Showing papers in "Journal of Applied Probability in 1996"

On the prediction of fractional Brownian motion

Abstract: Integration with respect to the fractional Brownian motion Z with Hurst parameter H∈ (1/2, 1) is discussed. The predictor E[Z a | Z s , s ∈ (- T, 0)] is represented as an integral with respect to Z, solving a weakly singular integral equation for the prediction weight function.

Buffer overflow asymptotics for a buffer handling many traffic sources

Abstract: As a model for an ATM switch we consider the overflow frequency of a queue that is served at a constant rate and in which the arrival process is the superposition of N traffic streams We consider an asymptotic as N → ∞ in which the service rate Nc and buffer size Nb also increase linearly in N In this regime, the frequency of buffer overflow is approximately exp(–NI(c, b)), where I(c, b) is given by the solution to an optimization problem posed in terms of time-dependent logarithmic moment generating functions Experimental results for Gaussian and Markov modulated fluid source models show that this asymptotic provides a better estimate of the frequency of buffer overflow than ones based on large buffer asymptotics

Multivariate normal approximations by Stein's method and size bias couplings

Abstract: Stein's method is used to obtain two theorems on multivariate normal approximation. Our main theorem, Theorem 1.2, provides a bound on the distance to normality for any non-negative random vector. Theorem 1.2 requires multivariate size bias coupling, which we discuss in studying the approximation of distributions of sums of dependent random vectors. In the univariate case, we briefly illustrate this approach for certain sums of nonlinear functions of multivariate normal variables. As a second illustration, we show that the multivariate distribution counting the number of vertices with given degrees in certain random graphs is asymptotically multivariate normal and obtain a bound on the rate of convergence. Both examples demonstrate that this approach may be suitable for situations involving non-local dependence. We also present Theorem 1.4 for sums of vectors having a local type of dependence. We apply this theorem to obtain a multivariate normal approximation for the distribution of the random p-vector, which counts the number of edges in a fixed graph both of whose vertices have the same given color when each vertex is colored by one of p colors independently. All normal approximation results presented here do not require an ordering of the summands related to the dependence structure. This is in contrast to hypotheses of classical central limit theorems and examples, which involve for example, martingale, Markov chain or various mixing assumptions.

A Pollaczek–Khintchine formula for M/G/1 queues with disasters

Abstract: A disaster occurs in a queue when a negative arrival causes all the work (and therefore customers) to leave the system instantaneously. Recent papers have addressed several issues pertaining to queueing networks with negative arrivals under the i.i.d. exponential service times assumption. Here we relax this assumption and derive a Pollaczek–Khintchine-like formula for M / G /1 queues with disasters by making use of the preemptive LIFO discipline. As a byproduct, the stationary distribution of the remaining service time process is obtained for queues operating under this discipline. Finally, as an application, we obtain the Laplace transform of the stationary remaining service time of the customer in service for unstable preemptive LIFO M / G /1 queues.

Time series models with univariate margins in the convolution-closed infinitely divisible class

Abstract: A unified way of obtaining stationary time series models with the univariate margins in the convolution-closed infinitely divisible class is presented. Special cases include gamma, inverse Gaussian, Poisson, negative binomial, and generalized Poisson margins. ARMA time series models obtain in the special case of normal margins, sometimes in a different stochastic representation. For the gamma and Poisson margins, some previously defined time series models are included, but for the negative binomial margin, the time series models are different and, in several ways, better than previously defined time series models. The models are related to multivariate distributions that extend a univariate

On a Markov chain approach for the study of reliability structures

Abstract: In this paper we consider a class of reliability structures which can be efficiently described through (imbedded in) finite Markov chains. Some general results are provided for the reliability evaluation and generating functions of such systems. Finally, it is shown that a great variety of well known reliability structures can be accommodated in this general framework, and certain properties of those structures are obtained on using their Markov chain imbedding description.

Stability and structural properties of stochastic storage networks

Abstract: We establish stability, monotonicity, concavity and subadditivity properties for open stochastic storage networks in which the driving process has stationary increments. A principal example is a stochastic fluid network in which the external inputs are random but all internal flows are deterministic. For the general model, the multi-dimensional content process is tight under the natural stability condition. The multi-dimensional content process is also stochastically increasing when the process starts at the origin, implying convergence to a proper limit under the natural stability condition. In addition, the content process is monotone in its initial conditions. Hence, when any content process with non-zero initial conditions hits the origin, it couples with the content process starting at the origin. However, in general, a tight content process need not hit the origin.

Stein's method for geometric approximation

Abstract: The Stein-Chen method for Poisson approximation is adapted to the setting of the geometric distribution. This yields a convenient method for assessing the accuracy of the geometric approximation to the distribution of the number of failures preceding the first success in dependent trials. The results are applied to approximating waiting time distributions for patterns in coin tossing, and to approximating the distribution of the time when a stationary Markov chain first visits a rare set of states. The error bounds obtained are sharper than those obtainable using related Poisson approximations.

The censored markov chain and the best augmentation

Abstract: Computationally, when we solve for the stationary probabilities for a countable-state Markov chain, the transition probability matrix of the Markov chain has to be truncated, in some way, into a finite matrix. Different augmentation methods might be valid such that the stationary probability distribution for the truncated Markov chain approaches that for the countable Markov chain as the truncation size gets large. In this paper, we prove that the censored (watched) Markov chain provides the best approximation in the sense that, for a given truncation size, the sum of errors is the minimum and show, by examples, that the method of augmenting the last column only is not always the best.

Convergence properties of simulated annealing for continuous global optimization

Abstract: In this paper conditions for the convergence of a class of simulated annealing algorithms for continuous global optimization are given. The previous literature about the subject gives results for the convergence of algorithms in which the next candidate point is generated according to a probability distribution whose support is the whole feasible set. A class of possible cooling schedules has been introduced in order to remove this restriction.

Economies of scale in queues with sources having power-law large deviation scalings

Abstract: We analyse the queue Q L at a multiplexer with L sources which may display long-range dependence. This includes, for example, sources modelled by fractional Brownian motion (FBM). The workload processes W due to each source are assumed to have large deviation properties of the form P[W,Ia(t)>x]exp[-v(t)K(x)] for appropriate scaling functions a and v, and rate-function K. Under very general conditions lim L→ ∞ L -1 log P[Q L > Lb] = -I(b), provided the offered load is held constant, where the shape function I is expressed in terms of the cumulant generating functions of the input traffic. For power-law sealings v(t) = t v , a(t) = t a (such as occur in FBM) we analyse the asymptotics of the shape function lim b→ ∞ b -u/a (I(b) - (δb v/a ) = v for some exponent u and constant v depending on the sources. This demonstrates the economies of scale available though the multiplexing of a large number of such sources, by comparison with a simple approximation P[Q L > Lb] exp[ - δLb v/a ] based on the asymptotic decay rate δ alone. We apply this formula to Gaussian processes, in particular FBM, both alone. and also perturbed by an Ornstein-Uhlenbeck process. This demonstrates a richer potential structure than occurs for sources with linear large deviation scalings.

Comparing sums of exchangeable Bernoulli random variables

Abstract: The paper is first concerned with a comparison of the partial sums associated with two sequences of n exchangeable Bernoulli random variables. It then considers a situation where such partial sums are obtained through an iterative procedure of branching type stopped at the first-passage time in a linearly decreasing upper barrier. These comparison results are illustrated with applications to certain urn models, sampling schemes and epidemic processes. A key tool is a non-standard hierarchical class of stochastic orderings between discrete random variables valued in {0, 1,..., n}.

On the asymptotic joint distribution of the sum and maximum of stationary normal random variables

Abstract: Let X 1 , X 2 , ·· ·be stationary normal random variables with ρ n = cov( X 0 , X n ). The asymptotic joint distribution of and is derived under the condition ρ n log n → γ [0,∞). It is seen that the two statistics are asymptotically independent only if γ = 0.

Dynamic bandwidth allocation for atm switches

Abstract: We explore a dynamic approach to the problems of call admission and resource allocation for communication networks with connections that are differentiated by their quality of service requirements. In a dynamic approach. the amount of spare resources is estimated on-line based on feedbacks fromm the network's quality of service monitoring mechanism. The schemes we propose remove the dependence on accurate traffic models and thus simplify the tasks of supplying traffic statistics required of network users. In this paper we present two dynamic algorithms. The objective of these algorithms is to find the minimum bandwidth necessary to satisfy a cell loss probability constraint at an asynchronous transfer mode (ATM) switch. We show that in both schemes the bandwidth chosen by the algorithm approaches the optimal value almost surely. Furthermore, in the second scheme. which determines the point closest to the optimal bandwidth from a finite number of choice, the expected learning time is finite.

First passage time distribution of a Wiener process with drift concerning two elastic barriers

Abstract: We solve the Fokker-Planck equation for the Wiener process with drift in the presence of elastic boundaries and a fixed start point. An explicit expression is obtained for the first passage density. The cases with pure absorbing and/or reflecting barriers arise for a special choice of a parameter constellation. These special cases are compared with results in Darling and Siegert [5] and Sweet and Hardin [15].

On the full information best-choice problem

Abstract: We introduce the optimal stopping problem of an infinite sequence of records associated with a planar Poisson process. This problem serves as a limiting form of the classical full information best-choice problem. A link between the finite problem and its limiting form is established via embedding n i.i.d. observations into the planar process.

On effects of discretization on estimators of drift parameters for diffusion processes

Abstract: In this paper statistical properties of estimators of drift parameters for diffusion processes are studied by modern numerical methods for stochastic differential equations. This is a particularly useful method for discrete time samples, where estimators can be constructed by making discrete time approximations to the stochastic integrals appearing in the maximum likelihood estimators for continuously observed diffusions. A review is given of the necessary theory for parameter estimation for diffusion processes and for simulation of diffusion processes. Three examples are studied.

On identifiability and order of continuous-time aggregated Markov chains, Markov-modulated Poisson processes, and phase-type distributions

Abstract: An aggregated Markov chain is a Markov chain for which some states cannot be distinguished from each other by the observer. In this paper we consider the identifiability problem for such processes in continuous time, i.e. the problem of determining whether two parameters induce identical laws for the observable process or not. We also study the order of a continuous-time aggregated Markov chain, which is the minimum number of states needed to represent it. In particular, we give a lower bound on the order. As a by-product, we obtain results of this kind also for Markov-modulated Poisson processes, i.e. doubly stochastic Poisson processes whose intensities are directed by continuous-time Markov chains, and phase-type distributions, which are hitting times in finite-state Markov chains.

On the limit behaviour of a superadditive bisexual Galton–Watson branching process

Abstract: The asymptotic behaviour of a superadditive bisexual Galton-Watson branching process studied. Sufficient conditions for the almost sure and L l convergence of the suitably normed process are given. Finally, a first approach to the study of the L' convergence for a superadditive bisexual Galton-Watson branching process under the Z log + Z condition is considered.

Peaks and eulerian numbers in a random sequence

Abstract: We consider the exact distribution of the number of peaks in 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 of ascents (which has been thoroughly analysed) and both our problem of peaks and similar problems on the circle. This link then permits easy deduction of certain general properties, such as linearity in n of 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 of peaks and troughs.

Tail of compound distributions and excess time

Abstract: 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.

Solution of finite QBD processes

Abstract: We provide an explicit matrix analytic solution for finite quasi birth and death (QBD) processes, directly expressed in terms of process parameters. We show that this solution has the same asymptotic complexity of previously proposed non-explicit solutions and is more general than some of them. Moreover, it can be easily extended to the case of generalized QBD processes.

Principles for modelling financial markets

Abstract: The paper introduces an approach focused towards the modelling of dynamics of financial markets. It is based on the three principles of market clearing, exclusion of instantaneous arbitrage and minimization of increase of arbitrage information. The last principle is equivalent to the minimization of the difference between the risk neutral and the real world probability measures. The application of these principles allows us to identify various market parameters, e.g. the risk-free rate of return. The approach is demonstrated on a simple financial market model, for which the dynamics of a virtual risk-free rate of return can be explicitly computed.

On the distribution of distances in recursive trees

Abstract: Recursive trees have been used to model such things as the spread of epidemics, family trees of ancient manuscripts, and pyramid schemes. A tree T n with n labeled nodes is recursive tree if n = 1, or n >1 and T n can be constructed by joining node n to a node of some recursive tree T n-1 . For arbitrary nodes i < n in a random recursive tree we give the exact distribution of X i,n , the distance between nodes i and n. We characterize this distribution as the convolution or the law of X i,i+1 and n - i - 1 Bernoulli distributions. We further characterize the law of X i,i+1 as a mixture of sums of Bernoulli. For i = i n growing as a function of n, we show that X in,n is asymptotically normal in several settings.

Exact results for a secretary problem

Abstract: We consider the following secretary problem: items ranked from 1 to n are randomly selected without replacement, one at a time, and to 'win' is to stop at an item whose overall rank is less than or equal to s, given only the relative ranks of the items drawn so far. Our method of analysis is based on the existence of an imbedded Markov chain and uses the technique of backwards induction. In principal the approach can be used to give exact results for any value of s; we do the working for s = 3. We give exact results for the optimal strategy, the probability of success and the distribution of T, and the total number of draws when the optimal strategy is implemented. We also give some asymptotic results for these quantities as n -+ o.

Rate modulation in dams and ruin problems

Abstract: We consider a dam in which the release rate depends both on the state and some modulating process. Conditions for the existence of a limiting distribution are established in terms of an associated risk process. The case where the release rate is a product of the state and the modulating process is given special attention, and in particular explicit formulas are obtained for a finite state space Markov modulation.

Busy period in GIx/G/∞

Abstract: Busy period problems in infinite server queues are studied systematically, starting from the batch service time. General relations are given for the lengths of the busy cycle, busy period and idle period, and for the number of customers served in a busy period. These relations show that the idle period is the most difficult while the busy cycle is the simplest of the four random variables. Renewal arguments are used to derive explicit results for both general and special cases.

A modified block replacement policy with two variables and general random minimal repair cost

Abstract: This paper considers a modified block replacement with two variables and general random minimal repair cost. Under such a policy, an operating system is preventively replaced by new ones at times kT (k= 1, 2, ···) independently of its failure history. If the system fails in [(k − 1)T, (k − 1)T+ T0 ) it is either replaced by a new one or minimally repaired, and if in [(k − 1) T + T 0 , kT) it is either minimally repaired or remains inactive until the next planned replacement. The choice of these two possible actions is based on some random mechanism which is age-dependent. The cost of the ith minimal repair of the system at age y depends on the random part C(y) and the deterministic part ci (y). The expected cost rate is obtained, using the results of renewal reward theory. The model with two variables is transformed into a model with one variable and the optimum policy is discussed.

Convergence rates for M/G/1 queues and ruin problems with heavy tails

Abstract: The time-dependent virtual waiting time in a M/G/1 queue converges to a proper limit when the traffic intensity is less than one. In this paper we give precise rates on the speed of this convergence when the service time distribution has a heavy regularly varying tail. The result also applies to the classical ruin problem. We obtain the exact rate of convergence for the ruin probability after time t for the case where claims arrive according to a Poisson process and claim sizes are heavy tailed. Our result supplements similar theorems on exponential convergence rates for relaxation times in queueing theory and ruin probabilities in risk theory.

Performance analysis of the discrete-time GI / Geom /1/ N queue

Abstract: This paper presents an analysis of the single-server discrete-time finite-buffer queue with general interarrival and geometric service time, GI/GeomIl//N. Using the supplementary variable technique, and considering the remaining interarrival time as a supplementary variable, two variations of this model, namely the late arrival system with delayed access (LAS-DA) and early arrival system (EAS), have been examined. For both cases, steady-state distributions for outside observers as well as at random and prearrival epochs have been obtained. The waiting time analysis has also been carried out. Results for the GeomIG/1/N queue with LAS-DA have been obtained from the GIIGeomIl//N queue with EAS. We also give various performance measures. An algorithm for computing state probabilities is given in an appendix.