bluuebunny/crossref_metadata_2025 huggingface dataset:
Deviations from monotonicity of a Wiener process with drift
Authors: Brockwell P. J.
Published: July 2016

If X(t) is a Wiener process with EX(t) = vt and var X(t) = σ
2
t (where v > 0) and if M(t) = max0≦τ≦t

X(τ) and Ta
is the time of first passage through level a (where a > 0) we show that

where is the coefficient of variation of Ta
. Applications of the result to the “maturity-time” representation of cell-growth and to queues with heavy traffic are discussed.
On a theorem of Ahsanullah and Rahman
Authors: Huang J. S.
Published: July 2016

A characterization theorem of the exponential distribution by Ahsanullah and Rahman is examined in contrast to existing results by several other authors.
Modes of growth of counting processes with increasing arrival rates
Authors: Waugh W. A. O'N.
Published: July 2016

Jumping processes which grow by unit jumps, with decreasing sojourn times between successive jumps, are studied. Markovian birth processes, non-Markovian branching processes, and some generalizations of these are special cases. Three classes are described, in one of which growth is explosive, in the second asymptotically continuous, and in the third oscillatory. A theorem is proved which gives an explicit functional expression in the asymptotically continuous case, and borderline cases between the classes are investigated.
Optimal and adaptive stopping based on capture times
Authors: Starr Norman
Published: July 2016

Independent exponential capture times are assumed for each of N prey. The payoff is the number of prey caught less a linear time cost. The optimal stopping time and value are obtained. When N is known an adaptive statistical procedure is proposed for which the expected payoff differs from the value by at most one-half plus a term which tends to zero (of order N
–α, a < 1) as N → ∞.
The recursive estimation of a Markov chain
Authors: Blight B. J. N. , Devore J. L.
Published: July 2016

For every hth member of a two-state Markov chain the value of a random variable Y is observed where the distribution of Y is conditional on the state of the corresponding member of the chain. A recursive set of equations is derived giving the posterior probabilities for both the observed and unobserved members. The use of this recursive solution to investigate the optimality of certain simple classification rules is discussed, and a “classification by runs” is also presented.
Linear birth and death processes with killing
Authors: Karlin Samuel , Tavaré Simon
Published: July 2016

We analyze a class of linear birth and death processes X(t) with killing. The generator is of the form λ i
= bi + θ, µi = ai, γ i
= ci, where γ i
is the killing rate. Then P{killed in (t, t + h) | X(t) = i} = γ ih + o(h), h ↓ 0. A variety of explicit results are found, and an example from population genetics is given.
Extreme-value properties of the explosion-time distribution in a pure birth process
Authors: Feigin Paul D. , Yashchin Emmanuel
Published: July 2016

In each of a large number N of independent cells a breakdown mechanism is under way and proceeds until the first of the cells actually fails. At such a time, in each cell, the situation reverts to some initial state and the mechanism restarts. In this paper we consider those mechanisms for which breakdown may be modelled as the explosion of a pure birth process. Of interest is the distribution of time between failures and the possibility of estimating N and/or model parameters by observing a sequence of failure times. Saddlepoint approximation methods are used in the relevant extreme-value theory analysis for two important cases.
Asymptotic properties of subcritical Galton-Watson processes
Authors: Bagley J. H.
Published: July 2016

Results connecting Yaglom's limit and Z1are obtained. For example, if α > 1,E[Zα1] andconverge or diverge together.
Coverage problems and random convex hulls
Authors: Jewell Nicholas P. , Romano Joseph P.
Published: July 2016

Consider the placement of a finite number of arcs on the circle of circumference 2π where the midpoint and length of each arc follows an arbitrary bivariate distribution. In the case where each arc has lengthπ, the probability that the circle is completely covered is equal to the probability that the convex hull of a finite random sample of points, chosen according to a certain bivariate distribution in the plane contains the origin. In general, we show that evaluating the probability that the random convex hull contains a fixed disc is equivalent to solving the general coverage problem where the midpoint and length of each arc follows an arbitrary bivariate distribution. Exact formulae for the above probabilities are obtained and some examples are considered.
Random coverage of a circle with applications to a shadowing problem
Authors: Yadin M. , Zacks S.
Published: July 2016

The coverage problem on the circle is considered from the shadowing process point of view. A random number of shadow arcs are distributed on a circle. The length of each arc is a random variable which depends on the random diameter of a shadowing disk and its random location. Formulae are derived for the numerical determination of the moments of the measure of vacancy of arcs on the circle, for a special example. An approximation to the distribution of the measure of vacancy is also provided.
Almost sure limit results for the supercritical Bellman-Harris process
Authors: Kuczek Thomas
Published: July 2016

As a consequence of the convergence of the age distribution, several other limit results follow for supercritical Bellman-Harris processes. These concern the distribution of split times, the growth ‘rate' of the process, and the distribution of ancestral lifelengths.
On the number of terminal vertices in certain random trees with an application to stemma construction in philology
Authors: Najock D. , Heyde C. C.
Published: July 2016

A fundamental task of philologists is to construct the family tree (stemma) of preserved copies of ancient manuscripts. A simple probabilistic model based on random rooted trees is proposed to assist in the identification of the number of terminal copies. The model provides the distribution of the number of terminal vertices in a random tree. An application to stemma construction is given.
Random polytopes in a ball
Authors: Buchta C. , Müller J.
Published: July 2016

The convex hull of n random points chosen independently and uniformly from a d-dimensional ball is a convex polytope. Its expected surface area, its expected mean width and its expected number of facets are explicitly determined.
Insensitivity of blocking probabilities in a circuit-switching network
Authors: Burman D. Y. , Lehoczky J. P. , Lim Y.
Published: July 2016

Consider a network of nodes (switches) and connecting links. Each link consists of a group of channels (trunks). A call instantaneously seizes channels along a route between the originating and terminating node, holds them for a randomly distributed length of time and frees them instantaneously at the end of the call. If no channels are available, the call is blocked. For special networks with exponential call holding times, Erlang has shown that the steady-state probabilities are in product form. In this paper, we extend this work to general networks and show that if for each pair of nodes there is a unique route, then the blocking probabilities are in product form and are insensitive to the call holding-time distribution, which means that they depend on the call duration only through its mean.
Asymptotic analysis of a queueing system by a two-dimensional state space
Authors: Blanc J. P. C.
Published: July 2016

A technique is developed for the analysis of the asymptotic behaviour in the long run of queueing systems with two waiting lines. The generating function of the time-dependent joint queue-length distribution is obtained with the aid of the theory of boundary value problems of the Riemann–Hilbert type and by introducing a conformal mapping of the unit disk onto a given domain. In the asymptotic analysis an extensive use is made of theorems on the boundary behaviour of such conformal mappings.
Some moment results for certain tandem and multiple-server queues
Authors: Wolfson Barry
Published: July 2016

It is known that for a GI/G/1 queue, if the interarrival time has finite first moment, then the (k + 1)th moment of the service time is finite if the kth moment of the stationary workload is finite. This result is extended to certain tandem and multiple-server queues.
Moments of the maximum in a critical branching process
Authors: Weiner Howard
Published: July 2016

Let Zn
denote the number of cells at time n in a critical discrete-time Galton–Watson branching process with finite offspring variance. Let

Martingale arguments are used to show that for some 0<a≦b<∞
A characterization of order statistic point processes that are mixed Poisson processes and mixed sample processes simultaneously
Authors: Deffner A. , Haeusler E.
Published: July 2016

The results of Nawrotzki (1962), Feigin (1979) and Puri (1982) show that the class of all point processes (on the real line) with the order statistic property consists of all mixed Poisson processes up to a time-scale transformation, and of all mixed sample processes. The present note characterizes those order statistic point processes that are mixed Poisson processes and mixed sample processes simultaneously.
Estimation and optimal stopping in a debugging model with masking
Authors: Siegrist Kyle
Published: July 2016

A system has an irremovable failure source and a number of removable flaws. The system undergoes a sequence of trials designed to detect and remove the flaws. On each trial, the irremovable failure source may cause failure which in turn may block the detection of any flaws. If not, then each flaw in the system, independently, is detected on that trial with a certain probability and each detected flaw, independently, is removed from the system with a certain probability before the next trial. Distributions of the outcomes of the trials are obtained. Point estimates of the parameters, based on accumulated trial data, are given. Assuming that certain costs are associated with trials, optimal stopping rules and a cost-benefit analysis are given.
The intensity conservation law for queues with randomly changed service rate
Authors: Miyazawa Masakiyo
Published: July 2016

König et al. (1978) have derived the so-called intensity conservation law in a stationary process connected with a marked point process (PMP). That law has been shown to be useful in obtaining invariance relations in queues (cf. Franken et al. (1981)). In this paper, somewhat different versions of the intensity conservation laws are derived for a stationary process with jump points. These laws are applied to queues with randomly changed service rate. As special cases, most of equations obtained by König et al.'s law can be derived from this law. Also, we derive some inequalities between characteristic quantities in a queue with a simple type of randomly changed service rate.
A note on stochastic decomposition in a GI/G/1 queue with vacations or set-up times
Authors: Doshi Bharat T.
Published: July 2016

In this note we prove some stochastic decomposition results for variations of theGI/G/1 queue. Our main model is aGI/G/1 queue in which the server, when it becomes idle, goes on a vacation for a random length of time. On return from vacation, if it finds customers waiting, then it starts serving the first customer in the queue. Otherwise it takes another vacation and so on. Under fairly general conditions the waiting time of an arbitrary customer, in steady state, is distributed as the sum of two independent random variables: one corresponding to the waiting time without vacations and the other to the stationary forward recurrence time of the vacation. This extends the decomposition result of Gelenbe and Iasnogorodski [5]. We use sample path arguments, which are also used to prove stochastic decomposition in aGI/G/1 queue with set-up time.
A coupling proof of weak convergence
Authors: Guttorp Peter , Kulperger Reg , Lockhart Richard
Published: July 2016

Weak convergence to reflected Brownian motion is deduced for certain upwardly drifting random walks by coupling them to a simple reflected random walk. The argument is quite elementary, and also gives the right conditions on the drift. A similar argument works for a corresponding continuous-time problem.
Optimal control of a finite dam: average-cost case
Authors: Yeh Lam
Published: July 2016

We consider the problem of minimizing the long-run average cost per unit time of operating a finite dam in the class of the policies of the following type. Assume that the dam is initially empty, the release rate is kept at 0 until the dam storage increases to λ, and as soon as this occurs, water is released at rate M, then the output rate is kept at M as long as the dam storage is more than τ and it must be decreased to 0 if the dam storage becomes τ. We assume that the input of water into the finite dam is a Wiener process with non-negative drift μ and variance parameter σ
2. There is a cost in increasing the output rate from 0 to M as well as in decreasing the rate from M to 0 and whenever the dam storage is below level a, there is a penalty cost per unit time depending on the level. A reward is given for each unit of water released. In this paper, the long-run average cost per unit time is determined, and therefore the optimal policy can be found numerically.
Poisson theorem for non-homogeneous Markov chains
Authors: Pawłowski Janusz
Published: July 2016

This paper gives necessary and sufficient conditions for the convergence in distribution of sums of the 0–1 Markov chains to a compound Poisson distribution.
Notes on the stability of closed queueing networks
Authors: Sigman Karl
Published: July 2016

A new proof of the stability of closed Jackson-type queueing networks (with general service-time distributions) is given and sufficient conditions are given for obtaining Cesaro, weak and total variation convergence of the continuous-time joint queue length and residual service-time process to a limiting distribution. The result weakens the sufficient conditions (for stability) of Borovkov (1986) by allowing more general service-time distributions.
Optimal stationary strategies in leavable Markov decision processes
Authors: Fassbender Matthias
Published: July 2016

This paper establishes the existence of an optimal stationary strategy in a leavable Markov decision process with countable state space and undiscounted total reward criterion.

Besides assumptions of boundedness and continuity, an assumption is imposed on the model which demands the continuity of the mean recurrence times on a subset of the stationary strategies, the so-called ‘good strategies'. For practical applications it is important that this assumption is implied by an assumption about the cost structure and the transition probabilities. In the last part we point out that our results in general cannot be deduced from related works on bias-optimality by Dekker and Hordijk, Wijngaard or Mann.
The distribution and moments of the number of components of a random function
Authors: Kupka Joseph
Published: July 2016

A relatively simple formula is presented for the probability distribution of the number K of components of a random function. This formula facilitates the (computer) calculation of the factorial moments of K and yields new expressions for the mean and variance of K.
A representation of a discrete distribution by its binomial moments
Authors: Brandt Andreas , Brandt Manfred , Sulanke Hannelore
Published: July 2016

Let pk, k ≧ 0, be a probability distribution having finite binomial moments Br, r ≧ 0, and the probability generating function U(z) with a radius of convergence α (≧ 1). In this note explicit and recursive formulae are derived allowing computation of the pk
in terms of the Br
if α > 1. Ch. Jordan's formula, which holds if α > 2, turns out to be a special case.
A note on the classification of Q-processes when Q is not regular
Authors: Pollett P. K.
Published: July 2016

We shall provide a classification of states for processes other than the Feller minimal process. In particular, we shall study the regularity, recurrence, transience, etc. of these processes. Our results are based on the observation that certain honest Q-processes exhibit regenerative behaviour. We shall also comment on the problem of determining stationary distributions directly from the transition rates.
An entropy concentration theorem: applications in artificial intelligence and descriptive statistics
Authors: Robert Claudine
Published: July 2016

The maximum entropy principle is used to model uncertainty by a maximum entropy distribution, subject to some appropriate linear constraints. We give an entropy concentration theorem (whose demonstration is based on large deviation techniques) which is a mathematical justification of this statistical modelling principle. Then we indicate how it can be used in artificial intelligence, and how relevant prior knowledge is provided by some classical descriptive statistical methods. It appears furthermore that the maximum entropy principle yields to a natural binding between descriptive methods and some statistical structures.
An integer-valued pth-order autoregressive structure (INAR(p)) process
Authors: Alzaid A. A. , Al-osh M.
Published: July 2016

An extension of the INAR(1) process which is useful for modelling discrete-time dependent counting processes is considered. The model investigated here has a form similar to that of the Gaussian AR(p) process, and is called the integer-valued pth-order autoregressive structure (INAR(p)) process. Despite the similarity in form, the two processes differ in many aspects such as the behaviour of the correlation, Markovian property and regression. Among other aspects of the INAR(p) process investigated here are the limiting as well as the joint distributions of the process. Also, some detailed discussion is given for the case in which the marginal distribution of the process is Poisson.
On arrivals that see time averages: a martingale approach
Authors: Melamed Benjamin , Whitt Ward
Published: July 2016

This paper is a sequel to our previous paper investigating whenarrivals see time averages(ASTA) in a stochastic model; i.e., when the steady-state distribution of an embedded sequence, obtained by observing a continuous-time stochastic process just prior to the points (arrivals) of an associated point process, coincides with the steady-state distribution of the observed process. The relation between the two distributions was also characterized when ASTA does not hold. These results were obtained using the conditional intensity of the point process given the present state of the observed process (assumed to be well defined) and basic properties of Riemann–Stieltjes integrals. Here similar results are obtained using the stochastic intensity associated with the martingale theory of point processes, as in Brémaud (1981). In the martingale framework, the ASTA result is almost an immediate consequence of the definition of a stochastic intensity. In a stationary framework, the results characterize the Palm distribution, but stationarity is not assumed here. Watanabe's (1964) martingale characterization of a Poisson process is also applied to establish a general version of anti–PASTA: if the points of the point process are appropriately generated by the observed process and the observed process is Markov with left-continuous sample paths, then ASTA implies that the point process must be Poisson.
A note on a D/G/K loss system with retrials
Authors: Pourbabai Behnam
Published: July 2016

An algorithm is suggested for approximating the performance of a D/G/K loss system with deterministic input, generally distributed processing time, K heterogeneous servers, the random access processing discipline, and retrials in steady state. In loss systems with retrials, the units which at the instants of their arrival at the system find all the servers busy, are not lost: those units retry to be processed by merging with the incoming arrival units. In this system, a fraction of the units which have not initially been processed will be allowed to leave the system. The performance of this system in steady state is approximated by a recursive technique.
A non-exponential queueing system with independent arrivals and batch servicing
Authors: Van Dijk Nico M. , Smeitink Eric
Published: July 2016

We study a queueing system with a finite number of input sources. Jobs are individually generated by a source but wait to be served in batches, during which the input of that source is stopped. The service speed of a server depends on the mode of other sources and thus includes interdependencies. The input and service times are allowed to be generally distributed. A classical example is a machine repair system where the machines are subject to shocks causing cumulative damage. A product-form expression is obtained for the steady state joint queue length distribution and shown to be insensitive (i.e. to depend on only mean input and service times). The result is of both practical and theoretical interest as an extension of more standard batch service systems.
On extremal service disciplines in single-stage queueing systems
Authors: Righter Rhonda , Shanthikumar J. George , Yamazaki Genji
Published: July 2016

It is shown that among all work-conserving service disciplines that are independent of the future history, the first-come-first-served (FCFS) service discipline minimizes [maximizes] the average sojourn time in a G/GI/1 queueing system with new better [worse] than used in expectation (NBUE[NWUE]) service time distribution. We prove this result using a new basic identity of G/GI/1 queues that may be of independent interest. Using a relationship between the workload and the number of customers in the system with different lengths of attained service it is shown that the average sojourn time is minimized [maximized] by the least-attained-service time (LAST) service discipline when the service time has the decreasing [increasing] mean residual life (DMRL[IMRL]) property.
Dispersive ordering by dilation
Authors: Muñoz-Perez J. , Sanchez-Gomez A.
Published: July 2016

In this paper a necessary and sufficient condition for the dispersive ordering in dilation sense is given by a convex function which is called the dispersive function and characterizes the distribution function. Some interesting properties of the ordering follow from this result.
Reliability of a consecutive-k-out-of-n system in a random environment
Authors: Chryssaphinou Ourania , Papastavridis Stavros
Published: July 2016

We consider a consecutive-k-out-of-n system situated in a random environment where shocks occur and cause components to fail with certain probabilities. Under certain reasonable assumptions we compute the moment generating function of the time of failure of the system and the distribution of the number of shocks required to cause failure of the system.
On interchangeability for exponential single-server queues in tandem
Authors: Kijima Masaaki , Makimoto Naoki
Published: July 2016

Consider two exponential single-server queues in tandem and suppose that service rates of customer n are λ n
and μ n
respectively. In this note, a simple and direct proof is given of the fact that the departure process from the tandem queue is statistically unaffected when the service rates are interchanged if λ n – μn
is independent of n. The proof is based only on the memoryless property of exponential distributions.
Convexity results for single-server queues and for multiserver queues with constant service times
Authors: Harel Arie
Published: July 2016

We show that the waiting time in queue and the sojourn time of every customer in the G/G/1 and G/D/c queue are jointly convex in mean interarrival time and mean service time, and also jointly convex in mean interarrival time and service rate. Counterexamples show that this need not be the case, for the GI/GI/c queue or for the D/GI/c queue, for c ≧ 2. Also, we show that the average number of customers in the M/D/c queue is jointly convex in arrival and service rates.

These results are surprising in light of the negative result for the GI/GI/2 queue (Weber (1983)).
Poisson approximation for some epidemic models
Authors: Ball Frank , Barbour A. D.
Published: July 2016

The Daniels' Poisson limit theorem for the final size of a severe general stochastic epidemic is extended to the Martin-Löf epidemic, and an order of magnitude for the error in the approximation is also given. The argument consists largely of showing that the number of survivors of a severe epidemic is essentially the same as the number of isolated vertices in a random directed graph. Poisson approximation for the latter quantity is proved using the Stein–Chen method and a suitable coupling.
Geometric bounds on iterative approximations for nearly completely decomposable Markov chains
Authors: Louchard Guy , Latouche Guy
Published: July 2016

We consider a finite Markov chain with nearly-completely decomposable stochastic matrix. We determine bounds for the error, when the stationary probability vector is approximated via a perturbation analysis.
On certain aspects of non-homogeneous Markov systems in continuous time
Authors: Gerontidis Ioannis I.
Published: July 2016

In the present paper we study three aspects in the theory of non-homogeneous Markov systems under the continuous-time formulation. Firstly, the relationship between stability and quasi-stationarity is investigated and conditions are provided for a quasi-stationary structure to be stable. Secondly, the concept of asymptotic attainability is studied and the possible regions of asymptotically attainable structures are determined. Finally, the cyclic case is considered, where it is shown that for a system in a periodic environment, the relative structure converges to a periodic vector, independently of the initial distribution. Two numerical examples illustrate the above theoretical results.
A convergence rate in extreme-value theory
Authors: Balkema A. A. , De Haan L.
Published: July 2016

A uniform convergence rate is determined for maxima of i.i.d. random variables from a distribution in the domain of attraction of the double-exponential distribution. The result is proved under a second-order condition on the underlying distribution parallelling the one given in Smith (1982) for the domain of attraction of the bounded-below and bounded-above families of limit distributions.
Strong approximations for multiple channel queues in heavy traffic
Authors: Hanqin Zhang , Guanghui Hsu , Rongxin Wang
Published: July 2016

We prove strong approximations for multiple channel queues in heavy traffic. Strong approximations are given for the waiting time and queue length processes, and for embedded sequences associated with the length process.
On the GIX/G/∞ system
Authors: Liu L. , Kashyap B. R. K. , Templeton J. G. C.
Published: July 2016

By using a shot noise process, general results on system size in continuous time are given both in transient state and in steady state with discussion on some interesting results concerning special cases. System size before arrivals is also discussed.
Throughput maximization in a loss queueing system with heterogeneous servers
Authors: Sobel Matthew J.
Published: July 2016

Assigning each arriving customer to the fastest idle server is shown to maximize throughput (equivalently, minimize blocking probability) in a queueing model with Poisson arrivals, heterogeneous exponential servers, and no waiting room. If a cost structure is imposed on this model, under specified conditions the same policy minimizes the expected discounted cost and the long-run average cost per unit time.
Wicksell's corpuscle problem in spherical spaces
Authors: Zähle M.
Published: July 2016

Wicksell's problem of unfolding ball size distributions is solved for the case of spherical spaces of curvature K > 0. The well-known formulas for the euclidean case result as limits if K → 0. The paper is completed by a brief discussion of some computer-aided statistical procedures.
A carrier-borne epidemic with multiple stages of infection
Authors: Gani J. , Michaletzky Gy.
Published: July 2016

This paper considers a carrier-borne epidemic in continuous time with m + 1 > 2 stages of infection. The carriers U(t) follow a pure death process, mixing homogeneously with susceptibles X
0(t), and infectives Xi
(t) in stages 1≦i≦m of infection. The infectives progress through consecutive stages of infection after each contact with the carriers. It is shown that under certain conditions {X
0(t), X
1(t), · ··, Xm
(t) U(t); t≧0} is an (m + 2)-variate Markov chain, and the partial differential equation for its probability generating function derived. This can be solved after a transfomation of variables, and the probability of survivors at the end of the epidemic found.
The Galton-Watson predator-prey process
Authors: Coffey John , Bühler Wolfgang J.
Published: July 2016

A probabilistic predator-prey model is constructed using linked discrete-time branching-type processes. A necessary and sufficient condition for positive probability of survival of both populations is given.
The joint limiting distribution of sums and maxima of stationary sequences
Authors: Anderson C. W. , Turkman K. F.
Published: July 2016

The joint limiting distribution of suitably normalized partial sums and maxima in a stationary strong mixing sequence with finite variance is derived. It is found that in the limit the two components are independent. This generalizes Chow and Teugels' result for independent sequences. Motivation for the present study comes from a statistical problem in the analysis of extreme winds.
Minification processes and their transformations
Authors: Lewis Peter A. W. , McKenzie Ed
Published: July 2016

It is shown that the stationary, autoregressive, Markovian minification processes introduced by Tavares and Sim can be extended to give processes with marginal distributions other than the exponential and Weibull distributions. Necessary and sufficient conditions on the hazard rate of the marginal distributions are given for a minification process to exist. Results are given for the derivation of the autocorrelation function; these correct the expression for the Weibull given by Sim. Monotonic transformations of the minification processes are also discussed and generate a whole new class of autoregressive processes with fixed marginal distributions. Stationary processes generated by a maximum operation are also introduced and a comparison of three different Markovian processes with uniform marginal distributions is given.
On Beurling's inequality in terms of thermal power
Authors: Kalpazidou S.
Published: July 2016

Beurling's inequality is proved to have an interpretation in terms of ingredients that do not depend on the effective numerical contribution of transition probabilities of circuit processes (those processes whose transition law is expressed by a denumerable class of directed weighted circuits). Connections with reversibility for Sk
-state, k≧2, circuit processes, where S is a denumerable set, are revealed.
Finite capacity vacation models with non-renewal input
Authors: Blondia C.
Published: July 2016

This paper studies a single server queue with finite waiting room where the server takes vacations according to two different strategies: (i) an exhaustive service discipline, where the server takes a vacation whenever the system becomes empty and these vacations are repeated as long as there are no customers in the system upon return from a vacation, i.e. a repeated vacation strategy; (ii) a limited service discipline, where the server begins a vacation either if K customers have been served in the same busy period or if the system is empty and then a repeated vacation strategy is followed. The input process is a general Markovian arrival process introduced by Lucantoni, Meier-Hellstern and Neuts, which as special cases includes the Markov modulated Poisson process and the phase-type renewal process. The service times and vacation times each are generally distributed random variables. For both models, we obtain the queue length distribution at departures, at an arbitrary time instant and at arrival time. We also derive the loss probability of an arriving customer. We obtain formulae for the LST of the virtual waiting time distribution and for the LST of the waiting time distribution at arrival epochs.
A conservation law for single-server queues and its applications
Authors: Yamazaki Genji , Sakasegawa Hirotaka , Shanthikumar J. George
Published: July 2016

We establish a conservation law forG/G/1 queues with any work-conserving service discipline using the equilibrium equations, also called the basic equations. We use this conservation law to prove an extremal property of the first-come firstserved (FCFS) service discipline: among all service disciplines that are work-conserving and independent of remaining service requirements for individual customers, the FCFS service discipline minimizes [maximizes] the mean sojourn time in aG/G/1 queue with independent (but not necessarily identical) service times with a common mean and new better [worse] than used (NBUE[NWUE]) distributions. This extends recent results of Halfin and Whitt (1990), Righter et al. (1990) and Yamazaki and Sakasegawa (1987a,b). In addition we use the conservation law to obtain an approximation for the mean queue length in aGI/GI/1 queue under the processor-sharing service discipline with finite degree of multiplicity, called LiPS discipline. Several numerical examples are presented which support the practical usefulness of the proposed approximation.
Queues with negative arrivals
Authors: Gelenbe Erol , Glynn Peter , Sigman Karl
Published: July 2016

We study single-server queueing models where in addition to regular arriving customers, there are negative arrivals. A negative arrival has the effect of removing a customer from the queue. The way in which this removal is specified gives rise to several different models. Unlike the standard FIFO GI/GI/1 model, the stability conditions for these new models may depend upon more than just the arrival and service rates; the entire distributions of interarrival and service times may be involved.
A result on the infinitely many neutral alleles diffusion model
Authors: Schmuland B.
Published: July 2016

The stationary version of the infinitely many neutral alleles diffusion model, as studied by Ethier and Kurtz, is constructed and analyzed using the theory of Dirichlet forms. We prove that the ‘boundary' of the state space is hit by the process iffθ< 1, whereθis the mutation parameter.
Optimal maintenance strategies for repairable systems with general degree of repair
Authors: Stadje Wolfgang , Zuckerman Dror
Published: July 2016

In this study we examine repairable systems with random lifetime. Upon failure, a maintenance action, specifying the degree of repair, is taken by a controller. The objective is to determine an age-dependent maintenance strategy which minimizes the total expected discounted cost over an infinite planning horizon. Using several properties of the optimal policy which are derived in this study, we propose analytical and numerical methods for determining the optimal maintenance strategy. In order to obtain a better insight regarding the structure and nature of the optimal policy and to illustrate computational procedures, a numerical example is analysed. The proposed maintenance model outlines a new research channel in the area of reliability with interesting theoretical issues and a wide range of potential applications in various fields such as product design, inventory systems for spare parts, and management of maintenance crews.
Sojourn times in vacation and polling systems with Bernoulli feedback
Authors: Takine T. , Takagi H. , Hasegawa T.
Published: July 2016

We study sojourn times in M/G/1 multiple vacation systems and multiqueue cyclic-service (polling) systems with instantaneous Bernoulli feedback. Three service disciplines, exhaustive, gated, and 1-limited, are considered for both M/G/1 vacation and polling systems. The Laplace-Stieltjes transforms of the sojourn time distributions in the three vacation systems are derived. For polling systems, we provide explicit expressions for the mean sojourn times in symmetric cases. Furthermore a pseudo-conservation law with respect to the mean sojourn times is derived for a polling system with a mixture of the three service disciplines.
A martingale approach to the slow server problem
Authors: Stockbridge Richard H.
Published: July 2016

A Markov queueing system having heterogeneous servers under a long-run average criterion is analyzed. A direct proof of the optimality of a stationary, Markov policy is given using martingale methods. Simultaneously, the problem is reduced to a linear programming problem. Analysis of the LP for a system having finite queueing length shows the optimal policy is not always of threshold type.
Hot metal to hard disc
Authors: Lyle Kathleen M.
Published: July 2016

The effects of computer technology on various aspects of mathematical printing are discussed. A brief account is given of the use of a microcomputer and commercial software in a publishing office environment.
Stochastic models in cell kinetics
Authors: Brockwell Peter J.
Published: July 2016

We discuss the role of stochastic processes in modelling the life-cycle of a biological cell and the growth of cell populations. Results for multiphase age-dependent branching processes have proved invaluable for the interpretation of many of the basic experimental studies of the life-cycle. Moreover problems from cell kinetics, in particular those related to diurnal rhythm in cell-growth, have stimulated research into ‘periodic' renewal theory, and the asymptotic behaviour of populations of cells with periodic death rate.
Estimation of inbreeding from population samples
Authors: Smith Cedric A. B. , Thomson Robert
Published: July 2016

Some simple methods of estimating inbreeding from tests on a random sample of individuals are discussed. These are mostly iterative procedures, converging to the maximum likelihood estimate, with algorithms for finding standard errors. But there are problems of interpretation, both from the statistical and biological points of view.
Some discrete distributions associated with orthogonal polynomials
Authors: Watson Geoffrey S.
Published: July 2016

Five distributions on the non-negative integers are given which use the orthogonal polynomials of Hermite, Legendre, Laguerre and Chebyshev.
Theory of semiregenerative phenomena
Authors: Prabhu N. U.
Published: July 2016

We develop a theory of semiregenerative phenomena. These may be viewed as a family of linked regenerative phenomena, for which Kingman [6], [7] developed a theory within the framework of quasi-Markov chains. We use a different approach and explore the correspondence between semiregenerative sets and the range of a Markov subordinator with a unit drift (or a Markov renewal process in the discrete-time case). We use techniques based on results from Markov renewal theory.
The width of a spectral window
Authors: Hannan E. J. , Rissanen J.
Published: July 2016

A method is derived for choosing the width of a spectral window in spectrum estimation. The method is based upon the idea of minimising complexity, i.e. the length, in bits, of a code for the data.
On entrance times of a homogeneous N-dimensional random walk: an identity
Authors: Cohen J. W.
Published: July 2016

Present developments in computer performance evaluation require detailed analysis of N-dimensional random walks on the set of lattice points in the first 2
N
-ant of Recent research has shown that for the two-dimensional case the inherent mathematical problem can often be formulated as a boundary value problem of the Riemann–Hilbert type. The paper is concerned with a derivation and analysis of an identity for the first entrance times distributions into the boundary of such random walks. The identity formulates a relation between these distributions and the zero-tuples of the kernel of the random walk; the kernel contains all the information concerning the structure of the random walk in the interior of its stage space. For the two-dimensional case the identity is resolved and explicit expressions for the entrance times distributions are obtained.
A correlated random walk for the transport and sedimentation of particles
Authors: Gani J.
Published: July 2016

This paper considers a bivariate random walk modelon a rectangular lattice for a particle injected into a fluid flowing in a tank. The numbers of jumps of the particle in thexandydirections in this particular model are correlated. It is shown that when the random walk forms a bivariate Markov chain in continuous time, it is possible to obtain the state probabilitiespxy(t) through their Laplace transforms. Two exit rules are considered and results for both of them derived.
Subexponential distributions and integrated tails
Authors: Klüppelberg Claudia
Published: July 2016

Let F be a distribution function on [0,∞) with finite expectation. In terms of the hazard rate of F several conditions are given which simultaneously imply subexponentiality of F and of its integrated tail distribution F
1. These conditions apply to a wide class of longtailed distributions, and they can also be used in connection with certain random walks which occur in risk theory and queueing theory.
The steady-state distribution of spent service times present in the M/G/1 foreground–background processor-sharing queue
Authors: Schassberger R.
Published: July 2016

The steady-state distribution of spent service times present in the M/G/1 foreground-background processor-sharing queue is described by a random measure, whose generating functional is obtained.
Random coverings of the circle by arcs with restricted endpoints
Authors: Janson Svante
Published: July 2016

A circle is covered by random arcs with a given length a and endpoints chosen (independently and uniformly) among m equispaced points on the circle. The asymptotic distribution as a → 0 and m → ∞of the number of arcs required for complete coverage is given. The result connects earlier results for the cases ma = 1 (a discrete problem) and m = ∞ (the continuous limiting case).
A note on cumulative shock models
Authors: Anderson Kevin K.
Published: July 2016

A shock model in which the time intervals between shocks are in the domain of attraction of a stable law of order less than 1 or relatively stable is considered. Weak limit theorems are established for the cumulative magnitude of the shocks and the first time the cumulative magnitude exceeds z without any assumption on the dependence between the intershock interval and shock magnitude.
A convexity result for single-server exponential loss systems with non-stationary arrivals
Authors: Svoronos Antony , Green Linda
Published: July 2016

We consider single-server loss systems with exponential service times and non-stationary Poisson input. We prove that if the arrival rate is given by a periodic function, the proportion of lost customers is convex increasing in the amplitude.
A note on the equilibrium M/G/1 queue length
Authors: Willmot Gordon E.
Published: July 2016

This note concerns the distribution of the equilibrium M/G/1 queue length. A representation for the probability generating function is given which allows for an explicit finite sum representation of the associated probabilities. The radius of convergence of the probability generating function and an asymptotic formula for the right tail of the distribution also follow from this representation, as well as infinite divisibility of the queue-length distribution when the service distribution is infinitely divisible. Extension of these results to the bulk arrival case is straightforward.
Some multivariate distributions derived from a non-fatal shock model
Authors: Savits Thomas H.
Published: July 2016

A non-homogeneous Poisson shock model has a continuous mean function Λ(t). The kth shock Sk
causes simultaneous failure of the components j ∊ J ∊ {1, ···, n} with probability pJ
(Sk
). If Tj
is the lifetime of component j, it is shown that (T
1, · ··, Tn
) belongs to various multivariate non-parametric life classes depending on the life class of .
On higher moments of the population size in a subcritical branching process
Authors: Willekens Eric
Published: July 2016

Let {Z(t), t ≧ 0} be an age-dependent subcritical branching process. In this paper we show that if the lifetime distribution is subexponential, EZα
(t) ~ EZ(t) (t →∞) for every α ≧ 1. If furthermore the lifetime distribution has a subexponential density, a rate of convergence result in the above relation is established.
Optimality of myopic stopping times for geometric discounting
Authors: Fristedt Bert , Berry Donald A.
Published: July 2016

Consider a sequence of conditionally independent Bernoulli random variables taking on the values 1 and − 1. The objective is to stop the sequence in order to maximize the discounted sum. Suppose the Bernoulli parameter has a beta distribution with integral parameters. It is optimal to stop when the conditional expectation of the next random variable is negative provided the discount factor is less than or equal to . Moreover, is best possible. The case where the parameters of the beta distribution are arbitrary positive numbers is also treated.
Honest bernoulli excursions
Authors: Smith Laurel , Diaconis Persi
Published: July 2016

For simple random walk on the integers, consider the chance that the walk has traveled distance k from its start given that its first return is at time 2n. We derive a limiting approximation accurate to order 1/n. We give a combinatorial explanation for a functional equation satisfied by the limit and show this yields the functional equation of Riemann's zeta function.
Maximal, non-uniform spacings and the coverage problem
Authors: Hüsler Jürg
Published: July 2016

The spacings have been widely applied to many situations, such as for example the coverage problem of the circle or the line, under the hypothesis of uniformity. The paper examines the asymptotic behaviour of the maximal spacings under non-uniformity, assuming that the existing density f has a minimum point m with f(m) = 0. The results are applied to the coverage of the circle by randomly placed arcs.
On berry–esseen rate for queue length of the GI/G/K system in heavy traffic
Authors: Jin Xing
Published: July 2016

This paper provides Berry–Esseen rate of limit theorem concerning the number of customers in a GI/G/K queueing system observed at arrival epochs for traffic intensity ρ > 1. The main method employed involves establishing several equalities about waiting time and queue length.
Extinction times for certain predator–prey processes
Authors: Ridler-Rowe C. J.
Published: July 2016

Finiteness of mean extinction times for certain predator-prey models has been established by Hitchcock (1986) with the aid of a criterion of Reuter (1957). Using this criterion and a ‘minimisation' lemma this note shows that the mean extinction times tend to zero as the combined initial population of predators and prey becomes large.
Chords through a convex body generated from within an embedded body
Authors: Enns E. G. , Ehlers P. F.
Published: July 2016

A body E is completely embedded within a convex body G. A line segment is generated by a measure depending only on E or on both E and G. This line segment is then projected to the surface of G in one or both directions. The distributions of selected line segments of this type are derived.
Inverting the statistics of the circle transform
Authors: Waksman Peter
Published: July 2016

For a plane domain the circle transform assigns to each circle its length of intersection with the domain. The problem is to determine the geometry of the domain given the mean of these intersection lengths as the circles' centers vary and given how that mean varies with circle radius. We also consider the mean square and higher moments of the intersection lengths — as a function of the circle radius. The discussion includes an identification of the geometric content of the mean when centers are inside of a disk and of the second moment when the centers are arbitrary points in the plane. The latter is equivalent to the distribution of distance between pairs of points of the domain.