bluuebunny/crossref_metadata_2025 huggingface dataset:
On the identification of Poisson arrivals in queues with coinciding time-stationary and customer-stationary state distributions
Authors: König Dieter , Miyazawa Masakiyo , Schmidt Volker
Published: July 2016

For several queueing systems, sufficient conditions are given ensuring that from the coincidence of some time-stationary and customer-stationary characteristics of the number of customers in the system such as idle or loss probabilities it follows that the arrival process is Poisson.
A discrete-time queueing network
Authors: Walrand Jean
Published: July 2016

A discrete-time network with multiple transitions is defined. It is characterized by the independence of the service processes at different nodes. That network is shown to have the familiar quasi-reversibility properties.
Some limit theorems for a subcritical branching process by immigration
Authors: Chow Y. S. , Yu K. F.
Published: July 2016

The strong law of large numbers of the Marcinkiewicz–Zygmund type is established for the total population in a subcritical branching process with immigration. The moment convergence of the total population is obtained under appropriate moment conditions on the offspring distribution and the immigration distribution.
Hitting spheres by straight-line motion or Brownian motion
Authors: Enns E. G. , Smith B. R. , Ehlers P. F.
Published: July 2016

Points randomly placed inRnare independently assigned a speed and direction of motion. The time between collisions with a centraln-sphere is obtained. If instead of straight-line motion the points perform Brownian motion, the analogous inter-collision-time distribution is obtained.
A service model in which the server is required to search for customers
Authors: Neuts Marcel F. , Ramalhoto M. F.
Published: July 2016

Customers enter a pool according to a Poisson process and wait there to be found and processed by a single server. The service times of successive items are independent and have a common general distribution. Successive services are separated by seek phases during which the server searches for the next customer. The search process is Markovian and the probability of locating a customer in (t, t + dt) is proportional to the number of customers in the pool at time t. Various stationary probability distributions for this model are obtained in explicit forms well-suited for numerical computation.

Under the assumption of exponential service times, corresponding results are obtained for the case where customers may escape from the pool.
Population processes allowing emigration of families
Authors: Tessera Abebe
Published: July 2016

In the familiar immigration–birth–death process the events of immigration, birth and death relate to the individual. There are processes in which the whole family and not just an individual migrates. Such population growth models are studied in some detail.
A threshold AR(1) model
Authors: Petruccelli Joseph D. , Woolford Samuel W.
Published: July 2016

We consider the model where φ
1, φ
2 are real coefficients, not necessarily equal, and the at
,'s are a sequence of i.i.d. random variables with mean 0. Necessary and sufficient conditions on the φ 's are given for stationarity of the process. Least squares estimators of the φ 's are derived and, under mild regularity conditions, are shown to be consistent and asymptotically normal. An hypothesis test is given to differentiate between an AR(1) (the case φ
1 = φ
2) and this threshold model. The asymptotic behavior of the test statistic is derived. Small-sample behavior of the estimators and the hypothesis test are studied via simulated data.
On the moments of a self-correcting process
Authors: Vere-Jones D. , Ogata Y.
Published: July 2016

The existence of ordinary and exponential moments of a point process with conditional intensity of the form

is deduced from a Markov chain representation for t – ρN(t). These results form an application of recent theorems of Tweedie (1983a, b) and are used to obtain laws of large numbers for a range of functionals of the process.
Some aspects of the stochastic model for the attachment of phages to bacteria
Authors: Srivastava R. C.
Published: July 2016

Recently Gani (1965) has considered a stochastic model for the attachment of phages to bacteria. In this paper we study some of the basic properties of this model. These results will be used in a subsequent paper dealing with the estimation of the parameter which occurs in this model.
A note on the black box problem
Authors: Keats R. G.
Published: July 2016

It is well known (Lee 1960), especially to communications and control systems engineers, that the weighting function (unit impulse function) of a linear causal system may be obtained by cross correlating its output with its input. In general a Wiener-Hopf integral equation must be solved for the weighting function; but if the input were “white noise” of unit spectral density then, as shown by Lee (1950), the weighting function would be equal to this cross correlation function for positive values of its argument. Although a perfectly “white” input cannot be obtained in practice, a number of modifications of this technique have been used; Anderson, Buland and Cooper (1959), for example, describe the use of specially selected samples of “discrete-interval binary noise” in a problem in adaptive control.
Biharmonic functions and Brownian motion
Authors: Helms L. L.
Published: July 2016

Let R be a bounded open subset of N-dimensional Euclidean space EN,
N ≧ 1, let {xt
: t ≧ 0} be a separable Brownian motion starting at a point x ɛ R, and let τ = τR
be the first time the motion hits the complement of R. It is known [1] that if g is a bounded measurable function on the boundary ∂R of R, then h(x) = Ex
[g(xτ
)] is a harmonic function of x ɛ R which “solves” the Dirichlet problem for the boundary function g; i.e., Δh = 0 on R, where Δ is the Laplacian. In elastic plate problems, one must solve the biharmonic equation

subject to certain boundary conditions. For the more important applications, these boundary conditions involve the values of u and the normal derivative of u at points of ∂R. Even though a treatment of this Neumann type problem is not available at this time, some things can be said about biharmonic functions and their relationship to Brownian motion. We will show, in fact, that u(x)= Ex[τ(xτ)] is a biharmonic function on R which “satisfies” the boundary conditions (i) u=0 on ∂R and (ii) Δu= −2g on ∂R, provided g satisfies certain hypotheses. More generally, we will show that u(x)=Ex[Δk
g(X
Δ)] is polyharmonic of order k + 1 on R (i.e., Δk + 1
u = Δ(Δk

u) = 0 on R) and that it satisfies certain boundary conditions. A treatment of the special case g ≡ 1 on ∂R can be found in [3].
Some passage-time generating functions for discrete-time and continuous-time finite Markov chains
Authors: Darroch J. N. , Morris K. W.
Published: July 2016

Let T denote a subset of the possible transitions between the states of a finite Markov chain and let Yk
denote the time of the kth occurrence of a T-transition. Formulae are derived for the generating functions of Yk
, of Yj + k — Yj
and of Yj + k — Yj
in the limit as j → ∞, for both discrete-time and continuoustime chains. Several particular cases are briefly discussed.
The range of the (n + 1)th moment for distributions on [0, 1]
Authors: Skibinsky Morris
Published: July 2016

Let p denote the class of all probability measures defined on the Borel subsets of the unit interval I = [0, 1]. For each positive integer n, take

Mn is convex, closed, bounded, and n-dimensional; the convex hull of the space curve {(t,t2, …, tn
): 0 ≦ t ≦ 1}; e.g., see Theorems 7.2, 7.3 of [1]. At each point (c1, C2, …, cn
) of Mn
, define

Note that v
−, v
+ depend only on C1, C2, …, Cn− 1; Vm
only on cn
; We shall as notational convenience dictates and as will be apparent from the context regard v
±

n
as functions on Mn− 1
or on higher order moment spaces and also regard Vn
as a function on moment spaces of order higher than n.
Sequential decisions in the control of a space-ship (finite fuel)
Authors: Bather John , Chernoff Herman
Published: July 2016

This paper is a sequel to [1] and considers a more realistic formulation of the same question: that of finding an optimal policy for controlling the path of a space-ship as it moves towards its target. The difference here is that we no longer suppose there is an infinite quantity of fuel, always available at a fixed price, for modifying the current direction of motion. This complicates the problem of reducing the final miss distance, by introducing an extra variable. As before, we shall be particularly concerned to find a control procedure which always minimizes the mean square terminal miss. From the theoretical point of view we are also interested to see whether the techniques used to approximate the optimal policy can be extended, and how far we shall be forced to adopt a new approach. Results are derived which provide bounds on the form of the optimal policy. The derivation depends on a comparison technique whose validity is intuitively obvious, but which is still only a conjecture. However, further confirmation is obtained in the quadratic case from asymptotic expansions giving the form of the solution both when the space-ship is far away from its target and during its final approach.
Poisson recurrence times
Authors: Dwass Meyer
Published: July 2016

Let Y1, Y
2, … be a sequence of independent and identically distributed Poisson random variables with parameter λ. Let Sn = Y
1 + … + Yn, n = 1,2,…, S
0 = 0. The event Sn = n is a recurrent event in the sense that successive waiting times between recurrences form a sequence of independent and identically distributed random variables. Specifically, the waiting time probabilities are

(Alternately, the fn
can be described as the probabilities for first return to the origin of the random walk whose successive steps are Y1 − 1, Y2
− 1, ….)
Some corrections to my paper “A stochastic calculus and its application to some fundamental theorems of natural selection”
Authors: Mode Charles J.
Published: July 2016

The purpose of this note is to point out some difficulties which arise in connection with the stochastic calculus used in the previous paper and to suggest some ways in which these difficulties may be overcome. It turns out that if one wishes to restrict the application of the stochastic calculus to stochastic processes whose sample functions are step-functions with probability one, then difficulties of the following sort arise. Suppose X(ω, t) is a stochastic process whose sample functions are, with probability one, right-continuous step-functions in t on the interval [a, b). Then for every ω outside a null set N and every t ɛ [a, b), there is a δ(ω, t) such that < 5(w, t) implies X(ω, t + h) = X(ω, t) which in turn implies (X(ω, t + h) — X(ω, t))/h → 0 with probability one as Since convergence with probability one implies convergence in probability, it follows that the right-hand derivative of X(ω, t) in probability is zero at every t ɛ [a, b). Consequently, if the left-hand t ɛ [a, b), then it is also zero, but in general left-hand derivatives in probability on [a, b) may not even exist as finite-valued random functions. It is also worthwhile to observe that if a process is not differentiable in probability, then it is not differentiable in quadratic mean. From these observations it follows that the claims in Section 6 of the previous paper regarding the existence of non-zero derivatives in quadratic mean are in error, for if a derivative in quadratic mean exists, then it is a derivative in probability and is therefore zero. Incidentally, it turns out that the Poisson process has a zero derivative in probability but is not differentiable in quadratic mean. The fact that the Poisson process has a derivative in probability seems to depend rather crucially on the fact that it is also a process with independent increments.
Generalizations of the elementary renewal theorem to distributions defined by concave recurrence relations
Authors: Reh W.
Published: July 2016

The paper examines the renewal function associated with a sequence of probability distributions, which is defined by concave recurrence relations or by an even more general procedure. The elementary renewal theorem is generalized to such sequences. The results can be used to establish renewal theorems for first death in branching processes, if only the possibly generation dependent probability generating functions converge to a limit.
Remarks on the spatial distribution of a reproducing population
Authors: Kingman J. F. C.
Published: July 2016

Several authors have noted that simple models for the evolution of a reproducing and spatially distributed population have no limiting distribution, although a Poisson process in statistical equilibrium has sometimes been implicitly assumed. It is shown that, even when a mechanism for restricting population density is postulated, a Poisson process is usually impossible to achieve, essentially because of an assumption of independent displacements. When this assumption is abandoned, a Poisson process is possible, at least for some highly idealised models.
A note on the limiting behaviour of discrete-time Markovian manpower models with inhomogeneous independent Poisson input
Authors: Mehlmann Alexander
Published: July 2016

Using the expectation recurrence equation of discrete-time Markovian manpower models, an asymptotic relation for the population numbers in various grades is derived.
The second moments of the absorption times in the Markov renewal branching process
Authors: Neuts Marcel F.
Published: July 2016

We derive explicit formulas for the second moments of the absorption time matrices in the Markov renewal branching process. These formulas may easily be computationally implemented and are useful in the iterative computation of the semi-Markov matrices, which give the distributions of the duration and of the number of customers served in a busy period of a great variety of complex queueing models.
An example of Wold's point processes by Markov-dependent intervals
Authors: Lai C. D.
Published: July 2016

Wold's point process is briefly introduced and its forward equation is derived in terms of an integro-differential equation which is used to obtain the ‘renewal' function. An example of this family of processes is given in which each interval is exponentially distributed. The theory of diagonal expansion for a bivariate distribution is used to obtain the numerical estimate of the spectrum of counts.
A Markov chain model for the kinetics of reactant isolation
Authors: Olson William H.
Published: July 2016

In the chemistry of some chain molecules, adjacent monomeric side groups can form dimers. Here a Markov chain model is developed for irreversible formation of dimers. Closed form expressions for the finite time expected numbers of sequences of unreacted monomers of various lengths (number of monomers) are given. The relation between this model and the parking problem is discussed.
On road traffic with free overtaking
Authors: Thedéen Torbjörn
Published: July 2016

The cars are considered as points on an infinite road with no intersections. They can overtake each other without any delay and they travel at constant speeds. These are independent identically distributed random variables also independent of the initial positions of the cars. The main purpose of the paper is the study of the asymptotic distribution for the number of overtakings (and/or meetings) in increasing rectangles in the time-road plane. Under the assumption of (weighted) Poisson distributed cars along the time-axis we deduce the asymptotic distribution of the standardized number of overtakings (and/or meetings) for large rectangles in the time-road plane. Lastly we shall indicate an application of the results.
The total waiting time in a busy period of a stable single-server queue, I.
Authors: Daley D. J.
Published: July 2016

A quantity of particular interest in the study of (road) traffic jams is the total waiting time X of all vehicles involved in a given hold-up (Gaver (1969): see note following (2.3) below and the first paragraph of Section 5). With certain assumptions on the process this random variable X is the same as the sum of waiting times of customers in a busy period of a GI/G/1 queueing system, and it is the object of this paper and its sequel to study the random variable in the queueing theory context.
The total waiting time in a busy period of a stable single-server queue, II
Authors: Daley D. J. , Jacobs D. R.
Published: July 2016

This paper is a continuation of Daley (1969), referred to as (I), whose notation and numbering is continued here. We shall indicate various approaches to the study of the total waiting time in a busy period2 of a stable single-server queue with a Poisson arrival process at rate λ, and service times independently distributed with common distribution function (d.f.) B(·). Let X'i
denote3 the total waiting time in a busy period which starts at an epoch when there are i (≧ 1) customers in the system (to be precise, the service of one customer is just starting and the remaining i − 1 customers are waiting for service). We shall find the first two moments of X'i, prove its asymptotic normality for i → ∞ when B(·) has finite second moment, and exhibit the Laplace-Stieltjes transform of X'i
in M/M/1 as the ratio of two Bessel functions.
A recursion theorem on solving differential-difference equations and applications to some stochastic processes
Authors: Severo Norman C.
Published: July 2016

A recursive procedure leading to the solution of triangular systems of differential-difference equations is presented. It is shown that the equations for special families of multi-dimensional pure birth processes and pure death processes can be represented as such a system. The theory is used to resolve a difficult problem of stochastic cross-infection among groups. Additional applications to problems in epidemic theory are discussed.
A note on the waiting times between record observations
Authors: Holmes Paul T. , Strawderman William E.
Published: July 2016

Let X
1,
X
2,
X
3,··· be independent, identically distributed random variables with a continuous distribution function and let the sequence of indices {Vr
} be defined as follows:

and for r ≧ 1,

V
r is the trial on which the rth (upper) record observation occurs. {V
r} will be an infinite sequence of random variables since the underlying distribution function of the X's is continuous. It is well known that the expected value of V
r. is infinite for every r (see, for example, Feller [1], page 15). Also define

and for r > 1

δr is the number of trials between the (r - l)th and the rth record. The distributions of the random variables Vr
and δ
r
do not depend on the distribution of the original random variables. It can be shown (see Neuts [2], page 206 or Tata 1[4], page 26) that

The following theorem is due to Neuts [2].
Extended criterion for comparison of empirical distributions
Authors: Percus Ora Engleberg , Percus Jerome K.
Published: July 2016

A generating function technique is used to determine the probability that the deviation between two empirical distributions drawn from the same population lies within a given band a specified number of times. We also treat the asymptotic problem of very large sample size, and obtain explicit expressions when the relative number of failures is very small or very large.
A generalized unimodality
Authors: Olshen Richard A. , Savage Leonard J.
Published: July 2016

Khintchine (1938) showed that a real random variable Z has a unimodal distribution with mode at 0 iff Z ~ U X (that is, Z is distributed like U X), where U is uniform on [0, 1] and U and X are independent. Isii ((1958), page 173) defines a modified Stieltjes transform of a distribution function F for w complex thus:

Apparently unaware of Khintchine's work, he proved (pages 179-180) that F is unimodal with mode at 0 iff there is a distribution function Φ for which . The equivalence of Khintchine's and Isii's results is made vivid by a proof (due to L. A. Shepp) in the next section.
A stochastic process related to language change
Authors: Brainerd Barron
Published: July 2016

The purpose of this note is two-fold. First, to introduce the mathematical reader to a group of problems in the study of language change which has received little attention from mathematicians and probabilists. Secondly, to introduce a birth and death process, arising naturally out of this group of problems, which has received little attention in the literature. This process can be solved using the standard methods and the solution is exhibited here.
On the fluctuations of election returns
Authors: Takács Lajos
Published: July 2016

Suppose that in a ballot candidateAscoresavotes and candidateBscoresbvotes and all the possiblevoting records are equally probable. Denote byαrandβrthe number of votes registered forAandBrespectively among the firstrvotes counted. Forj= 0, 1,···,a+bdefinePj(a, b) as the probability that the inequalityαr>aβr/bholds for exactlyjsubscriptsr= 1, 2, ..,a+b.
On a generalized entropy and a coding theorem
Authors: Rathie Pushpa N.
Published: July 2016

Let P= {p
1,···, PN
} be a finite discrete probability distribution. Then the entropy of the distribution P, introduced by Shannon [12], is defined as

Throughout this paper ∑ will stand for and logarithms will be taken to the base D (D > 1).
Mean life of series and parallel systems
Authors: Marshall Albert W. , Proschan Frank
Published: July 2016

Some inequalities are obtained which yield bounds for the mean life of series and of parallel systems in the case where component life distributions have properties such as a monotone failure rate, monotone failure rate average, or decreasing density. These bounds are based on comparisons with systems of exponential or uniform components. Similar comparisons are obtained when components have Weibull or Gamma distributions with different shape parameters. Some inequalities are also obtained for convolutions of life distributions helpful in the study of replacement policies.
Two results in the theory of queues
Authors: Ali H.
Published: July 2016

In this paper it is shown that the distribution of the instant of service of a customer is symmetric as between the distributions of service and interarrival time. Also U(t), the expected number of departures in (0, t), is a delayed renewal function for the GI/M/1 queue.
Inflow-Stock-Outflow time series, stochastic lag and disaggregation: a note
Authors: Benishay Haskel
Published: July 2016

This note describes a model for inflow-stock-outflow time series which are formed as sums of random numbers of subcomponents of randomly determined sizes and are related via a probabilisticly distributed lag. Such time series may be illustrated by population processes in connection with which not only numbers of units are of interest but also the sizes of the units and the sums of the sizes. Such series may also be illustrated by economic phenomena like new loans made, loans outstanding and loans repaid of a commercial bank; new capital investments, capital stock and depreciated capital for a group of firms as well as accessions to the labor force, labor force outstanding, and secessions from the labor force.
The fixed cycle traffic light problem: a note on a paper by Mcneil
Authors: Siskind Victor
Published: July 2016

Both paper and author (D. R. McNeil, (1968)) will be referred to below as DRM. The said paper deals with the following situation: an intersection is controlled by a traffic light with a fixed cycle time, T; the possibility of other delays, e.g., due to turning vehicles, is ignored; arrivals at the light form a compound Poisson process; if vehicles arrive to find the light green and the queue empty they are not delayed, while in the contrary case they depart when they reach the head of the queue, providing the light is green, each vehicle taking a constant time to move off. The length of the effective red period is R. For further details and discussion, DRM may be consulted.
An extremal markovian sequence
Authors: Alpuim M. Teresa
Published: July 2016

In this paper we consider an independent and identically distributed sequence {Yn
} with common distribution function F(x) and a random variable X
0, independent of the Yi
's, and define a Markovian sequence {Xn
} as Xi
= X
0, if i = 0, Xi
= k max{Xi

− 1, Yi
}, if i ≧ 1, k ∈ R, 0 < k < 1. For this sequence we evaluate basic distributional formulas and give conditions on F(x) for the sequence to possess a stationary distribution. We prove that for any distribution function H(x) with left endpoint greater than or equal to zero for which log H(ex
) is concave it is possible to construct such a stationary sequence with marginal distributions equal to it. We study the limit laws for extremes and kth order statistics.
Multitype finite mean supercritical age-dependent branching processes
Authors: Cohn Harry
Published: July 2016

Let {Zφ
(t)} be a multitype finite-mean supercritical (CMJ) process with general characteristic φ and malthusian parameter α. It is shown that there exist some constants {c(t} such that {c
–1(t)Zφ
(t)} converges almost surely to a non-degenerate limit Wφ
as t →∞. Characterizations of {c(t)} and Wφ
are also given.
Joint simulation of backward and forward recurrence times in a renewal process
Authors: Winter B. B.
Published: July 2016

It is shown that, in a renewal process with inter-arrival distributionF,an observation from the asymptotic (whent→∞) joint distribution of backward and forward recurrence times attcan be simulated by simulating an observation of the pair (UW, (1 –U)W), whereUandWare independent random variables withU~ uniform(0, 1) andWdistributed according to the length-biased version ofF.
Finiteness of moments of virtual work for GI/G/C queues
Authors: Ghahramani Saeed
Published: July 2016

Necessary and sufficient conditions for the finiteness of moments of virtual work for GI/G/c queues are presented.
Some explicit results for correlated random walks
Authors: Lal Ram , Bhat U. Narayan
Published: July 2016

In a correlated random walk (CRW) the probabilities of movement in the positive and negative direction are given by the transition probabilities of a Markov chain. The walk can be represented as a Markov chain if we use a bivariate state space, with the location of the particle and the direction of movement as the two variables. In this paper we derive explicit results for the following characteristics of the walk directly from its transition probability matrix: (i) n -step transition probabilities for the unrestricted CRW, (ii) equilibrium distribution for the CRW restricted on one side, and (iii) equilibrium distribution and first-passage characteristics for the CRW restricted on both sides (i.e., with finite state space).
The rate of convergence of the vector of variances and covariances in non-homogeneous Markov systems
Authors: Vassiliou P.-C. G. , Tsaklidis G.
Published: July 2016

The aims of this paper are twofold. The first is to study the rate of convergence of the sequence of vectors of expectations, variances and covariances in NHMS. It is proved that under certain conditions easily met in practice the rate is geometric. The second is to study the rate of convergence of the same sequence when the NHMS is under cyclic behaviour and it is also proved that the rate of convergence of the d subsequences (d being the period of the cycle) in which the sequence splits is geometric.
Limit theorems for jump shock models
Authors: Yamada Keigo
Published: July 2016

We consider an additive shock process where shocks occur according to a Poisson point process and they are accumulated in an appropriate way to the damage. It is shown that suitably normalized shock processes converge weakly to a process which is represented as a sum of a stable process and a deterministic process.
Interchange arguments in stochastic scheduling
Authors: Nain Philippe , Tsoucas Pantelis , Walrand Jean
Published: July 2016

Interchange arguments are applied to establish the optimality of priority list policies in three problems. First, we prove that in a multiclass tandem of two ·/M/1 queues it is always optimal in the second node to serve according to the cµ rule. The result holds more generally if the first node is replaced by a multiclass network consisting of ·/M/1 queues with Bernoulli routing. Next, for scheduling a single server in a multiclass node with feedback, a simplified proof of Klimov's result is given. From it follows the optimality of the index rule among idling policies for general service time distributions, and among pre-emptive policies when the service time distributions are exponential. Lastly, we consider the problem of minimizing the blocking in a communication link with lossy channels and exponential holding times.
NBUE and NWUE properties of increasing Markov processes
Authors: Karasu Izi , Özekici Süleyman
Published: July 2016

Consider a device that deteriorates in time according to an increasing Markov process so that it fails as soon as a critical threshold is exceeded. NBUE and NWUE properties of the lifetime of the device are identified to extend the existing literature on the PF
r
, IFR, IFRA, and NBU cases. In particular, it is shown that NBUE and NWUE characterizations can be made through a monotonicity property on the potential operator of the Markov process.
A certain configuration of random points on a circle associated with a generalized Lotka-Volterra equation
Authors: Itoh Yoshiaki
Published: July 2016

Invariant integrals of a Lotka-Volterra system with infinitely many species are introduced. The values of these integrals are given by the probabilities of certain configurations of random points on a circle when the probability density on the circle satisfies a certain symmetry condition.
Stochastic dispatching of multi-priority jobs to heterogeneous processors
Authors: Xu Susan H. , Mirchandani Pitu B. , Kumar Srikanta P. R. , Weber Richard R.
Published: July 2016

A number of multi-priority jobs are to be processed on two heterogeneous processors. Of the jobs waiting in the buffer, jobs with the highest priority have the first option of being dispatched for processing when a processor becomes available. On each processor, the processing times of the jobs within each priority class are stochastic, but have known distributions with decreasing mean residual (remaining) processing times. Processors are heterogeneous in the sense that, for each priority class, one has a lesser average processing time than the other. It is shown that the non-preemptive scheduling strategy for each priority class to minimize its expected flowtime is of threshold type. For each class, the threshold values, which specify when the slower processor is utilized, may be readily computed. It is also shown that the social and the individual optimality coincide.
Identifiability problems in coherent systems
Authors: Nowik Shmuel
Published: July 2016

Given a coherent system, let Z be the age of the machine at breakdown, and I the set of parts failed by time Z. Assume that the component lifetimes are independent. Assume further that the lifetime distributions are mutually absolutely continuous and that each possesses a single positive atom at the common essential infimum. We prove that the joint distribution of (Z, I) identifies the lifetime distribution of each part if and only if there is at most one component belonging to all cut sets. If we relax the mutual absolute continuity assumption by allowing isolated intervals of constancy, then a necessary and sufficient condition for identifiability is that no two parts be in parallel.
Bounded Recurrence times and lead time in the design of a repetitive screening program
Authors: Prorok Philip C.
Published: July 2016

An unsolved problem in the design of a repetitive screening trial is the determination of a procedure which yields the optimal number of screens for efficient evaluation. An approach to this problem described in this paper is to use lead time properties of cases detected at screening to define a stopping rule. Previous results are generalized by allowing the intervals between screens to be arbitrary, and the probability of detecting preclinical disease to vary among screens.
Optimal stopping in a partially observable binary-valued markov chain with costly perfect information
Authors: Monahan George E.
Published: July 2016

The problem of optimal stopping in a Markov chain when there is imperfect state information is formulated as a partially observable Markov decision process. Properties of the optimal value function are developed. It is shown that under mild conditions the optimal policy is well structured. An efficient algorithm, which uses the structural information in the computation of the optimal policy, is presented.
Some duality results for a class of multivariate semi-markov processes
Authors: Janssen J. , Reinhard J. M.
Published: July 2016

The duality results well known for classical random walk and generalized by Janssen (1976) for (J-X) processes (or sequences of random variables defined on a finite Markov chain) are extended to a class of multivariate semi-Markov processes. Just as in the classical case, these duality results lead to connections between some models of risk theory and queueing theory.
Scheduling jobs by stochastic processing requirements on parallel machines to minimize makespan or flowtime
Authors: Weber Richard R.
Published: July 2016

A number of identical machines operating in parallel are to be used to complete the processing of a collection of jobs so as to minimize either the jobs' makespan or flowtime. The total processing required to complete each job has the same probability distribution, but some jobs may have received differing amounts of processing prior to the start. When the distribution has a monotone hazard rate the expected value of the makespan (flowtime) is minimized by a strategy which always processes those jobs with the least (greatest) hazard rates. When the distribution has a density whose logarithm is concave or convex these strategies minimize the makespan and flowtime in distribution. These results are also true when the processing requirements are distributed as exponential random variables with different parameters.
On the equivalence of flows in networks of queues
Authors: Walrand J.
Published: July 2016

Consider a network of queues in equilibrium. When are the flows observed on two given links of that network equivalent, i.e., when do they have the same law?

The verification of the equivalence of two such point processes is first reduced to an algebraic problem by a technique based on the filtering theory.

This method is then used to show that the arrival and departure processes at an M/M/1 node in a Jackson network are not always equivalent, thereby contradicting a conjecture made in [8]. An example where that equivalence holds is also given; it provides new results on the time reversibility of some familiar processes.
Point process limits of lattice processes
Authors: Besag Julian , Milne Robin , Zachary Stan
Published: July 2016

Starting from a suitable sequence of auto-Poisson lattice schemes, it is shown that (almost) any purely inhibitory pairwise-interaction point process can be obtained in the limit. Further pairwise-interaction processes are obtained as limits of sequences of auto-logistic lattice schemes.
A note on the measurement of randomness
Authors: Hickey Raymond J.
Published: July 2016

The properties of a measure of randomness on the class of discrete probability distributions are discussed. A result concerning finiteness is obtained for a family of measures which includes the entropy function.
Record values and extreme value distributions
Authors: Nagaraja H. N.
Published: July 2016

The limit distribution of thekth maximum from a random sample of sizenwhenn →∞ is identified as the distribution of thekth lower record value from one of three extreme value distributions. This fact is used in giving a different canonical representation and new proofs of the results of Hall (1978) for this limiting random variable. A characterization of the exponential distribution based on upper record values is given.
Kac's formula, levy's local time and brownian excursion
Authors: Louchard G.
Published: July 2016

Kac's formula for Brownian functionals and Levy's local time decomposition are shown to be useful tools in analysing Brownian excursion properties. These tools are applied to maximum, local time and area distributions. Some curious connections between some of these distributions are explained by simple probabilistic arguments.
Density factorizations for brownian motion, meander and the three-dimensional bessel process, and applications
Authors: Imhof J.-P.
Published: July 2016

Joint densities concerning in particular the value and time of the maximum over a fixed time interval, or the behavior over intervals determined by some first- and last-passage times, are determined for Brownian motion, the three-dimensional Bessel process and Brownian meander. Simple change of measure formulas permit easy passage from one process to the other. Examples are given.
An M/G/c queue in which the number of servers required is random
Authors: Federgruen Awi , Green Linda
Published: July 2016

Many queueing situations such as computer, communications and emergency systems have the feature that customers may require service from several servers at the same time. They may thus be delayed until the required number of servers is available and servers may be idle when customers are waiting. We consider general server-completion-time distributions and derive approximation methods for the computation of the steady-state distribution of the number of customers in queue as well as the moments of the waiting-time distribution. Extensive computational results are reported.
Evolutionarily stable strategies with two types of player I. Two-species haploid or randomly mating diploid
Authors: Cressman R. , Dash A. T.
Published: July 2016

The evolution of strategies in animal contests is examined where the dynamical equation takes into account population growth rates. This leads to a different definition of evolutionary stable strategy (ESS) from the usual one. Consequences for independent haploid species are then contrasted with the previous theory. Inheritance patterns for male–female contests with sex-dependent payoffs are considered. In particular, if males and females evolve independently to the same ESS, then so does the diploid species under random mating. Finally, the evolution of diploid populations where strategies are determined at a diallelic locus is investigated.
A self-excited migration process
Authors: Branford A. J.
Published: July 2016

Processes whose entities would be independent but whose behaviour is influenced by environmental variables, which they in turn affect, are processes with mediated interactions. The independent open migration process with which is associated a random variable, the excitation, is such a process. This self-excited (conditionally) independent open migration process, with finite excitation, is related to a derived process, its piecewise-deterministic analogue. The correspondence gives insight into the process, as well as an equilibrium independence result. A simple example illustrates the presence of paroxysmal phenomena.
The stereological estimation of moments of particle volume
Authors: Jensen E. B. , Gundersen H. J. G.
Published: July 2016

In stereology or applied geometric probability quantitative characterization of aggregates of particles from information on lower-dimensional sections plays a major role. Most stereological methods developed for particle aggregates are based on the assumption that the particles are of the same, known (simple) shape. Information on the volume-weighted distribution of particle size may, however, be obtained under fairly general assumptions about particle shape if particle volume is chosen as size parameter. In fact, there exists in this case an unbiased stereological estimator of the first moment under the sole assumption that the particles are convex. In the present paper, we consider a particle aggregate in ℝ and derive estimators of the q th moment of the volume-weighted distribution of particle volume, based on point-sampling of particles and measurements on q -flats through sampled particles. The estimators are valid for arbitrarily shaped particles but if the particles are non-convex it is necessary for the determination of the estimators to be able to identify the different separated parts on a q-flat through the particle aggregate which belong to the same particle. Explicit forms of the estimators are given for q = 1. For q = 2, an explicit form of one of the estimators is derived for an aggregate of triaxial ellipsoids in three-dimensional space.
First-passage times with PFr densities
Authors: Assaf David , Shared Moshe , shanthikumar J. George
Published: July 2016

It is shown that if a finite-state continuous-time Markov process can be uniformized such that the embedded Markov chain has a TPr (totally positive of order r) transition matrix, then the first-passage time from state 0 to any other state has a PF
r
(Polya frequency of order r) density. As a consequence, results of Keilson (1971), Esary, Marshall and Proschan (1973), Ghosh and Ebrahimi (1982) and Derman, Ross and Schechner (1983) are strengthened. It is also shown that some cumulative damage shock models, with an underlying compound Poisson process and ‘damages' which are not necessarily non-negative, are associated with wear processes having PF
r
first-passage times to any threshold. First-passage times with completely monotone densities are also discussed.
Renewal theory for several patterns
Authors: Breen Stephen , Waterman Michael S. , Zhang Ning
Published: July 2016

Discrete renewal theory is generalized to study the occurrence of a collection of patterns in random sequences, where a renewal is defined to be the occurrence of one of the patterns in the collection which does not overlap an earlier renewal. The action of restriction enzymes on DNA sequences provided motivation for this work. Related results of Guibas and Odlyzko are discussed.
A queueing system with Markov-dependent arrivals
Authors: Tin Pyke
Published: July 2016

This paper considers a single-server queueing system with Markov-dependent interarrival times, with special reference to the serial correlation coefficient of the arrival process. The queue size and waiting-time processes are investigated. Both transient and limiting results are given.
Upper (lower) bounds on the mean of the maximum (minimum) of a number of random variables
Authors: Aven Terje
Published: July 2016

In this paper, we give some upper (lower) bounds on where the Xi
's are real-valued random variables. Some applications are given.
Optimal estimation for semimartingales
Authors: Thavaneswaran A. , Thompson M. E.
Published: July 2016

This paper extends a result of Godambe's theory of parametric estimation for discrete-time stochastic processes to the continuous-time case. LetP={P} be a family of probability measures such that (Ω,F, P) is complete, (Ft, t≧0) is a standard filtration, andX = (XtFt, t ≧ 0)is a semimartingale for everyP ∈ P. For a parameterθ(Ρ), supposeXt=Vt,θ+Ht,θwhere theVθprocess is predictable and locally of bounded variation and theHθprocess is a local martingale. Consider estimating equations forθof the formprocess is predictable. Under regularity conditions, an optimal form forαθin the sense of Godambe (1960) is determined. WhenVt,θis linear inθthe optimal, corresponds to certain maximum likelihood or least squares estimates derived previously in special cases. Asymptotic properties of, are discussed.
Approximation of the queue-length distribution of an M/GI/s queue by the basic equations
Authors: Miyazawa Masakiyo
Published: July 2016

We give a unified way of obtaining approximation formulas for the steady-state distribution of the queue length in theM/GI/squeue. The approximations of Hokstad (1978) and Case A of Tijms et al. (1981) are derived again. The main interest of this paper is in considering the theoretical meaning of the assumptions given in the literature. Having done this, we derive new approximation formulas. Our discussion is based on one version of the steady-state equations, called the basic equations in this paper. The basic equations are derived forM/GI/s/kwith finite and infinitek.Similar approximations are possible forM/GI/s/k(k< +∞).
Characterizations of inverse problems in queueing and their stability
Authors: Kalashnikov V. V. , Rachev S. T.
Published: July 2016

Two inverse problems in queueing are studied, namely: (i) a restoration of system characteristics, and (ii) a restoration of model classes. These problems are discussed in the framework of a general concept of characterization problems given by Kalashnikov and Rachev (1985).
Boundary domination and the distribution of the largest nearest-neighbor link in higher dimensions
Authors: Steele J. Michael , Tierney Luke
Published: July 2016

For a sample of points drawn uniformly from either the d-dimensional torus or the d-cube, d ≧ 2, we give limiting distributions for the largest of the nearest-neighbor links. For d ≧ 3 the behavior in the torus is proved to be different from the behavior in the cube. The results given also settle a conjecture of Henze (1982) and throw light on the choice of the cube or torus in some probabilistic models of computational complexity of geometrical algorithms.
On multitype processes based on progeny length of particles of a supercritical Galton-Watson process
Authors: Gadag V. G. , Rajarshi M. B.
Published: July 2016

Based on infinite and finite line of descent of particles of a one-dimensional supercritical Galton-Watson branching process (GWBP), we construct an associated bivariate process. We show that this bivariate process is a two-dimensional, supercritical GWBP. We also show that this process retains its branching property on appropriate probability spaces, when conditioned on set of non-extinction and set of extinction. Some asymptotic and weak convergence results for this process have been established. A generalisation of these results to a multitype p-dimensional GWBP has also been carried out.
A modified repair strategy for two-component systems with revealed and unrevealed faults
Authors: Johnson Norman L. , Kotz Samuel
Published: July 2016

Some consequences of a modified repair system for Phillips' (1981a, b) model for a two-component system are discussed. In the original model, both components are repaired whenever a revealed fault occurs; in the modified model only faulty components are repaired. Specifically (i) the distribution of time from the initial state up to discovery of an unrevealed fault, (ii) the expected proportion of time during which there exists an unrepaired fault, and (iii) the distribution of number of revealed faults up to and including the one which leads to a discovery of an unrevealed fault, are obtained. The theory is illustrated by examples, based on specific distributions for the times between repairs and occurrences of the two types of faults. A characterization of the exponential distribution is indicated.
A Markov chain approach to periodic queues
Authors: Asmussen Søren , Thorisson Hermann
Published: July 2016

We consider GI/G/1 queues in an environment which is periodic in the sense that the service time of the nth customer and the next interarrival time depend on the phase θ
n

at the arrival instant. Assuming Harris ergodicity of {θ
n

} and a suitable condition on the traffic intensity, various Markov chains related to the queue are then again Harris ergodic and provide limit results for the standard customer- and time-dependent processes such as waiting times and queue lengths. As part of the analysis, a result of Nummelin (1979) concerning Lindley processes on a Markov chain is reconsidered.