bluuebunny/crossref_metadata_2025 huggingface dataset:
Optimal list order under partial memory constraints
Authors: Kan Y. C. , Ross S. M.
Published: July 2016

Suppose that we are given a set of n elements which are to be arranged in some order. At each unit of time a request is made to retrieve one of these elements — the ith being requested with probability Pi. We show that the rule which always moves the requested element one closer to the front of the line minimizes the average position of the element requested among a wide class of rules for all probability vectors of the form P
1 = p, P
2= · ·· = Pn = (1 – p)/(n − 1). We also consider the above problem when the decision-maker is allowed to utilize such rules as ‘only make a change if the same element has been requested k times in a row', and show that as k approaches infinity we can do as well as if we knew the values of the Pi.
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: July 2016

Given n events A
1, A
2, · ··, 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 S
1 and S
2 where S
1 = Σi P(Ai
) and S2
= Σ
i<j

Ρ (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 S
1 and S
2.
A note on the ergodicity of Markov chains
Authors: Rosberg Zvi
Published: July 2016

For an aperiodic, irreducible Markov chain with the non-negative integers as state space, a criterion for ergodicity is given. This criterion generalizes the criteria appearing in Foster (1953), Pakes (1969) and Marlin (1973), in the sense that any test function (Liapunov function) which satisfies their conditions also satisfies ours. Applications are presented through some examples.
An upper bound for the velocity of first-passage percolation
Authors: Janson Svante
Published: July 2016

An upper bound for the asymptotic velocity in various directions of first-passage percolation on the square lattice is derived. In particular this gives a lower bound for the so-called time constant. The result is generalized to other lattices. Numerical examples are included.
Generalized two-dimensional bounded birth and death processes and some applications
Authors: Billard L.
Published: July 2016

Two-dimensional bounded birth and death processes are considered. State probabilities are given for the generalized case in which the transition rates are arbitrary time-independent functions of the population sizes. Some applications in epidemic theory and species interaction processes are briefly described.
Countable-state average-cost regenerative stopping problems
Authors: Miller Bruce L.
Published: July 2016

Regenerative stopping problems are stopping problems which recommence from the initial state upon stopping. An algorithm is presented which solves semi-Markov regenerative stopping problem with a finite number of continue actions by solving a sequence of stopping problems. New results for the optimal stopping problem are obtained as well as for the regenerative stopping problem. A model in the literature is used as a detailed example of the algorithm.
The transient state probabilities for a queueing model where potential customers are discouraged by queue length
Authors: Van Doorn Erik A.
Published: July 2016

Exact expressions are derived for the transition probabilities of the birth-death process with parametersandwhich serves as a queueing model where potential customers are discouraged by queue length.
The algebraic equivalence of two measures of genetic distance
Authors: Dawson Jeremy E.
Published: July 2016

A formal proof is given of the equivalence, conjectured by Rostron, of two measures of the genetic difference between two populations.
An invariance principle for solutions to stochastic difference equations
Authors: Guess Harry A.
Published: July 2016

In recent papers, McLeish and others have obtained invariance principles for weak convergence of martingales to Brownian motion. We generalize these results to prove that solutions of discrete-time stochastic difference equations defined in terms of martingale differences converge weakly to continuous-time solutions of Ito stochastic differential equations. Our proof is based on a theorem of Stroock and Varadhan which characterizes the solution of a stochastic differential equation as the unique solution of an associated martingale problem. Applications to mathematical population genetics are discussed.
Multivariate Hawkes processes: an application to financial data
Authors: Embrechts Paul , Liniger Thomas , Lin Lu
Published: July 2016

A Hawkes process is also known under the name of a self-exciting point process and has numerous applications throughout science and engineering. We derive the statistical estimation (maximum likelihood estimation) and goodness-of-fit (mainly graphical) for multivariate Hawkes processes with possibly dependent marks. As an application, we analyze two data sets from finance.
Exponential upper bounds via martingales for multiplexers with Markovian arrivals
Authors: Buffet E. , Duffield N. G.
Published: July 2016

We obtain explicit upper bounds in closed form for the queue length in a slotted time FCFS queue in which the service requirement is a sum of independent Markov processes on the state space {0, 1}, with integral service rate. The bound is of the form [queue length for any where c < 1 and y > 1 are given explicitly in terms of the parameters of the model. The model can be viewed as an approximation for the burst-level component of the queue in an ATM multiplexer. We obtain heavy traffic bounds for the mean queue length and show that for typical parameters this far exceeds the mean queue length for independent arrivals at the same load. We compare our results on the mean queue length with an analytic expression for the case of unit service rate, and compare our results on the full distribution with computer simulations.
A bivariate optimal replacement policy for a repairable system
Authors: Zhang Yuan Lin
Published: July 2016

In this paper, a repairable system consisting of one unit and a single repairman is studied. Assume that the system after repair is not as good as new. Under this assumption, a bivariate replacement policy (T, N), where T is the working age and N is the number of failures of the system is studied. The problem is to determine the optimal replacement policy (T, N)∗such that the long-run average cost per unit time is minimized. The explicit expression of the long-run average cost per unit time is derived, and the corresponding optimal replacement policy can be determined analytically or numerically. Finally, under some conditions, we show that the policy (T, N)∗ is better than policies N∗ or T∗.
More on n-point, win-by-k games
Authors: Haigh John
Published: July 2016

When Siegrist (1989) derived an expression for the probability that player A wins a game that consists of a sequence of Bernoulli trials, the winner being the first player to win n trials and have a lead of at least k, he noted the desirability of giving a direct probabilistic argument. Here we present such an argument, and extend the domain of applicability of the results beyond Bernoulli trials, including cases (such as the tie-break in lawn tennis) where the probability of winning each trial cannot reasonably be taken as constant, and to where there is Markov dependence between successive trials.
A modified block replacement policy with two variables and general random minimal repair cost
Authors: Sheu Shey-Huei
Published: July 2016

This paper considers a modified block replacement with two variables and general random minimal repair cost. Under such a policy, an operating system is preventively replaced by new ones at timeskT(k=1, 2, ···) independently of its failure history. If the system fails in [(k − 1)T,(k− 1)T+T0) it is either replaced by a new one or minimally repaired, and if in [(k− 1)T+T0, kT) it is either minimally repaired or remains inactive until the next planned replacement. The choice of these two possible actions is based on some random mechanism which is age-dependent. The cost of theith minimal repair of the system at ageydepends on the random partC(y) and the deterministic partci(y). The expected cost rate is obtained, using the results of renewal reward theory. The model with two variables is transformed into a model with one variable and the optimum policy is discussed.
On distribution tails and expectations of maxima in critical branching processes
Authors: Borovkov K. A. , Vatutin V A.
Published: July 2016

We derive the limit behaviour of the distribution tail of the global maximum of a critical Galton–Watson process and also of the expectations of partial maxima of the process, when the offspring law belongs to the domain of attraction of a stable law. Thus the Lindvall (1976) and Athreya (1988) results are extended to the infinite variance case. It is shown that in the general case these two asymptotics are closely related to each other, and the latter follows readily from the former. We also discuss a related problem from the theory of general branching processes.
On the full information best-choice problem
Authors: Gnedin Alexander V.
Published: July 2016

We introduce the optimal stopping problem of an infinite sequence of records associated with a planar Poisson process. This problem serves as a limiting form of the classical full information best-choice problem. A link between the finite problem and its limiting form is established via embedding n i.i.d. observations into the planar process.
Finite-time optimal control of a process leaving an interval
Authors: Mcbeth Douglas W. , Weerasinghe Ananda P. N.
Published: July 2016

Consider the optimal control problem of leaving an interval (– a, a) in a limited playing time. In the discrete-time problem, a is a positive integer and the player's position is given by a simple random walk on the integers with initial position x. At each time instant, the player chooses a coin from a control set where the probability of returning heads depends on the current position and the remaining amount of playing time, and the player is betting a unit value on the toss of the coin: heads returning +1 and tails − 1. We discuss the optimal strategy for this discrete-time game. In the continuous-time problem the player chooses infinitesimal mean and infinitesimal variance parameters from a control set which may depend upon the player's position. The problem is to find optimal mean and variance parameters that maximize the probability of leaving the interval [— a, a] within a finite time T > 0.
Multivariate Bonferroni-type lower bounds
Authors: Chen Tuhao , Seneta E.
Published: July 2016

We derive multivariate Sobel–Uppuluri–Galambos-type lower bounds for the probability that at least a
1 and at least a
2, and for the probability that exactly a
1 and a
2, out of n and N events, occur. The lower bound presented here reduces to a sharper bound than that of Galambos and Lee (1992). Our approach is by way of indicator functions and bivariate binomial moments. A new concept, marginal Bonferroni summation, is introduced in this paper.
A complete convergence theorem for an epidemic model
Authors: Andjel Enrique , Schinazi Rinaldo
Published: July 2016

We use an interacting particle system on ℤ to model an epidemic. Each site of ℤ can be in either one of three states: empty, healthy or infected. An empty site x gets occupied by a healthy individual at a rate βn
1(x) where n
1(x) is the number of healthy nearest neighbors of x. A healthy individual at x gets infected at rate αn
2(x) where n
2(x) is the number of infected nearest neighbors of x. An infected individual dies at rate δ independently of everything else. We show that for all α, β and δ> 0 and all initial configurations, all the sites of a fixed finite set remain either all empty or all healthy after an almost surely finite time. Moreover, if the initial configuration has infinitely many healthy individuals then the process converges almost surely (in the sense described above) to the all healthy state. We also consider a model introduced by Durrett and Neuhauser where healthy individuals appear spontaneously at rate β > 0 and for which coexistence of 1's and 2's was proved in dimension 2 for some values of α and β. We prove that coexistence may occur in any dimension.
On the distribution of distances in recursive trees
Authors: Dobrow Robert P.
Published: July 2016

Recursive trees have been used to model such things as the spread of epidemics, family trees of ancient manuscripts, and pyramid schemes. A tree Tn
with n labeled nodes is a recursive tree if n = 1, or n > 1 and Tn
can be constructed by joining node n to a node of some recursive tree Tn–

1. For arbitrary nodes i < n in a random recursive tree we give the exact distribution of Xi,n
, the distance between nodes i and n. We characterize this distribution as the convolution of the law of Xi,j+

1 and n – i – 1 Bernoulli distributions. We further characterize the law of Xi,j+

1 as a mixture of sums of Bernoullis. For i = in
growing as a function of n, we show that is asymptotically normal in several settings.
Dynamic bandwidth allocation for ATM switches
Authors: Hsu Ivy , Walrand Jean
Published: July 2016

We explore a dynamic approach to the problems of call admission and resource allocation for communication networks with connections that are differentiated by their quality of service requirements. In a dynamic approach, the amount of spare resources is estimated on-line based on feedbacks from the network's quality of service monitoring mechanism. The schemes we propose remove the dependence on accurate traffic models and thus simplify the tasks of supplying traffic statistics required of network users. In this paper we present two dynamic algorithms. The objective of these algorithms is to find the minimum bandwidth necessary to satisfy a cell loss probability constraint at an asynchronous transfer mode (ATM) switch. We show that in both schemes the bandwidth chosen by the algorithm approaches the optimal value almost surely. Furthermore, in the second scheme, which determines the point closest to the optimal bandwidth from a finite number of choices, the expected learning time is finite.
Busy period in GIX/G/∞
Authors: Liu Liming , Shi Ding-Hua
Published: July 2016

Busy period problems in infinite server queues are studied systematically, starting from the batch service time. General relations are given for the lengths of the busy cycle, busy period and idle period, and for the number of customers served in a busy period. These relations show that the idle period is the most difficult while the busy cycle is the simplest of the four random variables. Renewal arguments are used to derive explicit results for both general and special cases.
Models of superinfection and acquired immunity to multiple parasite strains
Authors: Milligan P. J. M. , Downham D. Y.
Published: July 2016

Individuals in communities in which different strains of pathogen are circulating can acquire resistance by accumulating immunity to each strain. After considering susceptibility, models of infection and immunity are defined for vector-borne diseases such as malaria and trypanosomiasis. For these models the prevalence of infection, the number of infections per individual, and the mean duration of infection, increase rapidly in young individuals, but decrease in older individuals as immunity is acquired to the various strains of pathogen; the mean interval between successive infections lengthens with age. The bivariate Poisson distribution is shown to be a close approximation to some stochastic processes. The models explain observed cross-sectional patterns of age prevalence, and longitudinal patterns in which individuals typically continue to become infected as they age, albeit with decreasing frequency. In these models the time spent infected depends on parasite diversity, as well as the inoculation and recovery rates. It is shown that control measures can cause an increase in the number of infections and the prevalence of infection in older individuals, and in the average prevalence in the community, even when strain-specific immunity is life-long.
The iterated Galton–Watson process
Authors: Gawel Boleslaw , Kimmel Marek
Published: July 2016

We consider the mathematical properties of a time-discrete stochastic process describing explosive proliferation of DNA repeats in human genetic diseases. The process is constructed using a cascade of Galton–Watson branching processes. The main results concern the probability of absorption and the supergeometric growth of the process in the supercritical case. Examples of simulations are provided.
On the limit behaviour of a superadditive bisexual Galton–Watson branching process
Authors: Gonzalez M. , Molina M.
Published: July 2016

The asymptotic behaviour of a superadditive bisexual Galton–Watson branching process is studied. Sufficient conditions for the almost sure and L
1 convergence of the suitably normed process are given. Finally, a first approach to the study of the L
1 convergence for a superadditive bisexual Galton–Watson branching process under the Z log+
Z condition is considered.
Solution of finite QBD processes
Authors: de Nitto Personè Vittoria , Grassi Vincenzo
Published: July 2016

We provide an explicit matrix analytic solution for finite quasi birth and death (QBD) processes, directly expressed in terms of process parameters. We show that this solution has the same asymptotic complexity of previously proposed non-explicit solutions and is more general than some of them. Moreover, it can be easily extended to the case of generalized QBD processes.
Compound Poisson limit theorems for Markov chains
Authors: Hsiau Shoou-Ren
Published: July 2016

This paper establishes a compound Poisson limit theorem for the sum of a sequence of multi-state Markov chains. Our theorem generalizes an earlier one by Koopman for the two-state Markov chain. Moreover, a similar approach is used to derive a limit theorem for the sum of the k th-order two-state Markov chain.
Non-explosivity of limits of conditioned birth and death processes
Authors: Roberts G. O. , Jacka S. D. , Pollett P. K.
Published: July 2016

Let X be a birth and death process on with absorption at zero and suppose that X is suitably recurrent, irreducible and non-explosive. In a recent paper, Roberts and Jacka (1994) showed that as T → ∞ the process conditioned to non-absortion until time T converges weakly to a time-homogeneous Markov limit, X∞
, which is itself a birth and death process. However the question of the possibility of explosiveness of X∞
remained open. The major result of this paper establishes that X∞
is always non-explosive.
Weak convergence of conditioned birth-death processes in discrete time
Authors: Schrijner Pauline , Van Doorn Erik A.
Published: July 2016

We consider a discrete-time birth-death process on the non-negative integers with −1 as an absorbing state and study the limiting behaviour asn →∞ of the process conditioned on non-absorption until timen.By proving that a condition recently proposed by Martinez and Vares is vacuously true, we establish that the conditioned process is always weakly convergent when all self-transition probabilities are zero. In the aperiodic case we obtain a necessary and sufficient condition for weak convergence.
Boundary crossing probability for Brownian motion and general boundaries
Authors: Wang Liqun , Pötzelberger Klaus
Published: July 2016

An explicit formula for the probability that a Brownian motion crosses a piecewise linear boundary in a finite time interval is derived. This formula is used to obtain approximations to the crossing probabilities for general boundaries which are the uniform limits of piecewise linear functions. The rules for assessing the accuracies of the approximations are given. The calculations of the crossing probabilities are easily carried out through Monte Carlo methods. Some numerical examples are provided.
The M/M/1 queue with mass exodus and mass arrivals when empty
Authors: Chen Anyue , Renshaw Eric
Published: July 2016

An M/M/1 queue is subject to mass exodus at rate β and mass immigration at rate when idle. A general resolvent approach is used to derive occupation probabilities and high-order moments. This powerful technique is not only considerably easier to apply than a standard direct attack on the forward p.g.f. equation, but it also implicitly yields necessary and sufficient conditions for recurrence, positive recurrence and transience.
Orthogonal representations of random fields and an application to geophysics data
Authors: Ruiz-Medina M. D. , Valderrama M. J.
Published: July 2016

We present a brief summary of some results related to deriving orthogonal representations of second-order random fields and its application in solving linear prediction problems. In the homogeneous and/or isotropic case, the spectral theory provides an orthogonal expansion in terms of spherical harmonics, called spectral decomposition (Yadrenko 1983). A prediction formula based on this orthogonal representation is shown. Finally, an application of this formula in solving a real-data problem related to prospective geophysics techniques is presented.
Heavy traffic analysis for continuous polling models
Authors: Kroese Dirk P.
Published: July 2016

We consider a continuous polling system in heavy traffic. Using the relationship between such systems and age-dependent branching processes, we show that the steady-state number of waiting customers in heavy traffic has approximately a gamma distribution. Moreover, given their total number, the configuration of these customers is approximately deterministic.
Sengupta's invariant relationship and its application to waiting time inference
Authors: Toyoizumi Hiroshi
Published: July 2016

This paper presents a new proof of Sengupta's invariant relationship between virtual waiting time and attained sojourn time and its application to estimating the virtual waiting time distribution by counting the number of arrivals and departures of a G/G/1 FIFO queue. Since this relationship does not require any parametric assumptions, our method is non-parametric. This method is expected to have applications, such as call processing in communication switching systems, particularly when the arrival or service process is unknown.
Last round betting
Authors: S. Ferguson Thomas , Melolidakis C.
Published: July 2016

Two players with differing amounts of money simultaneously choose an amount to bet on an even-money win-or-lose bet. The outcomes of the bets may be dependent and the player who has the larger amount of money after the outcomes are decided is the winner. This game is completely analyzed. In nearly all cases, the value exists and optimal strategies for the two players that give weight to a finite number of bets are explicitly exhibited. In a few situations, the value does not exist.
Preservation of multivariate stochastic orders under multivariate Poisson shock models
Authors: Wong Tityik
Published: July 2016

Consider two systems, labeled system 1 and system 2, each with m components. Suppose component i in system k, k = 1, 2, is subjected to a sequence of shocks occurring randomly in time according to a non-explosive counting process {Γ
i
(t), t > 0}, i = 1, ···, m. Assume that Γ1, · ··, Γ
m
are independent of Mk
= (Mk,

1, · ··, Mk,m
), the number of shocks each component in system k can sustain without failure. Let Zk,i
be the lifetime of component i in system k. We find conditions on processes Γ1, · ··, Tm
such that some stochastic orders between M
1 and M
2 are transformed into some stochastic orders between Z
1 and Z2. Most results are obtained under the assumption that Γ1, · ··, Γ
m
are independent Poisson processes, but some generalizations are possible and can be seen from the proofs of theorems.
Multivariate age replacement
Authors: Ebrahimi Nader
Published: July 2016

Age replacement policy is commonly used in order to reduce the number of in-service failures. In this paper we define a multivariate version of this policy and develop some of its desirable properties. We also obtain an optimal age replacement policy.
Response time distributions in tandem G-networks
Authors: Harrison Peter G. , Pitel Edwige
Published: July 2016

The Laplace transform of the probability distribution of the end-to-end delay in tandem networks is obtained where the first and/or second queue are G-queues, i.e. they have negative arrivals. For the most general case the method is based on the solution of a boundary value problem on a closed contour in the complex plane, which itself reduces to the solution of a Fredholm integral equation of the second kind. We also consider the dependence or independence of the sojourn times at each queue in the two special cases where only one of the queues is a G-queue, the other having no negative arrivals.
The mean waiting time of a GI/G/1 queue in light traffic via random thinning
Authors: Nananukul Soracha , Gong Wei-Bo
Published: July 2016

In this paper, we derive the MacLaurin series of the mean waiting time in light traffic for a GI/G/1 queue. The light traffic is defined by random thinning of the arrival process. The MacLaurin series is derived with respect to the admission probability, and we prove that it has a positive radius of convergence. In the numerical examples, we use the MacLaurin series to approximate the mean waiting time beyond light traffic by means of Padé approximation.
Runs on a circle
Authors: Koutras M. V. , Papadopoulos G. K. , Papastavridis S. G.
Published: July 2016

In the present paper we study the number of occurrences of non-overlapping success runs of length in a sequence of (not necessarily identical) Bernoulli trials arranged on a circle. An exact formula is given for the probability function, along with some sharp bounds which turn out to be very useful in establishing limiting (Poisson convergence) results. Certain applications to statistical run tests and reliability theory are also discussed.
Randomly perturbed vibrations
Authors: Elshamy M.
Published: July 2016

Let u
ε(t, x) be the position at time t of a point x on a string, where the time variable t varies in an interval I: = [0, T], T is a fixed positive time, and the space variable x varies in an interval J. The string is performing forced vibrations and also under the influence of small stochastic perturbations of intensity ε. We consider two kinds of random perturbations, one in the form of initial white noise, and the other is a nonlinear random forcing which involves the formal derivative of a Brownian sheet. When J has finite endpoints, a Dirichlet boundary condition is imposed for the solutions of the resulting non-linear wave equation. Assuming that the initial conditions are of sufficient regularity, we analyze the deviations u
ε(t, x) from u
0(t, x), the unperturbed position function, as the intensity of perturbation ε ↓ 0 in the uniform topology. We also discuss some continuity properties of the realization of the solutions u
ε(t, x).
Changes of numéraire, changes of probability measure and option pricing
Authors: Geman Hélyette , El Karoui Nicole , Rochet Jean-Charles
Published: July 2016

The use of the risk-neutral probability measure has proved to be very powerful for computing the prices of contingent claims in the context of complete markets, or the prices of redundant securities when the assumption of complete markets is relaxed. We show here that many other probability measures can be defined in the same way to solve different asset-pricing problems, in particular option pricing. Moreover, these probability measure changes are in fact associated with numéraire changes, this feature, besides providing a financial interpretation, permits efficient selection of the numéraire appropriate for the pricing of a given contingent claim and also permits exhibition of the hedging portfolio, which is in many respects more important than the valuation itself.

The key theorem of general numéraire change is illustrated by many examples, among which the extension to a stochastic interest rates framework of the Margrabe formula, Geske formula, etc.
On the saturation rule for the stability of queues
Authors: Baccelli François , Foss Serguei
Published: July 2016

This paper focuses on the stability of open queueing systems under stationary ergodic assumptions. It defines a set of conditions, the monotone separable framework, ensuring that the stability region is given by the following saturation rule: ‘saturate' the queues which are fed by the external arrival stream; look at the ‘intensity' μ of the departure stream in this saturated system; then stability holds whenever the intensity of the arrival process, say λ satisfies the condition λ < μ, whereas the network is unstable if λ > μ. Whenever the stability condition is satisfied, it is also shown that certain state variables associated with the network admit a finite stationary regime which is constructed pathwise using a Loynes-type backward argument. This framework involves two main pathwise properties, external monotonicity and separability, which are satisfied by several classical queueing networks. The main tool for the proof of this rule is subadditive ergodic theory. It is shown that, for various problems, the proposed method provides an alternative to the methods based on Harris recurrence and regeneration; this is particularly true in the Markov case, where we show that the distributional assumptions commonly made on service or arrival times so as to ensure Harris recurrence can in fact be relaxed.
Product forms based on backward traffic equations
Authors: Boucherie Richard J.
Published: July 2016

This paper introduces a new form of local balance and the corresponding product-form results. It is shown that these new product-form results allow capacity constraints at the stations of a queueing network without reversibility assumptions and without special blocking protocols. In particular, exact product-form results for heavily loaded queueing networks are obtained.
A multitype decomposable age-dependent branching process and its applications
Authors: Lee Chinsan , Yang Grace L.
Published: July 2016

Asymptotic formulas for means and variances of a multitype decomposable age-dependent supercritical branching process are derived. This process is a generalization of the Kendall–Neyman–Scott two-stage model for tumor growth. Both means and variances have exponential growth rates as in the case of the Markov branching process. But unlike Markov branching, these asymptotic moments depend on the age of the original individual at the start of the process and the life span distribution of the progenies.
A piecewise Markovian model for simulated annealing with stochastic cooling schedules
Authors: Kolonko M.
Published: July 2016

We introduce a stochastic process with discrete time and countable state space that is governed by a sequence of Markov matrices . Each Pm
is used for a random number of steps Tm
and is then replaced by Pm

+1. Tm
is a randomized stopping time that may depend on the most recent part of the state history. Thus the global character of the process is non-Markovian.

This process can be used to model the well-known simulated annealing optimization algorithm with randomized, partly state depending cooling schedules. Generalizing the concept of strong stationary times (Aldous and Diaconis [1]) we are able to show the existence of optimal schedules and to prove some desirable properties. This result is mainly of theoretical interest as the proofs do not yield an explicit algorithm to construct the optimal schedules.
Minification processes with discrete marginals
Authors: Kalamkar V. A.
Published: July 2016

We investigate the stationarity of minification processes when the marginal is a discrete distribution. There is a close relationship between the problem considered by Arnold and Isaacson (1976) and the stationarity in minification processes. We give a necessary and sufficient condition for a discrete distribution to be the marginal of a stationary minification process. Members of the Poisson and negative binomial families can be the marginals of stationary minification processes. The geometric minification process is studied in detail, and two characterizations of it based on the structure of the innovation process are given.
Weak limit results for the extremes of a class of shot noise processes
Authors: Homble Patrick , McCormick William P.
Published: July 2016

Shot noise processes form an important class of stochastic processes modeling phenomena which occur as shocks to a system and with effects that diminish over time. In this paper we present extreme value results for two cases — a homogeneous Poisson process of shocks and a non-homogeneous Poisson process with periodic intensity function. Shocks occur with a random amplitude having either a gamma or Weibull density and dissipate via a compactly supported impulse response function. This work continues work of Hsing and Teugels (1989) and Doney and O'Brien (1991) to the case of random amplitudes.
An optimal stopping problem for random walks with non-zero drift
Authors: Roters Markus
Published: July 2016

In this paper we give a solution of an optimal stopping problem concerning random walks with non-zero drift, thereby proving the necessity of the existence of ESτ
for Wald's equation ES
τ = ES
1 · Ετ to hold, even if attention is restricted to non-randomized stopping times τ. This answers a question of Robbins and Samuel (1966).
Records from a power model of independent observations
Authors: Weissman Ishay
Published: July 2016

Records from are analyzed, where {Yj
} is an independent sequence of random variables. Each Yj
has a continuous distribution function Fj = Fλj
for some distribution F and some λ j > 0. We study records, record times and related quantities for this sequence. Depending on the sequence of powers , a wide spectrum of behaviour is exhibited.
Error bounds for deterministic approximations to Markov processes, with applications to epidemic models
Authors: Gani J. , Yakowitz Sid
Published: July 2016

The computer age and the phenomenological complexity of the AIDS/HIV epidemic have engendered a rich profusion of deterministic and stochastic time series models for the development of an epidemic. The present study examines the reliability of deterministic approximations of fundamentally random processes. Through numerical analysis and probabilistic considerations, we derive absolute and simultaneous confidence interval bounding techniques, and offer a practical procedure based on these developments. A heartening aspect of the computational study presented at the close of this paper indicates that when the population size is in the thousands, the deterministic version to the classical logistic epidemic is a good approximation.
The performance of the move-to-front scheme under some particular forms of Markov requests
Authors: Rodrigues Eliane R.
Published: July 2016

The move-to-front scheme is studied taking into account some forms of Markov dependence for the way items are requested. One of the dependences specifically rules out two consecutive requests for the same item. The other is the so-called p-correlation. An expression for the stationary distribution of the sequence of arrangements of items is given in each case. A necessary and sufficient condition for these distributions to belong to a particular class of distributions is also given. The mean search time for an item is calculated for each form of dependence and these are compared with the value obtained in the case of independent requests. Some properties of the sequence of requests are given. Finally, an expression for the variance of the search time is obtained.
Large deviations and the Bayesian estimation of higher-order Markov transition functions
Authors: Papangelou F.
Published: July 2016

In the Bayesian estimation of higher-order Markov transition functions on finite state spaces, a prior distribution may assign positive probability to arbitrarily high orders. If there are n observations available, we show (for natural priors) that, with probability one, as n → ∞ the Bayesian posterior distribution ‘discriminates accurately' for orders up to β log n, if β is smaller than an explicitly determined β
0. This means that the ‘large deviations' of the posterior are controlled by the relative entropies of the true transition function with respect to all others, much as the large deviations of the empirical distributions are governed by their relative entropies with respect to the true transition function. An example shows that the result can fail even for orders β log n if β is large.
Spectral representations of the transition probability matrices for continuous time finite Markov chains
Authors: Peng Nan Fu
Published: July 2016

Using an easy linear-algebraic method, we obtain spectral representations, without the need for eigenvector determination, of the transition probability matrices for completely general continuous time Markov chains with finite state space. Comparing the proof presented here with that of Brown (1991), who provided a similar result for a special class of finite Markov chains, we observe that ours is more concise.
Peaks and Eulerian numbers in a random sequence
Authors: Warren Di , Seneta E.
Published: July 2016

We consider the exact distribution of the number of peaks in a random permutation of the integers 1, 2, ···, n. This arises from a test of whether n successive observations from a continuous distribution are i.i.d. The Eulerian numbers, which figure in the p.g.f., are then shown to provide a link between the simpler problem of ascents (which has been thoroughly analysed) and both our problem of peaks and similar problems on the circle. This link then permits easy deduction of certain general properties, such as linearity in n of the cumulants, in the more complex settings. Since the focus of the paper is on exact distributional results, a uniform bound on the deviation from the limiting normal is included. A secondary purpose of the paper is synthesis, beginning with the more familiar setting of peaks and troughs.
Tail of compound distributions and excess time
Authors: Lin Xiaodong
Published: July 2016

Bounds on the tail of compound distributions are considered. Using a generalization of Wald's fundamental identity, we derive upper and lower bounds for various compound distributions in terms of new worse than used (NWU) and new better than used (NBU) distributions respectively. Simple bounds are obtained when the claim size distribution is NWUC, NBUC, NWU, NBU, IMRL, DMRL, DFR and IFR. Examples on how to use these bounds are given.
Increasing the reliability of a machine reduces the period of its work
Authors: Sonin Isaac M.
Published: July 2016

The comparison of optimal strategies in a simple stochastic replacement model for two types of machines with identical cost characteristics when one of them is more reliable than the other is conducted. It is proven that the scheduled replacement period for the more reliable type is always less than for the less reliable one. An example is presented when even the expected period of use of a more reliable machine is less than the expected period for the less reliable one. Some related problems are briefly discussed.
Random record models
Authors: Gaver Donald P.
Published: July 2016

We study record times, mainly, and sizes in the following context. LetXndenote the size of thenth event occurring in a point stochastic pacing process 𝒫. TheXnis i.i.d. and 𝒫 is, variously, Poisson, negative binomial, renewal and Furry. Explicit distributions of first-record times are found, domains of attraction studied and the asymptotic lognormality of thenth-record time is shown for Poisson 𝒫.
Limiting diffusions for population-size dependent branching processes
Authors: Lipow Carla
Published: July 2016

A classical theorem on the convergence to a diffusion of a sequence of branching processes will be proved in the case where the branching processes are modified to allow dependence on population size of the generating function governing reproduction. Some recent semigroup convergence theorems due to Kurtz (1969), (1975) provide the main tools.
Subdivided populations: the characteristic values and rate of loss of alleles
Authors: Gladstien Keith
Published: July 2016

The characteristic values of certain matrices (underlying multiple allele models in population genetics) are investigated and related to the asymptotic behavior of Pr(at leastkdistinct alleles in the population at timet).
Poisson limits for a clustering model of strauss
Authors: Saunders Roy , Funk Gerald M.
Published: July 2016

In this article we present a limiting result for the random variable Yn
(r) which arises in a clustering model of Strauss (1975). The result is that under some sparseness-of-points conditions the process {Yn
(r): 0 ≦ r ≦ r
∞} converges weakly to a non-homogeneous Poisson process {Y(r): 0 ≦ r ≦ r
∞} when n → ∞. Simulation results are given to indicate the accuracy of the approximation when n is moderate and applications of the limiting result to tests for clustering are discussed.
The M/M
/∞ queue in a random environment
Authors: O'Cinneide C. A. , Purdue P.
Published: July 2016

The M/M/∞ queue in a random environment is an infinite-server queue where arrival and service rates are stochastic processes. Here we study the steady-state behavior of such a system. Explicit results are obtained for the factorial moments, the impossibility of a ‘matrix-Poisson' steady-state distribution is demonstrated and two numerical examples are presented.
On the waiting-time process in a single-line queue with repeated calls
Authors: Falin G. I.
Published: July 2016

Waiting time in a queueing system is usually measured by a period from the epoch when a subscriber enters the system until the service starting epoch. For repeated orders queueing systems it is natural to measure the waiting time by the number of repeated attempts,R, which have to be made by a blocked primary call customer before the call enters service. We study this problem for theM/M/1/1 retrial queue and derive expressions for mean, variance and generating function ofR.Limit theorems are stated for heavy- and light-traffic cases.
Asymptotic reliability of consecutive k-out-of-n systems
Authors: Canfield E. R. , Mccormick W. P.
Published: July 2016

By studying the minimum of moving maxima, that is the maxima taken over a sliding window of length k in an i.i.d. sequence, we obtain new results on the reliability of consecutive k-out-of-n systems. In particular, we give the reliability asymptotically with both k and n varying. The underlying method of our approach is to analyze the singularities of a generating function.
LRU is better than FIFO under the independent reference model
Authors: van den Berg J. , Gandolfi A.
Published: July 2016

Consider a two-level storage system operating with the least recently used (LRU) or the first-in, first-out (FIFO) replacement strategy. Accesses to the main storage are described by the independent reference model (IRM). Using the FKG inequality, we prove that the miss ratio for LRU is smaller than or equal to the miss ratio for FIFO.
Further studies of bivariate Bonferroni-type inequalities
Authors: Galambos Janos , Lee Min-Young
Published: July 2016

Let A
1, A
2, …, An
and B
1, B
2,. ., BN
be two sequences of events on the same probability space. Let mn
(A) and mN
(B), respectively, be the number of those Aj
and Bj
which occur. Let Si,j
denote the joint ith binomial moment of mn
(A) and jth binomial moment of mN
(B), 0 ≤ i ≤ n, 0 ≤ j ≤ N. For fixed non-negative integers a and b, we establish both lower and upper bounds on the distribution P(mn
(A) = r, mN
(B) = u) by linear combinations of Si,j
, 0 ≤ i ≤ a, 0 ≤ j ≤ b. When both a and b are even, all mentioned S¡,j are utilized in both the upper and the lower bound. In a set of remarks the results are analyzed and their relation to the existing literature, including the univariate case, is discussed.
On a class of two-dimensional nearest-neighbour random walks
Authors: Cohen J. W.
Published: July 2016

For positive recurrent nearest-neighbour, semi-homogeneous random walks on the lattice {0, 1, 2, …} X {0, 1, 2, …} the bivariate generating function of the stationary distribution is analysed for the case where one-step transitions to the north, north-east and east at interior points of the state space all have zero probability. It is shown that this generating function can be represented by meromorphic functions. The construction of this representation is exposed for a variety of one-step transition vectors at the boundary points of the state space.
Some joint distributions in Bernoulli excursions
Authors: Csáki Endre
Published: July 2016

Some exact and asymptotic joint distributions are given for certain random variables defined on the excursions of a simple symmetric random walk. We derive appropriate recursion formulas and apply them to get certain expressions for the joint generating or characteristic functions of the random variables.
On termination time processes
Authors: Dshalalow Jewgeni H.
Published: July 2016

In this paper we study the behavior of a delayed compound renewal process, S, about some fixed level, L. Normally, a jump process S increases at random times τ
1, τ
2, …, in random increments until it crosses L. S would then be terminated in a random number v of phases at time τ
v. In many applications, a more general termination scenario assumes that S may evolve either through v or σ random phases, whichever of the two is smaller (denoted by T). The number T of actual phases is called the termination index, and we evaluate a joint functional of T, the termination time τ T
and the termination level ST. We also seek information about the process S a step before its termination, and derive a joint functional for all relevant processes. Examples of these processes and their applications to various stochastic models are discussed.
Exact and ordinary lumpability in finite Markov chains
Authors: Buchholz Peter
Published: July 2016

Exact and ordinary lumpability in finite Markov chains is considered. Both concepts naturally define an aggregation of the Markov chain yielding an aggregated chain that allows the exact determination of several stationary and transient results for the original chain. We show which quantities can be determined without an error from the aggregated process and describe methods to calculate bounds on the remaining results. Furthermore, the concept of lumpability is extended to near lumpability yielding approximative aggregation.
The critical contact process on a homogeneous tree
Authors: Morrow Gregory J. , Schinazi Rinaldo B. , Zhang Yu
Published: July 2016

We prove that the expected number of particles of the critical contact process on a homogeneous tree is bounded above. This is the first graph for which the behavior of the expected number of particles of the critical contact process is known. As an easy corollary of our result we get that the critical contact process dies out on any homogeneous tree. This completes the work of Pemantle (1992).
Notes on information in population genetics
Authors: Holgate P.
Published: July 2016

The change with respect to time of the information in a population is studied, when the population is classified according to genetic type. The cases considered are (i) an infinite population classified at two loci, (ii) a large finite population classified at one locus, subject to sampling fluctuations and (iii) the same, subject to the effect of mutation.
A note on coupling of birth and death processes
Authors: Lindvall Torgny
Published: July 2016

The purpose of this note is to show how well the coupling device is fitted for use in the study of birth and death process asymptotics: known achievements get new proofs, and new results on rate of ‘forgetfulness of initial state' and stochastic domination are established.