bluuebunny/crossref_metadata_2025 huggingface dataset:
Some results for repairable systems with general repair
Authors: Kijima Masaaki
Published: April 2006

In this paper, we develop general repair models for a repairable system by using the idea of the virtual age process of the system. If the system has the virtual age Vn –1 = y immediately after the (n – l)th repair, the nth failure-time Xn is assumed to have the survival function where is the survival function of the failure-time of a new system. A general repair is represented as a sequence of random variables An taking a value between 0 and 1, where An denotes the degree of the nth repair. For the extremal values 0 and 1, An = 1 means a minimal repair and An= 0 a perfect repair. Two models are constructed depending on how the repair affects the virtual age process: Vn = Vn– 1+ AnXn as Model 1 and Vn = An(Vn– 1 + Xn) as Model II. Various monotonicity properties of the process with respect to stochastic orderings of general repairs are obtained. Using a result, an upper bound for E[Sn] when a general repair is used is derived.
Monotonicity of throughput in non-Markovian networks
Authors: Tsoucas Pantelis , Walrand Jean
Published: April 2006

Monotonicity of throughput is established in some non-Markovian queueing networks by means of pathwise comparisons. In a series of · /GI/s/N queues with loss at the first node it is proved that increasing the waiting room and/or the number of servers increases the throughput. For a closed network of · /GI/s queues it is shown that the throughput increases as the total number of jobs increases. The technique used for these results does not apply to blocking systems with finite buffers and feedback. Using a stronger coupling argument we prove throughput monotonicity as a function of buffer size for a series of two ·/M/1/N queues with loss and feedback from the second to the first node.
Linear dynamic errors-in-variables models
Authors: Deistler M.
Published: April 2006

Linear dynamical systems where both inputs and outputs are contaminated by errors are considered. A characterization of the sets of all observationally equivalent transfer functions is given, the role of the causality assumption is investigated and conditions for identifiability in the case of Gaussian as well as non-Gaussian observations are derived.
The ‘derived’ moving-average model and its role in causality
Authors: Penm J. H. W. , Terrell R. D.
Published: April 2006

In the situations where restrictions on the multivariate subset AR model are known, we propose methods of providing suitable standard errors of estimate and prediction which assist in assessing the importance of the coefficients appearing the ‘derived' moving-average (MA) model. The coefficient patterns of the derived moving-average model are proposed as an alternative basis for detecting Granger-causality.
Identification of non-linear systems using general state-dependent models
Authors: Priestley M. B. , Heravi S. M.
Published: April 2006

This paper describes an extension of the SDM scheme introduced by Priestley (1980) to the problem of the identification of non-linear systems using only input/output records. It is shown that, under quite general assumptions on the structure of the system, it is possible to identify a non-linear relationship between the input and output processes via a modified version of the Kalman-type algorithm previously applied to the study of non-linear time series models. The paper includes a numerical study of data generated from various types of non-linear systems.
On weak lumpability in Markov chains
Authors: Rubino Gerardo , Rubino Gerardo
Published: April 2006

We analyse the conditions under which the aggregated process constructed from an homogeneous Markov chain over a given partition of the state space is also Markov homogeneous. The past work on the subject is revised and new properties are obtained.
A discrete-time storage process with a general release rule
Authors: Glynn John E.
Published: April 2006

A discrete-time storage system with a general release rule and stationary nonnegative inflows is examined. A simple condition is found for the existence of a stationary storage and outflow for a general possibly non-monotone release function. It is also shown that in the Markov case (i.e. independent inflows) these distributions are unique under certain conditions. It is demonstrated that under these conditions the stationary behaviour in the Markov case varies continuously with parametric changes in the release rule. This result is used to prove convergence of a finite state space approximation for the Markov storage system.
Poisson theorem for non-homogeneous Markov chains
Authors: Pawłowski Janusz
Published: April 2006

This paper gives necessary and sufficient conditions for the convergence in distribution of sums of the 0–1 Markov chains to a compound Poisson distribution.
Conditions for fixation of an allele in the density-dependent wright–Fisher models
Authors: Klebaner Fima C.
Published: April 2006

A density-dependent Wright–Fisher model is a model where the population size changes randomly depending on the genetic composition process. If population sizesMnvary without density dependence then the condition ΣMn–1= ∞ is necessary and sufficient for fixation. It is shown that the above condition is no longer necessary for fixation in the density dependent models. Another necessary condition for fixation is given. Some known results on series of functions of sums of i.i.d. random variables are generalized to weighted sums.
Size-biased filtering of Poisson–Dirichlet samples with an application to partition structures in genetics
Authors: Hoppe Fred M.
Published: April 2006

A characteristic property of the Ewen sampling formula is shown to follow from the invariance under size-biased sampling of the Poisson–Dirichlet distribution.
Normal approximations to discrete unimodal distributions
Authors: Quinn B. G. , MacGillivray H. L.
Published: April 2006

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.
Some results involving HNBUE distributions
Authors: Basu A. P. , Kirmani S. N. U. A.
Published: April 2006

A characterization of the exponential distribution in the class of all distributions which are HNBUE or HNWUE is proved. An upper (a lower) bound is obtained on the renewal function of a renewal process when the underlying distribution is HNBUE (HNWUE).
Time-changing and truncatingK-capacity queues from oneKto another
Authors: Glasserman Paul , Gong Wei-Bo
Published: April 2006

For, we obtain aK′-capacity queue from aK-capacity queue through a random time change and a truncation, provided arrivals are Poisson or service is exponential. In the case of anM/G/1/Kqueue, the time change erases service intervals that begin with more thanK′customers in the systems. This construction yields a straightforward sample path proof of Keilson's result on the proportionality of the ergodic queue length probabilities inM/G/1/Kqueues. The same approach proves a strengthened result for ‘detailed' state probabilities. It also reproduces a proportionality result for a vacation model, due to Keilson and Servi. A ‘dual' construction yields a different kind of proportionality for theG/M/1/Kqueue.
Random mappings with an attracting center: Lagrangian distributions and a regression function
Authors: Berg Sven , Mutafchiev Ljuben
Published: April 2006

For a random mapping model with a single attracting center (Stepanov (1971)) we study the relationship between the sizes of the central tree, the adjacent points, and the free points. Joint, marginal and conditional distributions are shown to be of well-known Lagrangian type. Exact and asymptotic moment properties are investigated with the aid of Riordan's (1968) Abel identities. In particular, the relationship between the size of the central tree and that of the central component is discussed in terms of a regression function.
Strong approximations for multiple channel queues in heavy traffic
Authors: Hanqin Zhang , Guanghui Hsu , Rongxin Wang
Published: April 2006

We prove strong approximations for multiple channel queues in heavy traffic. Strong approximations are given for the waiting time and queue length processes, and for embedded sequences associated with the length process.
Biometric and chromosome algebras
Authors: Holgate Philip
Published: April 2006

This note continues the development of the infinite-dimensional genetic algebra approach to problems of population genetics. Two algebras are studied. One describes the familiar problem of a quantitative characteristic, and the other provides a way of treating the whole chromosome as an entity.
On the number of times where a simple random walk reaches its maximum
Authors: Katzenbeisser W. , Panny W.
Published: April 2006

Let Qn denote the number of times where a simple random walk reaches its maximum, where the random walk starts at the origin and returns to the origin after 2n steps. Such random walks play an important role in probability and statistics. In this paper the distribution and the moments of Qn, are considered and their asymptotic behavior is studied.
Estimation of the derivative of a stationary measure with respect to a control parameter
Authors: Vázquez-Abad Felisa J. , Kushner Harold J.
Published: April 2006

The paper deals with a problem which arises in the Monte Carlo optimization of steady state or ergodic systems which can be modelled by Markov chains. The transition probability depends on a parameter, and one wishes to find the parameter value at which some performance function is minimum. The only available data are obtained from either simulation or actual operating information. For such a problem one needs good statistical estimates of the derivatives. Conditions are given for the existence of the derivative of the stationary measure with respect to the parameter, in the sense that the derivative is a signed measure, and is the limit of the natural approximating sequence. Some properties and a useful characterization of the derivative are obtained. It is also shown that, under appropriate conditions, the derivative of the n-step transition function converges to the derivative of the stationary measure as n tends to ∞. This latter result is of particular importance whether one is simply estimating or is actually optimizing via some sequential Monte Carlo procedure, since the basic observations are always taken over a finite time interval.
On strict stationarity and ergodicity of a non-linear ARMA model
Authors: Liu Jian , Susko Ed
Published: April 2006

Two recent papers by Petruccelli and Woolford (1984) and Chan et al. (1985) showed that the key element governing ergodicity of a threshold AR(1) model is the joint behavior of the two linear AR(1) pieces falling in the two boundary threshold regimes. They used essentially the necessary and sufficient conditions for ergodicity of a general Markov chain of Tweedie (1974), (1975) in a rather clever manner. However, it is difficult to extend the results to the more general threshold ARMA models. Besides, irreducibility is also required to apply Tweedie's results. In this paper, instead of pursuing the ideas in Tweedie's results, we shall develop a criterion similar in spirit to the technique used by Beneš (1967) in the context of continuous-time Markov chains. Consequently, we derive a necessary and sufficient condition for existence of a strictly stationary solution of a general non-linear ARMA model to be introduced in Section 2 of this paper. This condition is then applied to the threshold ARMA(1, q) model to yield a sufficient condition for strict stationarity which is identical to the condition given by Petruccelli and Woolford (1984) for the threshold AR(1). Hence, the conjecture that the moving average component does not affect stationarity is partially verified. Furthermore, under an additional irreducibility assumption, ergodicity of a non-linear ARMA model is established. The paper then concludes with a necessary condition for stationarity of the threshold ARMA(1, q) model.
Pathwise recurrence orders and simulated annealing
Authors: Borkar Vivek S.
Published: April 2006

The Connors–Kumar notion of recurrence orders for state and state transitions of time-inhomogeneous Markov chains is redefined ‘pathwise' and applied to the sample path analysis of the simulated annealing algorithm.
Non-negative time series models for dry river flow
Authors: Hutton Jane L.
Published: April 2006

Non-negative time series which are first-order autoregressive with a mixed exponential innovation process are studied. Properties and approximate marginal distributions for such series are found. Modifications to include exact zeroes and to increase variability so that the time series is a more realistic model of rivers which are dry for part of the year are discussed.
Procedures for the evaluation of strategies for resource allocation in a stochastic environment
Authors: Glazebrook K. D.
Published: April 2006

Existing procedures for strategy evaluation based on earlier work by the author have led (inter alia) to procedures for the assessment of heuristics for stochastic scheduling and to the development of an approach to sensitivity analysis. Recent theoretical work has now led to the development of more effective evaluation procedures. The key results are reported here.
Certain state-dependent processes for dichotomised parasite populations
Authors: Kemp A. W. , Newton J.
Published: April 2006

The paper re-examines Quinn and MacGillivray's (1986) stationary birth-death process for a population of fixed sizeNconsisting of two types of parasite, active and passive, and sets up a more elaborate model for the dichotomy between parasites on hosts with and without open wounds resulting from previous parasite attacks. The probability generating functions for the stationary count distributions are obtained, allowing limiting forms of the distributions to be investigated.
On the stationary distribution of some extremal Markovian sequences
Authors: Alpuim M. T. , Athayde E.
Published: April 2006

This paper is concerned with the Markovian sequence Xn = Zn max{Xn–1, Yn},n ≧ 1, where X0 is any random variable, {Zn} and {Yn} are independent sequences of i.i.d. random variables both independent of X0. We consider the problem of characterizing the class of stationary distributions arising in such a model and give criteria for a d.f. to belong to it. We develop further results when the Zn's are random variables concentrated on the interval [0, 1], namely having a beta distribution.
An entropy concentration theorem: applications in artificial intelligence and descriptive statistics
Authors: Robert Claudine
Published: April 2006

The maximum entropy principle is used to model uncertainty by a maximum entropy distribution, subject to some appropriate linear constraints. We give an entropy concentration theorem (whose demonstration is based on large deviation techniques) which is a mathematical justification of this statistical modelling principle. Then we indicate how it can be used in artificial intelligence, and how relevant prior knowledge is provided by some classical descriptive statistical methods. It appears furthermore that the maximum entropy principle yields to a natural binding between descriptive methods and some statistical structures.
First-order autoregressive models for gamma and exponential processes
Authors: Sim C. H.
Published: April 2006

In this paper we propose an autoregressive representation for a particular type of stationary Gamma(θ–1,v) process whosen-dimensional joint distributions have Laplace transform |In+θSnVn|–v, whereSn=diag(s1, · ··,sn),Vnis ann×npositive definite matrix with elementsυij= p|i–j|i2,i, j =1, ···,nandpis the lag-1 autocorrelation of the gamma process. We also generalize the two-parameter NEAR(1) model of Lawrance and Lewis (1981) to an exponential first-order autoregressive model with three parameters. The correlation structure and higher-order properties of the two proposed models are also given.
Minimizing the learning loss in adaptive control of Markov chains under the weak accessibility condition
Authors: Agrawal Rajeev
Published: April 2006

We consider the adaptive control of Markov chains under the weak accessibility condition with a view to minimizing the learning loss. A certainty equivalence control with a forcing scheme is constructed. We use a stationary randomized control scheme for forcing and compute a maximum likelihood estimate of the unknown parameter from the resulting observations. We obtain an exponential upper bound on the rate of decay of the probability of error. This allows us to choose the rate of forcing appropriately, whereby we achieve a o(f(n) log n) learning loss for any function as .
A simple proof for optimality of (s, S) policies in infinite-horizon inventory systems
Authors: Zheng Yu-Sheng
Published: April 2006

The classical proofs for the existence of a stationary (s, S) inventory policy that minimizes the total discounted or average cost over an infinite horizon are lengthy because they depend heavily on the optimality results for corresponding finite-horizon models. This note presents a simpler alternative. Since optimal stationary (s, S) policies are relatively simple to characterize, it is easy to construct a solution to the optimality equation which is satisfied by an (s, S) policy or an equivalent variant thereof. For the discounted model, the proof characterizes an (s, S) policy that is optimal for all initial inventory positions. This policy can be generated by a simple existing algorithm. For the average-cost model, the optimality proof is completed with some additional arguments, which are simple but novel, to overcome the normal difficulties encountered in models with unbounded one-step expected costs.
Poisson approximation of intermediate empirical processes
Authors: Kaufmann E. , Reiss R.-D.
Published: April 2006

We investigate the asymptotic behaviour of empirical processes truncated outside an interval about the (1 –s(n)/n)-quantile wheres(n) → ∞ ands(n)/n→ 0 as the sample sizentends to ∞. It is shown that extreme value (Poisson) processes and, alternatively, the homogeneous Poisson process may serve as approximations if certain von Mises conditions hold.
Approximating kth-order two-state Markov chains
Authors: Wang Y. H.
Published: April 2006

In this paper, we consider kth-order two-state Markov chains {Xi} with stationary transition probabilities. For k = 1, we construct in detail an upper bound for the total variation d(Sn, Y) = Σx |𝐏(Sn = x) − 𝐏(Y = x)|, where Sn = X1 + · ··+ Xn and Y is a compound Poisson random variable. We also show that, under certain conditions, d(Sn, Y) converges to 0 as n tends to ∞. For k = 2, the corresponding results are given without derivation. For general k ≧ 3, a conjecture is proposed.
New models for Markov random fields
Authors: Cressie Noel , Lele Subhash
Published: April 2006

The Hammersley–Clifford theorem gives the form that the joint probability density (or mass) function of a Markov random field must take. Its exponent must be a sum of functions of variables, where each function in the summand involves only those variables whose sites form a clique. From a statistical modeling point of view, it is important to establish the converse result, namely, to give the conditional probability specifications that yield a Markov random field. Besag (1974) addressed this question by developing a one-parameter exponential family of conditional probability models. In this article, we develop new models for Markov random fields by establishing sufficient conditions for the conditional probability specifications to yield a Markov random field.
Alternating projections and interpolation of stationary processes
Authors: Pourahmadi Mohsen
Published: April 2006

By using the alternating projection theorem of J. von Neumann, we obtain explicit formulae for the best linear interpolator and interpolation error of missing values of a stationary process. These are expressed in terms of multistep predictors and autoregressive parameters of the process. The key idea is to approximate the future by a finite-dimensional space.
Optimality of index policies for stochastic scheduling with switching penalties
Authors: Van Oyen Mark P. , Pandelis Dimitrios G. , Teneketzis Demosthenis
Published: April 2006

We investigate the impact of switching penalties on the nature of optimal scheduling policies for systems of parallel queues without arrivals. We study two types of switching penalties incurred when switching between queues: lump sum costs and time delays. Under the assumption that the service periods of jobs in a given queue possess the same distribution, we derive an index rule that defines an optimal policy. For switching penalties that depend on the particular nodes involved in a switch, we show that although an index rule is not optimal in general, there is an exhaustive service policy that is optimal.
Extremal properties of the FIFO discipline in queueing networks
Authors: Righter Rhonda , Shanthikumar J. George
Published: April 2006

We show that using the FIFO service discipline at single server stations with ILR (increasing likelihood ratio) service time distributions in networks of monotone queues results in stochastically earlier departures throughout the network. The converse is true at stations with DLR (decreasing likelihood ratio) service time distributions. We use these results to establish the validity of the following comparisons:

(i) The throughput of a closed network of FIFO single-server queues will be larger (smaller) when the service times are ILR (DLR) rather than exponential with the same means.

(ii) The total stationary number of customers in an open network of FIFO single-server queues with Poisson external arrivals will be stochastically smaller (larger) when the service times are ILR (DLR) rather than exponential with the same means.

We also give a surprising counterexample to show that although FIFO stochastically maximizes the number of departures by any time t from an isolated single-server queue with IHR (increasing hazard rate, which is weaker than ILR) service times, this is no longer true for networks of more than one queue. Thus the ILR assumption cannot be relaxed to IHR.

Finally, we consider multiclass networks of exponential single-server queues, where the class of a customer at a particular station determines its service rate at that station, and show that serving the customer with the highest service rate (which is SEPT — shortest expected processing time first) results in stochastically earlier departures throughout the network, among all preemptive work-conserving policies. We also show that a cµ rule stochastically maximizes the number of non-defective service completions by any time t when there are random, agreeable, yields.
Laplace ordering and its applications
Authors: Alzaid Abdulhamid , Kim Jee Soo , Proschan Frank
Published: April 2006

Two arbitrary life distributionsFandGcan be ordered with respect to their Laplace transforms. We sayis Laplace-smaller thanfor alls> 0. Interpretations of this ordering concept in reliability, operations research, and economics are described. General preservation properties are presented. Using these preservation results we derive useful inequalities and discuss their applications toM/G/1 queues, time series, coherent systems, shock models and cumulative damage models.
A conservation law for single-server queues and its applications
Authors: Yamazaki Genji , Sakasegawa Hirotaka , Shanthikumar J. George
Published: April 2006

We establish a conservation law for G/G/1 queues with any work-conserving service discipline using the equilibrium equations, also called the basic equations. We use this conservation law to prove an extremal property of the first-come firstserved (FCFS) service discipline: among all service disciplines that are work-conserving and independent of remaining service requirements for individual customers, the FCFS service discipline minimizes [maximizes] the mean sojourn time in a G/G/1 queue with independent (but not necessarily identical) service times with a common mean and new better [worse] than used (NBUE[NWUE]) distributions. This extends recent results of Halfin and Whitt (1990), Righter et al. (1990) and Yamazaki and Sakasegawa (1987a,b). In addition we use the conservation law to obtain an approximation for the mean queue length in a GI/GI/1 queue under the processor-sharing service discipline with finite degree of multiplicity, called LiPS discipline. Several numerical examples are presented which support the practical usefulness of the proposed approximation.
Queues with negative arrivals
Authors: Gelenbe Erol , Glynn Peter , Sigman Karl
Published: April 2006

We study single-server queueing models where in addition to regular arriving customers, there are negative arrivals. A negative arrival has the effect of removing a customer from the queue. The way in which this removal is specified gives rise to several different models. Unlike the standard FIFOGI/GI/1 model, the stability conditions for these new models may depend upon more than just the arrival and service rates; the entire distributions of interarrival and service times may be involved.
The probability that the largest observation is censored
Authors: Maller R. A. , Zhou S.
Published: April 2006

Suppose n possibly censored survival times are observed under an independent censoring model, in which the observed times are generated as the minimum of independent positive failure and censor random variables. A practical difficulty arises when the largest observation is censored since then the usual non-parametric estimator of the distribution of the survival time is improper. We calculate the probability that this occurs and give necessary and sufficient conditions for this probability to converge to 0 as n →∞. As an application, we show that if this probability is 0, asymptotically, then a consistent estimator for the mean failure time can be found. An almost sure version of the problem is also considered.
Optimal second-order product probability bounds
Authors: Block Henry W. , Costigan Timothy M. , Sampson Allan R.
Published: April 2006

Let P(c) = P(X1 ≦ c1, · ··, Xp ≦ cp) for a random vector (X1, · ··, Xp). Bounds are considered of the form

where T is a spanning tree corresponding to the bivariate probability structure and di is the degree of the vertex i in T. An optimized version of this inequality is obtained. The main result is that alwayṡ dominates certain second-order Bonferroni bounds. Conditions on the covariance matrix of a N(0,Σ) distribution are given so that this bound applies, and various applications are given.
The extremal index for a Markov chain
Authors: Smith Richard L.
Published: April 2006

The paper presents a method of computing the extremal index for a discrete-time stationary Markov chain in continuous state space. The method is based on the assumption that bivariate margins of the process are in the domain of attraction of a bivariate extreme value distribution. Scaling properties of bivariate extremes then lead to a random walk representation for the tail behaviour of the process, and hence to computation of the extremal index in terms of the fluctuation properties of that random walk. The result may then be used to determine the asymptotic distribution of extreme values from the Markov chain.
LRU is better than FIFO under the independent reference model
Authors: van den Berg J. , Gandolfi A.
Published: April 2006

Consider a two-level storage system operating with the least recently used (LRU) or the first-in, first-out (FIFO) replacement strategy. Accesses to the main storage are described by the independent reference model (IRM). Using the FKG inequality, we prove that the miss ratio for LRU is smaller than or equal to the miss ratio for FIFO.
Control of asymptotic variability in non-homogeneous Markov systems
Authors: Vassiliou P.-C. G. , Georgiou A. C. , Tsantas N.
Published: April 2006

In this paper we provide two basic results. First, we find the set of all the limiting vectors of expectations, variances and covariances in an NHMS which are possible provided that we control the limit vector of the sequence of vectors of input probabilities. Secondly, under certain conditions easily met in practice we find the distribution of the limiting vector of expectations, variances and covariances to be multinomial with probabilities the corresponding limiting expected populations in the various states of the NHMS.
Derivatives of the expected delay in the GI/G/1 Queue
Authors: Chen Hong , Yao David D.
Published: April 2006

Consider the GI/G/1 queue in which the interarrival/service times are parameterized. The derivatives (with respect to the parameter) of the expected delay of the nth job and its limit are derived and expressed in terms of the expectation of certain stochastic derivatives. Applications of these derivatives in stochastic optimization and random walk are illustrated.
Behavior of the supercritical phase of a continuum percolation model on ℝd
Authors: Tanemura Hideki
Published: April 2006

A continuum percolation model onis considered. Using a renormalization technique developed by Grimmett and Marstrand, we show a continuum analogue of their results. We prove the critical value of the percolation equals the limit of the critical value of a slice as the thickness of the slice tends to infinity. We also prove that the effective conductivity in the model is bounded from below by a positive constant in the supercritical case.
Saddlepoint approximations in conditional inference
Authors: Wang Suojin
Published: April 2006

Saddlepoint approximations are derived for the conditional cumulative distribution function and density ofwhereis the sample mean ofni.i.d. bivariate random variables andg(x, y) is a non-linear function. The relative error of orderO(n–1) is retained. The results extend the important work of Skovgaard (1987), and are useful in conditional inference, especially in the case of small or moderate sample sizes. Generalizations to higher-dimensional random vectors are also discussed. Some examples are demonstrated.
Optimal buffer allocation in a two-stage queueing system
Authors: Sparaggis Panayotis D. , Gong Wei-Bo
Published: April 2006

We study the buffer allocation problem in a two-stage cyclic queueing system. First, we show that transposing the number of buffers assigned to each queue does not affect the throughput. Second, we prove that the optimal buffer allocation scheme, in the sense of maximizing the system's throughput, is the one for which the absolute difference between the number of buffers, assigned to each queue, is minimized, i.e., it becomes either 0 or 1. This optimal allocation is insensitive to the general-type service-time distributions. These two distributions may be different and service times may even be correlated.
Semi-Pareto processes
Authors: Pillai R. N.
Published: April 2006

Semi-Pareto processes, of which Pareto processes form a proper sub-class, are discussed here. A semi-Pareto process has semi-Pareto inputs. Asymptotic properties of the maximum and minimum of the first n observations are examined as well as the geometric maximum and geometric minimum. A characterization of the semi-Pareto distribution is given. A canonical representation of a special class of Pareto process is also given.
Eigenstructure of the infinitely-many-neutral-alleles diffusion model
Authors: Ethier S. N.
Published: April 2006

The complete set of eigenvalues is found for the (unlabeled) infinitely-many-neutral-alleles diffusion model. The transition density for the process, originally derived by Griffiths, is rederived as an eigenfunction expansion.
On record values and the exponent of a distribution with regularly varying upper tail
Authors: Berred Mohamed
Published: April 2006

Let {Xn, n≧ 1} be an i.i.d. sequence of positive random variables with a continuous distribution function having a regularly varying upper tail. Denote by {X(n),n≧ 1} the corresponding sequence of record values. We introduce two statistics based on the sequence of successive record values and investigate their asymptotic behaviour. We also give some numerical results.
On equivalence of Markov properties over undirected graphs
Authors: Matúš F.
Published: April 2006

The dependence of coincidence of the global, local and pairwise Markov properties on the underlying undirected graph is examined. The pairs of these properties are found to be equivalent for graphs with some small excluded subgraphs. Probabilistic representations of the corresponding conditional independence structures are discussed.
Markov-modulated single-server queueing systems
Authors: Prabhu N. U. , Tang L. C
Published: April 2006

We consider single-server queueing systems that are modulated by a discrete-time Markov chain on a countable state space. The underlying stochastic process is a Markov random walk (MRW) whose increments can be expressed as differences between service times and interarrival times. We derive the joint distributions of the waiting and idle times in the presence of the modulating Markov chain. Our approach is based on properties of the ladder sets associated with this MRW and its time-reversed counterpart. The special case of a Markov-modulated M/M/1 queueing system is then analysed and results analogous to the classical case are obtained.
The moments of FIND
Authors: Paulsen Volkert
Published: April 2006

To study the limiting behaviour of the random running-time of the FIND algorithm, the so-called FIND process was introduced by Grübel and Rösler [1]. In this paper an approach for determining the nth moment function is presented. Applied to the second moment this provides an explicit expression for the variance.
Forward and backward processes in bisexual models with fixed population sizes
Authors: Möhle M.
Published: April 2006

This paper introduces exchangeable bisexual models with fixed population sizes and non-overlapping generations. In each generation there are N pairs of individuals consisting of a female and a male. The N pairs of a generation produce N daughters and N sons altogether, and these 2N children form the N pairs of the next generation at random.

First the extinction of the lines of descendants of a fixed number of pairs is studied, when the population size becomes large. Under suitable conditions this structure can be approximately treated in the framework of a Galton-Watson process. In particular it is shown for the Wright-Fisher model that the discrepancy between the extinction probabilities in the model and in the approximating Galton-Watson process is of order N.

Next, the process of the number of ancestor-pairs of all pairs of a generation is analysed. Under suitable conditions this process, properly normed, has a weak limit as N becomes large. For the Wright-Fisher model this limit is an Ornstein–Uhlenbeck process (restricted to a discrete time-set). The corresponding stationary distributions of the backward processes converge to the normal distribution, as expected.
The evolution of the attainable structures of a homogeneous Markov system by fixed size
Authors: Tsaklidis George M.
Published: April 2006

In order to describe the evolution of the attainable structures of a homogeneous Markov system (HMS) with fixed size, we evaluate the volume of the sets of the attainable structures in Euclidean space as they are changing in time and we find the value of the volume asymptotically. We also estimate the evolution of the distance of two (attainable) structures of the system as it changes following the transformations of the structures; extensions are obtained concerning results from the Perron–Frobenius theory referring to Markov systems.
Homogeneous Markov chains by bounded transition matrix
Authors: Hartfiel D. J.
Published: April 2006

Let A be a stochastic matrix and ε a positive number. We consider all stochastic matrices within ε of A and their corresponding stochastic eigenvectors. A convex polytope containing these vectors is described. An efficient algorithm for computing bounds on the components of these vectors is also given. The work is compared to previous such work done by the author and by Courtois and Semai.
On transient behaviour of a nearest-neighbour birth-death process on a lattice
Authors: Granovsky Boris L. , Rozov Liat
Published: April 2006

We provide the explicit expression for the mean coverage function of a generalized voter model on a regular lattice and establish a characterization of the class of the above processes. As a result, we derive the exact rate of convergence of the considered processes to the steady state. We also prove the existence of different processes with the same mean coverage function on a given lattice.
Continuity of percolation probability in ∞ + 1 dimensions
Authors: Zhang Yu
Published: April 2006

We consider percolation on the graph of the product of a regular treeTwith degreedand the lineℤ, in which each tree edge is open with probability 1 – exp(–JTß) and each line edge is open with probability. LetC(o, 0) be the open cluster for. Denote byθ(β) the percolation probability. Here we show thatθ(β) is continuous whenß>ßc, whereßc= sup{ß:θ(β) = 0}.
Characterization of life distributions under some generalized stochastic orderings
Authors: Cai Jun , Wu Yanhong
Published: April 2006

In this paper we investigate the characterizations of life distributions under four stochastic orderings, <p, <(p), <(p)and <L, by a unified method. Conditions for the stochastic equality of two non-negative random variables under the four stochastic orderings are derived. Many previous results are consequences. As applications, we provide characterizations of life distributions by a single value of their Laplace transforms under orderings <pand <(p)and their moment generating functions under orderings<pand <(p). Under ordering <L, a characterization is given by the expected value of a strictly completely monotone function. The conditions for the stochastic equality of two non-negative vectors under the stochastic orderings <(p), <(p)and <Lare presented in terms of the Laplace transforms and moment generating functions of their extremes and sample means. Characterizations of the exponential distribution amongLandLlife distribution classes are also given.
Likelihood ratios for the infinite alleles model
Authors: Joyce Paul
Published: April 2006

The stationary distribution for the population frequencies under an infinite alleles model is described as a random sequence (x1,x2, · ··) such that Σxi= 1. Likelihood ratio theory is developed for random samples drawn from such populations. As a result of the theory, it is shown that any parameter distinguishing an infinite alleles model with selection from the neutral infinite alleles model cannot be consistently estimated based on gene frequencies at a single locus. Furthermore, the likelihood ratio (neutral versus selection) converges to a non-trivial random variable under both hypotheses. This shows that if one wishes to test a completely specified infinite alleles model with selection against neutrality, the test will not obtain power 1 in the limit.
Functional approximation theorems for controlled renewal processes
Authors: Konstantopoulos Takis , Papadakis Spyros N. , Walrand Jean
Published: April 2006

We prove a functional law of large numbers and a functional central limit theorem for a controlled renewal process, that is, a point process which differs from an ordinary renewal process in that the ith interarrival time is scaled by a function of the number of previous i arrivals. The functional law of large numbers expresses the convergence of a sequence of suitably scaled controlled renewal processes to the solution of an ordinary differential equation. Likewise, the functional central limit theorem establishes that the error in the law of large numbers converges weakly to the solution of a stochastic differential equation. Our proofs are based on martingale and time-change arguments.
Steady state solution of a single-server queue with linear repeated requests
Authors: Artalejo J. R. , Gomez-Corral A.
Published: April 2006

Queueing systems with repeated requests have many useful applications in communications and computer systems modeling. In the majority of previous work the repeat requests are made individually by each unsatisfied customer. However, there is in the literature another type of queueing situation, in which the time between two successive repeated attempts is independent of the number of customers applying for service. This paper deals with the M/G/1 queue with repeated orders in its most general setting, allowing the simultaneous presence of both types of repeat requests. We first study the steady state distribution and the partial generating functions. When the service time distribution is exponential we show that the performance characteristics can be expressed in terms of hypergeometric functions.
Stochastic vector difference equations with stationary coefficients
Authors: Glasserman Paul , Yao David D.
Published: April 2006

We give a unified presentation of stability results for stochastic vector difference equationsbased on various choices of binary operationsand, assuming thatare stationary and ergodic. In the scalar case, under standard addition and multiplication, the key condition for stability isE[log |A0|] < 0. In the generalizations, the condition takes the form γ< 0, where γis the limit of a subadditive process associated with. Under this and mild additional conditions, the processhas a unique finite stationary distribution to which it converges from all initial conditions.

The variants of standard matrix algebra we consider replace the operations + and × with (max, +), (max,×), (min, +), or (min,×). In each case, the appropriate stability condition parallels that for the standard recursions, involving certain subadditive limits. Since these limits are difficult to evaluate, we provide bounds, thus giving alternative, computable conditions for stability.
Random walks in a quarter plane with zero drifts: transience and recurrence
Authors: Asymont I. M. , Fayolle G. , Menshikov Μ. V.
Published: April 2006

In this paper we continue the study of the classification problem for random walks in the quarter plane, with zero drifts in the interior of the domain. The necessary and sufficient conditions for these random walks to be ergodic were found earlier in [3]. Here we obtain necessary and sufficient conditions for transience by constructing suitable Lyapounov functions.
Error bounds for deterministic approximations to Markov processes, with applications to epidemic models
Authors: Gani J. , Yakowitz Sid
Published: April 2006

The computer age and the phenomenological complexity of the AIDS/HIV epidemic have engendered a rich profusion of deterministic and stochastic time series models for the development of an epidemic. The present study examines the reliability of deterministic approximations of fundamentally random processes. Through numerical analysis and probabilistic considerations, we derive absolute and simultaneous confidence interval bounding techniques, and offer a practical procedure based on these developments. A heartening aspect of the computational study presented at the close of this paper indicates that when the population size is in the thousands, the deterministic version to the classical logistic epidemic is a good approximation.
The performance of the move-to-front scheme under some particular forms of Markov requests
Authors: Rodrigues Eliane R.
Published: April 2006

The move-to-front scheme is studied taking into account some forms of Markov dependence for the way items are requested. One of the dependences specifically rules out two consecutive requests for the same item. The other is the so-calledp-correlation. An expression for the stationary distribution of the sequence of arrangements of items is given in each case. A necessary and sufficient condition for these distributions to belong to a particular class of distributions is also given. The mean search time for an item is calculated for each form of dependence and these are compared with the value obtained in the case of independent requests. Some properties of the sequence of requests are given. Finally, an expression for the variance of the search time is obtained.
Exact and ordinary lumpability in finite Markov chains
Authors: Buchholz Peter
Published: April 2006

Exact and ordinary lumpability in finite Markov chains is considered. Both concepts naturally define an aggregation of the Markov chain yielding an aggregated chain that allows the exact determination of several stationary and transient results for the original chain. We show which quantities can be determined without an error from the aggregated process and describe methods to calculate bounds on the remaining results. Furthermore, the concept of lumpability is extended to near lumpability yielding approximative aggregation.
Peaks and Eulerian numbers in a random sequence
Authors: Warren Di , Seneta E.
Published: April 2006

We consider the exact distribution of the number ofpeaksin a random permutation of the integers 1, 2, ···,n. This arises from a test of whether n successive observations from a continuous distribution are i.i.d. The Eulerian numbers, which figure in the p.g.f., are then shown to provide a link between the simpler problem ofascents(which has been thoroughly analysed) and both our problem ofpeaksand similar problems on thecircle. This link then permits easy deduction of certain general properties, such as linearity innof the cumulants, in the more complex settings. Since the focus of the paper is on exact distributional results, a uniform bound on the deviation from the limiting normal is included. A secondary purpose of the paper is synthesis, beginning with the more familiar setting ofpeaks and troughs.
Tail of compound distributions and excess time
Authors: Lin Xiaodong
Published: April 2006

Bounds on the tail of compound distributions are considered. Using a generalization of Wald's fundamental identity, we derive upper and lower bounds for various compound distributions in terms of new worse than used (NWU) and new better than used (NBU) distributions respectively. Simple bounds are obtained when the claim size distribution is NWUC, NBUC, NWU, NBU, IMRL, DMRL, DFR and IFR. Examples on how to use these bounds are given.
Lundberg inequalities for a Cox model with a piecewise constant intensity
Authors: Schmidli Hanspeter
Published: April 2006

A Cox risk process with a piecewise constant intensity is considered where the sequence (Li) of successive levels of the intensity forms a Markov chain. The durationσiof the levelLiis assumed to be only dependent viaLi. In the small-claim case a Lundberg inequality is obtained via a martingale approach. It is shown furthermore by a Lundberg bound from below that the resulting adjustment coefficient gives the best possible exponential bound for the ruin probability. In the case where the stationary distribution ofLicontains a discrete component, a Cramér–Lundberg approximation can be obtained. By way of example we consider the independent jump intensity model (Björk and Grandell 1988) and the risk model in a Markovian environment (Asmussen 1989).
The distribution of Brownian quantiles
Authors: Yor Marc
Published: April 2006

The distribution of Brownian quantiles is determined, simplifying related integral expressions obtained by Lévy [9], [10] and more recently by Miura [11]. Three proofs are given, two of them involving last-passage times of Brownian motion, before time 1, at a given level.