bluuebunny/crossref_metadata_2025 huggingface dataset:
Generalised mixtures of exponential distributions
Authors: Roy Dilip , Mukherjee S. P.
Published: July 2016

An attempt has been made to generalise the concept of exponential mixture to the multivariate case through the random environment model. Some important properties of this class of mixture distributions have been studied, along with characterising properties. A further generalisation for a type of dependent exponential distribution has also been made.
Oil exploration: sequential decisions in the face of uncertainty
Authors: Benkherouf L. , Bather J. A.
Published: July 2016

A simple Bayesian model for oil exploration is suggested to investigate strategies for drilling. A condition on the way successes and failures affect the prior distribution implies a certain form of the detection mechanism. It is shown that the problem of finding strategies for drilling reduces to an optimal stopping problem. Two new families of distributions are obtained with generating functions related to classical work on partitions of integers. By using such distributions and simple mixtures of them as priors, the stopping problem can be solved explicitly. This leads to the construction of simple strategies and their effectiveness is demonstrated by evaluating suitable operating characteristics.
Waiting times for multiple occurrences of several events
Authors: Scheinberg Stephen
Published: July 2016

Consider an ‘experiment' which can be repeated indefinitely often resulting in independent random outcomes. Fix attention on a finite number of possible (sets of) outcomes E
1, E
2, … and define W = W(N
1, N
2, …) to be the expected number of repetitions needed to ensure that E
1 has occurred (at least) N
1 times, E
2 has occurred (at least) N
2 times, etc. This article examines the asymptotic behavior of W as a function of the sum Σ
j
N
j, as the latter grows without bound.
A note on random intensities and conditional survival functions
Authors: Yashin Anatoli , Arjas Elja
Published: July 2016

Failure intensities in which the evaluation of hazard is based on the observation of an auxiliary random process have become very popular in survival analysis. While their definition is well known, either as the derivative of a conditional failure probability or in the counting process and martingale framework, their relationship to conditional survival functions does not seem to be equally well understood. This paper gives a set of necessary and sufficient conditions for the so-called exponential formula in this context.
Comparisons for carrier-borne epidemics in heterogeneous and homogeneous populations
Authors: Lefevre Claude , Malice Marie-Pierre
Published: July 2016

Two specific models for the spread of a carrier-borne epidemic are considered which allow for individual variability. The implications of differences in the infection or removal rates are investigated by comparing the propagation of the disease for heterogeneous and homogeneous populations. This is achieved by means of the stochastically larger and more variable order relations for random variables. The results obtained extend earlier ones and are illustrated with some numerical examples.
First passage times and lumpability of semi-Markov processes
Authors: Sumita Ushio , Rieders Maria
Published: July 2016

A necessary and sufficient condition of Serfozo (1971) for lumpability of semi-Markov processes is reinterpreted in terms of first-exit times. Furthermore, a new necessary and sufficient condition is developed by establishing relationships between first-passage times and lumpability of semi-Markov processes. The approach taken in this paper is entirely based on the Laplace-transform domain.
On the convex hull of random points in a polytope
Authors: Dwyer Rex A.
Published: July 2016

The convex hull of n points drawn independently from a uniform distribution on the interior of a d-dimensional polytope is investigated. It is shown that the expected number of vertices is O(log
d–1
n) for any polytope, the expected number of vertices is Ω(log
d–1
n) for any simple polytope, and the expected number of facets is O(log
d–1
n) for any simple polytope. An algorithm is presented for constructing the convex hull of such sets of points in linear average time.
On the reliability function of a system of components sharing a common environment
Authors: Currit Allen , Singpurwalla Nozer D.
Published: July 2016

A multivariate distribution for describing the life-lengths of the components of a system which operates in an environment that is different from the test bench environment has been proposed by Lindley and Singpurwalla (1986). In this paper, the properties of the reliability function of such a system are studied and comparisons made with the reliability function obtained under the assumption of independence. It is interesting to note that the reliability function of parallel redundant systems whose components share a common unknown environment cannot be characterized by any of the well-known classes of distributions that have been proposed in the mathematical theory of reliability. This observation suggests the need for defining a new class of failure distributions. A formula for making Bayesian inferences for the reliability function is also given.
Formulas for the reliability of a consecutive-K-Out-of-n: F system
Authors: Papastavridis S. , Hadzichristos I.
Published: July 2016

General exact formulas are established to calculate the reliability of a consecutive-k-out-of-n: F system. The optimality of the system, under permutation of the components, is examined too.
Stability and continuity for slotted ALOHA with stationary non-independent input traffic
Authors: Sharma Vinod
Published: July 2016

Slotted ALOHA with a finite number of users, each with infinite buffer, is considered. For stationary, metrically transitive, non-independent input, the problem of existence of stationary queue length distributions is solved. Results are obtained for zero and arbitrary (finite a.s.) initial conditions. Continuity, in probability, of queue lengths with respect to input sequence is proved.
Multitype branching processes based on exact progeny lengths of particles in a Galton-Watson branching process
Authors: Gadag V. G. , Rajarshi M. B.
Published: July 2016

In Gadag and Rajarshi (1987), we studied a bivariate (multitype) branching process based on infinite and finite lines of descent, of particles of a supercritical one-dimensional (multitype) Galton-Watson branching process (GWBP). In this paper, we discuss a few more meaningful and interesting univariate and multitype branching processes, based on exact progeny lengths of particles in a GWBP. Our constructions relax the assumption of supercriticality made in Gadag and Rajarshi (1987). We investigate some finite-time and asymptotic results of these processes in some details and relate them to the original process. These results are then used to propose new and better estimates of the offspring mean. An illustration based on the branching process of the white male population of the USA is also given. We believe that our work offers a rather finer understanding of the branching property.
The limit distribution of the largest nearest-neighbour link in the unit d-cube
Authors: Dette Holger , Henze Norbert
Published: July 2016

Let·
p
denote the p-norm on . For n points drawn independently and uniformly from the unit d-cube, we obtain the limit distribution of the largest nearest-neighbour link (with respect to | · |
p
) in the cases ; (2) d = 3, p = 1, 2, ∞and (3) 4, p =∞.
Modelling dependence in simple and indirect majority systems
Authors: Boland Philip J. , Proschan Frank , Tong Y. L.
Published: July 2016

Majority systems are encountered in both decision theory and reliability theory. In decision theory for example a jury or committee employing a majority rule will make the ‘correct' decision if a majority of the individuals do so. In reliability theory some coherent systems function if and only if a majority of the components work properly. In this paper results concerning the reliability of majority systems are developed which are applicable in both areas. Two models incorporating dependence between individuals or components in majority systems are introduced, and various monotonicity results for their reliability functions are established. Comparisons are also made between direct (or simple) and indirect majority systems.
A proof of simple insensitive bounds for a pure overflow system
Authors: Van Dijk Nico M.
Published: July 2016

In [1], [4] and [6] simple and intuitively obvious bounds were suggested for various overflow systems and claimed to be insensitive, that is to be valid also in the non-exponential case. This insensitivity was formalized in [4] and [6] when the service distributions have a monotone failure rate. This technical note provides another proof which for the case of a pure overflow system with one customer class extends the insensitivity to the general case.
On some characterizations of the poisson process
Authors: Huang Wen-Jang
Published: July 2016

In this article we give some characterizations of Poisson processes, the model which we consider is inspired by Kimeldorf and Thall (1983) and we generalize the results of Chandramohan and Liang (1985). More precisely, we consider an arbitrarily delayed renewal process, at each arrival time we allow the number of arrivals to be i.i.d. random variables, also the mass of each unit atom can be split into k new atoms with the ith new atom assigned to the process Di, i = 1, ···, k. We shall show that the existence of a pair of uncorrelated processes Di, Dj, i ≠ j, implies the renewal process is Poisson. Some other related characterization results are also obtained.
Time series formed from the superposition of discrete renewal processes
Authors: Blight P. A.
Published: July 2016

The superposition of independent, discrete, renewal processes produces a counting process which is also a discrete time series. The conditional distribution and correlation structure of this kind of time series may be obtained. In suitable conditions the conditional distribution has a spectrum which is exactly or approximately rational. When this is so, an ARMA can be found which matches the spectrum of the superposition.
A Gaussian Markovian process on a square lattice
Authors: Moran P. A. P.
Published: July 2016

A definition of the Markovian property is given for a lattice process and a Gaussian Markovian lattice process is constructed on a torus lattice. From this a Gaussian Markovian process is constructed for a lattice in the plane and its properties are studied.
Weak convergence of probability measures and random functions in the function space D[0,∞)
Authors: Lindvall Torgny
Published: July 2016

This paper extends the theory of weak convergence of probability measures and random functions in the function space D[0,1] to the case D [0,∞), elaborating ideas of C. Stone and W. Whitt. 7)[0,∞) is a suitable space for the analysis of many processes appearing in applied probability.
An algebraic approach to the waiting time process in GI/M/S
Authors: Loris-Teghem Jacqueline
Published: July 2016

The transient behaviour of the waiting time process in GI/M/S is studied algebraically by means of a two-dimensional Markovian process {(vn, ln
} , where the variables vn
denote the times of full occupation immediately after the arrival instants Tn
and where ln
= max {0,– 1 + number of idle servers at (Tn
+ 0)}.
The simple branching process with infinite mean. I
Authors: Seneta E.
Published: July 2016

The simple branching process {Zn
} with mean number of offspring per individual infinite, is considered. Conditions under which there exists a sequence {pn
} of positive constants such that pn
log (1 +Zn
) converges in law to a proper limit distribution are given, as is a supplementary condition necessary and sufficient for pn~ constant cn
as n→∞, where 0 < c < 1 is a number characteristic of the process. Some properties of the limiting distribution function are discussed; while others (with additional results) are deferred to a sequel.
Some ratio identities for a class of skip free random walks
Authors: Pegg P. A.
Published: July 2016

We prove a ratio identity giving a transform of the distribution of the position of a two barrier random walk as a ratio of the equivalent transforms for the one barrier random walk. This ratio identity holds for skip free random walks where the increments are dependent on both the previous increment and also the position of the random walk.
A note on random walks. II
Authors: Speed T. P.
Published: July 2016

The various duality results existing for two-barrier random walks are unified. Particular cases in the literature involving reflecting and reversing are deduced as corollaries.
The extinction probability of descendants in bisexual models of fixed population size
Authors: Kämmerle K.
Published: July 2016

In this paper exchangeable bisexual models with fixed population size and non-overlapping generations are introduced. Each generation consists of N pairs of individuals. The pairs of a generation have altogether 2N children. These individuals form randomly the N pairs of the next generation. The extinction probability of the descendants of a fixed number of pairs of generation 0 is discussed. Under suitable conditions it can be approximately described by the extinction probability of a Galton–Watson process, if the population size is large. Special examples are a bisexual Wright–Fisher model and models with a uniformly bounded number of children of a pair.
A model for random instantaneous growth on an interval
Authors: Quine M. P.
Published: July 2016

Points arrive in succession on an interval and immediately ‘cover' a region of length ½ to each side (less if they are close to the boundary or to a covered part). The location of a new point is uniformly distributed on the uncovered parts. We study the mean and variance of the total number of points ever formed, in particular as a → 0, in which case we also establish asymptotic normality.
Invariant stochastic properties of a class of directed circuits
Authors: Kalpazidou S.
Published: July 2016

Invariant stochastic properties are studied for circuit processes, i.e. processes whose sample paths can be represented by a denumerable set of overlapping directed circuits c in a denumerable set S and a set of positive weights , when the family of the weights varies.
A note on optimal inspection policy for stochastically deteriorating series systems
Authors: Qiu Yuping
Published: July 2016

A counterexample is given to demonstrate that a previously proposed characterization of optimal inspection policy for series systems is not correct in the discounted case. A set of valid necessary conditions is provided which leads to the characterization of optimality for a special case. The limiting case of these necessary conditions is also examined.
An elementary proof of Sengupta's invariance relation and a remark on Miyazawa's conservation principle
Authors: Brémaud P.
Published: July 2016

In this short note we derive Sengupta's (1989) invariance relation using elementary arguments and we show that Miyazawa's (1983), (1985) conservation principle, on which Sengupta's proof is based, admits the Palm inversion formula as a consequence. This contrasts with Miyazawa's first proof based on the inversion formula. We also show that Neveu's (1976) cycle formula is a direct consequence of Miyazawa's principle.
On ratios of random variables and generalised mortality rates
Authors: Roberts Leigh A.
Published: July 2016

We find the distribution function of a ratio of dependent random variables which can represent a generalised mortality rate in a demographic or life insurance context. Each death in the numerator and each unit of exposure in the denominator are weighted by a random sum at risk, which is assumed to follow a gamma distribution. General results on the existence of moments of ratios of random variables are established, and applied to show that the moments of the rates considered here depend in a simple way on the minimum number of entrants into the mortality investigation.
On the weak convergence of sequences of circuit processes: a probabilistic approach
Authors: Kalpazidou Sophia
Published: July 2016

The asymptotic behaviour of sequences of Markov processes whose finite distributions depend upon the sample paths ω of a positive recurrent Markov chain ξ is studied. The existence of such sequences depends upon the existence of a unique class of directed weighted circuits having a probabilistic interpretation in terms of the directed circuits occurring along the sample paths of ξ. An application to multiple Markov chains is given.
On the system queue length distributions of LCFS-P queues with arbitrary acceptance and restarting policies
Authors: Miyazawa Masakiyo
Published: July 2016

Shanthikumar and Sumita (1986) proved that the stationary system queue length distribution just after a departure instant is geometric forGI/GI/1 with LCFS-P/H service discipline and with a constant acceptance probability of an arriving customer, where P denotes preemptive and H is a restarting policy which may depend on the history of preemption. They also got interesting relationships among characteristics. We generalize those results forG/G/1 with an arbitrary restarting LCFS-P and with an arbitrary acceptance policy. Several corollaries are obtained. Fakinos' (1987) and Yamazaki's (1990) expressions for the system queue length distribution are extended. For a Poisson arrival case, we extend the well-known insensitivity for LCFS-P/resume, and discuss the stationary distribution for LCFS-P/repeat.
Further counterexamples to the monotonicity property of t-step maintainable structures
Authors: Haigh John
Published: July 2016

Until Guerry's (1990) counterexample to a conjecture of Davies about three-state hierarchical organisations kept at constant size via annual promotion, wastage and recruitment, it was easy to believe that such structures maintainable in t steps would also be maintainable in t + 1 steps. Here we present further counterexamples, which show that t-step maintainability does not imply (t + 1)-step maintainability, for astonishingly large values of t.
The maximum and mean of a random length sequence
Authors: Haas Peter J.
Published: July 2016

We obtain a limit theorem for the joint distribution of the maximum value and sample mean of a random length sequence of independent and identically distributed random variables. This extends a previous bivariate convergence result for fixed length sequences and incidentally yields a new proof of Berman's classical limit theorem for the maximum value of a random number of random variables. Our approach uses a property of record time sequences and leads to probabilistically intuitive proofs. We also consider the partition of a finite interval into a random number of subintervals by the points of a non-delayed renewal process. Using the bivariate convergence result for random length sequences, we establish a limit theorem for the joint distribution of the number and maximum length of the subintervals as the interval length becomes large. This leads to limiting results for the ratio of the maximum to the mean subinterval length. Such results are of interest in connection with a simple model of parallel processing.
Contact and chord length distributions of the Poisson Voronoi tessellation
Authors: Muche Lutz , Stoyan Dietrich
Published: July 2016

This paper presents the form of some characteristics of the Voronoi tessellation which is generated by a stationary Poisson process in . Expressions are given for the spherical and linear contact distribution functions. These formulae lead to numerically tractable double-integral formulae for chord length probability density functions.
A note on a one-compartment model with clustering
Authors: Booth James G.
Published: July 2016

The classical one-compartment model with no input or pure death process is shown to be a limiting case of a ‘binomial cascade' model which has the same mean and in which particles exit the compartment in binomial clusters. The transition probabilities of the binomial cascade process are derived in closed form. The model is easily modified to allow Poisson input into the compartment. Distributional results are given for this model also. In particular, it is shown that the M/M/∞ queue is a limiting case.
On measures of non-degeneracy
Authors: Athreya K. B.
Published: July 2016

If φ is a convex function and X a random variable then (by Jensen's inequality) ψ φ
(X) = Eφ
(X) – φ (EX) is non-negative and 0 iff either φ is linear in the range of X or X is degenerate. So if φ is not linear then ψ φ
(X) is a measure of non-degeneracy of the random variable X. For φ (x) = x
2, ψ φ
(X) is simply the variance V(X) which is additive in the sense that V(X + Y) = V(X) + V(Y) if X and Y are uncorrelated. In this note it is shown that if φ ″(·) is monotone non-increasing then ψ φ
is sub-additive for all (X, Y) such that EX ≧ 0, P(Y ≧ 0) = 1 and E(X | Y) = EX w.p.l, and is additive essentially only if φ is quadratic. Thus, it confirms the unique role of variance as a measure of non-degeneracy. An application to branching processes is also given.
On the volume distribution of the typical Poisson–Delaunay cell
Authors: Rathie P. N.
Published: July 2016

A method of obtaining the distribution of the volume of the typical cell of a Delaunay tessellation generated by a Poisson process in is developed and used to derive the density when d = 1, 2, 3.
On equivalence of Markov properties over undirected graphs
Authors: Matúš F.
Published: July 2016

The dependence of coincidence of the global, local and pairwise Markov properties on the underlying undirected graph is examined. The pairs of these properties are found to be equivalent for graphs with some small excluded subgraphs. Probabilistic representations of the corresponding conditional independence structures are discussed.
Extreme values of the cyclostationary Gaussian random process
Authors: Konstant D. G. , Piterbarg V.I.
Published: July 2016

In this paper the class of cyclostationary Gaussian random processes is studied. Basic asymptotics are given for the class of Gaussian processes that are centered and differentiable in mean square. Then, under certain conditions on the non-degeneration of the centered cyclostationary Gaussian process with integrable covariance functions, the Gnedenko-type limit formula

is established for and all x > 0.
On permutation policies for the scheduling of deteriorating stochastic jobs on a single machine
Authors: Glazebrook K. D.
Published: July 2016

A single machine is available to process a collection of stochastic jobs. Processing is preemptive and so (for example) the machine is allowed to switch away from a job before completion, should that prove advantageous. The jobs are deteriorating in the sense that their processing requirements grow (at job-specific rates) as they await processing. This phenomenon might be expected to enhance the status of non-preemptive policies. The primary objective of the paper is to find conditions which are sufficient to ensure the existence of a permutation policy to minimise the expected makespan. We also derive results for a weighted flowtime criterion. Applications of such models to the control of queues and to communication systems have been cited by other authors.
A martingale approach to the PASTA property
Authors: Scheutzow Michael
Published: July 2016

It is known (Weizsäcker and Winkler (1990)) that for bounded predictable functions H and a Poisson process with jump times

exists almost surely, and that in this case both limits are equal. Here we relax the boundedness condition on H. Our tool is a law of large numbers for local L
2-martingales. We show by examples that our condition is close to optimal. Furthermore we indicate a generalization to point processes on more general spaces. The above property is called PASTA (‘Poisson arrivals see time averages') and is heavily used in queueing theory.
Lumpability and marginalisability for continuous-time Markov chains
Authors: Ball Frank , Yeo Geoffrey F.
Published: July 2016

We consider lumpability for continuous-time Markov chains and provide a simple probabilistic proof of necessary and sufficient conditions for strong lumpability, valid in circumstances not covered by known theory. We also consider the following marginalisability problem. Let {X{t)} = {(X
1(t), X
2(t), · ··, Xm
(t))} be a continuous-time Markov chain. Under what conditions are the marginal processes {X
1(t)}, {X
2(t)}, · ··, {Xm
(t)} also continuous-time Markov chains? We show that this is related to lumpability and, if no two of the marginal processes can jump simultaneously, then they are continuous-time Markov chains if and only if they are mutually independent. Applications to ion channel modelling and birth–death processes are discussed briefly.
Partially observable semi-Markov reward processes
Authors: Masuda Yasushi
Published: July 2016

The main objective of this paper is to investigate the conditional behavior of the multivariate reward process given the number of certain signals where the underlying system is described by a semi-Markov process and the signal is defined by a counting process. To this end, we study the joint behavior of the multivariate reward process and the multivariate counting process in detail. We derive transform results as well as the corresponding real domain expressions, thus providing clear probabilistic interpretation.
A general recursive discrete-time filter
Authors: Elliott Robert J.
Published: July 2016

A reference probability is explicitly constructed under which the signal and observation processes are independent. A simple, explicit recursive form is then obtained for the conditional density of the signal given the observations. Both non-linear and linear filters are considered, as well as two different information patterns.
Transient behavior of coverage processes by applications to the infinite-server queue
Authors: Browne Sid , Steele J. Michael
Published: July 2016

We obtain the distribution of the length of a clump in a coverage process where the first line segment of a clump has a distribution that differs from the remaining segments of the clump. This result allows us to provide the distribution of the busy period in an M/G/∞ queueing system with exceptional first service, and applications are considered. The result also provides the tool necessary to analyze the transient behavior of an ordinary coverage process, namely the depletion time of the ordinary M/G/∞ system.
The probability that the largest observation is censored
Authors: Maller R. A. , Zhou S.
Published: July 2016

Suppose n possibly censored survival times are observed under an independent censoring model, in which the observed times are generated as the minimum of independent positive failure and censor random variables. A practical difficulty arises when the largest observation is censored since then the usual non-parametric estimator of the distribution of the survival time is improper. We calculate the probability that this occurs and give necessary and sufficient conditions for this probability to converge to 0 as n →∞. As an application, we show that if this probability is 0, asymptotically, then a consistent estimator for the mean failure time can be found. An almost sure version of the problem is also considered.
On Klimov's model by two job classes and exponential processing times
Authors: Chao Xiuli
Published: July 2016

We consider the Klimov model for an open network of two types of jobs. Jobs of type i arrive at station i, have processing times that are exponentially distributed with parameter µi
, and when processed either go on to station j with probability pij
, or depart the network with probability pi

0. Costs are charged at a rate that depends on the number of jobs of the two types in the system. It is shown that for arbitrary arrival processes the policy that gives priority to those jobs for whom the rate of change of the cost function is greatest minimizes the expected cost rate at every time t. This result is stronger than the Klimov result in two ways: arrival processes are arbitrary, and the minimization is at each time t. But the result holds for only two types.
G-networks by triggered customer movement
Authors: Gelenbe Erol
Published: July 2016

The generalized queueing networks (G-networks) which we introduce in this paper contain customers and signals. Both customers and signals can be exogenous, or can be obtained by a Markovian movement of a customer from one queue to another after service transforming itself into a signal or remaining a customer. A signal entering a queue forces a customer to move instantaneously to another queue according to a Markovian routing rule, or to leave the network, while customers request service. This synchronised or triggered motion is useful in representing the effect of tokens in Petri nets, in modelling systems in which customers and work can be instantaneously moved from one queue to the other upon certain events, and also for certain behaviours encountered in parallel computer system modelling. We show that this new class of network has product-form stationary solution, and establish the non-linear customer flow equations which govern it. Network stability is discussed in this new context.
The density of interfaces: a new first-passage problem
Authors: Chayes L. , Winfield C.
Published: July 2016

We introduce and study a novel type of first-passage percolation problem onwhere the associated first-passage time measures the density of interface between two types of sites. If the types, designated + and –, are independently assigned their values with probabilitypand (1 —p) respectively, we show that the density of interface is non-zero provided that both species are subcritical with regard to percolation, i.e.pc>p> 1 –pc.Furthermore, we show that asp↑pcorp↓ (1 –pc), the interface density vanishes with scaling behavior identical to the correlation length of the site percolation problem.
Queueing networks by negative customers and negative queue lengths
Authors: Henderson W.
Published: July 2016

A number of papers have recently appeared in the literature in which customers, in moving from node to node in the network arrive as either a positive customer or as a batch of negative customers. A positive customer joining its queue increases the number of customers at the queue by 1 and each negative customer decreases the queue length by 1, if possible. It has been shown that the equilibrium distribution for these networks assumes a geometric product form, that certain partial balance equations prevail and that the parameters of the geometric distributions are, as in Jackson networks, the service facility throughputs of customers. In this paper the previous work is generalised by allowing state dependence into both the service and routing intensities and by allowing the possibility, although not the necessity, for negative customers to build up at the nodes.
Generalized Bonferroni inequalities
Authors: Spencer Thomas H.
Published: July 2016

Consider a number of events in a probability space. Let X be a random variable that is the number of events that occur. Given some of the moments of the distribution of X, it is possible to obtain bounds on the probability that at least one event occurs. The best possible bounds are given here. If there are many equiprobable events that are d- wise independent, and d is even, then the probability that at least one event happens is at least 1 — O(µ–d

/2), where μ = E(X).
On the clique number of a random overlap graph
Authors: Łuczak Tomasz , Palka Zbigniew
Published: July 2016

The behaviour of the size of the largest complete subgraph contained in an overlap graph of a random acyclic digraph is presented.
On dispersion of stable random vectors and its application in the prediction of multivariate stable processes
Authors: Soltani A. Reza , Moeanaddin R.
Published: July 2016

Our aim in this article is to derive an expression for the best linear predictor of a multivariate symmetric α stable process based on many past values. For this purpose we introduce a definition of dispersion for symmetric α stable random vectors and choose the linear predictor which minimizes the dispersion of the error vector.
Another look at the moment bounds on reliability
Authors: Sengupta Debasis
Published: July 2016

In this paper a unified derivation of the upper and lower bounds (in terms of the mean) of an IFR, DFR, IFRA, DFRA, NBU or NWU reliability function is presented. The method of proof provides a simpler alternative to the various proofs known in the literature. Moreover, this method can be used to generalize the existing results in two ways, as demonstrated here. First, the bounds for the reliability function are obtained in terms of any finite moment in all these cases. Subsequently we provide a moment bound on a reliability function which ages faster or slower than a known one in some sense. The existing literature does not offer any of these generalizations, except for a few results which are available in an unnecessarily complicated form.
The optimality of LEPT in parallel machine scheduling
Authors: Chang Cheng-Shang , Righter Rhonda
Published: July 2016

We consider preemptive scheduling on parallel machines where the number of available machines may be an arbitrary, possibly random, function of time. Processing times of jobs are from a family of DLR (decreasing likelihood ratio) distributions, and jobs may arrive at random agreeable times. We give a constructive coupling proof to show that LEPT stochastically minimizes the makespan, and that it minimizes the expected cost when the cost function satisfies certain agreeability conditions.
Regenerative rare events simulation via likelihood ratios
Authors: Asmussen Søren , Rubinstein Reuven Y. , Wang Chia-Li
Published: July 2016

In this paper we obtain some new theoretical and numerial results on estimation of small steady-state probabilities in regenerative queueing models by using the likelihood ratio (score function) method, which is based on a change of the probability measure. For simple GI/G/1 queues, this amounts to simulating the regenerative cycles by a suitable change of the interarrival and service time distribution, typically corresponding to a reference traffic intensity ρ
0 which is < 1 but larger than the given one ρ. For the M/M/1 queue, the resulting gain of efficiency is calculated explicitly and shown to be considerable. Simulation results are presented indicating that similar conclusions hold for gradient estimates and in more general queueing models like queueing networks.
On an elementary characterization of the increasing convex ordering, by an application
Authors: Makowski Armand M.
Published: July 2016

In this short note, we present a simple characterization of the increasing convex ordering on the set of probability distributions on ℝ. We show its usefulness by providing a very short proof of a comparison result for M/GI/1 queues due to Daley and Rolski, and obtained by completely different means.
Return times in nearly-completely decomposable stochastic processes
Authors: Latouche G. , Louchard G.
Published: July 2016

Consider a finite irreducible aperiodic Markov chain with nearly-completely decomposable stochastic matrix: i.e. a Markov chain for which the states can be grouped into disjoint aggregates, in such a way that the probabilities of transition between states of the same aggregate are large compared to the probabilities of transition between states belonging to different aggregates. Let Ω be a subset of one of the aggregates. Second-order approximations are determined for the first and second moments of the time to reach Ω and the return time to Ω.
Asymptotic properties of discrete duels
Authors: Kimeldorf George , Lang John Patrick
Published: July 2016

For fixed accuracy functions, letGnmbe a discrete-fire (silent, noisy, or silent-noisy) duel in which the amounts of ammunition available to the two players arenandm, respectively. Asnandmboth become large, a class of ‘ε-optimal' pure strategies (with support (anm, 1)) is derived. Ifvnmdenotes the value ofGnm, the asymptotic relationships among the quantitiesn, m, vnm, andanmare investigated.
An improved model for signalised intersections with vehicle-actuated control
Authors: Cowan Richard
Published: July 2016

Explicit results are found for average delays and average lengths of red and green phases for a realistic continuous-time model of a two-street intersection operating with vehicle-actuated controls. The assumed arrival processes satisfactorily model both the random ‘bunch-gap' structure of traffic and the fact that vehicles have non-zero length.
On the output theorem of queueing theory, via filtering
Authors: Brémaud P.
Published: July 2016

We prove and extend Burke's output theorem for queues in equilibrium with the help of a filtering formula giving the time evolution of the least square estimate of the state with respect to the output.
On the optimal assignment of customers to parallel servers
Authors: Weber Richard R.
Published: July 2016

We consider a queuing system with several identical servers, each with its own queue. Identical customers arrive according to some stochastic process and as each customer arrives it must be assigned to some server's queue. No jockeying amongst the queues is allowed. We are interested in assigning the arriving customers so as to maximize the number of customers which complete their service by a certain time. If each customer's service time is a random variable with a non-decreasing hazard rate then the strategy which does this is one which assigns each arrival to the shortest queue.
The variance of duration of stay in an absorbing Markov process
Authors: Rust Philip F.
Published: July 2016

Given a stationary Markov process with s transient states and r absorbing states, a matrix infinite series solution is presented for the variance of duration of stay in state j within the interval [0, t), given initial state i. Closed forms are derived for absorbing states, and for transient states if eigenvalues are real and distinct. Several relationships among Markov matrices are presented.
On the quantiles of the gamma and F distributions
Authors: Saunders I. W. , Moran P. A. P.
Published: July 2016

We show, in the context of a neurobiological problem, that ζ a
(α)−ζ a
(β) increases with a, while ζ a
(α)/ζ a
(β) decreases with a, where 0 < β < α < 1 and ζ a
(α) is the α th quantile of either the gamma distribution with shape parameter a, or a scaled F distribution with parameters 2a and 2m for any m.
The weak convergence of a class of estimators of the variance function of a two-dimensional Poisson process
Authors: Liebetrau A. M.
Published: July 2016

Results of a previous paper (Liebetrau (1977a)) are extended to higher dimensions. An estimator V∗(t
1, t
2) of the variance function V(t
1, t
2) of a two-dimensional process is defined, and its first- and second-moment structure is given assuming the process to be Poisson. Members of a class of estimators of the form where and for 0 < α i
< 1, are shown to converge weakly to a non-stationary Gaussian process. Similar results hold when the t′i
are taken to be constants, when V is replaced by a suitable estimator and when the dimensionality of the underlying Poisson process is greater than two.
On competition between two species
Authors: Ridler-Rowe C. J.
Published: July 2016

A stochastic model for the competition between two species is considered and a limit theorem is obtained for the size of the surviving species. A limit theorem is also obtained for the probability that a given species should survive the other.
Cumulative constrained sojourn times in semi-Markov processes with an application to pensionable service
Authors: Sahin Izzet
Published: July 2016

This paper is concerned with the characterization of the cumulative pensionable service over an individual's working life that is made up of random lengths of service in different employments in a given industry, under partial coverage, transferability, and a uniform vesting rule. This characterization uses some results that are developed in the paper involving a functional and cumulative constrained sojourn times (constrained in the sense that if a sojourn time is less than a given constant it is not counted) in semi-Markov processes.
The discrete-time single-server queue with time-inhomogeneous compound Poisson input and general service time distribution
Authors: Minh Do Le
Published: July 2016

This paper studies a discrete-time, single-server queueing model having a compound Poisson input with time-dependent parameters and a general service time distribution.

All major transient characteristics of the system can be calculated very easily. For the queueing model with periodic arrival function, some explicit results are obtained.
Average delay in queues with non-stationary Poisson arrivals
Authors: Ross Sheldon M.
Published: July 2016

One of the major difficulties in attempting to apply known queueing theory results to real problems is that almost always these results assume a time-stationary Poisson arrival process, whereas in practice the actual process is almost invariably non-stationary. In this paper we consider single-server infinite-capacity queueing models in which the arrival process is a non-stationary process with an intensity function ∧(t), t ≧ 0, which is itself a random process. We suppose that the average value of the intensity function exists and is equal to some constant, call it λ, with probability 1.

We make a conjecture to the effect that ‘the closer {∧(t), t ≧ 0} is to the stationary Poisson process with rate λ ' then the smaller is the average customer delay, and then we verify the conjecture in the special case where the arrival process is an interrupted Poisson process.
Representations and limit theorems for extreme value distributions
Authors: Hall Peter
Published: July 2016

LetXn1≦Xn2≦ ··· ≦Xnndenote the order statistics from a sample ofnindependent, identically distributed random variables, and suppose that the variablesXnn, Xn,n–1, ···, when suitably normalized, have a non-trivial limiting joint distributionξ1,ξ2, ···, asn → ∞. It is well known that the limiting distribution must be one of just three types. We provide a canonical representation of the stochastic process {ξn,n≧ 1} in terms of exponential variables, and use this representation to obtain limit theorems forξnasn →∞.
On random divisions of a convex set
Authors: Janson Svante
Published: July 2016

This paper gives an elementary proof that, under some general assumptions, the number of parts a convex set in Rd
is divided into by a set of independent identically distributed hyperplanes is asymptotically normally distributed. An example is given where the distribution of hyperplanes is ‘too singular' to satisfy the assumptions, and where a different limiting distribution appears.
A characterization of the exponential distribution by spacings
Authors: Ahsanullah M.
Published: July 2016

Let X
1, X
2, ···, Xn
be a random sample of size n from a population with probability density function f(x), x > 0, and let X
1, n
< X
2, n
< ··· < Xn, n
be the associated order statistics. A characterization of the exponential distribution is shown by considering identical distribution of the random variables (n − i + 1)(Xi, n − X
i−1, n
) and (n − i)(X
i+1, n
− X
i, n
) for one i and one n with 2 ≦ i ˂ n.
Waveforms and velocities for non-nearest-neighbour contact distributions
Authors: Renshaw Eric
Published: July 2016

This paper examines a model for ecological and epidemiological spread. Expressions are derived for mean waveforms and expectation velocities for two specific contact distributions. Whilst one distribution may be bounded above by a negative exponential function the other may not, and these two situations respectively give rise to finite and infinite asymptotic expectation velocities.
Evolutionarily stable strategies with two types of player
Authors: Taylor Peter D.
Published: July 2016

A definition of ESS (evolutionarily stable strategy) is suggested for games in which there are two types of player, each with its own set of strategies, and the fitness of any strategy depends upon the strategy mix, of both types, in the population as a whole. We check that the standard ESS results hold for this definition and discuss a mate-desertion model which has appeared in the literature in which the two types are male and female.
The volume of an isotropic random parallelotope
Authors: Ruben Harold
Published: July 2016

The p-content of the p-parallelotope ∇
p, n
determined by p independent isotropic random points z
1, …, zp
in ℝ
n
(1 < p ≦ n) can be expressed as a product of independent variates in two ways, by successive orthogonal projection onto linear subspaces and by radial projection of the points, enabling calculation of the actual distribution as well as the moments of ∇
p, n
. This is done explicitly in several cases. The results also have interest in multivariate statistics.
Least absolute deviation estimates in autoregression with infinite variance
Authors: Gross S. , Steiger W. L.
Published: July 2016

We consider an L
1 analogue of the least squares estimator for the parameters of stationary, finite-order autoregressions. This estimator, the least absolute deviation (LAD), is shown to be strongly consistent via a result that may have independent interest. The striking feature is that the conditions are so mild as to include processes with infinite variance, notably the stationary, finite autoregressions driven by stable increments in Lα, α > 1. Finally, sampling properties of LAD are compared to those of least squares. Together with a known convergence rate result for least squares, the Monte Carlo study provides evidence for a conjecture on the convergence rate of LAD.
An enumerative approach to a problem in probability
Authors: Devitt John S.
Published: July 2016

Sequences over the finite alphabet {1, …, m} are used to model the various job queues of a machine, classifying each job according to which one of a set of m resources is used by the job. Known results concerning the enumeration of such sequences according to the number of rises, levels and falls are then used to obtain the various moments of the distribution of the random variable for the number of changes of resources occurring in a sequence of n jobs given the conditional probabilities of changing resources between independent jobs. The techniques involved form a useful approach to problems where such enumerative information is available.
Level crossings of a constructed process
Authors: Sharpe Ken
Published: July 2016

A method is presented for constructing a continuous, stationary process with a given rate of upcrossings at all levels and for which it is possible to find the exact distribution of the number of upcrossings, in an interval, at all levels.
On the variance of a trimmed Poisson distribution
Authors: Kemperman J. H. B.
Published: July 2016

Consider a queue for a special performance or sale. A person with arrival number i will decide to join the queue with probability pi. Here the pi
denote fixed constants. Let X denote the number of arrivals, X″ the number of customers leaving the queue and X′ = X – X″ the number of customers who stay with the queue. For the case that p

i+1 ≦ p

i
and X has a Poisson distribution, it is shown that Var (Xʺ) ≧ E(Xʺ) and Var (X′) ≦ E(X′). There are also results for the case where {p
i
} and X are rather arbitrary.
A branching process showing a phase transition
Authors: Moore T. , Snell J. L.
Published: July 2016

It is shown that a phase transition for the Ising model on a tree graph can be detected by the limiting behavior of the magnetization. The relevant limit theorems are special cases of limit theorems for multitype branching processes.
A simple condition for regularity in negative programming
Authors: Whittle P.
Published: July 2016

A simple condition (the ‘bridging condition') is given for a Markov decision problem with non-negative costs to enjoy the regularity properties enunciated in Theorem 1. The bridging condition is sufficient for regularity, and is not far from being necessary, in a sense explained in Section 2. In Section 8 we consider the different classes of terminal loss functions (domains of attraction) associated with different solutions of (14). Some conjectures concerning these domains of attraction are either proved, or disproved by counter-example.