Showing papers in "Advances in Applied Probability in 1984"
TL;DR: An efficient computational approach to the analysis of finite birth-and-death models in a Markovian environment is given in this paper, where the emphasis is on obtaining numerical methods for evaluating stationary distributions and moments of first-passage times.
Abstract: An efficient computational approach to the analysis of finite birth-and-death models in a Markovian environment is given. The emphasis is upon obtaining numerical methods for evaluating stationary distributions and moments of first-passage times.
270 citations
TL;DR: In this article, the authors considered a single-server queueing system in which a customer gives up whenever his waiting time is larger than a random threshold, his patience time, and established the extensions of the classical GI/GI/1 formulae concerning the stability condition and the relation between actual and virtual waiting time distribution functions.
Abstract: We consider a single-server queueing system in which a customer gives up whenever his waiting time is larger than a random threshold, his patience time. In the case of a GI/GI/1 queue with i.i.d. patience times, we establish the extensions of the classical GI/GI/1 formulae concerning the stability condition and the relation between actual and virtual waiting-time distribution functions. We also prove that these last two distribution functions coincide in the case of a Poisson input process and determine their common law.
217 citations
TL;DR: This paper discusses the asymptotic properties of the algorithm that depend on uniform rates of convergence being established for covariances up to some lag increasing indefinitely with the length of record, T.
Abstract: This paper is in three parts. The first deals with the algebraic and topological structure of spaces of rational transfer function linear systems—ARMAX systems, as they have been called. This structure theory is dominated by the concept of a space of systems of order, or McMillan degree, n, because of the fact that this space, M(n), can be realised as a kind of high-dimensional algebraic surface of dimension n(2s + m) where s and m are the numbers of outputs and inputs. In principle, therefore, the fitting of a rational transfer model to data can be considered as the problem of determining n and then the appropriate element of M(n). However, the fact that M(n) appears to need a large number of coordinate neighbourhoods to cover it complicates the task. The problems associated with this program, as well as theory necessary for the analysis of algorithms to carry out aspects of the program, are also discussed in this first part of the paper, Sections 1 and 2. The second part, Sections 3 and 4, deals with algorithms to carry out the fitting of a model and exhibits these algorithms through simulations and the analysis of real data. The third part of the paper discusses the asymptotic properties of the algorithm. These properties depend on uniform rates of convergence being established for covariances up to some lag increasing indefinitely with the length of record, T. The necessary limit theorems and the analysis of the algorithms are given in Section 5. Many of these results are of interest independent of the algorithms being studied.
175 citations
TL;DR: In this article, the authors present a survey of the exponential growth of general branching populations, in the supercritical case, and the asymptotic stability of composition according to very general ways of counting.
Abstract: A single-type general branching population develops by individuals reproducing according to i.i.d. point processes on R +, interpreted as the individuals' ages. Such a population can be measured or counted in many different ways: those born, those alive or in some sub-phase of life, for example. Special choices of reproduction point process and counting yield the classical Galton–Watson or Bellman–Harris process. This reasonably self-contained survey paper discusses the exponential growth of such populations, in the supercritical case, and the asymptotic stability of composition according to very general ways of counting. The outcome is a strict definition of a stable population in exponential growth, as a probability space, a margin of which is the well-known stable age distribution.
174 citations
TL;DR: A wide variety of results pertinent to (diffusion approximations of) the classical multi-allele single-locus WrightFisher model and its relatives are unified by this approach as mentioned in this paper.
Abstract: This paper reviews a variety of results for genealogical (or line-of-descent) processes that arise in connection with the theory of some classical selectively neutral haploid population genetics models. While some new results and derivations are included, the principal aim of the paper is to demonstrate the central importance and simplicity of genealogical Markov chains in this theory. Considerable attention is given to ‘diffusion time scale’ approximations of such genealogical processes. A wide variety of results pertinent to (diffusion approximations of) the classical multi-allele single-locus WrightFisher model and its relatives are unified by this approach. Other examples where the genealogical process plays an explicit role (for example, the infinite-sites models) are discussed.
125 citations
TL;DR: For the haploid genetic model of Moran, the joint distribution of the numbers of distinct ancestors of a collection of nested subsamples is derived and these results are shown to apply to the diffusion approximations of a wide variety of other genetic models, including the Wright–Fisher process.
Abstract: For the haploid genetic model of Moran, the joint distribution of the numbers of distinct ancestors of a collection of nested subsamples is derived. These results are shown to apply to the diffusion approximations of a wide variety of other genetic models, including the Wright-Fisher process. The results allow us to relate the ancestries of populations sampled at different times. Analogous results for a line-of-descent process that incorporates the effect of mutation are given. Some results about the ages of alleles in an infinite-alleles model are described.
110 citations
TL;DR: Many queues and related stochastic models, and in particular those that have a matrix-geometric stationary probability vector, have steady-state queue-length densities that are asymptotically geometric.
Abstract: Many queues and related stochastic models, and in particular those that have a matrix-geometric stationary probability vector, have steady-state queue-length densities that are asymptotically geometric. The graph of the asymptotic rate η of these densities as a function of the traffic intensity ρ is the caudal characteristic curve. This is an informative graph from which a number of qualitative inferences about the behavior of the queue may be drawn. The caudal characteristic curve may be computed (by elementary algorithms) for several useful models for which a complete exact numerical solution is not practically feasible. These include queues with certain types of superimposed arrival processes and/or multiple non-exponential servers. The necessary theorems which lead to the algorithmic procedures as well as the interpretation of several numerical examples are discussed.
94 citations
TL;DR: In this article, the main index of interest in reliability is the time to the first system failure, and the assumption that element failure rates are low allows to obtain an expression for the main term in the asymptotic representation of system reliability function.
Abstract: : Section 1 of this paper reviews some works related to reliability evaluation of nonrenewable systems. The assumption that element failure rates are low allows to obtain an expression for the main term in the asymptotic representation of system reliability function. Section 2 is devoted to renewable systems. The main index of interest in reliability is the time to the first system failure. A typical situation in reliability is that the repair time is much smaller than the element lifetime. This fast repair property leads to an interesting phenomenon that for many renewable systems the time to system failure converges in probability, under appropriate norming, to exponential distribution. Some basic theorems explaining this fact are presented and a series of typical examples is considered. Special attention is paid to reviewing the works describing the exponentiality phenomenon in the birth-and-death processes. Some important aspects of computing the normalizing constants are considered, among them, the role played by so-called main event. Section 2 contains also a review on various bounds on the deviation from exponentiality. Sections 3 , 4 describe some additional aspects of asymptotics in reliability. It is typical for the probabilistic models considered in these sections, that a small parameter is introduced in an explicit form into the characteristic of the random processes. A considerable part of this review is based on the sources which were originally published in Russian and are available in the English translation. (Author)
94 citations
TL;DR: In this paper, a stochastic model for the development in time of a population {Z n } where the law of offspring distribution depends on the population size was considered, and it was shown that if m 1 not slower than n-α, α, α > 0, and do not grow to ∞ faster than ns, β < 1 then Zn /mn converges almost surely and in L 2 to a non-degenerate limit.
Abstract: We consider a stochastic model for the development in time of a population {Z n } where the law of offspring distribution depends on the population size. We are mainly concerned with the case when the mean mk and the variance of offspring distribution stabilize as the population size k grows to ∞, The process exhibits different asymptotic behaviour according to m l; moreover, the rate of convergence of mk to m plays an important role. It is shown that if m 1 not slower than n– α, α > 0, and do not grow to ∞ faster than ns , β <1 then Zn /mn converges almost surely and in L 2 to a non-degenerate limit. A number of general results concerning the behaviour of sums of independent random variables are also given.
89 citations
TL;DR: In this paper, the M/G/1 processor-sharing queue is studied by way of an approximating sequence of models featuring a round-robin discipline and operating in discrete time.
Abstract: The M/G/1 processor-sharing queue is studied by way of an approximating sequence of models featuring a round-robin discipline and operating in discrete time. In particular, residence-time distributions of jobs are derived.
86 citations
TL;DR: Perturbation bounds for the stationary distribution of a finite indecomposable Markov chain are discussed and new bounds are presented which more completely exploit the stochastic features of the perturbation and which also are easily computable.
Abstract: This paper discusses perturbation bounds for the stationary distribution of a finite indecomposable Markov chain. Existing bounds are reviewed. New bounds are presented which more completely exploit the stochastic features of the perturbation and which also are easily computable. Examples illustrate the tightness of the bounds and their application to bounding the error in the Simon-Ando aggregation technique for approximating the stationary distribution of a nearly completely decomposable Markov chain. NEARLY COMPLETELY DECOMPOSABLE SYSTEMS; STOCHASTIC MATRICES
TL;DR: In this article, densities of additive functionals φ of stationary random sets X are defined and formulas for these densities are derived when K is a compact convex set in.
Abstract: For certain stationary random sets X, densities D φ (X) of additive functionals φ are defined and formulas for are derived when K is a compact convex set in . In particular, for the quermassintegrals and motioninvariant X, these formulas are in analogy with classical integral geometric formulas. The case where X is the union set of a Poisson process Y of convex particles is considered separately. Here, formulas involving the intensity measure of Y are obtained.
TL;DR: In this paper, the authors studied the distribution properties of the system failure time in general shock models associated with correlated renewal sequences (X n, Y n ), depending on whether the magnitude of the nth shock X n is correlated to the length Y n of the interval since the last shock, or the length of the subsequent interval to the next shock.
Abstract: In this paper we study some distribution properties of the system failure time in general shock models associated with correlated renewal sequences ( X n , Y n ) . Two models, depending on whether the magnitude of the nth shock X n is correlated to the length Y n of the interval since the last shock, or to the length of the subsequent interval to the next shock, are considered. Sufficient conditions under which the system failure time is completely monotone, new better than used, new better than used in expectation, and harmonic new better than used in expectation are given for these two models.
TL;DR: In this paper, the authors consider an inventory system to which arrival of items stored is a renewal process and the demand is a Poisson process, and compute ergodic limits for the lost demands and the lost items processes.
Abstract: We consider an inventory system to which arrival of items stored is a renewal process, and the demand is a Poisson process. Items stored have finite and fixed lifetimes. The blood-bank model inspired this study. Three models are studied. In the first one, we assume that each demand is for one unit and unsatisfied demands leave the system immediately. Using results on this model one is able to study a model in which arrival of items is Poisson but demands are for several units, and a model in which demands are willing to wait. We compute ergodic limits for the lost demands and the lost items processes and the limiting distribution of the number of items stored. The main tool in this analysis is an analogy to M/G/1 queueing systems with impatient customers.
TL;DR: For M/GI/1/∞ queues with instantaneous Bernoulli feedback time and customer-stationary characteristics of the number of customers in the system and of the waiting time are investigated in this paper.
Abstract: For M/GI/1/∞ queues with instantaneous Bernoulli feedback time- and customer-stationary characteristics of the number of customers in the system and of the waiting time are investigated. Customer-stationary characteristics are thereby obtained describing the behaviour of the queueing processes, for example, at arrival epochs, at feedback epochs, and at times at which an arbitrary (arriving or fed-back) customer enters the waiting room. The method used to obtain these characteristics consists of simple relationships between them and the time-stationary distribution of the number of customers in the system at an arbitrary point in time. The latter is obtained from the wellknown Pollaczek–Khinchine formula for M/GI/1/∞ queues without feedback.
TL;DR: A local limit theorem for P{Ta = n, S7 - a 5 x} is obtained in this article, where Ta is the first time a random walk S with positive drift exceeds a.
Abstract: A local limit theorem for P{Ta = n, S7 - a 5 x} is obtained, where Ta is the first time a random walk S, with positive drift exceeds a. Applications to large-deviation probabilities and to the crossing of a non-linear boundary are given.
TL;DR: In this paper, the authors consider a production-inventory problem in which the production rate can be continuously controlled in order to cope with random fluctuations in the demand and derive practically useful approximations for the switchover levels of the control rule such that a pre-specified value of the service level is achieved.
Abstract: We consider a production-inventory problem in which the production rate can be continuously controlled in order to cope with random fluctuations in the demand. The demand process for a single product is a compound Poisson process. Excess demand is backlogged. Two production rates are available and the inventory level is continuously controlled by a switch-over rule characterized by two critical numbers. In accordance with common practice, we consider service measures such as the average number of stockouts per unit time and the fraction of demand to be met directly from stock on hand. The purpose of the paper is to derive practically useful approximations for the switch-over levels of the control rule such that a pre-specified value of the service level is achieved.
TL;DR: In this article, a group of n independent identical machines with exponential lifetimes is considered and the optimal repair policy is of the following type: repair when the number of failed machines reaches some prescribed number.
Abstract: A group of n independent identical machines with exponential lifetimes is considered. Repair is made simultaneously for all failed machines. It is possible to observe the ‘state parameter', i.e. the number of operating machines. It is proved that for two types of critieria (minimal cost per time unit and maximal return per time unit) the optimal repair policy is of the following type: repair when the number of failed machines reaches some prescribed number .
TL;DR: In this article, a simple and efficient method of obtaining sufficient conditions for the existence of a stationary regime for multichannel fully available queueing systems with repeated calls was proposed, where the stationary regime is defined in terms of the number of repeated calls.
Abstract: We propose a simple and efficient method of obtaining sufficient conditions for the existence of a stationary regime for multichannel fully available queueing systems with repeated calls.
TL;DR: Shaked as discussed by the authors gave a characterisation of dispersive ordering for distributions which are absolutely continuous with interval supports, but there is a technical error in his proof of the second part of Theorem 2.1, although the result is still true.
Abstract: Shaked (1982) gives a characterisation of dispersive ordering for distributions which are absolutely continuous with interval supports. However there is a technical error in his proof of the second part of Theorem 2.1, although the result is still true. The difficulty arises because zero terms are discarded in the definition of the sign change operator S . Therefore the equality of and x′ does not imply that
TL;DR: In this paper, a general class of Markovian network queueing models is presented, where each node has a heterogeneous class of customers arriving at their own Poisson rate, ultimately to receive their own exponential service requirements.
Abstract: We develop the mathematical machinery in this paper to construct a very general class of Markovian network queueing models. Each node has a heterogeneous class of customers arriving at their own Poisson rate, ultimately to receive their own exponential service requirements. We add to this a very general type of service discipline as well as class (node) switching. These modifications allow us to model in the limit, service with a general distribution. As special cases for this model, we have the product-form networks formulated by Kelly, as well as networks with priority scheduling. For the former, we give an algebraic proof of Kelly's results for product-form networks. This is an approach that motivates the form of the solution, and justifies the various needs of local and partial balance conditions. For any network that belongs to this general model, we use the operator representation to prove stochastic dominance results. In this way, we can take the transient behavior for very complicated networks and bound its joint queue-length distribution by that for M / M /1queues.
TL;DR: In this paper, the authors give a description of the asymptotic behaviour (CLT and ILL) of the random walks associated with Gegenbauer's polynomials.
Abstract: Random walks on N associated with orthogonal polynomials have properties similar to classical random walks on . In fact such processes have independent increments with respect to a hypergroup structure on with a convolution and a Fourier transform which is the basic tool for their study. We illustrate these ideas by giving a description of the asymptotic behaviour (CLT and ILL) of the random walks associated with Gegenbauer's polynomials. Moreover we can then use these random walks as a reference scale to deduce asymptotic properties of other Markov chains on via a comparison theorem which is of independent interest.
TL;DR: In this article, it was shown that as n tends to ∞, the distribution functions of the normalized maxima of the processes {X n(λ), (Y n (λ)}, (I n(ϵ)}, {I n (ϵ)) over the interval λ∈ [0,π] each converge to the extremal distribution function exp [−e-x ], where ∞ < x <∞.
Abstract: Let {e t, t = 1, 2, ···, n} be a sequence of mutually independent standard normal random variables. Let X n(λ) and Y n(λ) be respectively the real and imaginary parts of exp iλ t, and let . It is shown that as n tends to∞, the distribution functions of the normalized maxima of the processes {X n(λ)}, (Y n(λ)}, {I n(λ)} over the interval λ∈ [0,π] each converge to the extremal distribution function exp [–e–x ], —∞ < x <∞. It is also shown that these results can be extended to the case where {e t} is a stationary Gaussian sequence with a moving-average representation.
TL;DR: In this article, it was shown that in a closed cycle of exponential queues where the first and last nodes are multiserver queues while the other nodes are single-server queues, the cycletime distribution has a simple product form.
Abstract: In a closed cycle of exponential queues where the first and the last nodes are multiserver queues while the other nodes are single-server queues, the cycletime distribution has a simple product form. The same result holds for passage-time distributions on overtake-free paths in Gordon-Newell networks. In brief, we prove Burke's theorem on passage times in closed networks. From a customer's point of view possibly the most important performance measure in distributed systems (local area networks, telecommunication systems, computer centers, etc.) may be the passage time, i.e. the time between entering the system (or a particular part of it) and the moment of departure. But the problem of determining passage-time distributions in networks of queues (which are the natural models for distributed systems) resists solution insofar as properties beyond expected values are requested. (The latter are easily obtained by Little's theorem when steady-state distributions are known.) Clearly, passage-time expectations are important in projecting a distributed system, but a typical request of an employer is that 'at most five percent of all executed jobs may need more than a given value of execution time in travelling through the system': this is a question about the distribution of passage times in general networks. But for a long time the famous theorems of Reich (1957), (1963) and Burke (1964), (1968), concerning open tandems of multiserver queues, were the only results available (for a review see Burke (1972)). Their results on passage times can be summarized as follows. In a sequence of exponential systems, the first and last of which are multiserver queues, while all other systems are single-server queues, with jobs arriving in a Poisson stream, the sojourn times at different nodes are independent. So the passage-time distribution can be obtained by simple convolutions.
TL;DR: In this article, the Gaussian likelihood function for multivariable ARMA models is studied and its behavior at the boundary of the parameter space is described; its continuity properties as well as the question of the existence of a maximum are discussed.
Abstract: The paper deals with some properties of the (Gaussian) likelihood function for multivariable ARMA models. Its behaviour at the boundary of the parameter space is described; its continuity properties as well as the question of the existence of a maximum are discussed. We have not been able to show in general the existence of the maximum over the usual parameter spaces. However, the maximum always exists over a suitably enlarged parameter space (given that the data are non-degenerate), which includes parameters corresponding to processes with discrete spectral components.
TL;DR: In this paper, a state-space model is presented for a queueing system where two classes of customer compete in discrete-time for the service attention of a single server with infinite buffer capacity.
Abstract: A state-space model is presented for a queueing system where two classes of customer compete in discrete-time for the service attention of a single server with infinite buffer capacity. The arrivals are modelled by an independent identically distributed random sequence of a general type while the service completions are generated by independent Bernoulli streams; the allocation of service attention is governed by feedback policies which are based on past decisions and buffer content histories. The cost of operation per unit time is a linear function of the queue sizes. Under the model assumptions, a fixed prioritization scheme, known as the μ c -rule, is shown to be optimal when the expected long-run average criterion and the expected discounted criterion, over both finite and infinite horizons, are used. This static prioritization of the two classes of customers is done solely on the basis of service and cost parameters. The analysis is based on the dynamic programming methodology for Markov decision processes and takes advantage of the sample-path properties of the adopted state-space model.
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TL;DR: In this paper, a body receives a sequence of t right-handed rotations, each rotation being through a fixed angle A about an axis of rotation that is completely random in direction.
Abstract: A body receives a sequence of rotations through a fixed angle about an axis whose direction is arbitrary. The probability distribution governing the resulting orientation of the body is determined. The problem is generalized to the case where the axis of each individual rotation makes the same angle, 0, with an axis fixed in the body but is otherwise random. The resulting distribution is shown, in the case 0 = , to reduce to the Roberts-Ursell distribution for random walk on a sphere. Some diffusion limits are examined. RANDOM WALK; QUATERNIONS; HYPERSPHERICAL HARMONICS; DIFFUSION DISTRIBUTION 1. Statement of problem Suppose that a probability distribution is given for the initial orientation of a rigid body. Suppose that body receives a sequence of t right-handed rotations, each rotation being through a fixed angle A about an axis of rotation that is completely random in direction. We shall seek the probability distribution, Pt, for the orientation of the body after these t steps. The solution to the problem as just defined also provides Pt when the individual steps are rotations of 2ir A rather than A, for the orientation after a rotation A about one axis is the same as for 27r A about the opposite axis. It is, however, possible to obtain a solution, P,, which distinguishes between these two possibilities. In more technical language, we shall seek a solution in the double-covering space, Spin (3), of SO(3) rather than in SO (3) itself, so obtaining a solution, p,, that depends on the homotopy class in SO (3) of the path followed by the rotation in its random walk. In more mundane terms, after five random rotations of 1' there is a finite probability that the final rotational state of the body could have been reached from the initial configuration by a single rotation, A, about some axis, with 2? -A _3'. That final Received 1 August 1983. * Postal address: Department of Applied Mathematics, The University of Newcastle upon Tyne, NE1 7RU, U.K. ** Postal address: Department of Applied Mathematics, The University of Sydney, NSW 2006, Australia.
TL;DR: In this paper, upper and lower bounds for the distribution of max s v(x, s) and max x v[x, t] are presented by adapting Levy-type inequalities and exploiting a connection of v[, t) with the Ornstein-Uhlenbeck process through Slepian's theorem.
Abstract: Let v(x, t) denote the displacement of an infinitely long, idealized string performing damped vibrations caused by white noise. Upper and lower bounds for the distribution of max s v(x, s) and max x v(x, t) are presented. The results are obtained by adapting Levy-type inequalities and exploiting a connection of v(x, t) with the Ornstein-Uhlenbeck process through Slepian's theorem. The case of forced-damped vibrations is also analysed. Finally, a section is devoted to the case of a semi-infinite string performing damped vibrations.
TL;DR: In this article, sufficient conditions under which (Z, R) is a new better than used (NBU) process were presented and several examples of NBu processes satisfying these conditions were given.
Abstract: Let Z {Z(t), t R+} with Z(O)= 0 be a random process under investigation and N be a point process associated with Z. Both Z and N are defined on the same probability space. Let R -A{R,, n = 0, 1, 2, } with Ro = 0 denote the consecutive positions of points of N on the half-line R+. In this paper we present sufficient conditions under which (Z, R) is a new better than used (NBU) process and give several examples of NBu processes satisfying these conditions. In particular we consider the processes in which N is a renewal and a general point process. The NBU property of some semi-Markov processes is also presented. QUEUES; DAMS; INVENTORY MODELS; SHOCK MODELS; SEMI-MARKOV PROCESSES
TL;DR: In this article, the authors consider the case where a traveler starts at a given city and in each flight, independently of the past journey, chooses a city at random as the destination.
Abstract: Let K be a finite graph with vertex set V= {x0, x,, ., x_1} and automorphism group G. It is assumed that G acts transitively on V. We can imagine that the vertices of K represent or cities and a traveler visits the cities in a series of random flights. The traveler starts at a given city and in each flight, independently of the past journey, chooses a city at random as the destination. Denote by v, (n = 1, 2, - - -) the location of the traveler at the end of the nth flight, and by vo the initial location. It is assumed that the transition probabilities P{v, = xi n-= xi}, xi V, xi E V, do not depend on n and are invariant under the action of G on V. The main result of this paper consists in determining p(n), the probability that the traveler returns to the initial