics and Probability ,Discrete mathematics ,Queueing theory ,Markov chain ,General Mathematics ,Variable-order Markov model ,010102 general mathematics ,01 natural sciences ,Continuous-time Markov chain ,010104 statistics & probability ,symbols.namesake ,Tree structure ,Matrix analytic method ,Jacobian matrix and determinant ,symbols ,Examples of Markov chains ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this paper, we study the classification of matrix GI/M/1-type Markov chains with a tree structure. We show that the Perron–Frobenius eigenvalue of a Jacobian matrix provides information for classifying these Markov chains. A fixed-point approach is utilized. A queueing application is presented to show the usefulness of the classification method developed in this paper.
- Published
- 2003
7. The delay distribution of a type k customer in a first-come-first-served MMAP[K]/PH[K]/1 queue
- Author
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Chris Blondia and B. Van Houdt
- Subjects
Discrete mathematics ,Statistics and Probability ,Queueing theory ,mmap ,M/G/k queue ,General Mathematics ,010102 general mathematics ,01 natural sciences ,010104 statistics & probability ,First-come, first-served ,Probability distribution ,Phase-type distribution ,Markovian arrival process ,0101 mathematics ,Statistics, Probability and Uncertainty ,Queue ,Algorithm ,Mathematics - Abstract
This paper presents an algorithmic procedure to calculate the delay distribution of a type k customer in a first-come-first-served (FCFS) discrete-time queueing system with multiple types of customers, where each type has different service requirements (the MMAP[K]/PH[K]/1 queue). First, we develop a procedure, using matrix analytical methods, to handle arrival processes that do not allow batch arrivals to occur. Next, we show that this technique can be generalized to arrival processes that do allow batch arrivals to occur. We end the paper by presenting some numerical examples.
- Published
- 2002
8. Approximate entropy for testing randomness
- Author
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Andrew L. Rukhin
- Subjects
Discrete mathematics ,Statistics and Probability ,Random number generation ,General Mathematics ,010102 general mathematics ,Approximate entropy ,Joint entropy ,01 natural sciences ,Rényi entropy ,010104 statistics & probability ,Maximum entropy probability distribution ,Randomness tests ,Statistical physics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Randomness ,Joint quantum entropy ,Mathematics - Abstract
This paper arose from interest in assessing the quality of random number generators. The problem of testing randomness of a string of binary bits produced by such a generator gained importance with the wide use of public key cryptography and the need for secure encryption algorithms. All such algorithms are based on a generator of (pseudo) random numbers; the testing of such generators for randomness became crucial for the communications industry where digital signatures and key management are vital for information processing. The concept of approximate entropy has been introduced in a series of papers by S. Pincus and co-authors. The corresponding statistic is designed to measure the degree of randomness of observed sequences. It is based on incremental contrasts of empirical entropies based on the frequencies of different patterns in the sequence. Sequences with large approximate entropy must have substantial fluctuation or irregularity. Alternatively, small values of this characteristic imply strong regularity, or lack of randomness, in a sequence. Pincus and Kalman (1997) evaluated approximate entropies for binary and decimal expansions of e, π, √2 and √3 with the surprising conclusion that the expansion of √3 demonstrated much less irregularity than that of π. Tractable small sample distributions are hardly available, and testing randomness is based, as a rule, on fairly long strings. Therefore, to have rigorous statistical tests of randomness based on this approximate entropy statistic, one needs the limiting distribution of this characteristic under the randomness assumption. Until now this distribution remained unknown and was thought to be difficult to obtain. To derive the limiting distribution of approximate entropy we modify its definition. It is shown that the approximate entropy as well as its modified version converges in distribution to a χ2-random variable. The P-values of approximate entropy test statistics for binary expansions of e, π and √3 are plotted. Although some of these values for √3 digits are small, they do not provide enough statistical significance against the randomness hypothesis.
- Published
- 2000
9. A correspondence between product-form batch-movement queueing networks and single-movement networks
- Author
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W. Henderson, J. L. Coleman, Peter G. Taylor, and Charles E. M. Pearce
- Subjects
Statistics and Probability ,Discrete mathematics ,Queueing theory ,Mathematical optimization ,General Mathematics ,010102 general mathematics ,BCMP network ,Loss network ,01 natural sciences ,010104 statistics & probability ,Evolving networks ,Product (mathematics) ,Jackson network ,Layered queueing network ,State space ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
A number of recent papers have exhibited classes of queueing networks, with batches of customers served and routed through the network, which have generalised product-form equilibrium distributions. In this paper we look at these from a new viewpoint. In particular we show that, under standard assumptions, for a network to possess an equilibrium distribution that factorises into a product form over the nodes of the network for all possible transition rates, it is necessary and sufficient that it be equivalent to a suitably-defined single-movement network. We consider also the form of the state space for such networks.
- Published
- 1997
10. On the extension of signature-based representations for coherent systems with dependent non-exchangeable components
- Author
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Jorge Navarro and Juan Fernández-Sánchez
- Subjects
Statistics and Probability ,Discrete mathematics ,Independent and identically distributed random variables ,Class (set theory) ,General Mathematics ,Reliability (computer networking) ,010102 general mathematics ,Copula (linguistics) ,Extension (predicate logic) ,01 natural sciences ,Signature (logic) ,010104 statistics & probability ,0101 mathematics ,Statistics, Probability and Uncertainty ,Representation (mathematics) ,Mathematics - Abstract
The signature representation shows that the reliability of the system is a mixture of the reliability functions of the k-out-of-n systems. The first representation was obtained for systems with independent and identically distributed (IID) components and after it was extended to exchangeable (EXC) components. The purpose of the present paper is to extend it to the class of systems with identically distributed (ID) components which have a diagonal-dependent copula. We prove that this class is much larger than the class with EXC components. This extension is used to compare systems with non-EXC components.
- Published
- 2020
11. The N-star network evolution model
- Author
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István Fazekas, Attila Perecsényi, and Csaba Noszály
- Subjects
Statistics and Probability ,Star network ,Random graph ,Discrete mathematics ,General Mathematics ,010102 general mathematics ,Joins ,Preferential attachment ,01 natural sciences ,Power law ,Doob–Meyer decomposition theorem ,010104 statistics & probability ,Asymptotic power ,0101 mathematics ,Statistics, Probability and Uncertainty ,Unit (ring theory) ,Mathematics - Abstract
A new network evolution model is introduced in this paper. The model is based on cooperations of N units. The units are the nodes of the network and the cooperations are indicated by directed links. At each evolution step N units cooperate, which formally means that they form a directed N-star subgraph. At each step either a new unit joins the network and it cooperates with N − 1 old units, or N old units cooperate. During the evolution both preferential attachment and uniform choice are applied. Asymptotic power law distributions are obtained both for in-degrees and for out-degrees.
- Published
- 2019
12. Multiple drawing multi-colour urns by stochastic approximation
- Author
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Olfa Selmi, Cécile Mailler, and Nabil Lasmar
- Subjects
Statistics and Probability ,Mathematics(all) ,General Mathematics ,Markov process ,Stochastic approximation ,01 natural sciences ,Multiple drawing Pólya urn ,010104 statistics & probability ,symbols.namesake ,Polya urn ,stochastic approximation ,FOS: Mathematics ,0101 mathematics ,Mathematics ,Discrete mathematics ,discrete-time martingale ,Probability (math.PR) ,010102 general mathematics ,Ball (bearing) ,symbols ,limit theorem ,Statistics, Probability and Uncertainty ,reinforced process ,Mathematics - Probability - Abstract
A classical P��lya urn scheme is a Markov process whose evolution is encoded by a replacement matrix $(R_{i,j})_{1\leq i,j\leq d}$. At every discrete time-step, we draw a ball uniformly at random, denote its colour $c$, and replace it in the urn together with $R_{c,j}$ balls of colour $j$ (for all $1\leq j\leq d$). We are interested in multi-drawing P��lya urns, where the replacement rule depends on the random drawing of a set of $m$ balls from the urn (with or without replacement). This generalisation has already been studied in the literature, in particular by Kuba & Mahmoud (ArXiv:1503.09069 and 1509.09053), where second order asymptotic results are proved for $2$-colour urns under the balanced and the affinity assumptions. The main idea of this work is to apply stochastic approximation methods to this problem, which enables us to remove the affinity hypothesis of Kuba & Mahmoud and generalise the result to more-than-two-colour urns. We also give some partial results in the two-colour non-balanced case., This new arxiv version (v6) corrects a mistake that we discovered in the previous versions of this paper (v1-5). The mistake was in Theorem 1$(a)$ and in the last sentence of Theorem 4. In this new version, Theorem 1$(a)$ has been corrected, and Theorem 4 has been deleted
- Published
- 2018
13. On the invariance principle for reversible Markov chains
- Author
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Sergey Utev and Magda Peligrad
- Subjects
Statistics and Probability ,60F05, 60F17, 60J05 ,Statistics::Theory ,Pure mathematics ,General Mathematics ,Markov chain ,Reversible process ,01 natural sciences ,Time reversibility ,Continuous-time Markov chain ,010104 statistics & probability ,60J05 ,Mathematics::Probability ,60F05 ,FOS: Mathematics ,0101 mathematics ,Mathematics ,Central limit theorem ,Discrete mathematics ,Markov chain mixing time ,functional central limit theorem ,Invariance principle ,Probability (math.PR) ,010102 general mathematics ,Statistics::Computation ,60F17 ,Statistics, Probability and Uncertainty ,Kolmogorov's criterion ,Mathematics - Probability - Abstract
In this paper, we investigate the functional central limit theorem for stochastic processes associated to partial sums of additive functionals of reversible Markov chains with general spate space, under the normalization standard deviation of partial sums. For this case, we show that the functional central limit theorem is equivalent to the fact that the variance of partial sums is regularly varying with exponent 1 and the partial sums satisfy the CLT. It is also equivalent to the conditional CLT., 10 pages
- Published
- 2016
14. Backward stochastic difference equations for dynamic convex risk measures on a binomial tree
- Author
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Tak Kuen Siu, Samuel N. Cohen, Robert J. Elliott, Elliott, Robert J, Siu, Tak Kuen, and Cohen, Samuel N
- Subjects
Statistics and Probability ,General Mathematics ,binomial tree ,dynamic convex risk measure ,Login ,Binomial theorem ,01 natural sciences ,stochastic distortion probability ,010104 statistics & probability ,Dynamic convex risk measure ,60H07 ,Probability theory ,91G80 ,0502 economics and business ,conditional nonlinear expectation ,0101 mathematics ,backward stochastic difference equation ,Link (knot theory) ,Mathematics ,Discrete mathematics ,050208 finance ,Stochastic difference equations ,05 social sciences ,Regular polygon ,EZproxy ,91G20 ,Binomial options pricing model ,Statistics, Probability and Uncertainty - Abstract
Using backward stochastic difference equations (BSDEs), this paper studies dynamic convex risk measures for risky positions in a simple discrete-time, binomial tree model. A relationship between BSDEs and dynamic convex risk measures is developed using nonlinear expectations. The time consistency of dynamic convex risk measures is discussed in the binomial tree framework. A relationship between prices and risks is also established. Two particular cases of dynamic convex risk measures, namely risk measures with stochastic distortions and entropic risk measures, and their mathematical properties are discussed. Refereed/Peer-reviewed
- Published
- 2015
15. A stochastic model for time lag in reporting of claims
- Author
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Jan-Erik Karlsson
- Subjects
Statistics and Probability ,Discrete mathematics ,General Mathematics ,010102 general mathematics ,Zero (complex analysis) ,Generating function ,Function (mathematics) ,01 natural sciences ,Point process ,010104 statistics & probability ,Interval (graph theory) ,Order (group theory) ,Renewal theory ,0101 mathematics ,Statistics, Probability and Uncertainty ,Random variable ,Mathematics - Abstract
We assume that the number of claims occur according to a renewal process and treat the number of claims that occur and are reported in a certain time interval as a renewal process with random displacements. We obtain a renewal equation for the mean value function and an integral equation for the Laplace transform of the distribution of the claims that are reported. We also give asymptotic expressions for the mean value function and calculate the generating function in the case where the renewal process is a Poisson process. This matter is a part of the IBNR-problem in insurance mathematics. INCURRED BUT NOT REPORTED CLAIMS; RENEWAL PROCESS WITH RANDOM DISPLACEMENTS ntroduction At a given point of time one has only partial knowledge of an insurance business. For instance, there is always a time lag between the occurrence of a road accident and the reporting of and full knowledge of the claim amount due to that accident. This lack of information generates many questions. One, which this paper touches, is: if we observe M(t) occurred and reported claims during the interval (0, t), how many claims occurred in that period? On the accounting date one has to answer this question and make an estimate of the number of claims which occurred during the last year. The starting point will be assumptions of the number of claims that occur in (0, t). Since the days of Filip Lundberg one has assumed that this is described by a Poisson process. Several authors have tried to generalize Lundberg's model. Without intending to make a complete list we mention Ove Lundberg, Sparre Andersen and Jan Grandell. We will pursue the course of Sparre Andersen who in 1957 presented a paper [1] where the epochs of claims form a renewal process. As to the distribution of the time lag between the occurrence of a claim and the report of that claim there are several possible ways. We will make the assumption that the lags are independent and equally distributed, which seems to be a reasonable assumption as a first approximation at least when the claim amounts are of a fairly equal size. Received in revised form 21 March 1973. 382 This content downloaded from 207.46.13.176 on Mon, 20 Jun 2016 05:40:22 UTC All use subject to http://about.jstor.org/terms A stochastic modelfor time lag in reporting of claims 383 A model We assume that Feller [3], ch. XI is known. Assume that claims occur according to a delayed renewal process on [0, oo). Number the claims 0, 1, 2, -.. in the order in which they occur. So is a stochastic variable representing the time between the starting point zero and the occurrence of the first claim and So has the distribution function Fo. T, is a stochastic variable representing the time between the occurrence of claim number n 1 and claim number n, n = 1,2, ---. The variables TI, T2, -" have the same non-arithmetic distribution function F, with F(O) = 0. Put P, = E T, and aF = var T. and assume that ao 1, This content downloaded from 207.46.13.176 on Mon, 20 Jun 2016 05:40:22 UTC All use subject to http://about.jstor.org/terms 384 JAN-ERIK KARLSSON A fundamental property of the renewal process is that N(t) > n + 1 if and only if S, n if and only if S, + X, . We omit the index Fo, if P(So = 0) = 1. Thus we have MFo(t) = I e,(t) 0 and VFo(t) = P((t) P((t) = 1) = (G*Fo*F*)(t) 0 0 or putting Go = G,*Fo VF,(t) = (Go * U) (t). Thus VFo(t) is the solution of the renewal equation VFo(t) = Go(t) + VFr(tx)F {dx}. Conditioning by So, we see that VFo(t) = (V* Fo)(t). We can also get the renewal equation for VFo(t) by the usual renewal reasoning: EMFo(t) = f E(MFo(t) lSo + X0 = S, So + T1 = y)Pso+xo{ds} Pso+r,{dy} where (0 if s > t, y > t, E(MF(t) So + Xo = s, S + T1 = ) = 1 if s t, y> t, EM(t y) if s > t, y t, 1 +EM(t y) if s ?t, y t. This content downloaded from 207.46.13.176 on Mon, 20 Jun 2016 05:40:22 UTC All use subject to http://about.jstor.org/terms A stochastic modelfor time lag in reporting of claims 385 This technique will be used later. According to the renewal theorem for a delayed renewal process, we have for h > 0 lim (VFo(t + h) VFo(t)) = h I/,. If we put z(t) = VFo(t) t /ip it can be shown that Z(t) satisfies Z(t) = z(t) + f (t x) F{dx} with z(t) = (1 F(y))dy (1 Go(t)). Since z(t) is directly integrable (we assume that Po = ESo All use subject to http://about.jstor.org/terms A stochastic model for time lag in reporting of claims 387 References [1] ANDERSEN, E. SPARRE (1957) On the collective theory risk in the case of contagion between claims. Transactions XVth International Congress of Actuaries, New York. Vol. II. [2] DALEY, D. J. AND VERE-JONES, D. (1972) A summary of the theory of point processes. Stochastic Point Processes: Statistical Analysis, Theory and Applications (P. A. W. Lewis, ed.), Wiley, New York. 299-383. [3] FELLER, W. (1970) An Introduction to Probability Theory and its Applications. Vol. 2, 2nd edition. Wiley, New York. [4] PRABHU, N. U. (1965) Queues and Inventories. Wiley, New York. [51 THEDtEN, T. (1967) On point .v.vtoms in R, under random motion. Doctor's thesis, Royal Institute of Technology, Stockholm. This content downloaded from 207.46.13.176 on Mon, 20 Jun 2016 05:40:22 UTC All use subject to http://about.jstor.org/terms
- Published
- 1974
16. Asymptotic inference for an Ising lattice III. Non-zero field and ferromagnetic states
- Author
-
David K. Pickard
- Subjects
Discrete mathematics ,Statistics and Probability ,General Mathematics ,010102 general mathematics ,Inference ,01 natural sciences ,Theoretical physics ,010104 statistics & probability ,Ferromagnetism ,Zero field ,Ising lattice ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In previous papers (1976), (1977) limit theorems were obtained for the classical Ising model in the absence of an external magnetic field, thereby providing a basis for asymptotic inference. The present paper extends these results to arbitrary external magnetic fields. Statistical inference for this model is important because its nearest-neighbour interactions provide a natural first approximation to spatial interaction among binary variables located on square lattices. The most interesting behaviour occurs in zero field and at or beyond the critical point. In this case, the central limit result for nearest-neighbour interactions requires an unusual norming, the limiting variances may depend on the nature of the boundary conditions, and there cannot be any central limit result for external magnetic field. The implications of these phenomena for statistical inference are also discussed. In particular, the maximum likelihood estimator of magnetic field is not consistent. Rather it appears to have a non-trivial asymptotic distribution.
- Published
- 1979
17. Multidimensional age-dependent branching processes allowing immigration: The limiting distribution
- Author
-
Norman Kaplan
- Subjects
Discrete mathematics ,Statistics and Probability ,media_common.quotation_subject ,General Mathematics ,Immigration ,010102 general mathematics ,Asymptotic distribution ,Age dependent ,Time model ,01 natural sciences ,Branching (linguistics) ,010104 statistics & probability ,Applied mathematics ,Renewal theorem ,0101 mathematics ,Statistics, Probability and Uncertainty ,media_common ,Mathematics - Abstract
This paper continues the author's study of age-dependent branching processes allowing immigration. In this paper the multidimensional case is considered. A sufficient condition is obtained for the existence of a legitimate limiting distribution. Several corollaries are obtained, which generalize many of the results of the discrete theory and those of the one-dimensional continuous time model.
- Published
- 1974
18. Approximation of the queue-length distribution of an M/GI/s queue by the basic equations
- Author
-
Masakiyo Miyazawa
- Subjects
Statistics and Probability ,Discrete mathematics ,M/G/k queue ,General Mathematics ,010102 general mathematics ,M/M/1 queue ,M/D/c queue ,G/G/1 queue ,01 natural sciences ,M/M/∞ queue ,010104 statistics & probability ,Burke's theorem ,M/G/1 queue ,M/M/c queue ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
We give a unified way of obtaining approximation formulas for the steady-state distribution of the queue length in theM/GI/squeue. The approximations of Hokstad (1978) and Case A of Tijms et al. (1981) are derived again. The main interest of this paper is in considering the theoretical meaning of the assumptions given in the literature. Having done this, we derive new approximation formulas. Our discussion is based on one version of the steady-state equations, called the basic equations in this paper. The basic equations are derived forM/GI/s/kwith finite and infinitek.Similar approximations are possible forM/GI/s/k(k< +∞).
- Published
- 1986
19. A multivariate IFR class
- Author
-
Thomas H. Savits
- Subjects
Discrete mathematics ,Statistics and Probability ,Class (set theory) ,Multivariate statistics ,Multivariate analysis ,Concave function ,General Mathematics ,010102 general mathematics ,Block (permutation group theory) ,Univariate ,Characterization (mathematics) ,01 natural sciences ,Mathematical theory ,010104 statistics & probability ,Calculus ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
Various univariate classes of life distributions have been introduced in the mathematical theory of reliability. Recently there has been much interest in obtaining multivariate versions of these classes. Although there have been many different approaches, this document mentions only two because of their nice closure properties: the multivariate IFRA class of Block and Savits (1980) and the multivariate NBU class of Marshall and Shaked (1982). Both of these classes are closed under deletion, conjunction, convolution and weak limits. This paper introduces a multivariate IFR class that has similiar closure properties. its spirit also closely parallels that found in the two previously mentioned papers. Other definitions of a multivariate IFR class have been proposed by Marshall (1975), but none possess all desirable closure properties. Section 2 presents some preliminary facts about log concave functions. Section 3 contains new characterization of the (univariate) IFR class. The multivariate genralization and properties thereof are given in Section 4. A useful alternative condition is delineated in Section 5. Finally, this class is compared with the multivariate IFRA class of Block and Savits (1980) in Section 6.
- Published
- 1985
20. Some explicit formulas and computational methods for infinite-server queues with phase-type arrivals
- Author
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Marcel F. Neuts and Vaidyanathan Ramaswami
- Subjects
Discrete mathematics ,Statistics and Probability ,Queueing theory ,Differential equation ,General Mathematics ,010102 general mathematics ,Erlang (unit) ,01 natural sciences ,Computer Science::Performance ,010104 statistics & probability ,Linear differential equation ,Applied mathematics ,Renewal theory ,Special case ,0101 mathematics ,Statistics, Probability and Uncertainty ,Finite set ,Queue ,Mathematics - Abstract
This paper discusses infinite server queues whose input is a Phase Type Renewal Process. The problems of obtaining the transient and steady-state distributions and moments of the queue length are reduced to the solution of certain well-behaved systems of linear differential equations. Sample computations are provided with as many as ten phases. The paper contains some useful explicit formulas and also discusses the interesting special case where the service time is also of phase type. The Phase Type Distributions include a wide variety of models such as generalized Erlang, hyperexponential (mixtures of a finite number of exponentials) as very special cases and possess great versatality in modeling a number of interesting qualitative features such as bimodality.
- Published
- 1980
21. Homogeneous row-continuous bivariate markov chains with boundaries
- Author
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J. Keilson and M. Zachmann
- Subjects
Discrete mathematics ,Statistics and Probability ,Markov chain mixing time ,Markov kernel ,Markov chain ,General Mathematics ,010102 general mathematics ,Row and column spaces ,01 natural sciences ,Continuous-time Markov chain ,010104 statistics & probability ,Ergodic theory ,Examples of Markov chains ,Markov property ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
The matrix-geometric results of M. Neuts are extended to ergodic row-continuous bivariate Markov processes [J(t), N(t)] on state space B = {(j, n)} for which: (a) there is a boundary level N for N(t) associated with finite buffer capacity; (b) transition rates to adjacent rows and columns are independent of row level n in the interior of B. Such processes are of interest in the modelling of queue-length for voice-data transmission in communication systems. One finds that the ergodic distribution consists of two decaying components of matrix-geometric form, the second induced by the finite buffer capacity. The results are obtained via Green's function methods and compensation. Passage-time distributions for the two boundary problems are also made available algorithmically. MATRIX-GEOMETRIC; GREEN'S FUNCTION; COMPENSATION In a previous paper [10] a theoretical treatment of finite row-continuous bivariate Markov chains B(t)= [J(t), N(t)] was developed, providing an algorithmic basis for finding their ergodic distributions and associated passagetime moments. The continuous-time chain B(t) with state space = {(j, n):'0 j J, 0 n N} was described as row-continuous in the sense that the marginal process N(t), indexed by row coordinate n, changed at transition epochs by at most 1. In the present paper we restrict our discussion to those row-continuous chains for which the transition rate matrices, v°, vn, v, describing rates local to row n, are independent of n for each 1 _ n 5 N 1. For n = 0, one has vo = 0, and for n = N, v = 0. Such processes may be described as row-homogeneous, row-continuous processes modified by two retaining boundaries, as for earlier similar univariate contexts [4]. This research was conducted at the MIT Laboratory for Information and Decision Systems with partial support provided by the USAF OSR Grant Number AFOSR-79-0043, U.S. Air Force Geophysics Laboratory Grant Number F19628-80-C-0003, and the National Science Foundation Grant Number NSF/ECS 79-19880. © Applied Probability Trust 1988
- Published
- 1988
22. Conditioned limit theorems for waiting-time processes of the M/G/1 queue
- Author
-
G. Hooghiemstra
- Subjects
Statistics and Probability ,Discrete mathematics ,Waiting time ,M/G/k queue ,General Mathematics ,010102 general mathematics ,G/G/1 queue ,01 natural sciences ,Combinatorics ,010104 statistics & probability ,Burke's theorem ,M/G/1 queue ,M/M/c queue ,Limit (mathematics) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
This paper is on conditioned weak limit theorems for imbedded waiting-time processes of an M/G/1 queue. More specifically we study functional limit theorems for the actual waiting-time process conditioned by the event that the number of customers in a busy period exceeds n or equals n. Attention is also paid to the actual waiting-time process with random time index. Combined with the existing literature on the subject this paper gives a complete account of the conditioned limit theorems for the actual waiting-time process of an M/G/1 queue for arbitrary traffic intensity and for a rather general class of service-time distributions. The limit processes that occur are Brownian excursion and meander, while in the case of random time index also the following limit occurs: Brownian excursion divided by an independent and uniform (0, 1) distributed random variable.
- Published
- 1983
23. On the comparison of point processes
- Author
-
Y. L. Deng
- Subjects
Statistics and Probability ,Discrete mathematics ,General Mathematics ,010102 general mathematics ,Regular polygon ,01 natural sciences ,Point process ,Combinatorics ,010104 statistics & probability ,Superposition principle ,Stopping time ,Limit (mathematics) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Focus (optics) ,Mathematics - Abstract
Several different orderings for the comparison of point processes have been introduced and their relationships discussed in Whitt [9], Daley [2] and Deng [4]. It is of some interest to know whether these orderings, in general, are preserved under various operations on point processes. Some results concerning limit operations were given in Deng [4]. In the present paper, we first further introduce some convex and concave orderings for counting processes, and survey the relationships among all orderings mentioned in [9], [4] and this paper. Then we focus our attention on the study of the conditions for the preservation of orderings under the operations of superposition, thinning, shift, and random change of time.
- Published
- 1985
24. A limit theorem for random walks with drift
- Author
-
C. C. Heyde
- Subjects
Discrete mathematics ,Statistics and Probability ,Heterogeneous random walk in one dimension ,General Mathematics ,Loop-erased random walk ,010102 general mathematics ,Random walk ,01 natural sciences ,010104 statistics & probability ,Scaling limit ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Quantum walk ,Limit (mathematics) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Random variable ,Donsker's theorem ,Mathematics - Abstract
M(x) = max[kj Mk _ x]. M(x) + 1 is then the first passage time out of the interval ( oo, x] for the random walk process S,. In this paper we shall concern ourselves with just those cases in which M(x) is a proper random variable with EM(x) < co and, when suitably normed, possesses a limit distribution as x -) co. It will be shown that M(x) can possess such a limit if and only if the random variables Xi belong to the domain of attraction of one of a certain group of stable laws and the limit law will be obtained under these circumstances. This result (Theorem 2) constitutes a generalization from the case of non-negative summands of various limit results of classical renewal theory (see for example Feller [6], 359-360). The significance of this type of generalization has previously been explored in Heyde [8]; Theorem 2 of this paper is in fact an extension of Theorem 4 of [8]. In order to obtain the limit distribution mentioned above we need a version of
- Published
- 1967
25. Two queues in series with a finite, intermediate waitingroom
- Author
-
Marcel F. Neuts
- Subjects
Statistics and Probability ,Discrete mathematics ,Independent and identically distributed random variables ,Service (business) ,Queueing theory ,Series (mathematics) ,General Mathematics ,010102 general mathematics ,Poisson distribution ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,symbols ,0101 mathematics ,Statistics, Probability and Uncertainty ,Unit (ring theory) ,Random variable ,Queue ,Mathematics - Abstract
A service unit I, with Poisson input and general service times is in series with a unit II, with negative-exponential service times. The intermediate waitingroom can accomodate at most k persons and a customer cannot leave unit I when the waitingroom is full. The paper shows that this system of queues can be studied in terms of an imbedded semi-Markov process. Equations for the time dependent distributions are given, but the main emphasis of the paper is on the equilibrium conditions and on asymptotic results. 1. Description of the model The system of queues, discussed in this paper, consists of two units. Customers arrive at a first unit (I) according to a homogeneous Poisson process of rate 2. Their service times in unit I are independent, identically distributed random variables with common distribution function H(.). We assume that H(.) has a positive, finite mean a and we will denote the Laplace-Stieltjes transform of H( ) by h(s), Re s 0. Upon completion of service in unit I, all customers go on to a second unit (II) via a finite waitingroom. We assume that there can be not more than k customers in unit II and in the waitingroom at any time. If upon completion of service in unit I a customer finds the waitingroom full, then the unit I blocks until a service in unit II is completed. At that time he is allowed to enter the waitingroom. We assume that the service times in unit II are independent, identically distributed random variables with a negative-exponential distribution with mean 1/a. The service times in unit II are also stochastically independent of those in unit I and of the arrival process. The case k = 1, i.e., when no customers can wait between the two units, was studied by B. Avi-Itzhak and M. Yadin[1], T. Suzuki [11] and N. U. Prabhu [7]. These authors allow the service times in the second unit to have a general distribution. In the case of general k, we impose the requirement that these service Received 3 July 1967. Research supported in part by Office of Naval Research Contract NONR 1100(26). 123 This content downloaded from 157.55.39.217 on Mon, 18 Apr 2016 07:16:32 UTC All use subject to http://about.jstor.org/terms
- Published
- 1968
26. Stochastic Monotonicity and Duality of kth Order with Application to Put-Call Symmetry of Powered Options
- Author
-
Vassili N. Kolokoltsov
- Subjects
Statistics and Probability ,Stochastic monotonicity ,General Mathematics ,Markov process ,Duality (optimization) ,97M30 ,Monotonic function ,Perturbation function ,01 natural sciences ,dual semigroup ,62P05 ,010104 statistics & probability ,symbols.namesake ,60J25 ,FOS: Mathematics ,Strong duality ,Wolfe duality ,stochastic duality ,0101 mathematics ,Mathematics ,60J60 ,Discrete mathematics ,put-call symmetry and reversal ,powered and digital options ,Computer Science::Information Retrieval ,010102 general mathematics ,Probability (math.PR) ,generators of dual processes ,straddle ,Weak duality ,Valuation of options ,symbols ,60J75 ,Statistics, Probability and Uncertainty ,Mathematical economics ,Mathematics - Probability - Abstract
We introduce a notion of $k$th order stochastic monotonicity and duality that allows one to unify the notion used in insurance mathematics (sometimes refereed to as Siegmund's duality) for the study of ruin probability and the duality responsible for the so-called put - call symmetries in option pricing. Our general $k$th order duality can be financially interpreted as put - call symmetry for powered options. The main objective of the present paper is to develop an effective analytic approach to the analysis of duality leading to the full characterization of $k$th order duality of Markov processes in terms of their generators, which is new even for the well-studied case of put -call symmetries., Comment: To appear in Journal of Applied Probability 52:1 (March 2015)
- Published
- 2015
27. Couplings for locally branching epidemic processes
- Author
-
A. D. Barbour, University of Zurich, and Barbour, A D
- Subjects
Statistics and Probability ,General Mathematics ,Closeness ,Branching (polymer chemistry) ,01 natural sciences ,Limit theory ,Coupling ,010104 statistics & probability ,510 Mathematics ,Deterministic approximation ,Exponential growth ,1804 Statistics, Probability and Uncertainty ,60J85 ,Statistical physics ,2613 Statistics and Probability ,0101 mathematics ,2600 General Mathematics ,Mathematics ,Branching process ,Discrete mathematics ,epidemic process ,010102 general mathematics ,92H30 ,First order ,10123 Institute of Mathematics ,60K35 ,deterministic approximation ,branching process approximation ,Statistics, Probability and Uncertainty - Abstract
The asymptotic behaviour of many locally branching epidemic models can, at least to first order, be deduced from the limit theory of two branching processes. The first is Whittle's (1955) branching approximation to the early stages of the epidemic, the phase in which approximately exponential growth takes place. The second is the susceptibility approximation; the backward branching process that approximates the history of the contacts that would lead to an individual becoming infected. The simplest coupling arguments for demonstrating the closeness of these branching process approximations do not keep the processes identical for quite long enough. Thus, arguments showing that the differences are unimportant are also needed. In this paper we show that, for some models, couplings can be constructed that are sufficiently accurate for this extra step to be dispensed with.
- Published
- 2014
28. Improving the Asmussen–Kroese-Type Simulation Estimators
- Author
-
Sheldon M. Ross and Samim Ghamami
- Subjects
65C05 ,Statistics and Probability ,General Mathematics ,efficient Monte Carlo estimation ,variance reduction ,Monte Carlo method ,Type (model theory) ,Control variates ,01 natural sciences ,010104 statistics & probability ,stop-loss transform ,control variate ,Statistics ,0101 mathematics ,Heavy-tailed random variable ,conditioning ,stratification ,Mathematics ,Event (probability theory) ,Discrete mathematics ,010102 general mathematics ,Estimator ,91G20 ,Variance reduction ,Statistics, Probability and Uncertainty ,Random variable ,rare event - Abstract
The Asmussen–Kroese Monte Carlo estimators of P(S n > u) and P(S N > u) are known to work well in rare event settings, where S N is the sum of independent, identically distributed heavy-tailed random variables X 1,…,X N and N is a nonnegative, integer-valued random variable independent of the X i . In this paper we show how to improve the Asmussen–Kroese estimators of both probabilities when the X i are nonnegative. We also apply our ideas to estimate the quantity E[(S N -u)+].
- Published
- 2012
29. Backward Coalescence Times for Perfect Simulation of Chains with Infinite Memory
- Author
-
Emilio De Santis and Mauro Piccioni
- Subjects
Statistics and Probability ,Coalescence (physics) ,Discrete mathematics ,Stationary process ,chains with complete connections ,General Mathematics ,010102 general mathematics ,68U20 ,01 natural sciences ,Past history ,010104 statistics & probability ,Simulation algorithm ,coupling ,perfect simulation ,60J10 ,Countable set ,A priori and a posteriori ,Perfect simulation ,60G99 ,Statistical physics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
This paper is devoted to the perfect simulation of a stationary process with an at most countable state space. The process is specified through a kernel, prescribing the probability of the next state conditional to the whole past history. We follow the seminal work of Comets, Fernández and Ferrari (2002), who gave sufficient conditions for the construction of a perfect simulation algorithm. We define backward coalescence times for these kind of processes, which allow us to construct perfect simulation algorithms under weaker conditions than in Comets, Fernández and Ferrari (2002). We discuss how to construct backward coalescence times (i) by means of information depths, taking into account some a priori knowledge about the histories that occur; and (ii) by identifying suitable coalescing events.
- Published
- 2012
30. On the Zagreb Index of Random Recursive Trees
- Author
-
Qunqiang Feng and Zhishui Hu
- Subjects
Statistics and Probability ,Random graph ,Discrete mathematics ,Stochastic process ,General Mathematics ,010102 general mathematics ,Loop-erased random walk ,Random tree ,Zagreb index ,Martingale central limit theorem ,recursive tree ,01 natural sciences ,Random binary tree ,Recursive tree ,Combinatorics ,05C05 ,010104 statistics & probability ,martingale central limit theorem ,60F05 ,60C05 ,0101 mathematics ,Statistics, Probability and Uncertainty ,Martingale (probability theory) ,Mathematics - Abstract
We investigate the Zagreb index, one of the topological indices, of random recursive trees in this paper. Through a recurrence equation, the first two moments of Zn, the Zagreb index of a random recursive tree of size n, are obtained. We also show that the random process {Zn − E[Zn], n ≥ 1} is a martingale. Then the asymptotic normality of the Zagreb index of a random recursive tree is given by an application of the martingale central limit theorem. Finally, two other topological indices are also discussed in passing.
- Published
- 2011
31. An Almost-Sure Renewal Theorem for Branching Random Walks on the Line
- Author
-
Matthias Meiners
- Subjects
Statistics and Probability ,Discrete mathematics ,General Mathematics ,010102 general mathematics ,Random walk ,01 natural sciences ,Combinatorics ,010104 statistics & probability ,Mathematics::Probability ,Probability theory ,Convergence of random variables ,Branching random walk ,0101 mathematics ,Statistics, Probability and Uncertainty ,Brouwer fixed-point theorem ,Martingale (probability theory) ,Real line ,Mathematics ,Branching process - Abstract
In the present paper an almost-sure renewal theorem for branching random walks (BRWs) on the real line is formulated and established. The theorem constitutes a generalization of Nerman's theorem on the almost-sure convergence of Malthus normed supercritical Crump-Mode-Jagers branching processes counted with general characteristic and Gatouras' almost-sure renewal theorem for BRWs on a lattice.
- Published
- 2010
32. Transient Asymptotics of Lévy-Driven Queues
- Author
-
Abdelghafour Es-Saghouani, Michel Mandjes, Krzysztof Dębicki, Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands, and Stochastics (KDV, FNWI)
- Subjects
Discrete mathematics ,Statistics and Probability ,Stationary process ,Sublinear function ,Lévy process ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Stable process ,Transient analysis ,010104 statistics & probability ,Large deviations ,Compound Poisson process ,Calculus ,Probability distribution ,Large deviations theory ,Queues ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics ,Event (probability theory) - Abstract
With (Q t ) t denoting the stationary workload process in a queue fed by a Lévy input process (X t ) t , this paper focuses on the asymptotics of rare event probabilities of the type P(Q 0 > pB, Q T B > qB) for given positive numbers p and q, and a positive deterministic function T B . We first identify conditions under which the probability of interest is dominated by the ‘most demanding event’, in the sense that it is asymptotically equivalent to P(Q > max{p, q}B) for large B, where Q denotes the steady-state workload. These conditions essentially reduce to T B being sublinear (i.e. T B /B → 0 as B → ∞). A second condition is derived under which the probability of interest essentially ‘decouples’, in that it is asymptotically equivalent to P(Q > pB)P(Q > qB) for large B. For various models considered in the literature, this ‘decoupling condition’ reduces to requiring that T B is superlinear (i.e. T B / B → ∞ as B → ∞). This is not true for certain ‘heavy-tailed’ cases, for instance, the situations in which the Lévy input process corresponds to an α-stable process, or to a compound Poisson process with regularly varying job sizes, in which the ‘decoupling condition’ reduces to T B / B 2 → ∞. For these input processes, we also establish the asymptotics of the probability under consideration for T B increasing superlinearly but subquadratically. We pay special attention to the case T B = RB for some R > 0; for light-tailed input, we derive intuitively appealing asymptotics, intensively relying on sample path large deviations results. The regimes obtained can be interpreted in terms of the most likely paths to overflow.
- Published
- 2010
33. Exponential Random Graphs as Models of Overlay Networks
- Author
-
Ayalvadi Ganesh, Moez Draief, and Laurent Massoulié
- Subjects
Statistics and Probability ,Dense graph ,General Mathematics ,Symmetric graph ,01 natural sciences ,law.invention ,Combinatorics ,010104 statistics & probability ,symbols.namesake ,Pathwidth ,law ,Line graph ,Random regular graph ,FOS: Mathematics ,0101 mathematics ,Mathematics ,Random graph ,Block graph ,Discrete mathematics ,Probability (math.PR) ,010102 general mathematics ,Planar graph ,symbols ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
In this paper, we give an analytic solution for graphs with n nodes and E edges for which the probability of obtaining a given graph G is specified in terms of the degree sequence of G. We describe how this model naturally appears in the context of load balancing in communication networks, namely Peer-to-Peer overlays. We then analyse the degree distribution of such graphs and show that the degrees are concentrated around their mean value. Finally, we derive asymptotic results on the number of edges crossing a graph cut and use these results $(i)$ to compute the graph expansion and conductance, and $(ii)$ to analyse the graph resilience to random failures., Comment: 18 pages
- Published
- 2009
34. On Nonoptimality of Bold Play for Subfair Red-And-Black with a Rational-Valued House Limit
- Author
-
Yi-Ching Yao, Shoou-Ren Hsiau, Pei-Shou Chung, and May-Ru Chen
- Subjects
Discrete mathematics ,Statistics and Probability ,General Mathematics ,010102 general mathematics ,Possession (law) ,01 natural sciences ,010104 statistics & probability ,Integer ,Irrational number ,Gambling and information theory ,Limit (mathematics) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In the subfair red-and-black gambling problem, a gambler can stake any amount in his possession, winning an amount equal to the stake with probability w and losing the stake with probability 1 − w, where 0 < w < ½. The gambler seeks to maximize the probability of reaching a fixed fortune (to be normalized to unity) by gambling repeatedly with suitably chosen stakes. In their classic work, Dubins and Savage (1965), (1976) showed that it is optimal to play boldly. When there is a house limit of l (0 < l < ½), so that the gambler can stake no more than l, Wilkins (1972) showed that bold play remains optimal provided that 1 / l is an integer. On the other hand, building on an earlier surprising result of Heath, Pruitt and Sudderth (1972), Schweinsberg (2005) recently showed that, for all irrational 0 < l < ½ and all 0 < w < ½, bold play is not optimal for some initial fortune. The purpose of the present paper is to present several results supporting the conjecture that, for all rational l with 1 / l not an integer and all 0 < w < ½, bold play is not optimal for some initial fortune. While most of these results are based on Schweinsberg's method, in a special case where his method is shown to be inapplicable, we argue that the conjecture can be verified with the help of symbolic-computation software.
- Published
- 2008
35. Continuous-State Branching Processes and Self-Similarity
- Author
-
Andreas E. Kyprianou and Juan Carlos Pardo
- Subjects
Statistics and Probability ,Discrete mathematics ,Self-similarity ,General Mathematics ,010102 general mathematics ,Markov process ,State (functional analysis) ,01 natural sciences ,Branching (linguistics) ,010104 statistics & probability ,symbols.namesake ,Transformation (function) ,symbols ,Statistical physics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics ,Branching process - Abstract
In this paper we study the α-stable continuous-state branching processes (for α ∈ (1, 2]) and the α-stable continuous-state branching processes conditioned never to become extinct in the light of positive self-similarity. Understanding the interaction of the Lamperti transformation for continuous-state branching processes and the Lamperti transformation for positive, self-similar Markov processes gives access to a number of explicit results concerning the paths of α-stable continuous-state branching processes and α-stable continuous-state branching processes conditioned never to become extinct.
- Published
- 2008
36. Some Blackwell-Type Renewal Theorems for Weighted Renewal Functions
- Author
-
Jianxi Lin
- Subjects
Discrete mathematics ,Statistics and Probability ,010104 statistics & probability ,Distribution (number theory) ,General Mathematics ,010102 general mathematics ,Applied mathematics ,Renewal theory ,0101 mathematics ,Type (model theory) ,Statistics, Probability and Uncertainty ,01 natural sciences ,Mathematics - Abstract
In this paper, a new approach is proposed to investigate Blackwell-type renewal theorems for weighted renewal functions systematically according to which of the tails of weighted renewal constants or the underlying distribution is asymptotically heavier. Some classical results are improved considerably.
- Published
- 2008
37. Epidemics on Random Graphs with Tunable Clustering
- Author
-
Maria Deijfen, Mathias Lindholm, Andreas Nordvall Lagerås, and Tom Britton
- Subjects
Statistics and Probability ,Discrete mathematics ,Random graph ,Group (mathematics) ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,Computer Science::Social and Information Networks ,Intersection graph ,Quantitative Biology::Other ,01 natural sciences ,92D30, 05C80 ,010104 statistics & probability ,Probability theory ,Statistics ,FOS: Mathematics ,Quantitative Biology::Populations and Evolution ,Graph (abstract data type) ,Probability distribution ,0101 mathematics ,Statistics, Probability and Uncertainty ,Cluster analysis ,Mathematics - Probability ,Branching process ,Mathematics - Abstract
In this paper, a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. It is investigated how these quantities varies with the clustering in the graph and it turns out for instance that, as the clustering increases, the epidemic threshold decreases. The network is modelled by a random intersection graph, in which individuals are independently members of a number of groups and two individuals are linked to each other if and only if they share at least one group., 17 pages, 1 figure
- Published
- 2008
38. Sums of Dependent Nonnegative Random Variables with Subexponential Tails
- Author
-
Qihe Tang and Bangwon Ko
- Subjects
Discrete mathematics ,Statistics and Probability ,Variables ,050208 finance ,media_common.quotation_subject ,General Mathematics ,05 social sciences ,01 natural sciences ,Copula (probability theory) ,Combinatorics ,010104 statistics & probability ,Probability theory ,0502 economics and business ,0101 mathematics ,Statistics, Probability and Uncertainty ,Random variable ,media_common ,Mathematics - Abstract
In this paper we study the asymptotic tail probabilities of sums of subexponential, nonnegative random variables, which are dependent according to certain general structures with tail independence. The results show that the subexponentiality of the summands eliminates the impact of the dependence on the tail behavior of the sums.
- Published
- 2008
39. Characterizations of Conditional Comonotonicity
- Author
-
Ka Chun Cheung
- Subjects
Discrete mathematics ,Statistics and Probability ,050208 finance ,Generalization ,Multivariate random variable ,Comonotonicity ,General Mathematics ,05 social sciences ,Conditional probability distribution ,01 natural sciences ,010104 statistics & probability ,Convergence of random variables ,Probability theory ,0502 economics and business ,Almost surely ,0101 mathematics ,Statistics, Probability and Uncertainty ,Representation (mathematics) ,Mathematics - Abstract
The notion of conditional comonotonicity was first used implicitly by Kaas, Dhaene, and Goovaerts (2000) and was formally introduced by Jouini and Napp (2004) as a generalization of the classical concept of comonotonicity. The objective of the present paper is to further investigate this relatively new concept. The main result is that a random vector is comonotonic conditional to a certain σ-field if and only if it is almost surely comonotonic locally on each atom of the conditioning σ-field. We also provide a new proof of a distributional representation and an almost sure representation of a conditionally comonotonic random vector.
- Published
- 2007
40. A matrix-analytic approach to the N-player ruin problem
- Author
-
Yvik Swan and F Thomas Bruss
- Subjects
Statistics and Probability ,Discrete mathematics ,Markov chain ,General Mathematics ,010102 general mathematics ,Applied probability ,Hitting time ,Markov process ,Ruin theory ,01 natural sciences ,Combinatorics ,010104 statistics & probability ,symbols.namesake ,Absorbing Markov chain ,symbols ,Phase-type distribution ,0101 mathematics ,Statistics, Probability and Uncertainty ,First-hitting-time model ,Mathematics - Abstract
Consider N players, respectively owning x1, x2, …, xN monetary units, who play a sequence of games, winning from and losing to each other integer amounts according to fixed rules. The sequence stops as soon as (at least) one player is ruined. We are interested in the ruin process of these N players, i.e. in the probability that a given player is ruined first, and also in the expected ruin time. This problem is called the N-player ruin problem. In this paper, the problem is set up as a multivariate absorbing Markov chain with an absorbing state corresponding to the ruin of each player. This is then discussed in the context of phase-type distributions where each phase is represented by a vector of size N and the distribution has as many absorbing points as there are ruin events. We use this modified phase-type distribution to obtain an explicit solution to the N-player problem. We define a partition of the set of transient states into different levels, and on it give an extension of the folding algorithm (see Ye and Li (1994)). This provides an efficient computational procedure for calculating some of the key measures.
- Published
- 2006
41. On stochastic recursive equations of sum and max type
- Author
-
Ludger Rüschendorf
- Subjects
Statistics and Probability ,Discrete mathematics ,Independent and identically distributed random variables ,General Mathematics ,010102 general mathematics ,Stochastic interpretation ,0102 computer and information sciences ,Type (model theory) ,01 natural sciences ,Combinatorics ,Moment (mathematics) ,010104 statistics & probability ,Distribution (mathematics) ,Probability theory ,010201 computation theory & mathematics ,Homogeneous differential equation ,Uniqueness ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this paper we consider stochastic recursive equations of sum type, , and of max type, , where Ai, bi, and b are random, (Xi) are independent, identically distributed copies of X, and denotes equality in distribution. Equations of these types typically characterize limits in the probabilistic analysis of algorithms, in combinatorial optimization problems, and in many other problems having a recursive structure. We develop some new contraction properties of minimal Ls-metrics which allow us to establish general existence and uniqueness results for solutions without imposing any moment conditions. As an application we obtain a one-to-one relationship between the set of solutions to the homogeneous equation and the set of solutions to the inhomogeneous equation, for sum- and max-type equations. We also give a stochastic interpretation of a recent transfer principle of Rösler from nonnegative solutions of sum type to those of max type, by means of random scaled Weibull distributions.
- Published
- 2006
42. On the infinite-horizon probability of (non)ruin for integer-valued claims
- Author
-
Vladimir K. Kaishev and Zvetan G. Ignatov
- Subjects
Statistics and Probability ,Discrete mathematics ,Independent and identically distributed random variables ,General Mathematics ,010102 general mathematics ,Convolution of probability distributions ,01 natural sciences ,Symmetric probability distribution ,010104 statistics & probability ,Probability theory ,Joint probability distribution ,Compound Poisson process ,Calculus ,Probability distribution ,0101 mathematics ,Statistics, Probability and Uncertainty ,Random variable ,Mathematics - Abstract
We consider a compound Poisson process whose jumps are modelled as a sequence of positive, integer-valued, dependent random variables, W 1,W 2,…, viewed as insurance claim amounts. The number of points up to time t of the stationary Poisson process which models the claim arrivals is assumed to be independent of W 1,W 2,…. The premium income to the insurance company is represented by a nondecreasing, nonnegative, real-valued function h(t) on [0,∞) such that lim t→∞ h(t) = ∞. The function h(t) is interpreted as an upper boundary. The probability that the trajectory of such a compound Poisson process will not cross the upper boundary in infinite time is known as the infinite-horizon nonruin probability. Our main result in this paper is an explicit expression for the probability of infinite-horizon nonruin, assuming that certain conditions on the premium-income function, h(t), and the joint distribution of the claim amount random variables, W 1,W 2,…, hold. We have also considered the classical ruin probability model, in which W 1,W 2,… are assumed to be independent, identically distributed random variables and we let h(t)=u + ct. For this model we give a formula for the nonruin probability which is a special case of our main result. This formula is shown to coincide with the infinite-horizon nonruin probability formulae of Picard and Lefèvre (2001), Gerber (1988), (1989), and Shiu (1987), (1989).
- Published
- 2006
43. Level Crossing Ordering of Skip-Free-to-the-Right Continuous-Time Markov Chains
- Author
-
António Pacheco and Fátima Ferreira
- Subjects
Statistics and Probability ,Discrete mathematics ,Markov chain ,General Mathematics ,010102 general mathematics ,Process (computing) ,Level crossing ,01 natural sciences ,Stochastic ordering ,Birth–death process ,Set (abstract data type) ,Continuous-time Markov chain ,010104 statistics & probability ,Probability theory ,0101 mathematics ,Statistics, Probability and Uncertainty ,Algorithm ,Mathematics - Abstract
As proposed by Irle and Gani in 2001, a process X is said to be slower in level crossing than a process Y if it takes X stochastically longer to exceed any given level than it does Y. In this paper, we extend a result of Irle (2003), relative to the level crossing ordering of uniformizable skip-free-to-the-right continuous-time Markov chains, to derive a new set of sufficient conditions for the level crossing ordering of these processes. We apply our findings to birth-death processes with and without catastrophes, and M/M/s/c systems.
- Published
- 2005
44. Explicit criteria for several types of ergodicity of the embedded M/G/1 and GI/M/n queues
- Author
-
Zhenting Hou and Yuanyuan Liu
- Subjects
Discrete mathematics ,Statistics and Probability ,Queueing theory ,Mathematics::Dynamical Systems ,Markov chain ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,Ergodicity ,Generating function ,01 natural sciences ,Combinatorics ,010104 statistics & probability ,Matrix analytic method ,Burke's theorem ,M/G/1 queue ,Probability distribution ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
This paper investigates the rate of convergence to the probability distribution of the embedded M/G/1 and GI/M/n queues. We introduce several types of ergodicity including l-ergodicity, geometric ergodicity, uniformly polynomial ergodicity and strong ergodicity. The usual method to prove ergodicity of a Markov chain is to check the existence of a Foster–Lyapunov function or a drift condition, while here we analyse the generating function of the first return probability directly and obtain practical criteria. Moreover, the method can be extended to M/G/1- and GI/M/1-type Markov chains.
- Published
- 2004
45. The almost sure central limit theorem for one-dimensional nearest-neighbour random walks in a space-time random environment
- Author
-
Jean Bérard
- Subjects
Discrete mathematics ,Statistics and Probability ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,Random function ,Random element ,Random walk ,01 natural sciences ,Combinatorics ,010104 statistics & probability ,Convergence of random variables ,Random compact set ,0101 mathematics ,Statistics, Probability and Uncertainty ,Donsker's theorem ,Randomness ,Mathematics ,Central limit theorem - Abstract
The central limit theorem for random walks on ℤ in an i.i.d. space-time random environment was proved by Bernabei et al. for almost all realization of the environment, under a small randomness assumption. In this paper, we prove that, in the nearest-neighbour case, when the averaged random walk is symmetric, the almost sure central limit theorem holds for an arbitrary level of randomness.
- Published
- 2004
46. Precise large deviations for sums of random variables with consistently varying tails
- Author
-
Jia-An Yan, Qihe Tang, Kai W. Ng, Hailiang Yang, and Actuarial Science & Mathematical Finance (ASE, FEB)
- Subjects
Statistics and Probability ,Independent and identically distributed random variables ,Discrete mathematics ,Sequence ,Counting process ,General Mathematics ,010102 general mathematics ,Function (mathematics) ,01 natural sciences ,Combinatorics ,Cox process ,010104 statistics & probability ,Heavy-tailed distribution ,Large deviations theory ,0101 mathematics ,Statistics, Probability and Uncertainty ,Random variable ,Mathematics - Abstract
Let {X k , k ≥ 1} be a sequence of independent, identically distributed nonnegative random variables with common distribution function F and finite expectation μ > 0. Under the assumption that the tail probability is consistently varying as x tends to infinity, this paper investigates precise large deviations for both the partial sums S n and the random sums S N(t), where N(·) is a counting process independent of the sequence {X k , k ≥ 1}. The obtained results improve some related classical ones. Applications to a risk model with negatively associated claim occurrences and to a risk model with a doubly stochastic arrival process (extended Cox process) are proposed.
- Published
- 2004
47. Ladder variables, internal structure of Galton–Watson trees and finite branching random walks
- Author
-
Jean-François Marckert and Abdelkader Mokkadem
- Subjects
Galton watson ,Discrete mathematics ,Statistics and Probability ,Heterogeneous random walk in one dimension ,General Mathematics ,010102 general mathematics ,Loop-erased random walk ,Random walk ,01 natural sciences ,Combinatorics ,Branching (linguistics) ,010104 statistics & probability ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this paper, we consider Galton–Watson trees conditioned by size. We show that the number of k-ancestors (ancestors that have k children) of a node u is (almost) proportional to its depth. The k, j-ancestors are also studied. The methods rely on the study of ladder variables on an associated random walk. We also give an application to finite branching random walks.
- Published
- 2003
48. On ordered series and later waiting time distributions in a sequence of Markov dependent multistate trials
- Author
-
Yung-Ming Chang and James C. Fu
- Subjects
Statistics and Probability ,Discrete mathematics ,Markov chain mixing time ,Markov chain ,General Mathematics ,Variable-order Markov model ,010102 general mathematics ,Discrete phase-type distribution ,Markov process ,Markov model ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Markov renewal process ,symbols ,Applied mathematics ,Markov property ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
The sooner and later waiting time problems have been extensively studied and applied in various areas of statistics and applied probability. In this paper, we give a comprehensive study of ordered series and later waiting time distributions of a number of simple patterns with respect to nonoverlapping and overlapping counting schemes in a sequence of Markov dependent multistate trials. Exact distributions and probability generating functions are derived by using the finite Markov chain imbedding technique. Examples are given to illustrate our results.
- Published
- 2003
49. On the integral of the workload process of the single server queue
- Author
-
Zbigniew Palmowski, Onno Boxma, A.A. Borovkov, Stochastic Operations Research, and Mathematics and Computer Science
- Subjects
D/M/1 queue ,Discrete mathematics ,Statistics and Probability ,M/G/k queue ,General Mathematics ,M/D/1 queue ,010102 general mathematics ,M/M/1 queue ,G/G/1 queue ,01 natural sciences ,010104 statistics & probability ,Burke's theorem ,M/G/1 queue ,M/M/c queue ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
This paper is devoted to a study of the integral of the workload process of the single server queue, in particular during one busy period. Firstly, we find asymptotics of the area 𝒜 swept under the workload process W(t) during the busy period when the service time distribution has a regularly varying tail. We also investigate the case of a light-tailed service time distribution. Secondly, we consider the problem of obtaining an explicit expression for the distribution of 𝒜. In the general GI/G/1 case, we use a sequential approximation to find the Laplace—Stieltjes transform of 𝒜. In the M/M/1 case, this transform is obtained explicitly in terms of Whittaker functions. Thirdly, we consider moments of 𝒜 in the GI/G/1 queue. Finally, we show asymptotic normality of .
- Published
- 2003
50. Losses per cycle in a single-server queue
- Author
-
Ronald W. Wolff
- Subjects
Statistics and Probability ,Discrete mathematics ,Service (business) ,M/G/k queue ,General Mathematics ,010102 general mathematics ,Single server queue ,Expected value ,01 natural sciences ,010104 statistics & probability ,Elementary proof ,Statistics ,M/G/1 queue ,Order (group theory) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Queue ,Mathematics - Abstract
Several recent papers have shown that for the M/G/1/n queue with equal arrival and service rates, the expected number of lost customers per busy cycle is equal to 1 for every n ≥ 0. We present an elementary proof based on Wald's equation and, for GI/G/1/n, obtain conditions for this quantity to be either less than or greater than 1 for every n ≥ 0. In addition, we extend this result to batch arrivals, where, for average batch size β, the same quantity is either less than or greater than β. We then extend these results to general ways that customers may be lost, to an arbitrary order of service that allows service interruption, and finally to reneging.
- Published
- 2002
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