bluuebunny/crossref_metadata_2025 huggingface dataset:
Characterizations of the normal distribution by suitable transformations
Authors: Beer S. , Lukacs E.
Published: April 2006

Yu. V. Linnik showed that certain transformations, given by Formulae (1.1), (1.6) and (1.7) transform a normal sample into itself. The transformations (1.1) and (1.7) apply to samples of size 2 while (1.6) admits an arbitrary sample size. It is also assumed that the population mean is zero.

In the present paper the converse theorems are proven so that characterizations of the normal distribution are obtained. The problem leads to the functional equations (2.3) and (2.13) whose solution yields the desired results.
Regular stochastic matrices and digraphs
Authors: Berman Elizabeth
Published: April 2006

This paper presents an algorithm to determine whether a stochastic matrix is regular. The main theorem is the following. Hypothesis: An n-by-n stochastic matrix has at least one positive entry off the main diagonal in every row and column. There is at most one row with n — 1 zeros and at most one column with n — 1 zeros. There are no j-by-k submatrices consisting entirely of zeros, where j and k are integers greater than 1, with j + k = n. Conclusion: The matrix is regular. Similar results hold for strongly connected digraphs.
Gaussian approximations to service problems: a communication system example
Authors: Gaver Donald P. , Lehoczky John P.
Published: April 2006

Messages arrive at a group of service channels in accordance with a time-dependent Poisson process. An arrival either (i) immediately beginsk-stage Markovian service if an empty channel is reached, or (ii) balks and enters a retrial population if the channel sought is busy. Diffusion approximations to the number of messages in service (each stage) and in the retrial population are derived by writing stochastic differential equations (I+ 0). Steady-state distributions are found and compared with certain simulation results.
Supercritical age-dependent branching processes with immigration
Authors: Athreya K. B. , Parthasarathy P. R. , Sankaranarayanan G.
Published: April 2006

A branching process with immigration of the following type is considered. For every i, a random number Ni of particles join the system at time . These particles evolve according to a one-dimensional age-dependent branching process with offspring p.g.f. and life time distribution G(t). Assume . Then it is shown that Z(t) e–αt converges in distribution to an extended real-valued random variable Y where a is the Malthusian parameter. We do not require the sequences {τi} or {Ni} to be independent or identically distributed or even mutually independent.
Random walks with negative drift conditioned to stay positive
Authors: Iglehart Donald L.
Published: April 2006

Let {Xk: k ≧ 1} be a sequence of independent, identically distributed random variables with EX1 = μ < 0. Form the random walk {Sn: n ≧ 0} by setting S0 = 0, Sn = X1 + … + Xn, n ≧ 1. Let T denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of X1) that Sn, conditioned on T > n converges weakly to a limit random variable, S∗, and to find the Laplace transform of the distribution of S∗. We also investigate a collection of random walks with mean μ < 0 and conditional limits S∗ (μ), and show that S∗ (μ), properly normalized, converges to a gamma distribution of second order as μ ↗ 0. These results have applications to the GI/G/1 queue, collective risk theory, and the gambler's ruin problem.
On the busy period of discrete-time queues
Authors: Gergely T. , Török T. L.
Published: April 2006

This paper obtains the probability and the expectation of the length of the busy and idle periods in a discrete-time service system by means of two-component Markov chains and their first passage times. The M/G/1 model is discussed as a special case.
Searching for a one-dimensional random walker
Authors: McCabe Bernard J.
Published: April 2006

Let {xk}k ≧ − r be a simple Bernoulli random walk with x–r = 0. An integer valued threshold ϕ = {ϕk}k≧1 is called a search plan if |ϕk+1−ϕk|≦1 for all k ≧ 1. If ϕ is a search plan let τϕ be the smallest integer k such that x and ϕ cross or touch at k. We show the existence of a search plan ϕ such that ϕ1 = 0, the definition of ϕ does not depend on r, and the first crossing time τϕ has finite mean (and in fact finite moments of all orders). The analogous problem for the Wiener process is also solved.
The linear cell-size-dependent branching process
Authors: Clifford Peter , Sudbury Aidan
Published: April 2006

In this paper is developed a theory of branching processes in which the division probability and rate of growth of cells depend only on their ‘size’ and in which ‘size’ is shared between daughters. Specific results are obtained in a linear case including the calculation of the correlation coefficient for the life spans of sisters.
Spectral analysis for a random process on the circle
Authors: Roy Roch
Published: April 2006

A random process on the circle is a family of random variables X(P,t) indexed by the position P on the unit circle and by the time t (t = 0, + 1, ···). For a homogeneous and stationary process, using its Fourier series expansion, we deduce a spectral representation of the covariance function. The purpose of the paper is to develop a spectral analysis when X(P,t) is observed at all the points on the circle at t = 0, 1, ···, T – 1. The asymptotic distribution of the family of finite Fourier transforms, of the family of periodograms and of the family of spectral densities estimates are obtained from results available for vector-valued time series. Also, a sample covariance function for X (P,t) is defined and its asymptotic distribution derived.
A low traffic approximation for queues
Authors: Bloomfield P. , Cox D. R.
Published: April 2006

A general procedure is outlined for obtaining lower bounds and approximations to the amount of congestion in queues with low traffic. Some detailed formulae are given for a number of single-server systems and compared with exact solutions where available. Results are also given for a discrete time system in which departures clash with new arrivals.
A note on the number of busy servers in a GI/G/s queue in light traffic
Authors: Kennedy Douglas P.
Published: April 2006

It is proved that in a GI/G/s queue with the traffic intensity ρ < (S – m + l)/S, m ≧ 1, at least m servers are simultaneously idle infinitely often with probability one.
Convergence of transport processes with radially symmetric direction changes, and chain molecules
Authors: Gorostiza Luis G.
Published: April 2006

When passing from two to more dimensions, the study of non-isotropic scattering transport processes, and chain molecules, which are both covered by the same mathematical model, becomes affected by the non-commutativity of rotations. The techniques developed in [2], together with results on radially symmetric direction changes, are used in this paper to obtain a functional central limit theorem for those random processes, with a suitable normalization, the limit being a Brownian motion process which is completely identified.
A quantum-mechanical central limit theorem for anti-commuting observables
Authors: Hudson R. L.
Published: April 2006

A quantum-mechanical central limit theorem for sums of pairwise anti-commuting representations of the canonical anti-commutation relations over a finite-dimensional space is formulated and proved.
The estimation of frequency
Authors: Hannan E. J.
Published: April 2006

Very general forms of the strong law of large numbers and the central limit theorem are proved for estimates of the unknown parameters in a sinusoidal oscillation observed subject to error. In particular when the unknown frequency θ0, is in fact 0 or <it is shown that the estimate, , satisfies for N ≧ N0 (ω) where N0 (ω) is an integer, determined by the realisation, ω, of the process, that is almost surely finite.
An example on geometric ergodicity of a finite Markov chain
Authors: Teugels Jozef L.
Published: April 2006

A discrete time Markov chain with (n + 1) states is constructed in which there are n states having different decay parameters for geometric ergodicity.
Determination of the order of a Markov chain by Akaike's information criterion
Authors: Tong H.
Published: April 2006

Using Akaike's information criterion, we have presented an objective procedure for the determination of the order of an ergodic Markov chain with a finite number of states. The procedure exploits the asymptotic properties of the maximum likelihood ratio statistics and Kullback and Leibler's mean information for the discrimination between two distributions. Numerical illustrations are given, using data from Bartlett (1966), Good and Gover (1967) and some weather records.
Conditional moments in a critical age-dependent branching process
Authors: Weiner Howard J.
Published: April 2006

Let X(t), N(t) respectively denote the number of cells alive at t and the total number of cells born by t in a critical age-dependent Bellman-Harris branching process.

The asymptotic behavior of the conditional moments, for 0 < α ≦ 1, E(Nn(αt) | X(t) > 0), E(Nn(t) |X(αt) > 0), is obtained.
The uncontrolled traffic merge
Authors: Cowan Richard
Published: April 2006

The merging of n independent traffic streams is studied. The arrival processes comprise random sized bunches of cars, the bunches being separated in time by random sized gaps. Within each bunch, cars are spaced at one time unit apart. Under a distributional assumption on the inter-bunch gaps exact formulae are found for (a) the mean delay to cars, (b) the first two moments of the post-merge bunch size, and (c) the distribution of the inter-bunch gaps in the post-merge process. It is shown that the distributional assumption is realistic and moreover that the distribution of (c) is of the same form. The arrival bunches have general size distribution with finite mean and variance.
Maximum likelihood estimates of incorrect Markov models for time series and the derivation of AIC
Authors: Ogata Yosihiko
Published: April 2006

The asymptotic behavior of the maximum likelihood estimators of Markov models or autoregressive models are given when the true distribution is not a member of the assumed parametric family. The derivation of Akaike's Information Criterion is reviewed for this case.
Scheduling tasks with exponential service times on non-identical processors to minimize various cost functions
Authors: Weiss Gideon , Pinedo Michael
Published: April 2006

We consider preemptive scheduling of N tasks on m processors; processors have different speeds, tasks require amounts of work which are exponentially distributed, with different parameters. The policies of assigning at every moment the task with shortest (longest) expected processing time among those not yet completed to the fastest processor available, second shortest (longest) to the second fastest etc., are examined, and shown to minimize expected values of various cost functions. As special cases we obtain policies which minimize expected flowtime, expected makespan and expected lifetime of a series system with m component locations and N spares.
The distribution of the convex hull of a Gaussian sample
Authors: Eddy William F.
Published: April 2006

The distribution of the convex hull of a random sample of d-dimensional variables is described by embedding the collection of convex sets into the space of continuous functions on the unit sphere. Weak convergence of the normalized convex hull of a circular Gaussian sample to a process with extreme-value marginal distributions is demonstrated. The proof shows that an underlying sequence of point processes converges to a Poisson point process and then applies the continuous mapping theorem. Several properties of the limit process are determined.
Some analyses on the control of queues using level crossings of regenerative processes
Authors: Shanthikumar J. G.
Published: April 2006

Some properties of the number of up- and downcrossings over level u, in a special case of regenerative processes are discussed. Two basic relations between the density functions and the expected number of upcrossings of this process are derived. Using these reults, two examples of controlledM/G/1 queueing systems are solved. Simple relations are derived for the waiting time distribution conditioned on the phase of control encountered by an arriving customer. The Laplace-Stieltjes transform of the distribution function of the waiting time of an arbitrary customer is also derived for each of these two examples.
Limiting results for arrays of binary random variables on rectangular lattices under sparseness conditions
Authors: Saunders Roy , Kryscio Richard J. , Funk Gerald M.
Published: April 2006

In this article we give limiting results for arrays {Xij(m, n) (i, j)Dmn} of binary random variables distributed as particular types of Markov random fields overmxnrectangular latticesDmn.Under some sparseness conditions which restrict the number ofXij(m, n)'s which are equal to one we show that the random variables (l= 1, ···,r)converge to independent Poisson random variables for 0 <d1< d2< · ·· <drwhenm→∞ nd∞.The particular types of Markov random fields considered here provide clustering (or repulsion) alternatives to randomness and involve several parameters. The limiting results are used to consider statistical inference for these parameters. Finally, a simulation study is presented which examines the adequacy of the Poisson approximation and the inference techniques when the lattice dimensions are only moderately large.
Some new integral geometric formulae, with stochastic applications
Authors: Miles R. E.
Published: April 2006

Alternative forms of the integral geometric density of an r-subspace [r-flat] containing q[q + 1] points in euclidean n-space Rn are given Stochastic applications in R3 include formulae for

(i) the mean area of intersection of a domain by an isotropic plane through the origin; and

(ii) the variance of the area of intersection of a domain by an isotropic uniform plane.
Optimal stopping in a semi-Markov shock model
Authors: Zuckerman Dror
Published: April 2006

We examine a failure model for a system existing in a random environment. The system accumulates damage through a shock process and the failure time depends on the accumulated damage in the system. The cumulative damage process is assumed to be a semi-Markov process. Upon failure the system must be replaced by a new identical one and a failure cost is incurred. If the system is replaced before failure, a smaller cost is incurred. We allow a controller to replace the system at any stopping time before failure time. We consider the problem of specifying a replacement rule which minimizes the total long-run average cost per unit time.
The expected population size in a cell-size-dependent branching process
Authors: Sudbury Aidan
Published: April 2006

In cell-size-dependent growth the probabilistic rate of division of a cell into daughter-cells and the rate of increase of its size depend on its size. In this paper the expected number of cells in the population at time t is calculated for a variety of models, and it is shown that population growths slower and faster than exponential are both possible. When the cell sizes are bounded conditions are given for exponential growth.
Stationary state probabilities at arrival instants for closed queueing networks with multiple types of customers
Authors: Lavenberg S. S. , Reiser M.
Published: April 2006

We consider closed networks of interconnected service centers with multiple types of customers and multiple classes, whose stationary state probabilities at arbitrary times have a product form. A customer can change its class but not its type as it traverses the network. We show that the stationary state probabilities at instants at which customers of a particular type arrive at a particular service center and enter a particular class are equal to the stationary state probabilities at arbitrary times for the network with one less customer of that type. Applications of this result are given.
Inequalities for the probability of the occurrence of at least m out of n events
Authors: Sathe Y. S. , Pradhan Meena , Shah S. P.
Published: April 2006

Given n events A1, A2, · ··, An, bounds are obtained for the probability of the occurrence of at least m out of the n events. The bounds are linear in terms of S1 and S2 where S1 = Σi P(Ai) and S2 = Σi<jΡ (Α i,Α j). The bounds are illustrated with an example. For m = 1 the bounds are the same as the best known linear bounds in terms of S1 and S2.
Stochastic stable population growth
Authors: Lange Kenneth , Holmes William
Published: April 2006

In classical demographic theory, the age structure of a population eventually stabilizes, and the population as a whole grows at a geometric rate. It is possible to prove stochastic analogues of these results if vital rates fluctuate according to a stationary stochastic process. The approach taken here is to study the action of random matrix products on random vectors. This permits the application of Hilbert's projective metric and leads to considerable simplification of the ergodic and central limit theory of population growth.
Survival under the pure birth shock model
Authors: Klefsjö Bengt
Published: April 2006

Suppose that a device is subjected to shocks and that denotes the probability of surviving k shocks. Then is the probability that the device will survive beyond t, where is the counting process which governs the arrival of shocks. A-Hameed and Proschan (1975) considered the survival function H(t) under what they called the pure birth shock model. In this paper we shall prove that is IFRA and DMRL under conditions which differ from those used by A-Hameed and Proschan (1975).
Dynamically optimized replacement with a Markovian renewal process
Authors: Siedersleben Johannes
Published: April 2006

We consider a system that gradually deteriorates. At random times Tn the system is inspected and may or may not be replaced, but it must be if its state is too bad. The process of deterioration is described by a Markovian renewal process. Under some monotonicity conditions we derive a policy for the finite and the infinite horizon which minimizes the total discounted costs. This paper is a generalization of a result of Feldman (1977).
On central limit and iterated logarithm supplements to the martingale convergence theorem
Authors: Heyde C. C.
Published: April 2006

Let {Sn, n ≧ 1} be a zero, mean square integrable martingale for which so that Sn → S∞ a.s., say, by the martingale convergence theorem. The paper is principally concerned with obtaining central limit and iterated logarithm results for Bn(Sn – S∞) where the multipliers Bn ↑ ∞ a.s. An example on the Pólya urn scheme is given to illustrate the results.
Two theorems on optimal stopping with backward solicitation
Authors: Karni Edi , Schwartz Aba
Published: April 2006

This paper deals with optimal stopping rules for a sampling process with uncertain recall, i.e. the probability that a past observation is currently available declines exponentially with the time elapsed since it was last observed. The main result of this paper is that for such sampling processes, and for any utility function, if a solicitation of a past observation incurs the same cost as a new draw, then it is never optimal to continue the sampling when the observation solicited is found to be available. This result applies to both bounded and unbounded sequential decision procedures.
Estimating the law of randomly moving particles by counting
Authors: McDunnough Philip
Published: April 2006

The fluctuation of the counts of particles inmregions are used to yield an estimate of a fundamental motion parameter. The effect of varyingmon the precision of the estimation method is considered. An exact analysis and simulation of the Fürth pedestrian vector process is presented.
Renewal processes with random numbers of delays: application to a conception and birth model
Authors: Lange Kenneth , Johnson Norman J.
Published: April 2006

Asymptotic formulas and Laplace–Stieltjes transforms are derived for the first two moments of a renewal process with a random number of delays. These are simplified when all the delays follow the same distribution. An asymptotic occupancy result is also derived for two-stage renewal processes with random numbers of delays. As an example, a demographic model of conception and birth is discussed. This model represents the sequence of live births to a woman as a renewal process. If the woman practises birth control after achieving her desired family composition, the renewal process has a random number of delays.
The weak convergence of a class of estimators of the variance function of a two-dimensional Poisson process
Authors: Liebetrau A. M.
Published: April 2006

Results of a previous paper (Liebetrau (1977a)) are extended to higher dimensions. An estimator V∗(t1, t2) of the variance function V(t1, t2) of a two-dimensional process is defined, and its first- and second-moment structure is given assuming the process to be Poisson. Members of a class of estimators of the form where and for 0 < α i < 1, are shown to converge weakly to a non-stationary Gaussian process. Similar results hold when the t′i are taken to be constants, when V is replaced by a suitable estimator and when the dimensionality of the underlying Poisson process is greater than two.
A two-dimensional ‘immigration–branching' model with application to earthquake occurrence times and energies
Authors: Lai C. D.
Published: April 2006

A two-dimensional Poisson cluster point process is formulated by the use of a probability generating functional. Moment measures of both the cluster centre and member processes are discussed. An example is provided, and the magnitude frequency law is proved in this case.
The asymptotic behaviour of an invasion process
Authors: Kelly F. P.
Published: April 2006

Black and white cells are positioned at the vertices of a rectangular lattice. When a cell division occurs, the daughter cells are of the same colour as the parent cell; one of them replaces an adjacent cell and the other remains in the position of the parent cell. In one variant of the model it is assumed that whenever a white cell appears at the origin it is transformed into a black cell; apart from this the black and white cells are equally competitive and in particular they divide at the same rate. Initially, only the cell at the origin is black. The asymptotic behaviour of the black clone is investigated.
Lanchester square-law battles: Transient and terminal distributions
Authors: Goldie Charles M.
Published: April 2006

For a battle governed by the stochastic version of Lanchester's square law the transient distribution of state is found; also various terminal distributions. The terminal distributions arise also in certain related random walks.
A restricted occupancy problem
Authors: Fang Kai-Tai
Published: April 2006

There are m urns each containing k cells; n balls are assigned to the m urns in such a way that each cell contains at most one ball. Let Mt be the number of urns containing exactly t balls (t = 0, 1, •••, k). In the paper, the distribution of Mt, its moments and moment-generating function are obtained. The derivation of the last two seems to be new.
Acute triangles in the n-ball
Authors: Hall Glen Richard
Published: April 2006

Using Baddeley's [1] extension of Crofton's differential equation we derive an elementary integral formula for the probability that three randomly chosen points in the unit n-ball in ℝn, with respect to Lebesgue measure, form an acute triangle. When the dimension is 2 this probability is 4/π2 − 1/8, while when the dimension is 3 it is 33/70.
Simple trigonometric models for narrow-band stationary processes
Authors: Hasofer A. M.
Published: April 2006

Over a finite interval, a Gaussian stationary process can be approximated by a finite trigonometric sum, and the error introduced by the approximation can be exactly bounded, as far as the distribution of the upper tail of the maximum is concerned. A simple case is exhibited, where a narrow band process is well approximated by means of a two-term trigonometric representation.
The robust estimation of autoregressive processes by functional least squares
Authors: Heathcote C. R. , Welsh A. H.
Published: April 2006

The stationary autoregressive model but with a long-tailed error distribution is analysed using the method of functional least squares. A family of estimators indexed by a real parameter is obtained and uniform consistency and weak convergence established. The optimum member of the family is chosen to have minimum variance with respect to the parameter, and the parameter value chosen detects and adjusts for long-tailed error distributions. Results of a simulation are given.
Permutation probabilities for gamma random variables
Authors: Henery Robert J.
Published: April 2006

The order statistics of a set of independent gamma variables, in general not identically distributed, may serve as a basis for ordering players in a hypothetical game. An alternative formulation in terms of negative binomial variables leads to an expression for the probability that the random gammas are in a given order. This expression may contain rather many terms and some approximations are discussed — firstly as the gamma parametersαitend to equality with allnithe same, and secondly when the probability of an inversion is small. In another interpretation the probabilities discussed arise in the statement of confidence limits for the ratios of population variances, and here the inversion probability is small enough usually that lower and upper bounds may be given for the probability that the sample variances occur in their expected order. These bounds are calculated from the probability that two variables are in expected order, and for gamma variables this probability is obtained from theF-distribution.
Extreme-point solutions in Markov decision processes
Authors: Assaf David
Published: April 2006

The paper presents sufficient conditions for certain functions to be convex. Functions of this type often appear in Markov decision processes, where their maximum is the solution of the problem. Since a convex function takes its maximum at an extreme point, the conditions may greatly simplify a problem. In some cases a full solution may be obtained after the reduction is made. Some illustrative examples are discussed.
Imperfect repair
Authors: Brown Mark , Proschan Frank
Published: April 2006

A device is repaired at failure. With probability p, it is returned to the ‘good-as-new' state (perfect repair), with probability 1 – p, it is returned to the functioning state, but it is only as good as a device of age equal to its age at failure (imperfect repair). Repair takes negligible time. We obtain the distribution Fp of the interval between successive good-as-new states in terms of the underlying life distribution F. We show that if F is in any of the life distribution classes IFR, DFR, IFRA, DFRA, NBU, NWU, DMRL, or IMRL, then Fp is in the same class. Finally, we obtain a number of monotonicity properties for various parameters and random variables of the stochastic process. The results obtained are of interest in the context of stochastic processes in general, as well as being useful in the particular imperfect repair model studied.
Optimal control of arrivals to multiserver queues in a random environment
Authors: Helm Werner E. , Waldmann Karl-Heinz
Published: April 2006

We study the problem of optimal customer admission to multiserver queues. These queues are assumed to live in an extraneous environment which changes in a semi-Markovian way. Arrivals, service mechanism and random reward/cost structure may all depend on these surroundings. Included as special cases areSM/M/cqueues, in particularG/M/cqueues, in a random environment. By a direct inductive approach we establish optimality of a generalized control-limit rule depending on the actual environment. Particular emphasis is laid on different applications that show the versatility of the proposed setup.
Mixed percolation on the square lattice
Authors: Wierman John C.
Published: April 2006

In a planar percolation model, faces of the underlying graph, as well as the sites and bonds, may be viewed as random elements. With this viewpoint, Whitney duality allows construction of a planar dual percolation model for each planar percolation model, which applies to mixed models with sites, bonds, and faces open or closed at random. Using self-duality for percolation models on the square lattice, information is obtained about the percolative region in the mixed model.
Geometric rate of growth in population-size-dependent branching processes
Authors: Klebaner F. C.
Published: April 2006

We consider a branching-process model {Zn}, where the law of offspring distribution depends on the population size. We consider the case when the meansmn(mnis the mean of offspring distribution when the population size is equal ton) tend to a limitm >1 asn→∞. For a certain class of processes {Zn} necessary conditions for convergence inL1andL2and sufficient conditions for almost sure convergence and convergence inL2ofWn= Zn/mnare given.
Multidimensional point processes and random closed sets
Authors: Baudin Michel
Published: April 2006

Multidimensional random point processes are shown to be a special case of Matheron's random closed sets. This theory, and in particular Choquet's theorem, are then used to prove a general result on heterogeneous Poisson processes and a uniqueness property of mixed Poisson processes with gamma mixing distributions.
Invariance in resource-sharing systems
Authors: Courcoubetis C. , Varaiya P. , Walrand J.
Published: April 2006

A method for proving the invariance of hitting-time distributions with respect to the control strategy of a Markov chain is presented. The method is applied to resource-sharing problems. It provides a new proof of a known result and extends it. Sufficient conditions for such an invariance are given and are illustrated by examples.
Some moment results for certain tandem and multiple-server queues
Authors: Wolfson Barry
Published: April 2006

It is known that for a GI/G/1 queue, if the interarrival time has finite first moment, then the (k + 1)th moment of the service time is finite if the kth moment of the stationary workload is finite. This result is extended to certain tandem and multiple-server queues.
An unbiased estimator for the drift of a stopped Wiener process
Authors: Ferebee Brooks
Published: April 2006

An unbiased estimator is given for the drift of a Wiener process which is observed up to a stopping time. For first-passage times across a moving boundary the estimator can be calculated by a simple modification of standard computer programs for computing the first-passage density. Examples and interpretations of the estimator are given.
The critical branching process with immigration stopped at zero
Authors: Ivanoff B. Gail , Seneta E.
Published: April 2006

Limit theorems for the Galton–Watson process with immigration (BPI), where immigration is not permitted when the process is in state 0 (so that this state is absorbing), have been studied for the subcritical and supercritical cases by Seneta and Tavaré (1983). It is pointed out here that, apart from a change of context, the corresponding theorem in the critical case has been obtained by Vatutin (1977). Extensions which follow from a more general form of initial distribution are sketched, including a new form of limit result (7).
Adaptive control of M/M/1 queues—continuous-time Markov decision process approach
Authors: Yeh Lam , Thomas L. C.
Published: April 2006

By considering continuous-time Markov decision processes where decisions can be made at any time, we show in the case of M/M/1 queues with discounted costs that there exists a monotone optimal policy among all the regular policies.
Quasi-stationary distribution of a two-unit warm-standby redundant system
Authors: Kalpakam S. , Hameed M. A. Shahul
Published: April 2006

This paper establishes the existence of the limit ast→ ∞ of the distribution of the residual lifetime at timetof a two-unit warm-standby reliability system supported by a single repair facility, conditioned on the event that the system has not been down at any time in (0,t). The conditioned limit distribution is proved to be exponential irrespective of the distributions of the lifetimes and repair times of the individual units, provided that their Laplace transforms are rational functions of their arguments.
On interpoint distances for planar Poisson cluster processes
Authors: Kryscio Richard J. , Saunders Roy
Published: April 2006

For stationary Poisson or Poisson cluster processesξonR2we study the distribution of the interpoint distances using the interpoint distance functionand the nearest-neighbor indicator function. HereSr(x) is the interior of a circle of radiusrhaving centerx, I(t) is that subset ofDwhich hasx∊DandSt(x) ⊂Dandχis the usual indicator function. We show that if the regionD⊂R2is large, then these functions are approximately distributed as Poisson processes indexed byand, whereµ(D) is the Lebesgue measure ofD.
Scheduling jobs by stochastic processing requirements on parallel machines to minimize makespan or flowtime
Authors: Weber Richard R.
Published: April 2006

A number of identical machines operating in parallel are to be used to complete the processing of a collection of jobs so as to minimize either the jobs' makespan or flowtime. The total processing required to complete each job has the same probability distribution, but some jobs may have received differing amounts of processing prior to the start. When the distribution has a monotone hazard rate the expected value of the makespan (flowtime) is minimized by a strategy which always processes those jobs with the least (greatest) hazard rates. When the distribution has a density whose logarithm is concave or convex these strategies minimize the makespan and flowtime in distribution. These results are also true when the processing requirements are distributed as exponential random variables with different parameters.
Point process limits of lattice processes
Authors: Besag Julian , Milne Robin , Zachary Stan
Published: April 2006

Starting from a suitable sequence of auto-Poisson lattice schemes, it is shown that (almost) any purely inhibitory pairwise-interaction point process can be obtained in the limit. Further pairwise-interaction processes are obtained as limits of sequences of auto-logistic lattice schemes.
An optimal betting strategy for repeated games
Authors: Gottlieb Gary
Published: April 2006

We consider the problem of finding a betting strategy for an infinite sequence of wagers where the optimality criterion is the minimization of the expected exit time of wealth from an interval. We add the side constraint that the right boundary is hit first with at least some specified probability. The optimal strategy is derived for a diffusion approximation.
On the range of a constrained random walk
Authors: Den Hollander W. Th. F. , Weiss G. H.
Published: April 2006

We study statistical properties of the range (= number of distinct sites visited) of a lattice random walk in discrete time constrained to visit a given site at a given time. In particular, we calculate the mean and obtain a bound on the variance of the range in the large time limit. The results are applied to a problem involving an unconstrained random walk in the presence of randomly distributed traps. A key role is played by the associated random walk that is obtained from the original random walk via a Cramer transform.
Response times in M/M/1 time-sharing schemes with limited number of service positions
Authors: Avi-Itzhak Benjamin , Halfin Shlomo
Published: April 2006

Two service schemes for an M/M/1 time-sharing system with a limited number of service positions are studied. Both schemes possess the equilibrium properties of symmetric queues; however, in the first one, a preempted job is placed at the end of the waiting line; while in the second one, it is placed at the head of the line. Methods for calculating the Laplace transforms and moments of the response times are presented. The variances of the response times are then compared numerically to indicate that the first scheme is superior to the second scheme. It is also indicated that in both cases the response time variance decreases when the number of service positions increases.
A note on random intensities and conditional survival functions
Authors: Yashin Anatoli , Arjas Elja
Published: April 2006

Failure intensities in which the evaluation of hazard is based on the observation of an auxiliary random process have become very popular in survival analysis. While their definition is well known, either as the derivative of a conditional failure probability or in the counting process and martingale framework, their relationship to conditional survival functions does not seem to be equally well understood. This paper gives a set of necessary and sufficient conditions for the so-called exponential formula in this context.
Stochastic models as functionals: some remarks on the renewal case
Authors: Grübel Rudolf
Published: April 2006

We regard the standard stochastic model of renewal theory as a functional which associates with a lifetime distribution μ the corresponding renewal measure v = Σ0∞μ ∗n. The behaviour of this functional near exponential distributions is investigated. The literature mainly deals with the asymptotic behaviour of v([0, t]) and v[t, t + 1]) as t → ∞— this new approach leads to expansions which are valid uniformly in t ∈ R.
The stability of infinite-server networks with random routing
Authors: Berezner S. A. , Malyshev V. A.
Published: April 2006

We consider networks with a very large or infinite number of nodes, linked by cable channels. The request which comes to a node is ordered to occupy a certain route of successive channels. The functioning of the system is regulated by the reserving of channels in order of the arrivals of the requests. Under some general conditions the existence of an ergodic region for such networks is proved. The practical value of the result lies in the fact that these conditions do not depend on the size of the graph.
Finiteness of moments of virtual work for GI/G/C queues
Authors: Ghahramani Saeed
Published: April 2006

Necessary and sufficient conditions for the finiteness of moments of virtual work for GI/G/c queues are presented.
Bivariate processes with positive or negative dependent structures
Authors: Ebrahimi Nader
Published: April 2006

In this paper various notions of positive and negative dependence for bivariate stochastic processes are introduced and their interrelationship is studied. Examples are given to illustrate these concepts.
Asymptotic behaviour of a stopping time related to cumulative sum procedures and single-server queues
Authors: Stadje Wolfgang
Published: April 2006

For a sequenceξ1,ξ2, · ·· of i.i.d. random variables letX0= 0 andXk= max(Xk–1+ξk, 0) fork= 1, 2, ···. Let. These stopping times are used in Page's (1954) one-sided cusum procedures and are also important in queueing theory. Various asymptotic properties ofNxare derived.
On random mappings with a single attracting centre
Authors: Mutafchiev Ljuben R.
Published: April 2006

We consider the random vector T = (T(0), ···, T(n)) with independent identically distributed coordinates such that Pr{T(i) = j} = Pj, j = 0, 1, ···, n, Σ . A realization of T can be viewed as a random graph GT with vertices {0, ···, n} and arcs {(0, T(0)), ···, (n, T(n))}. For each T we partition the vertex-set of GT into three disjoint groups and study the joint probability distribution of their cardinalities. Assuming that we observe the asymptotics of this distribution, as n → ∞, for all possible values of P0. It turns out that in some cases these cardinalities are asymptotically independent and identically distributed.
An improved Poisson limit theorem for sums of dissociated random variables
Authors: Barbour A. D. , Eagleson G. K.
Published: April 2006

A Poisson limit theorem for sums of dissociated 0–1 random variables is refined by deriving the first terms in an asymptotic expansion. The most natural refinement does not remove all the first-order error in a number of applications to tests of clustering, and a further approximation is derived which gives excellent results in practice. The proofs are based on the technique of Stein and Chen.
Typical polymorphisms maintained by selection at a single locus
Authors: Kingman J. F. C.
Published: April 2006

It is known that several different alleles can be maintained at a locus by selection, but only when the various genotypic fitnesses satisfy very special conditions. It is shown in this paper that a population with many possible mutations will evolve in such a way that these conditions arise naturally for most fitness regimes. The first steps are taken towards the assessment of the likely size and shape of the resulting stable polymorphisms.
Coincident probabilities and applications in combinatorics
Authors: Karlin Samuel
Published: April 2006

For a strong Markov process on the line with continuous paths the Karlin–McGregor determinant formula of coincidence probabilities for multiple particle systems is extended to allow the individual component processes to start at variable times and run for variable durations. The extended formula is applied to a variety of combinatorial problems including counts of non-crossing paths in the plane with variable start and end points, dominance orderings, numbers of dominated majorization orderings, and time-inhomogeneous random walks.
On the limit distribution of the number of cycles in a random graph
Authors: Takács Lajos
Published: April 2006

Let Γn(p) denote a random graph with n vertices in which any two vertices, independently of the others, are connected by an edge with probability p where . Denote by γ n(p) the total number of cycles in the graph Γn(p). The main object of this paper is to prove that the limit distribution of γ n(λ /n) is a Poisson distribution with expectation if n → ∞and 0 < λ< 1.
Markov chains with binomial time change
Authors: Siegrist Kyle
Published: April 2006

The effect of a binomial time change on a given Markov chain is studied. Results are obtained for the hitting times, potential operator, and transience and recurrence properties of the time-changed chain. The limiting behavior is considered as the binomial parameter approaches 0 and the time variable approaches∞.
On the number of leaves of a euclidean minimal spanning tree
Authors: Steele J. Michael , Shepp Lawrence A. , Eddy William F.
Published: April 2006

Let Vk,n be the number of vertices of degree k in the Euclidean minimal spanning tree of Xi, , where the Xi are independent, absolutely continuous random variables with values in Rd. It is proved that n–1Vk,n converges with probability 1 to a constant α k,d. Intermediate results provide information about how the vertex degrees of a minimal spanning tree change as points are added or deleted, about the decomposition of minimal spanning trees into probabilistically similar trees, and about the mean and variance of Vk,n.
Saddlepoint expansions for conditional distributions
Authors: Skovgaard Ib M.
Published: April 2006

A saddlepoint expansion is given for conditional probabilities of the formwhereis an average ofnindependent bivariate random vectors. A more general version, corresponding to the conditioning on ap –1-dimensional linear function of ap-dimensional variable is also included. A separate formula is given for the lattice case. The expansion is a generalization of the Lugannani and Rice (1980) formula, which reappears ifandare independent. As an example an approximation to the hypergeometric distribution is derived.
Non-linear filtering of rare events with large signal-to-noise ratio
Authors: Heunis A. J.
Published: April 2006

The theory of robust non-linear filtering in Clark (1978) and Davis (1980), (1982) is used to evaluate the limiting conditional distribution of a diffusion, given an observation of a ‘rare-event' sample-path of the diffusion, as the signal-to-noise ratio and the diffusion noise-intensity converge to infinity and zero respectively. Under mild conditions it is shown that the limiting conditional distribution is a Dirac measure concentrated at a trajectory which solves a variational problem parametrised by the sample-path of the observed signal.
Server sharing with a limited number of service positions and symmetric queues
Authors: Avi-Itzhak Benjamin , Halfin Shlomo
Published: April 2006

A class of M/G/1 time-sharing queues with a finite number of service positions and unlimited waiting space is described. The equilibrium distribution of symmetric queues belonging to this class is invariant under arbitrary service-independent reordering of the customers at instants of arrivals and departures. The delay time distribution, in the special case of one service position where preempted customers join the end of the line, is provided in terms of Laplace transforms and generating functions. It is shown that placing preempted customers at the end of the line rather than at the beginning of the line results in a reduction of the delay time variance. Comparisons with the delay time variance of the case of unlimited number of service positions (processor sharing system) are presented.
A convexity result for single-server exponential loss systems with non-stationary arrivals
Authors: Svoronos Antony , Green Linda
Published: April 2006

We consider single-server loss systems with exponential service times and non-stationary Poisson input. We prove that if the arrival rate is given by a periodic function, the proportion of lost customers is convex increasing in the amplitude.
On the solution of compartmental processes and their use in modelling stages of infection of a disease with reference to onchocerciasis
Authors: Saporu F. W. O.
Published: April 2006

The solution of a hierarchical n-compartmental homogeneous process with multiple-valued migration rates is obtained using Laplace transform techniques. Such models are suggested for modelling the infection process of endemic diseases because of their property of gamma-distributed compartmental residence times in each of n compartments. Its relevance to the onchocerciasis infection process (with and without the imposition of controls) is discussed and application to age-prevalence data (having only two infective states into which an individual can be classified) is undertaken.
Inverting the statistics of the circle transform
Authors: Waksman Peter
Published: April 2006

For a plane domain the circle transform assigns to each circle its length of intersection with the domain. The problem is to determine the geometry of the domain given the mean of these intersection lengths as the circles' centers vary and given how that mean varies with circle radius. We also consider the mean square and higher moments of the intersection lengths — as a function of the circle radius. The discussion includes an identification of the geometric content of the mean when centers are inside of a disk and of the second moment when the centers are arbitrary points in the plane. The latter is equivalent to the distribution of distance between pairs of points of the domain.
The inverses of thinned renewal processes
Authors: Yannaros Nikos
Published: April 2006

The class of inverses of a p-thinned renewal process is considered. It is shown that this class consists of renewal processes. It consists of Cox and renewal processes if and only if the given thinned process is Cox and renewal. In the non-Cox case, there exists a unique top renewal process, which by thinning generates all the possible inverses. Conditions for a renewal process to be a top process are given. Finally, a gamma renewal process is shown to be a top process when α > 1, where α is the shape parameter of the gamma distribution.
A reconciliation of two different expressions for the first-passage density of brownian motion to a curved boundary
Authors: Durbin J.
Published: April 2006

An expression for the first-passage density of Brownian motion to a curved boundary due to Daniels and Lerche is shown to give the same result as a different form due to the author. The equivalence is extended to continuous Gaussian Markov processes.