bluuebunny/crossref_metadata_2025 huggingface dataset:
On coupling of random walks and renewal processes
Authors: Lindvall Torgny , Rogers L. C. G.
Published: July 2016

The use of Mineka coupling is extended to a case with a continuous state space: an efficient coupling of random walks S and S' in can be made such that S' — S is virtually a one-dimensional simple random walk. This insight settles a zero-two law of ergodicity. One more proof of Blackwell's renewal theorem is also presented.
First passage time distribution of a Wiener process with drift concerning two elastic barriers
Authors: Dominé Marco
Published: July 2016

We solve the Fokker-Planck equation for the Wiener process with drift in the presence of elastic boundaries and a fixed start point. An explicit expression is obtained for the first passage density. The cases with pure absorbing and/or reflecting barriers arise for a special choice of a parameter constellation. These special cases are compared with results in Darling and Siegert [5] and Sweet and Hardin [15].
A sample path analysis of the delay in the M/G/C system
Authors: Seshadri Sridhar
Published: July 2016

Using sample path analysis we show that under the same load the mean delay in queue in the M/G/2 system is smaller than that in the corresponding M/G/1 system, when the service time has either the DMRL or NBU property and the service discipline is FCFS. The proof technique uses a new device that equalizes the work in a two server system with that in a single sterver system. Other interesting quantities such as the average difference in work between the two servers in the GI/G/2 system and an exact alternate derivation of the mean delay in the M/M/2 system from sample path analysis are presented. For the same load, we also show that the mean delay in the M/G/C system with general service time distribution is smaller than that in the M/G/1 system when the traffic intensity is less than 1/c.
On a stochastic differential equation modeling of prey-predator evolution
Authors: Gard T. C. , Kannan D.
Published: July 2016

We study a stochastic differential equation model of prey-predator evolution. To keep in line with a known deterministic model we include the social and interaction terms with the drift in our model, and the randomness arises as fluctuations in the ecosystem. The notion of equilibrium population level and special types of fluctuations force us to work with degenerate elliptic operators. We consider the propagation of the population system in a sufficiently large but bounded domain. This enables us to look at not only the population extinction, but also the explosion beyond a certain level. Both extinction and explosion are possible; and, when they are not, we show that the population asymptotically reaches the equilibrium level. We show that the extinction, explosion and saturation probabilities satisfy, as functions of the initial population size, an integral equation arising out of a Dirichlet problem for a non-degenerate elliptic equation; and these probabilities are also smooth solutions of the Dirichlet problems. They are also used to express the solution of another Dirichlet problem for a degenerate elliptic equation.
Malthusian behaviour of the critical and subcritical age-dependent branching processes with arbitrary state space
Authors: Saunders D. I.
Published: July 2016

For the age-dependent branching process with arbitrary state space let M(x, t, A) be the expected number of individuals alive at time t with states in A given an initial individual at x. Subject to various conditions it is shown that M(x, t, A)e–at
converges to a non-trivial limit where α is the Malthusian parameter (α = 0 for the critical case, and is negative in the subcritical case). The method of proof also yields rates of convergence.
A stochastic general epidemic in m sub-populations
Authors: Billard L.
Published: July 2016

Epidemic models have generally assumed that the population mixes homogeneously. More realistic models can be obtained by dividing the population into sub-populations within which mixing is homogeneous but between which mixing is heterogeneous. We consider such a model for the general epidemic process and present a formulation which permits inroads to be made towards obtaining explicit solutions to the state probabilities.
An Erdös-Rényi strong law for sample quantiles
Authors: Book Stephen A. , Truax Donald R.
Published: July 2016

From a random sample X
1, X
2, …, XN
there can be constructed N – K + 1 successive sample means of the form for 0 ≦ n ≦ N − K, where Erdös and Rényi (1970) studied the maximum Σ(N, K) of these N – K + 1 sample means. Under appropriate conditions, they showed that for a wide interval of λ's there exist constants C(λ), depending only on λ and the distribution from which the sample was selected, such that Σ(N, [C(λ) log N])→ λ a.s. as N→∞. In the present article, analogous results are developed for the maximum of the N – K + 1 successive sample medians and, more generally, for all sample quantiles.
A note on roots of Markov shifts
Authors: Rudolfer S. M.
Published: July 2016

Let Tv
be the two-sided shift operator associated with a finite Markov chain of period v; Using results of Krengel and Michel and Adler, Shields and Smorodinsky, necessary and sufficient conditions for the existence of an rth root of Tv
are obtained. In particular, if the Markov chain is irreducible, then Tv
has an rth root when and only when (r, v) = 1.
On an approximation made when analysing stochastic processes
Authors: Morgan Byron J. T. , Hinde John P.
Published: July 2016

We investigate the effect of a particular mode of approximation by means of four examples of its use; in each case the model approximated is a Markov process with discrete states in continuous time.
Motion of a particle in a velocity-dependent random force
Authors: Karmeshu
Published: July 2016

The motion of a particle is investigated in the presence of a velocity-dependent random force assumed to be proportional to velocity. Two different possibilities are considered, namely, the presence and absence of random driving force. In the absence of random driving force, the velocity and displacement auto-correlation function are calculated. The probability distribution in velocity space is also evaluated. It is found that in the absence of intrinsic damping, the energy of the particle increases without limit. The condition for the energetic stability of the particle in the presence of random driving force is obtained. The Fokker-Planck equation for the probability distribution in velocity space is derived from the stochastic Liouville equation for delta-correlated velocity-dependent random force.
Projected thick sections through multi-dimensional particle aggregates
Authors: Davy Pamela
Published: July 2016

Formulae relating properties of a two-phase structure to properties of a section through that structure are well known in stereology. An exact description of the error introduced into these formulae when the projection of a thick slice is observed rather than a true section (i.e. a slice of zero thickness) is presented for a model of opaque particles distributed according to a homogeneous Poisson point process in En.
Some asymptotic results for optimal stopping based on capture times
Authors: Wardrop Robert L.
Published: July 2016

A region contains n prey labeled 1, 2, …, n. Prey i is captured at the random time Z
i; where Z
1, Z
2, …, Z
n are i.i.d. with distribution function F. The statistician must decide when to stop searching, with the goal of maximizing the number of prey captured minus a linear time cost, c. The optimal strategy and its expected payoff are studied asymptotically as n, c →∞, for F a beta or Weibull distribution.
A note on the central limit theorem for doubly stochastic Poisson processes
Authors: Rootzén Holger
Published: July 2016

In this note, necessary and sufficient conditions for the central limit theorem for the number of events in a doubly stochastic Poisson process are given.
A note on left-continuous random walks
Authors: Letac Gérard , Mazet Pierre , Schiffmann Gérard
Published: July 2016

Let ℒ be the set of probability measures concentrated on { − 1, 0, 1, 2, …}. Then, if μ
0 is in ℒ, there exists at most one μ
1 in ℒ, with μ0≠ μ
1, such that for all n = 1, 2, ….
On the virtual and actual waiting time distributions of a GI/G/1 queue
Authors: Harrison J. Michael , Lemoine Austin J.
Published: July 2016

Consider a stable GI/G/1 queue with non-lattice interarrival time distribution. Let G and H be the limiting actual and virtual waiting time distributions respectively. Two separate statements of the relationship between G and H are found in a classical theorem of Takàcs and a more recent (and previously unpublished) theorem of Hooke. A simplified proof of Takàcs's theorem, based on a sample path relationship between the virtual and actual waiting time processes, has recently been advanced. This paper gives a similar proof of Hooke's theorem, based on the same sample path relationship, and demonstrates the utility of the result in analyzing the special case of Poisson input. In particular, by combining the Takàcs and Hooke results one can obtain the Pollaczek–Khintchine formula without any reference to the imbedded Markov chain.
Diffusion approximations to linear stochastic difference equations with stationary coefficients
Authors: Guess Harry A. , Gillespie John H.
Published: July 2016

It is shown that solutions to linear first-order stochastic difference equations with stationary autocorrelated coefficients converge weakly in D[0,1] to an Ito stochastic integral plus a correction term when the time scale is shifted so that the means, variances, and covariances of the coefficients all approach zero at the same rate. Other limit theorems applicable to different time scale shifts are also given. These results yield two different continuous time limits to a recent model of Roughgarden (1975) for population growth in stationary random environments. One limit, an Ornstein-Uhlenbeck process, is applicable in the presence of rapidly fluctuating autocorrelated environments; the other limit, which is not a diffusion process, applies to the case of slowly varying, highly autocorrelated environments. Other applications in population biology and genetics are discussed.
On the ordered partial sums of real random variables
Authors: Takács Lajos
Published: July 2016

In 1952 Pollaczek discovered a remarkable formula for the Laplace-Stieltjes transforms of the distributions of the ordered partial sums for a sequence of independent and identically distributed real random variables. In this paper Pollaczek's result is proved in a simple way and is extended for a semi-Markov sequence of real random variables.
On the weak convergence of a class of estimators of the variance-time curve of a weakly stationary point process
Authors: Liebetrau A. M.
Published: July 2016

The second-moment structure of an estimator V*(t) of the variance-time curve V(t) of a weakly stationary point process is obtained in the case where the process is Poisson. This result is used to establish the weak convergence of a class of estimators of the form Tβ
(V*(tTα
) – V(tTα
)), 0 < α < 1, to a non-stationary Gaussian process. Similar results are shown to hold when α = 0 and in the case where V(tTα
) is replaced by a suitable estimator.
The reversed ladder of a random walk
Authors: Stam A. J.
Published: July 2016

The reversed strict ascending ladder epoch v
1 + ··· +vm
of the random walk S(n) with drift to ∞ is the m th time n at which the event S(n) < S(k), k > n, occurs and the corresponding ladder height is H
1 + ··· + Hm
= S(v
1 + ··· + vm
). It is shown that the random vectors (vi, Hi
) are independent and for i ≧ 2 have the same distribution as the first strict ascending ladder time and height in the usual sense. This leads to an equality between the distribution of the first strict ladder height and P{min S(n) ≧ x} for x > 0. Reversed weak ladder points are defined analogously.
Distance distributions associated with poisson processes of geometric figures
Authors: Berman Mark
Published: July 2016

Consider a process of identically-shaped (but not necessarily equal-sized) figures (e.g. points, clusters of points, lines, spheres) embedded at random in n-dimensional space. A simple technique is derived for finding the distribution of the distance from a fixed point, chosen independently of the process of figures, to the k th nearest figure. The technique also shows that the distribution is independent of the distribution of the orientations of the figures. It is noted that the distribution obtained above (for equal-sized figures) is identical to the distribution of the distance from a fixed figure to the k th nearest of a random process of points.
Superposition of distinguishable point processes
Authors: Ito Yoshifusa
Published: July 2016

The properties of frequency distributions of inter-event intervals in the pooled output of several point processes where the points can be distinguished are investigated. In the case where the component processes are mutually independent, the p.d.f.'s of inter-event intervals in the pooled output are explicitly expressed, and the reciprocal relations on the p.d.f.'s and on the numbers of inter-event intervals are revealed.
Regenerative sets and subordinators
Authors: Gzyl Cristina
Published: July 2016

It has been shown that the probabilistic structure of a regular regenerative set can be described in terms of the exponent of an appropriate subordinator. We give a relation which expresses the distribution of two Markov processes associated with a regenerative set in terms of this exponent.
Gaussian approximation of some closed stochastic epidemic models
Authors: Wang Frank J. S.
Published: July 2016

A generalization of Bailey's general epidemic model is considered. In this generalized model, it is assumed that the probability of any particular susceptible becoming infected during the small time interval (t, t + Δt) is α(X(t))Δt + o(Δt), for some function a, where X(t) is the proportion of infected individuals in the entire population, the probability that an infected individual is infected for at least a length of time t is F(t), and recovered individuals are permanently immune from further attack. In this paper, central limit theorems are obtained for the proportion of infected individuals and the proportion of susceptibles in the entire population.
On two marked point processes
Authors: Rüschendorf Ludger
Published: July 2016

Two examples for marked point processes are discussed and some characteristic parameters of these models are calculated. Both examples are in some way modifications of the counter models which are well known and treated in several textbooks.
A controlled birth and death process model of optimal product pricing under stochastically changing demand
Authors: Deshmukh S. D. , Winston Wayne
Published: July 2016

We consider the problem of product pricing when the firm's market share is changing stochastically according to a birth and death process. The current market share together with the price prevailing determine the current rate of profit made as well as the birth and death rates. The optimal pricing policy must balance the immediate advantage of setting a high price in terms of increased current profit against the disadvantage in terms of a possible erosion of the future market share. We formulate a continuous-time Markov decision model and analyse it using a recent technique developed by Lippman [6] for optimization of exponential queueing systems. The optimal pricing policy is characterized as having a sort of monotonicity property. We also analyse the dependence of the optimal policy on the problem parameters and indicate further extensions of the model.
Dams with additive inputs revisited
Authors: Pegg P. A. , Phatarfod R. M.
Published: July 2016

In this paper we consider a finite dam with unit release. Ratio identities are derived for the various random walks involved, and from them are derived the p.g.f. of the time of first emptiness with and without overflow and the complete time-dependent distribution of the dam content.
On Lotka–Volterra predator prey models
Authors: Billard L.
Published: July 2016

Ever since Lotka (1925) and Volterra (1926), (1931) first considered mathematical formulations for prey-predator processes, the resultant equations have resisted attempts to solve them. However, over the intervening 50 years, standard techniques have allowed a few isolated results to be obtained for some simplified versions of the original process, but the classical equations for the stochastic model have remained unsolved. We give here solutions to the classical process for the case in which interactions occur over a sufficiently short period of time that no births occur. The technique used is one recently developed by Severo (1969a), (1969b), (1971). The approach can be easily generalised to allow solution for the case in which births do occur, as well as for the simplified versions of the original process.
On the convergence of the supercritical branching processes with immigration
Authors: Cohn Harry
Published: July 2016

It is shown for a supercritical branching process with immigration that if the log moment of the immigration distribution is infinite, then no sequence of positive constants {cn
} exists such that {Xn/cn
} converges in law to a proper limit distribution function F, except for the case F(0 +) = 1. Seneta's result [1] combined with the above-mentioned one imply that if 1 < m < ∞ then the finiteness of the log moment of the immigration distribution is a necessary and sufficient condition for the existence of some constants {cn
} such that {Xn
/c
n} converges in law to a proper limit distribution function F, with F(0 +) < 1.
The probability distribution of the duration of the game in the classical ruin problem
Authors: Karni Edi
Published: July 2016

This paper develops simple expressions for the exact probability distribution of the duration of the game, and the probability distribution of the duration of the game conditional upon its termination with the gambler's ruin, in the classical ruin problem.
Almost sure convergence in Markov branching processes with infinite mean
Authors: Grey D. R.
Published: July 2016

If {Zn} is a Galton–Watson branching process with infinite mean, it is shown that under certain conditions there exist constants {cn} and a functionL, slowly varying at 0, such thatconverges almost surely to a non-degenerate random variable, whose distribution function satisfies a certain functional equation. The method is then extended to a continuous-time Markov branching process {Zt} with infinite mean, where it is shown that there is always a function φ, slowly varying at 0, such thatconverges almost surely to a non-degenerate random variable, and a necessary and sufficient condition is given for this convergence to be equivalent to convergence offor some constant α > 0.
A curious binary lattice process
Authors: Pickard D. K.
Published: July 2016

A rigorous treatment is given for a construction via Markov chains of a binary (0–1) stationary homogeneous Markov random field on Z × Z. The resulting process possesses rather interesting properties. For example, its correlation structure is geometric and it may be easily simulated. Some of the results are rather unintuitive — indeed counter-intuitive — but their demonstration is straightforward involving only the most elementary properties of Markov chains.
A note on simple branching processes with infinite mean
Authors: Hudson Irene L. , Seneta E.
Published: July 2016

We consider the Bienaymé–Galton–Watson process without and with immigration, and with offspring distribution having infinite mean. For such a process, {Zn
} say, conditions are given ensuring that there exists a sequence of positive constants, {ρn
}, such that {ρnU(Zn
+ 1)} converges almost surely to a proper non-degenerate random variable, where U is a function slowly varying at infinity, defined on [1, ∞), continuous and strictly increasing, with U(1) = 0, U(∞) = ∞. These results subsume earlier ones with U(t) = log t.
A note on a condition satisfied by certain random walks
Authors: Doney R. A.
Published: July 2016

The problem considered is to elucidate under what circumstances the condition

holds, where and Xi
are independent and have common distribution function F. The main result is that if F has zero mean, and (*) holds with
F belongs to the domain of attraction of a completely asymmetric stable law of parameter 1/γ. The cases are also treated. (The case cannot arise in these circumstances.) A partial result is also given for the case when and the right-hand tail is ‘asymptotically larger’ than the left-hand tail. For 0 < γ < 1, (*) is known to be a necessary and sufficient condition for the arc-sine theorem to hold for Nn
, the number of positive terms in (S
1, S
2, …, Sn
). In the final section we point out that in the case γ = 1 a limit theorem of a rather peculiar type can hold for Nn.
A simple model for the balance between selection and mutation
Authors: Kingman J. F. C.
Published: July 2016

A model for the variation in time of the fitness distribution in a large haploid population is shown to have simple limiting properties which can be elucidated in fairly explicit terms. The novel feature is that mutation is not assumed to cause a small perturbation in fitness but to bring down the evolutionary ‘house of cards’. A threshold phenomenon appears: if a certain inequality holds the limiting distribution is a skewed version of the mutant fitness distribution, but otherwise an atom of probability builds up at the upper limit of fitness.
A stochastic model of lexical change
Authors: Dobson Annette J.
Published: July 2016

A model is given to describe the concurrent evolution of vocabularies of different languages belonging to the same family. Lexical change involves the replacement of old words by new words acquired by inheritance, borrowing or some other form of innovation. This process is described by a set of simultaneous differential equations involving the probabilities that, at any time, languages share similar words for any particular meaning. Various special cases of the model are compared using data for some Indo-European languages.
Branching processes with random absorbing processes
Authors: Bruss F. Thomas
Published: July 2016

A discrete Galton–Watson process is modified by an absorbing process, which, within each generation, eliminates a subset of the living particles without leaving offspring. The absorbing process will be only roughly specified by its expected efficiency on the associated process. We give a sharp sufficient condition for the joint process to be extinguished with probability one and give, after an easy generalization, an example for possible biomedical applications.
On entrance—exit distributions of Markov processes
Authors: Gzyl Cristina , Gzyl Henryk
Published: July 2016

We use a result on integration by parts for stochastic integrals together with a technique developed by Getoor in [6], to express entrance—exit distributions for a standard process X, and a set Φ which is the support of a continuous additive functional C, in terms of the infinitesimal generators of semigroups associated with the time-changed process (X

τ

t

), where (τ

t
) is the right-continuous inverse of C.
Uniform limit theorems for non-singular renewal and Markov renewal processes
Authors: Arjas Elja , Nummelin Esa , Tweedie Richard L.
Published: July 2016

We show that if the increment distribution of a renewal process has some convolution non-singular with respect to Lebesgue measure, then the skeletons of the forward recurrence time process are φ-irreducible positive recurrent Markov chains. Known convergence properties of such chains give simple proofs of uniform versions of some old and new key renewal theorems; these show in particular that non-singularity assumptions on the increment and initial distributions enable the assumption of direct Riemann integrability to be dropped from the standard key renewal theorem. An application to Markov renewal processes is given.
Optimal replacement of damaged devices
Authors: Abdel-Hameed M. , Shimi I. N.
Published: July 2016

A device is subject to a sequence of shocks occurring randomly at times n = 1, 2, ⃛. At each point in time, shocks occur according to a Poisson distribution with parameter λ. Shocks cause damage and damage accumulates additively. They can cause the device to fail, and the probability of such a failure depends on the accumulated damage. Failure occurs because of shocks and can occur only at times n = 1, 2, ⃛. The device can be replaced before or at failure. If the device fails it is immediately replaced at a fixed cost. Replacement before failure can only occur at times n = 1, 2, ⃛, and is done at a lower cost depending on the amount of accumulated damage at replacement. In this paper we determine the optimal replacement policy that minimizes the expected cost per unit time.
On the time to first overflow in dams with inputs forming a Markov chain
Authors: Kennedy Douglas P.
Published: July 2016

A finite dam is studied in which the net input in each period, i.e. the excess of input over demand, is a function of a Markov chain. By using a martingale, the joint distribution of the time to first overflow and the cumulative unsatisfied demand until overflow is investigated. The probability of overflow before the unsatisfied demand exceeds a fixed level is also considered.
The expected number of visits of a sequence of normal partial sums to each side of a boundary
Authors: Pollar Moshe
Published: July 2016

Let be the sequence of partial sums of independent N(μ, 1) random variables. The boundary in the {(n, Sn
)} plane which minimizes the expected number of times n that Sn
will be below a boundary when μ = θ > 0 subject to a given expected number of visits that Sn
will be above the same boundary when μ = 0 is shown to be linear and is also characterized. Similar results are shown for Brownian motion.
Run probabilities and the motion of a particle on a given path
Authors: Soltani A. Reza , Khodadadi A.
Published: July 2016

Let {Xn
} be a sequence of independent (or Markov dependent) trials taking values in a given set S. Let JR be a given path of length k in S, i.e. R is a run of length k whose elements come from S. {Xn
} may indicate the motion of a particle on S. We consider the problem of finding the probability that at trial m, the particle has for the first time moved length l ≦ k on R which is equivalent to finding the probability of the first occurrence of any subrun of length l ≦ k of R. In the case of l = k this gives the result of Schwager [6].
On the ordering of tandem queues with exponential servers
Authors: Lehtonen Tapani
Published: July 2016

We consider tandem queues which have a general arrival process. The queueing system consists of s (s ≧ 2) single-server service stations and the servers have exponential service-time distributions. Firstly we give a new proof for the fact that the departure process does not depend on the particular allocation of the servers to the stations. Secondly, considering the service rates, we prove that the departure process becomes stochastically faster as the homogeneity of the servers increases in the sense of a given condition. It turns out that, given the sum of the service rates, the departure process is stochastically fastest in the case where the servers are homogeneous.
On the probability that a random point is the jth nearest neighbour to its own kth nearest neighbour
Authors: Henze Norbert
Published: July 2016

In a homogeneous Poisson process in R
d
, consider an arbitrary point X and let Y be its kth nearest neighbour. Denote by Rk
the rank of X in the proximity order defined by Y, i.e., Rk
= j if X is the jth nearest neighbour to Y. A representation for Rk
in terms of a sum of independent random variables is obtained, and the limiting distribution of Rk,
as k →∞, is shown to be normal. This result generalizes to mixtures of Poisson processes.
Server utilization factors in queueing loss systems with ordered entry and heterogeneous servers
Authors: Pourbabai Behnam , Sonderman David
Published: July 2016

Approximation expressions for the server utilization factor of each server in a heterogeneous-server G/G/n queueing loss system with ordered entry are derived. The system is assumed to face a stationary counting process. Service times are generally distributed with possibly different service rates. The numerical results from this approximation method are then compared with those from a simulation study.
The extremal index for a Markov chain
Authors: Smith Richard L.
Published: July 2016

The paper presents a method of computing the extremal index for a discrete-time stationary Markov chain in continuous state space. The method is based on the assumption that bivariate margins of the process are in the domain of attraction of a bivariate extreme value distribution. Scaling properties of bivariate extremes then lead to a random walk representation for the tail behaviour of the process, and hence to computation of the extremal index in terms of the fluctuation properties of that random walk. The result may then be used to determine the asymptotic distribution of extreme values from the Markov chain.
A closure characterisation of phase-type distributions
Authors: Maier Robert S. , O'Cinneide Colm Art
Published: July 2016

We characterise the classes of continuous and discrete phase-type distributions in the following way. They are known to be closed under convolutions, mixtures, and the unary ‘geometric mixture' operation. We show that the continuous class is the smallest family of distributions that is closed under these operations and contains all exponential distributions and the point mass at zero. An analogous result holds for the discrete class.

We also show that discrete phase-type distributions can be regarded as ℝ+-rational sequences, in the sense of automata theory. This allows us to view our characterisation of them as a corollary of the Kleene–Schützenberger theorem on the behavior of finite automata. We prove moreover that any summable ℝ+-rational sequence is proportional to a discrete phase-type distribution.
Nested variational inequalities and related optimal starting–stopping problems
Authors: Sun M.
Published: July 2016

This paper introduces several versions of starting-stopping problem for the diffusion model defined in terms of a stochastic differential equation. The problem could be regarded as a stochastic differential game in which the player can only decide when to start the game and when to quit the game in order to maximize his fortune. Nested variational inequalities arise in studying such a problem, with which we are able to characterize the value function and to obtain optimal strategies.
Optimal block replacement policies with multiple choice at failure
Authors: Sheu Shey-Huei
Published: July 2016

A generalization of the block replacement policy (BRP) is proposed and analysed. Under such a policy, an operating system is preventively replaced at timeskT(k= 1, 2, 3, ···), independently of its failure history. At failure an operating system is either replaced by a new or a used one or minimally repaired or remains inactive until the next planned replacement. The cost of theith minimal repair of the new subsystem at ageydepends on the random partC(y) and the deterministic partci(y). The mathematical model is defined and general analytical results are obtained.
Stochastic convexity of sums of i.i.d. non-negative random variables with applications
Authors: Makowski Armand M. , Philips Thomas K.
Published: July 2016

We present some monotonicity and convexity properties for the sequence of partial sums associated with a sequence of non-negative independent identically distributed random variables. These results are applied to a system of parallel queues with Bernoulli routing, and are useful in establishing a performance comparison between two scheduling strategies in multiprocessor systems.
The MacLaurin series for the GI/G/1 queue
Authors: Gong Wei-Bo , Hu Jian-Qiang
Published: July 2016

We derive the MacLaurin series for the moments of the system time and the delay with respect to the parameters in the service time or interarrival time distributions in the GI/G/1 queue. The coefficients in these series are expressed in terms of the derivatives of the interarrival time density function evaluated at zero and the moments of the service time distribution, which can be easily calculated through a simple recursive procedure. The light traffic derivatives can be obtained from these series. For the M/G/1 queue, we are able to recover the formulas for the moments of the system time and the delay, including the Pollaczek–Khinchin mean-value formula.
Logarithmic asymptotics for steady-state tail probabilities in a single-server queue
Authors: Glynn Peter W. , Whitt Ward
Published: July 2016

We consider the standard single-server queue with unlimited waiting space and the first-in first-out service discipline, but without any explicit independence conditions on the interarrival and service times. We find conditions for the steady-state waiting-time distribution to have asymptotics of the form x
–1 log P(W
> x) → –θ ∗as x → ∞for θ ∗ > 0. We require only stationarity of the basic sequence of service times minus interarrival times and a Gärtner–Ellis condition for the cumulant generating function of the associated partial sums, i.e. n
–1 log E exp (θSn
) → ψ (θ) as n → ∞, plus regularity conditions on the decay rate function ψ. The asymptotic decay rate θ is the root of the equation ψ (θ) = 0. This result in turn implies a corresponding asymptotic result for the steady-state workload in a queue with general non-decreasing input. This asymptotic result covers the case of multiple independent sources, so that it provides additional theoretical support for a concept of effective bandwidths for admission control in multiclass queues based on asymptotic decay rates.
Burgers' equation by non-local shot noise data
Authors: Surgailis Donatas , Woyczynski Wojbor A.
Published: July 2016

We study the scaling limit of random fields which are solutions of a non-linear partial differential equation, known as the Burgers equation, under stochastic initial conditions. These are assumed to be of a non-local shot noise type and driven by a Cox process. Previous work by Bulinskii and Molchanov (1991), Surgailis and Woyczynski (1993a), and Funaki et al. (1994) concentrated on the case of local shot noise data which permitted use of techniques from the theory of random fields with finite range dependence. Those are not available for the non-local case being considered in this paper.

Burgers' equation is known to describe various physical phenomena such as non-linear and shock waves, distribution of self-gravitating matter in the universe, and other flow satisfying conservation laws (see e.g. Woyczynski (1993)).
AB percolation on plane triangulations is unimodal
Authors: Appel Martin J. B.
Published: July 2016

Let ℱ be a countable plane triangulation embedded in ℝ2in such a way that no bounded region contains more than finitely many vertices, and letPpbe Bernoulli (p) product measure on the vertex set of ℱ. LetEbe the event that a fixed vertex belongs to an infinite path whose vertices alternate states sequentially. It is shown that theAB percolation probability function θΑΒ(p) =Pp(E) is non-decreasing inpfor 0 ≦p≦ ½. By symmetry,θAΒ(p) is therefore unimodal on [0, 1]. This result partially verifies a conjecture due to Halley and stands in contrast to the examples of Łuczak and Wierman.
On simulation of random vectors by given densities in regions and on their boundaries
Authors: Borovkov K. A.
Published: July 2016

We suggest a new universal method of stochastic simulation, allowing us to generate rather efficiently random vectors with arbitrary densities in a connected open region or on its boundary. Our method belongs to the class of dynamic Monte Carlo procedures and is based on a special construction of a Markov chain on the boundary of the region. Its remarkable feature is that this chain admits a simple simulation, based on a universal (depending only on the dimensionality of the space) stochastic driver.
Asymptotic analysis and local lattice central limit theorems
Authors: Moran P.A.P.
Published: July 2016

Methods of evaluating the coefficients of high powers of functions defined by power series with positive coefficients are considered. Such methods, which were originally used by Laplace, can, for example, be used to obtain asymptotic formulae for Stirling numbers. They are equivalent to using local lattice central limit theorems. An alternative method using direct numerical integration on a contour integral giving the required coefficient is described. Exact bounds for the accuracy of this method can often be obtained by considerations of the unimodality of discrete distributions. The results are illustrated using convolutions of the rectangular and logarithmic distributions.
An invariance principle in k-dimensional extended renewal theory
Authors: Csenki Attila
Published: July 2016

Let ·be a sequence of k -dimensional i.i.d. random vectors and define the first-passage times for where (cvτ
)
v, τ= 1,· ··,
k is the covariance matrix of In this paper the weak convergence of Z
n
in (D[0, ∞))
k
is proved under the assumption (0,∞) for all v = 1, ···, k. We deduce the result from the Donsker invariance principle by means of Theorem 5.5 of Billingsley (1968). This method is also used to derive a limit theorem for the first-exit time Mn = min{Nnt for fixed t
1,···, tk
> 0. The second result is an extension of a theorem of Hunter (1974) whose method of proof applies only if Ρ (ξ
1
[0,∞)k) = 1 and μ ν = tv
for all v = 1, ···, k.
On estimation of parameters of Gaussian stationary processes
Authors: Taniguchi Masanobu
Published: July 2016

In fitting a certain parametric family of spectral densities fθ
(x) to a Gaussian stationary process with the true spectral density g (x), we propose two estimators of θ, say by minimizing two criteria D
1 (·), D
2(·) respectively, which measure the nearness of fθ
(x) to g (x). Then we investigate some asymptotic behavior of with respect to efficiency and robustness.
Some new integral geometric formulae, with stochastic applications
Authors: Miles R. E.
Published: July 2016

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

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

(ii) the variance of the area of intersection of a domain by an isotropic uniform plane.
New results in the theory of repeated orders queueing systems
Authors: Choo Qui Hoon , Conolly Brian
Published: July 2016

The repeated orders queueing system (ROO) permits no waiting or queue in the normal sense. Instead customers who find the service (or device, to use an engineering term) busy make reapplications at random intervals and in random order until their needs are met. Thus a second demand stream supplements the basic first arrival stream. Familiar examples are provided in a telephone communication setting, in particular in the context of a multiaccess computing system. Cohen [3] and Aleksandrov [1] made the first contributions to the theory of ROO. This paper complements their work with a steady-state analysis of system time (waiting time including service of a new arrival), of service idle time, and of system busy period.
A note on the expected number of survivors in supercritical carrier-borne epidemics
Authors: Saunders Roy , Lefèvre Claude , Kryscio Richard J.
Published: July 2016

We provide a formal proof of a conclusion due to Abakuks (1974) which states that the expected number of survivors in Downton's carrier-borne epidemic model approaches the limit (ρ /π)δ
as the initial number of susceptibles tends to infinity. Here ρ denotes the relative removal rate for carriers, π denotes the conditional probability that an infected susceptible will become a carrier, δ denotes the Kronecker delta function and denotes the initial number of carriers.
Some inequalities for moments and coefficients of variation for a large class of probability functions
Authors: Bronk Burt V.
Published: July 2016

Some inequalities for moments and coefficients of variation of probability densities over the positive real line are obtained by means of simple geometrical relationships. As an illustrative application rigorous bounds are obtained for the ratio of weight average to number average molecular weight for a large class of distributions of macromolecules, giving a more precise characterization of this empirical measure of heterogeneity.
Scheduling tasks with exponential service times on parallel processors
Authors: Glazebrook K. D.
Published: July 2016

We consider the problem of how to schedule n tasks on to several identical processors to meet the objective of minimising the expected flow-time. The strategy which always serves those tasks whose processing-time distributions have the highest hazard rates is shown to be optimal when these distributions are all exponential.
Non-linear ESS models and polymorphism
Authors: Hines W. G. S. , Haigh John
Published: July 2016

Extensions of ESS theory to situations well outside the classical formulation often assume, as a convenience, that the population being modelled is, in some sense, monomorphic. While this assumption is in keeping with the original approach used in developing the theory, it is rendered less plausible by the observation that the original models do not preclude the possibility of polymorphism, a potentially serious omission. We consider a generalisation of the classical bilinear fitness function, and examine the circumstances that will tend to favour monomorphism.
The visibility of stationary and moving targets in the plane subject to a Poisson field of shadowing elements
Authors: Yadin M. , Zacks S.
Published: July 2016

A methodology for an analytical derivation of visibility probabilities of n stationary target points in the plane is developed for the case when shadows are cast by a Poisson random field of obscuring elements. In addition, formulae for the moments of a measure of the total proportional visibility along a star-shaped curve are given.
A simple construction of an upper bound for the mean of the maximum of n identically distributed random variables
Authors: Gravey A.
Published: July 2016

This note gives a method of finding an upper bound for the mean of the maximum of n identically distributed non-negative random variables. The bound is explicitly given and numerically compared with the exact value of the mean of the maximum for some classical distributions (geometric, Poisson, Erlang, hyperexponential).
The shortest queue problem
Authors: Halfin Shlomo
Published: July 2016

A Poisson stream of customers arrives at a service center which consists of two single-server queues in parallel. The service times of the customers are exponentially distributed, and both servers serve at the same rate. Arriving customers join the shortest of the two queues, with ties broken in any plausible manner. No jockeying between the queues is allowed. Employing linear programming techniques, we calculate bounds for the probability distribution of the number of customers in the system, and its expected value in equilibrium. The bounds are asymptotically tight in heavy traffic.
Interconnected birth and death processes
Authors: Puri Prem S.
Published: July 2016

Two cases of multiple linearly interconnected linear birth and death processes are considered. It is found that in general the solution of the Kolmogorov differential equations for the probability generating function (p.g.f)gof the random variables involved is not obtainable by standard methods, although one can obtain moments of the random variables from these equations. A method is considered for obtaining an approximate solution forg.This is based on the introduction of a sequence of stochastic processes such that the sequence {f(n)} of their p.g.f.'s tends togasn → ∞in an appropriate manner. The method is applied to the simple case of two birth and death processes with birth and death rates λiandμi, i =1,2, interconnected linearly with transition rates v andδ(see Figure 2). For this case some limit theorems are established and the probability of ultimate extinction of both the processes is considered. In addition, for the special cases (i) λ1=δ= 0, with the remaining rates time dependent and (ii) λ2=δ= 0, with the remaining rates constant, explicit solutions forgare obtained and studied.
An indication of the asymptotic nature of the mendelian Markov process
Authors: Khazanie R. G.
Published: July 2016

Consider a finite Markov process {Xn
} described by the one-step transition probabilities

In describing the transition probabilities in the above manner we are adopting the convention that (0)0 = 1 so that the states 0 and M are absorbing, and the states 1,2,···,M-1 are transient.
Queues with time-dependent arrival rates I—the transition through saturation
Authors: Newell G. F.
Published: July 2016

Suppose that the arrival rate λ(t) of customers to a service facility increases with time at a nearly constant rate, dλ(t)/dt = a, so as to pass through the saturation condition, λ(t) = μ = service capacity, at some time which we label as t = 0. The stochastic properties of the queue are investigated here through use of the diffusion approximation (Fokker-Planck equation). It is shown that there is a characteristic time Tproportional to α
–2/3 such that if , then the queue distribution stays close to the prevailing equilibrium distribution associated with the λ(t) and μ, evaluated at time t. For |t| = O(T), however, the mean queue length is much less than the equilibrium mean, and is measured in units of some characteristic length L which is proportional to α
–1/3. For , the queue is approximately normally distributed with a mean of the order L larger than that predicted by deterministic queueing models. Numerical estimates are given for the mean and variance of the distribution for all t. The queue distributions are also evaluated in non-dimensional units.
Analytic characterization of the optimal control of a queueing system
Authors: Zacks S. , Yadin M.
Published: July 2016

In a recent paper [7] the authors studied the optimal control policy of the following queueing system. Customers arrive at a service station according to a time homogeneous Poisson process with a known arrival intensity, λ. The service time at the station is a random variable having a negative exponential distribution with intensity μ, which is under control and can be varied over a certain range, according to the management policy.
Average cost semi-markov decision processes
Authors: Ross Sheldon M.
Published: July 2016

The semi-Markov decision model is considered under the criterion of long-run average cost. A new criterion, which for any policy considers the limit of the expected cost incurred during the first n transitions divided by the expected length of the first n transitions, is considered. Conditions guaranteeing that an optimal stationary (non-randomized) policy exist are then presented. It is also shown that the above criterion is equivalent to the usual one under certain conditions.
A process with chain dependent growth rate
Authors: Keilson Julian , Rao S. Subba
Published: July 2016

Additive processes on finite Markov chains have been investigated by Miller ([8], [9]), Keilson and Wishart ([2], [3], [4]) and by Fukushima and Hitsuda [1]. These papers study a two-dimensional Markov Process {X(t),R(t)} whose state space isR1× {1, 2, ···,R} characterized by the following properties:

(i)

R(t) is an irreducible Markov chain on states 1,2, …,R governed by atransition probability matrixBo= {brs}.

(ii)

X(t) is a sumof random increments dependent on the chain, i.e., if the ith transition takes the chain from state r to state s, then the incrementhas the distribution

(iii)

Nt, is t in discrete time while in the continuous time case Nt, might be an independent Poisson process.