bluuebunny/crossref_metadata_2025 huggingface dataset:
Isolation by distance in a hierarchically clustered population
Authors: Sawyer Stanley , Felsenstein Joseph
Published: July 2016

A biological population with local random mating, migration, and mutation is studied which exhibits clustering at several different levels. The migration is determined by the clustering rather than actual geographic or physical distance. Darwinian selection is assumed to be absent, and population densities are such that nearby individuals have a probability of being related. An expression is found for the equilibrium probability of genetic relatedness between any two individuals as a function of their clustering distance. Asymptotics for a small mutation rateuare discussed for both a finite number of clustering levels (and of total population size), and for an infinite number of levels. A natural example is discussed in which the probability of heterozygosity varies asuto a power times a periodic function of log(l/u).
The limiting properties of population distributions with particular application to manpower planning
Authors: Woodward Mark
Published: July 2016

A model for predicting expected-value population distributions is developed, assuming that all movements are Markovian and time-homogeneous. Each individual is classified by the amount of time he has spent in the population and by which of a number of classes, of an unspecified nature, he inhabits. The limiting properties of the population distribution are derived, and, in particular, conditions for convergence to a stable distribution are given.

Some discussion of the relevance of the theory to practical applications is given, primarily to manpower planning when recruitment occurs purely to maintain a specified overall population size.
Random contact processes, snowball sampling and factorial series distributions
Authors: Berg Sven
Published: July 2016

A contact process first posed as a random mapping problem is discussed together with a model for the spread of a rumour through an archaic telephone system. Snowball sampling and the well-known Reed–Frost chain binomial model are also commented on. Our main interest and emphasis is on distributional properties of discrete-time models. Simple combinatorial arguments are used along with a class of discrete distributions originally proposed for catch–recapture sampling.
On a stopping problem involving refusal and forced stopping
Authors: Yasuda Masami
Published: July 2016

Although the usual optimal stopping problem is described as a Markov decision process with two decisions, stop and continue, we shall consider a model which distinguishes the observer's strategy from the system's two decisions. The observer can select a strategy defined on an action space, and the decision of the system to stop or continue is determined by a prescribed conditional probability. For this model, it may happen that the strategy (a) to stop is refused, or (b) to continue is forcibly stopped. This is a slight modification of the one-dimensional stopping problem by involving refusal and forced stopping. The model is motivated by the uncertain secretary choice problem of Smith (1975) and the multivariate stopping problem of Kurano, Yasuda and Nakagami (1980), (1982).
Random triangles in convex regions
Authors: Henze Norbert
Published: July 2016

Consider a random triangle formed by vertices which are independent and distributed uniformly inside a given bounded convex region K. Then a well-known result of W. Blaschke states that, among all regions of fixed area, the area AK
of this random triangle is stochastically bounded by A
circle (A
c) and A triangle (A
t). The distribution of the latter was obtained by Alagar (1977). We use the classical Crofton technique to derive an integral representation for the distribution function F
c of A
c from which F
c is calculated using numerical quadrature. The maximum deviation between Fc and the distribution function of A
t is less than 0.04. Additionally, we obtain an explicit form for the distribution of A
parallelogram which may be represented as a mixture of functions involving Dirichlet-distributed random vectors.
On the number of clumps resulting from the overlap of randomly placed figures in a plane
Authors: Kellerer A. M.
Published: July 2016

When two-dimensional figures, called laminae, are randomly placed on a plane domains result that can either be aggregates or individual laminae. The intersection of the union, U, of these domains with a specified field of view, F, in the plane is considered. The separate elements of the intersection are called clumps; they may be laminae, aggregates or partial laminae and aggregates. A formula is derived for the expected number of clumps minus enclosed voids. For bounded laminae homeomorphic to a closed disc with isotropic random direction the formula contains only their mean area and mean perimeter, the area and perimeter of F, and the intensity of the Poisson process.
Note on the determination of cluster centers from a realization of a multidimensional Poisson cluster process
Authors: Baudin Michel
Published: July 2016

This is the sequel to a previous paper (Baudin (1981)). The joint probability generating functional of two point processes is introduced as a tool to compute the conditional intensity of the process of cluster centers of a multidimensional Poisson cluster process when a realization is given in a bounded observation window. An explicit formula is derived but it is too complicated for actual use; a linear method for practical estimation is discussed.
A note on many-server queueing systems with ordered entry, with an application to conveyor theory
Authors: Nawijn W. M.
Published: July 2016

Consider a many-server queueing system in which the servers are numbered. If a customer arrives when two or more servers are idle he selects the server with lowest index (this is called the ordered entry selection rule). An explicit expression for the traffic handled by the various servers in a GI/M/s queueing system with ordered entry is derived. For the M/M/s queueing system the probability distribution of the number of busy servers among the first k(k = 1, 2,…, s) servers will be given. Finally, a formula for the traffic handled by the first server in an M/D/s system will be derived. All results are derived under steady-state conditions. As an application some numerical data for the server utilizations will be given and compared to data obtained from simulation studies of a closed-loop continuous belt-conveyor.
The spatial distribution of Tribolium confusum
Authors: Renshaw Eric
Published: July 2016

Neyman, Park and Scott (1956) describe an experiment which they performed to determine the spatial distribution ofTribolium confusumdeveloping within a closed container. To explain the concentration of beetles at the boundary a birth–death–migration model is developed in which the beetles may migrate over a set of lattice points, and this is shown to produce a distribution of the required shape. Not only is this distribution independent of the number of lattice points, but it is also indistinguishable from the associated diffusion process.
On the size distribution for some epidemic models
Authors: Watson Ray
Published: July 2016

We consider the standard epidemic model and several extensions of this model, including Downton's carrier-borne epidemic model. A random time-scale transformation is used to obtain equations for the size distribution and to derive asymptotic approximations for the size distribution for each of the models
On stochastic scheduling with precedence relations and switching costs
Authors: Glazebrook K. D.
Published: July 2016

A collection of jobs is to be processed by a single machine. The amount of processing required by each job is a random variable with a known probability distribution. The jobs must be processed in a manner which is consistent with a precedence relation but the machine is free to switch from one job to another at any time; such switches are costly, however. This paper discusses conditions under which there is an optimal strategy for allocating the machine to the jobs which is given by a fixed permutation of the jobs indicating in which order they should be processed. When this is so, existing algorithms may be helpful in giving the best job ordering.
Generalizations of the simple epidemic process
Authors: Billard L. , Lacayo H. , Langberg N. A.
Published: July 2016

Recently, Billard, Lacayo and Langberg (1979) considered the classical simple epidemic process from the viewpoint of interinfection times. However, this concept can be expanded to include arbitrary generalized simple epidemic processes as well as non-exponential interinfection times. This work finds the distribution of the number of infectives for these generalizations as well as the mean and variance.
Success probabilities for second guessers
Authors: Pittenger A. O.
Published: July 2016

Two people independently and with the same distribution guess the location of an unseen object in n-dimensional space, and the one whose guess is closer to the unseen object is declared the winner. The first person announces his guess, but the second modifies his unspoken idea by moving his guess in the direction of the first guess and as close to it as possible. It is shown that if the distribution of guesses is rotationally symmetric about the true location of the unseen object, ¾ is the sharp lower bound for the success probability of the second guesser. If the distribution is fixed and the dimension increases, then for a certain class of distributions, the success probability approaches 1.
Birth and death processes with random environments in continuous time
Authors: Cogburn Robert , Torrez William C.
Published: July 2016

A generalization to continuous time is given for a discrete-time model of a birth and death process in a random environment. Some important properties of this process in the continuous-time setting are stated and proved including instability and extinction conditions, and when suitable absorbing barriers have been defined, methods are given for the calculation of extinction probabilities and the expected duration of the process.
Transient distribution of the number of segregating sites in a neutral infinite-sites model with no recombination
Authors: Griffiths R. C.
Published: July 2016

The transient distribution of the number of segregating sites in a sample from a large population of 2Ngenes is found. Segregating sites are split into those in common with the sites segregating in the initial population and those segregating due to new mutations. The waiting time distribution for a population to lose all of its initial sites is also studied. A neutral infinite-sites model with no recombination is used. This paper extends the work of Watterson (1975), from the stationary case; and of Li (1977) from the transient distribution in a sample of 2.
The stochastic geyser problem for first-passage times
Authors: Steinebach Josef
Published: July 2016

Let X
1, X
2, · ·· be a sequence of independent, identically distributed (i.i.d.) random variables with positive mean. An analogue of Rényi's (1962) stochastic geyser problem is solved for the associated process of first-passage times. More precisely, it is shown that a single realization of the sequence determines the distribution function (d.f.) of the Xn
's almost surely (a.s.), even if the observations are erroneous up to an order o(log n).
The supremum distribution of another Gaussian process
Authors: Cressie Noel , Davis Robert W.
Published: July 2016

A formula is derived for the supremum of a stationary Gaussian process which has a correlation function that is tent-like in shape, until it flattens out at a constant negative value. Examples and graphs are presented in the last section.
Limit theorems for general size distributions
Authors: Chen Wen-Chen
Published: July 2016

Let X
1, X
2, · ··, Xn
, · ·· be independent and identically distributed non-negative integer-valued random variables with finite mean and variance. For any positive integer n and m we consider the random vector i.e., L has the same distribution as the conditional distribution of (X
1, · ··, Xm
) given the condition It is easy to see that our model includes the classical urn model, the Bose–Einstein urn model and the Pólya urn model as special cases. For any non-negative integer s define G(s) = the number of Li′s such that Li
= s, and U = the number of Li′s such that Li
is an even number; in this paper we study the asymptotic behaviour of the random variables considered above. Some central limit theorems and a multinormal local limit theorem are proved.
A non-random limit for the volume covered k times
Authors: Stam A. J.
Published: July 2016

Let n sets with volume ~ n
-–1 be placed in R
m
, independently and with the same distribution. As n →∞, the volume in V CR
m
, covered by exactly k of these sets under certain conditions converges to a non-random limit, which is the integral over V of a density that is of the type of a Poisson probability.
On a Markovian queue with weakly correlated interarrival times
Authors: Latouche Guy
Published: July 2016

A queueing system with exponential service and correlated arrivals is analysed. Each interarrival time is exponentially distributed. The parameter of the interarrival time distribution depends on the parameter for the preceding arrival, according to a Markov chain. The parameters of the interarrival time distributions are chosen to be equal to a common value plus a factor ofε, where ε is a small number. Successive arrivals are then weakly correlated.

The stability condition is found and it is shown that the system has a stationary probability vector of matrix-geometric form. Furthermore, it is shown that the stationary probabilities for the number of customers in the system, are analytic functions ofε, for sufficiently smallε, and depend more on the variability in the interarrival time distribution, than on the correlations.
Adaptive control of a production-inventory system
Authors: Doshi Bharat T.
Published: July 2016

We study a production-inventory system in which input is deterministic, and its rate is the controlled parameter. The output is a compound Poisson process with exponential jump-size distribution. The parameters of the output process are unknown. The following adaptive policy is used: At each time t the unknown parameters are estimated by maximum likelihood method and the input rate which would be optimal if these were the true values of the parameter is used. The issues investigated are (1) the convergence of maximum likelihood estimates to the true value, and (2) the asymptotic properties of the cost under the adaptive policy.
On a general branching process
Authors: Daly Fergus
Published: July 2016

A branching process is described which contains some better-known simple branching processes as special cases. Variables examined are the rate of development, the number of individuals currently alive, and the duration of the process. The apparent dynamics of collapsing supercritical realisations are isolated.
Clustering of events in a stochastic process
Authors: Glaz Joseph
Published: July 2016

In this paper we derive bounds for the expected waiting time of clustering of at least n events of a stochastic process within a fixed interval of length p. Using this approach of clustering, we derive bounds for the expected duration of the period of time that at least n servers are busy in an ∞-server queue with constant service time. For the case of Poisson arrivals we derive the exact distribution of the duration of that period.
Intercept lengths of random probes through boxes
Authors: Coleman Rodney
Published: July 2016

The distribution and its moments are given of the length of the intercept made by a random probe through a box having perpendicular edge lengths a, b and c. Three types of random probe are considered.
On snowball sampling, random mappings and related problems
Authors: Berg Sven
Published: July 2016

If a rumour is spread in a population through random contacts between its members, to how many people, on average, will the rumour be told? What can be said about the variations in the number of people who have heard the rumour? Does it matter very much whether a single person or a group of persons starts spreading the rumour? Questions such as these are discussed below, using the framework of a simple stochastic model for snowball sampling.
Weak convergence of an iterated renewal process
Authors: Sen Pranab Kumar
Published: July 2016

Asymptotic normality and the weak invariance principle are studied for an iterated renewal process of order A martingale decomposition is used in this context to provide a simple proof.
The use of prior information in the estimation of the optimal interpolator parameter
Authors: Kabaila Paul V.
Published: July 2016

We consider the estimation of the parameter of the best interpolator from a restricted class of linear, time-invariant interpolators for the weakly stationary process {xt
}. The estimation is supposed to be based on a part-realisation x
1, · ··, xn
and on certain prior information regarding the spectral density of the process. Two methods of estimation are proposed — one which uses x
1, · ··, xn
and the prior information and another which uses only x
1, · ··, xn. For the methods proposed it is reasonable to conjecture that the method which utilises the prior information as well as x
1, · ··, xn
, will be more efficient. Reasonable as this conjecture may seem, its proof is by no means obvious and is the main subject of the paper.
Best-choice problems involving uncertainty of selection and recall of observations
Authors: Petruccelli Joseph D.
Published: July 2016

This paper explores best choice problems which allow both recall of applicants and uncertainty of a current applicant accepting an offer of employment. Properties of optimal selection procedures are derived for the general case. Optimal procedures and the associated probabilities of obtaining the best applicant are found in two special cases. The results unify and extend those of Yang (1974) and Smith (1975).
On sequential occupancy problems
Authors: Holst Lars
Published: July 2016

Consider n cells into which balls are thrown at random until k cells contain at least l + 1 balls each. Let Y
l, · ··, Yn
be the number of balls in the cells when stopping. In this paper two representations are given for the characteristic functions of random variables of the form The usefulness of these representations are illustrated by two examples. In the first the number of cells with exactly one ball when each cell contains at least one ball is considered. In the second the waiting time until the ball-throwing process stops is discussed.
Some elaborations upon Gani's model for the type-token relationship
Authors: Brainerd Barron
Published: July 2016

This paper considers certain stochastic models for token and type counts in literary texts. Elaborating on some models of Gani, it is shown that reasonable fits can be obtained to some data of Yule on Macaulay's Essays.
Algorithms for a continuous-review (s, S) inventory system
Authors: Ramaswami V.
Published: July 2016

The steady-state distribution of the inventory position for a continuous-review (s, S) inventory system is derived in a computationally tractable form. Demands for items in inventory are assumed to form an N-process which is the ‘versatile Markovian point process' introduced by Neuts (1979). The N-process includes the phase-type renewal process, Markov-modulated Poisson process etc., as special cases and is especially useful in modelling a wide variety of qualitative phenomena such as peaked arrivals, interruptions, inhibition or stimulation of arrivals by certain events etc.
On a generalized storage model with moment assumptions
Authors: Puri Prem S. , Woolford Samuel W.
Published: July 2016

This paper considers a semi-infinite storage model, of the type studied by Senturia and Puri [13] and Balagopal [2], defined on a Markov renewal process, {(Xn, Tn
), n = 0, 1, ·· ·}, with 0 = T
0 < T
1 < · ··, almost surely, where Xn
takes values in the set {1, 2, ·· ·}. If at Tn, Xn
= j, then there is a random ‘input' Vn
(j) (a negative input implying a demand) of ‘type' j, having distribution function Fj
(·). We assume that {Vn
(j)} is an i.i.d. sequence of random variables, taken to be independent of {(Xn, Tn
)} and of {Vn
(k)}, for k ≠ j, and that Vn
(j) has first and second moments. Here the random variables Vn
(j) represent instantaneous ‘inputs' (a negative value implying a demand) of type j for our storage model. Under these assumptions, we establish certain limit distributions for the joint process (Z(t), L(t)), where Z(t) (defined in (2)) is the level of storage at time t and L(t) (defined in (3)) is the demand lost due to shortage of supply during [0, t]. Different limit distributions are obtained for the cases when the ‘average stationary input' ρ, as defined in (5), is positive, zero or negative.
Transient behaviour of queueing networks
Authors: Harrison P. G.
Published: July 2016

In most contemporary queueing network analysis, the assumption is made that a network is in a state of equilibrium. That is, the network's state space probabilities are assumed to be time independent. It is therefore important to be able to quantify precisely when this assumption is valid. Furthermore there are also situations in which it is desirable to model the transient behaviour of networks which occur in practice, such as computer and communication systems. For example, the immediate effects of component failure or instantaneous alteration of system status may be predicted.

In this paper an iterative solution is derived to the time-dependent Kolmogorov equations of queueing networks, and is shown to be convergent. From the solution, modelling of transient situations becomes possible and the time periods during which the equilibrium assumption can and should not be made may be identified; for example in terms of a time constant which is easily computed to a first-order approximation.
On Fibonacci (or lagged Bienaymé-Galton-Watson) branching processes
Authors: Heyde C. C.
Published: July 2016

This paper is concerned with a discrete-time population model in which a new individual entering the population at time t can produce offspring for the first time at time t + 2 and then subsequently at times t + 3, t + 4, ···. The numbers of offspring produced on each occasion are independent random variables each with the distribution of Z for which EZ = m <∞, and individuals have independent lines of descent. This model is contrasted with the corresponding Bienaymé-Galton-Watson one. If Xn
denotes the number of individuals in the population at time n, it is shown that z–nXn
almost surely converges to a random variable W, as n→∞, where Various properties of W are obtained, in particular W > 0 a.s. if and only if EZ | log Z | < ∞ Results are also given on the rate of convergence of to z when Var Z < ∞ and these display a surprising dependence on the size of z.
On the convergence of stochastically monotone sequences of random variables and some applications
Authors: Cohn Harry
Published: July 2016

The paper is concerned with the relationship between various modes of convergence for stochastically monotone sequences of random variables. A necessary and sufficient condition, as well as a sufficient condition, for convergence in probability of a vaguely convergent sequence are given. If, in addition, the sequence is assumed Markovian the same conditions are shown to pertain to almost sure convergence. A counterexample in the case when stochastic monotonicity fails is presented and some applications to branching processes are discussed.
On the distribution of random lines
Authors: Hall Peter
Published: July 2016

Draw two lines, L
1 and L
2, in a plane. Along L
1 place the points of a Poisson process, and through each point draw a line, the angles of intersection with L
1 being distributed independently and uniformly on (0, π) . The intercepts of these random lines with L
2 form a new process, which is Poisson if and only if L
1 and L
2 are parallel. Curiously, if L
1 and L
2 are inclined then, with probability 1, the new process forms a dense subset of L
2. It is not really even a point process. In the present paper we shall investigate this anomaly and some of its generalizations.
A hide–search game
Authors: Subelman Eduardo J.
Published: July 2016

We consider a game in which one player hides a ball in one of n boxes and the other player is allowed to search for it. There are known probabilities that the searcher will overlook the ball if he searches the correct box. The hider wishes to minimize and the searcher to maximize the probability that the ball will be found in m or fewer searches. We exhibit a procedure which allows efficient computation of optimal strategies for both players without solving the game as a linear program. The results are extended to a non-zero-sum game where the searcher's objective is to minimize the expected time until the ball is found.
An analysis of the fixed-cycle traffic-light problem
Authors: Cowan Richard
Published: July 2016

A realistic non-Poisson arrival process is used in a model for intersections controlled by fixed-cycle traffic lights. Average delays, queue sizes and percentage of delayed vehicles are derived. The distribution of the number of vehicles which pass through during the green phases is found. Certain model anomalies which are inherent in earlier work are eliminated by the use of this model.
Queues with paired customers
Authors: Latouche Guy
Published: July 2016

Queueing systems with a special service mechanism are considered. Arrivals consist of two types of customers, and services are performed for pairs of one customer from each type. The state of the queue is described by the number of pairs and the difference, called the excess, between the number of customers of each class. Under different assumptions for the arrival process, it is shown that the excess, considered at suitably defined epochs, forms a Markov chain which is either transient or null recurrent. A system with Poisson arrivals and exponential services is then considered, for which the arrival rates depend on the excess, in such a way that the excess is bounded. It is shown that the queue is stable whenever the service rate exceeds a critical value, which depends in a simple manner on the arrival rates. For stable queues, the stationary probability vector is of matrix-geometric form and is easily computable.
On the comparison of waiting times in tandem queues
Authors: Niu Shun-Chen
Published: July 2016

Using a definition of partial ordering of distribution functions, it is proven that for a tandem queueing system with many stations in series, where each station can have either one server with an arbitrary service distribution or a number of constant servers in parallel, the expected total waiting time in system of every customer decreases as the interarrival and service distributions becomes smaller with respect to that ordering. Some stronger conclusions are also given under stronger order relations. Using these results, bounds for the expected total waiting time in system are then readily obtained for wide classes of tandem queues.
On strong mixing and Leadbetter's D condition
Authors: Chernick Michael R.
Published: July 2016

Strong mixing is a condition which is often assumed to prove limit theorems for strictly stationary processes. Leadbetter's conditionD(un) is used to prove limit theorems for maxima of stationary processes.

A sufficient condition for strong mixing to hold is given for the case where the process satisfies apth-order Markov property. This condition can be easy to check for whenpis small. This point is illustrated by two examples of first-order autoregressive processes.

The conditionD(un) is shown to hold for any stationary Markov process.
Some inequalities on the distribution of ladder epochs
Authors: Downham D.Y. , Fotopoulos S.B.
Published: July 2016

An algorithm for calculating the probability distribution of ladder epochs is derived. Two theorems are given for bounds on the distribution function of ladder epoch probabilities.
Optimal service rates in the multiserver loss system with heterogeneous servers
Authors: Nath G. B. , Enns E. G.
Published: July 2016

A multichannel loss system with heterogeneous servers operating in parallel is analyzed. The sum of the service rates of all servers is assumed constant. The optimal service rates that minimize the probability of losing a customer are obtained, and are shown to be different from each other. The percentage reduction in the probability of losing a customer in the homogeneous and the best heterogeneous system for a few representative values are included.
Threshold strategies for risk processes and their relation to queueing theory
Authors: Boxma Onno J. , Löpker Andreas , Perry David
Published: July 2016

We consider a risk model with threshold strategy, where the insurance company pays off a certain percentage of the income as dividend whenever the current surplus is larger than a given threshold. We investigate the ruin time, ruin probability, and the total dividend, using methods and results from queueing theory.
On the excursions of reflected local-time processes and stochastic fluid queues
Authors: Konstantopoulos Takis , Kyprianou Andreas E. , Salminen Paavo
Published: July 2016

In this paper we extend our previous work. We consider the local-time process L of a strong Markov process X, add negative drift to L, and reflect it à la Skorokhod to obtain a process Q. The reflection of X, together with Q, is, in some sense, a macroscopic model for a service system with two priorities. We derive an expression for the joint law of the duration of an excursion, the maximum value of the process on it, and the time between successive excursions. We work with a properly constructed stationary version of the process. Examples are also given in the paper.
Loss rate for a general Lévy process with downward periodic barrier
Authors: Palmowski Zbigniew , Świa̧tek Przemysław
Published: July 2016

In this paper we consider a general Lévy processXreflected at a downward periodic barrierAtand a constant upper barrierK, giving a processVKt=Xt+LAt−LKt. We find the expression for a loss rate defined bylK=ELK1and identify its asymptotics asK→∞ whenXhas light-tailed jumps and EX1<0.
Population-size-dependent, age-structured branching processes linger around their carrying capacity
Authors: Jagers Peter , Klebaner Fima C.
Published: July 2016

Dependence of individual reproduction upon the size of the whole population is studied in a general branching process context. The particular feature under scrutiny is that of reproduction changing from supercritical in small populations to subcritical in large populations. The transition occurs when the population size passes a critical threshold, known in ecology as the carrying capacity. We show that populations either die out directly, never coming close to the carrying capacity, or grow quickly towards the carrying capacity, subsequently lingering around it for a time that is expected to be exponentially long in terms of a carrying capacity tending to infinity.
Asymptotic normality of M-estimators in nonhomogeneous hidden Markov models
Authors: Jensen Jens Ledet
Published: July 2016

Results on asymptotic normality for the maximum likelihood estimate in hidden Markov models are extended in two directions. The stationarity assumption is relaxed, which allows for a covariate process influencing the hidden Markov process. Furthermore, a class of estimating equations is considered instead of the maximum likelihood estimate. The basic ingredients are mixing properties of the process and a general central limit theorem for weakly dependent variables.
Decay rates for some quasi-birth-and-death processes with phase-dependent transition rates
Authors: Motyer Allan J. , Taylor Peter G.
Published: July 2016

Recently, there has been considerable interest in the calculation of decay rates for models that can be viewed as quasi-birth-and-death (QBD) processes with infinitely many phases. In this paper we make a contribution to this endeavour by considering some classes of models in which the transition function is not homogeneous in the phase direction. We characterize the range of decay rates that are compatible with the dynamics of the process away from the boundary. In many cases, these rates can be attained by changing the transition structure of the QBD process at level 0. Our approach, which relies on the use of orthogonal polynomials, is an extension of that in Motyer and Taylor (2006) for the case where the generator has homogeneous blocks.
Poisson hail on a hot ground
Authors: Baccelli Francois , Foss Sergey
Published: July 2016

We consider a queue where the server is the Euclidean space, and the customers are random closed sets (RACSs) of the Euclidean space. These RACSs arrive according to a Poisson rain and each of them has a random service time (in the case of hail falling on the Euclidean plane, this is the height of the hailstone, whereas the RACS is its footprint). The Euclidean space serves customers at speed 1. The service discipline is a hard exclusion rule: no two intersecting RACSs can be served simultaneously and service is in the first-in–first-out order, i.e. only the hailstones in contact with the ground melt at speed 1, whereas the others are queued. A tagged RACS waits until all RACSs that arrived before it and intersecting it have fully melted before starting its own melting. We give the evolution equations for this queue. We prove that it is stable for a sufficiently small arrival intensity, provided that the typical diameter of the RACS and the typical service time have finite exponential moments. We also discuss the percolation properties of the stationary regime of the RACS in the queue.
The general correlated random walk
Authors: Chen Anyue , Renshaw Eric
Published: July 2016

In this paper we provide further results on the generald-dimensional correlated random walk. In particular, we prove that then-step characteristic function of any correlated random walk satisfies a recurrence formula which enables both it and the total characteristic function to be obtained. Some examples are then considered.
On the validity of Wald's equation
Authors: Roters Markus
Published: July 2016

In this paper we review conditions under which Wald's equation holds, mainly if the expectation of the given stopping time is infinite. As a main result we obtain what is probably the weakest possible version of Wald's equation for the case of independent, identically distributed (i.i.d.) random variables. Moreover, we improve a result of Samuel (1967) concerning the existence of stopping times for which the expectation of the stopped sum of the underlying i.i.d. sequence of random variables does not exist. Finally, we show by counterexamples that it is impossible to generalize a theorem of Kiefer and Wolfowitz (1956) relating the moments of the supremum of a random walk with negative drift to moments of the positive part of X
1 to the case where the expectation of X
1 is —∞. Here, the Laplace–Stieltjes transform of the supremum of the considered random walk plays an important role.
Super-extremal processes and the argmax process
Authors: Resnick Sidney I. , Roy Rishin
Published: July 2016

In this paper, we develop the probabilistic foundations of the dynamic continuous choice problem. The underlying choice set is a compact metric space E such as the unit interval or the unit square. At each time point t, utilities for alternatives are given by a random function . To achieve a model of dynamic continuous choice, the theory of classical vector-valued extremal processes is extended to super-extremal processes
Y
= {Yt, t > 0}. At any t > 0, Y

t
is a random upper semicontinuous function on a locally compact, separable, metric space E. General path properties of Y are discussed and it is shown that Y is Markov with state-space US(E). For each t > 0, Y

t
is associated.

For a compact metric E, we consider the argmax process M = {Mt, t > 0}, where . In the dynamic continuous choice application, the argmax process M represents the evolution of the set of random utility maximizing alternatives. M is a closed set-valued random process, and its path properties are investigated.
Some results on the relative ageing of two life distributions
Authors: Sengupta Debasis , Deshpande Jayant V.
Published: July 2016

Kalashnikov and Rachev (1986) have proposed a partial ordering of life distributions which is equivalent to an increasing hazard ratio, when the ratio exists. This model can represent the phenomenon of crossing hazards, which has received considerable attention in recent years. In this paper we study this and two other models of relative ageing. Their connections with common partial orderings in the reliability literature are discussed. We examine the closure properties of the three orderings under several operations. Finally, we give reliability and moment bounds for a distribution when it is ordered with respect to a known distribution.
Drift vectors are not sufficient to determine recurrence of a Markov chain on ℤ3
+
Authors: Kotler Mitchell
Published: July 2016

For a Markov chain in
n = 2, drift vectors (conditional expected jumps) on the interior and the boundaries distinguish between recurrence and transience. The result of this paper is that the analogous proposition in the n = 3 case fails.
A note on the existence of regeneration times
Authors: Sigman Karl , Thorisson Hermann , Wolff Ronald W.
Published: July 2016

We rigorously prove that for a stochastic process, , the existence of a first regeneration time, R
1, implies the existence of an infinite sequence of such times, {R
1, R
2, · ·· }, and hence that the definition of regenerative process need only demand the existence of a first regeneration time. Here we include very general processes up to and including processes where cycles are stationary but not necessarily independent and identically distributed.
Simple random walk statistics. Part I: Discrete time results
Authors: Katzenbeisser W. , Panny W.
Published: July 2016

In a famous paper, Dwass (1967) proposed a method to deal with rank order statistics, which constitutes a unifying framework to derive various distributional results. In the present paper an alternative method is presented, which allows us to extend Dwass's results in several ways, namely arbitrary endpoints, horizontal steps and arbitrary probabilities for the three step types. Regarding these extensions the pertaining rank order statistics are extended as well to simple random walk statistics. This method has proved appropriate to generalize all results given by Dwass. Moreover, these discrete time results can be taken as a starting point to derive the corresponding results for randomized random walks by means of a limiting process.
Integral equations for compound distribution functions
Authors: Møller Christian Max
Published: July 2016

The aim of the present paper is to introduce some techniques, based on the change of variable formula for processes of finite variation, for establishing (integro) differential equations for evaluating the distribution of jump processes for a fixed period of time. This is of interest in insurance mathematics for evaluating the distribution of the total amount of claims occurred over some period of time, and attention will be given to such issues. Firstly we will study some techniques when the process has independent increments, and then a more refined martingale technique is discussed. The building blocks are delivered by the theory of marked point processes and associated martingale theory. A simple numerical example is given.
Continuity of percolation probability in ∞ + 1 dimensions
Authors: Zhang Yu
Published: July 2016

We consider percolation on the graph of the product of a regular tree T with degree d and the line ℤ, in which each tree edge is open with probability 1 – exp(–JTß) and each line edge is open with probability . Let C(o, 0) be the open cluster for . Denote by θ (β) the percolation probability. Here we show that θ (β) is continuous when ß > ßc
, where ßc
= sup{ß : θ(β) = 0}.
On Kingman's characteristic functional approach to Rényi's characterization of Poisson processes
Authors: Yao Yi-Ching
Published: July 2016

In his recent monograph on Poisson processes, Kingman generalized Rényi's characterization of Poisson processes and also suggested a characteristic functional approach. A direct proof is given along this line.
On heavy traffic approximations for transient characteristics of M/M/∞ queues
Authors: Guillemin Fabrice M. , Mazumdar Ravi R. , Simonian Alain D.
Published: July 2016

In this paper, transient characteristics related to excursions of the occupation process ofM/M/∞ queues are studied, when the excursion level is large and close to the mean offered load. We show that the classical diffusion approximation by an Ornstein–Uhlenbeck (OU) process captures well the average values of the transient variables considered, while the asymptotic distributions of these variables depart from those corresponding to the OU process. They exhibit, however, equivalent tail behaviour at infinity and numerical evidence shows that they are amazingly close to each other over the whole half-line.
From FIFO to LIFO: a functional ordering of service delay via arrival discipline
Authors: Li Jingwen
Published: July 2016

Vasicek (1977) proved that among all queueing disciplines that do not change the departure process of the queue, FIFO and LIFO yield, respectively, the smallest and the largest expectation of any given convex function of the service delay. In this note we further show that, if arriving customers join the queue stochastically ‘closer' to the server(s), then the expected value of any convex function of service delay is larger. As a more interesting result, we also show that if the function under consideration is concave, then the conclusion will be exactly the opposite. This result indicates that LIFO will be the best discipline if the delay cost is an increasing function but at a diminishing rate.
More on optimal allocation of components in coherent systems
Authors: Meng Fan C.
Published: July 2016

More applications of the principle for interchanging components due to Boland et al. (1989) in reliability theory are presented. In the context of active redundancy improvement we show that if two nodes are permutation equivalent then allocating a redundancy component to the weaker position always results in a larger increase in system reliability, which generalizes a previous result due to Boland et al. (1992). In the case of standby redundancy enhancement, we prove that a series (parallel) system is the only system for which standby redundancy at the component level is always more (less) effective than at the system level. Finally, the principle for interchanging components is extended from binary systems to the more complicated multistate systems.
Asymptotic expansions for functionals of dilation of point processes
Authors: Borovkov A. A.
Published: July 2016

This paper provides a direct approach to obtaining formulas for derivatives of functionals of point processes in rare perturbation analysis ([2], [6]). Results are obtained for arbitrary (not necessarily stationary) point processes in and

d
, d
2, under transparent conditions, close to minimal. Formulas for higher-order derivatives allow one to construct asymptotical expansions. The results can be useful in sensitivity analysis, in light traffic theory for queues and for computation by simulation of derivatives at positive intensity, while the computation of the derivatives via statistical estimation of the functional itself and its increments usually gives poor results.
Principles for modelling financial markets
Authors: Platen Eckhard , Rebolledo Rolando
Published: July 2016

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.
Exact results for a secretary problem
Authors: Quine M. P. , Law J. S.
Published: July 2016

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 → ∞.
On identifiability and order of continuous-time aggregated Markov chains, Markov-modulated Poisson processes, and phase-type distributions
Authors: Rydén Tobias
Published: July 2016

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.
Fast simulation of Markov fluid models
Authors: Ridder Ad
Published: July 2016

In this paper we study continuous flow finite buffer systems with input rates modulated by Markov chains. Discrete event simulations are applied for estimating loss probabilities. The simulations are executed under a twisted version of the original probability measure (importance sampling). We present a simple rule for determining a new measure, then show that the new measure matches the ‘most likely' empirical measure that we expect from large deviations arguments, and finally prove optimality of the new measure.
Rate of convergence of the fluid approximation for generalized Jackson networks
Authors: Chen Hong
Published: July 2016

It is known that a generalized open Jackson queueing network after appropriate scaling (in both time and space) converges almost surely to a fluid network under the uniform topology. Under the same topology, we show that the distance between the scaled queue length process of the queueing network and the fluid level process of the corresponding fluid network converges to zero in probability at an exponential rate.
A local proof of the Swiss Army formula of Palm calculus
Authors: Konstantopoulos Takis
Published: July 2016

The so-called ‘Swiss Army formula', derived by Brémaud, seems to be a general purpose relation which includes all known relations of Palm calculus for stationary stochastic systems driven by point processes. The purpose of this article is to present a short, and rather intuitive, proof of the formula. The proof is based on the Ryll–Nardzewski definition of the Palm probability as a Radon-Nikodym derivative, which, in a stationary context, is equivalent to the Mecke definition.