Showing papers in "Journal of Applied Probability in 1993"
TL;DR: A new, simple algorithm for the matrix-geometric rate matrix has quadratic convergence and is shown theoretically and through numerical examples that it converges very fast and provides extremely accurate results even for almost unstable models.
Abstract: Quasi-birth-death processes are commonly used Markov chain models in queueing theory, computer performance, teletraffic modeling and other areas. We provide a new, simple algorithm for the matrix-geometric rate matrix. We demonstrate that it has quadratic convergence. We show theoretically and through numerical examples that it converges very fast and provides extremely accurate results even for almost unstable models.
377 citations
TL;DR: In this paper, an estimator of σ based on discrete observation of the diffusion X throughout a given finite time interval is proposed and the asymptotic behavior of this estimator when the step of discretization tends to zero.
Abstract: This paper is concerned with the problem of estimation for the diffusion coefficient of a diffusion process on R, in a non-parametric situation. The drift function can be unknown and considered as a nuisance parameter. We propose an estimator of σ based on discrete observation of the diffusion X throughout a given finite time interval. We describe the asymptotic behaviour of this estimator when the step of discretization tends to zero. We prove consistency and asymptotic normality, the rate of convergence to the normal law being a random variable linked to the local time of the diffusion or to its suitable discrete approximation. This can also be interpreted as a convergence to a mixture of normal law.
283 citations
TL;DR: The generalized queueing networks (G-networks) introduced in this article contain customers and signals, and 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.
Abstract: 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.
256 citations
TL;DR: In this article, it is shown that to a given marked point process there is a sequence of such Markovian arrival streams with the property that as m → ∞, the state transitions of a given Markovians environmental process and Poisson arrival rates depend on the environment.
Abstract: A Markovian arrival stream is a marked point process generated by the state transitions of a given Markovian environmental process and Poisson arrival rates depending on the environment. It is shown that to a given marked point process there is a sequence of such Markovian arrival streams with the property that as m →∞. Various related corollaries (involving stationarity, convergence of moments and ergodicity) and counterexamples are discussed as well.
236 citations
TL;DR: In this article, the authors derived expressions for the Laplace transform of the sojourn time density in a single-server queue with exponential service times and independent Poisson arrival streams of both ordinary, positive customers and negative customers which eliminate a positive customer if present.
Abstract: We derive expressions for the Laplace transform of the sojourn time density in a single-server queue with exponential service times and independent Poisson arrival streams of both ordinary, positive customers and negative customers which eliminate a positive customer if present. We compare first-come first-served and last-come first-served queueing disciplines for the positive customers, combined with elimination of the last customer in the queue or the customer in service by a negative customer. We also derive the corresponding result for processor-sharing discipline with random elimination. The results show differences not only in the Laplace transforms but also in the means of the distributions, in contrast to the case where there are no negative customers. The various combinations of queueing discipline and elimination strategy are ranked with respect to these mean values.
73 citations
TL;DR: In this article, a simple probabilistic proof of necessary and sufficient conditions for strong lumpability for continuous-time Markov chains is provided. But this proof is only valid in circumstances not covered by known theory.
Abstract: 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.
64 citations
TL;DR: In this article, the authors generalize the previous work 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.
Abstract: 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.
59 citations
TL;DR: In this paper, a simple characterization of the Linnik distribution is used to construct discrete-time processes with a stationary LINI distribution, which are related to exponential processes introduced by Arnold (1989), Lawrance and Lewis (1981), and Gaver and Lewis(1980).
Abstract: Using a simple characterization of the Linnik distribution, discrete-time processes having a stationary Linnik distribution are constructed. The processes are structurally related to exponential processes introduced by Arnold (1989), Lawrance and Lewis (1981) and Gaver and Lewis (1980). Multivariate versions of the processes are also described. These Linnik models appear to be viable alternatives to stable processes as models for temporal changes in stock prices.
59 citations
TL;DR: In this article, it was shown that if one mixture distribution dominates another in a strong sense, the resulting mixture of the dominant distribution will have larger optimal burn-in time than the other mixture distributions.
Abstract: Burn-in is a procedure used for eliminating weak components in a mixed population. In this paper we focus on general mixed populations. Three types of results are established. First, it is shown that any mixed population displays a type of monotonicity property which is appropriate for burn-in. Second, it is shown that if, asymptotically, components have constant failure rates, then the mixed population will also asymptotically have a constant failure rate and this will correspond to the rate of the strongest subpopulation of the mixture. Finally, it is shown for a reasonable cost function, that if one mixture distribution dominates another in a strong sense, the resulting mixture of the dominant distribution will have larger optimal burn-in time.
59 citations
TL;DR: Comparison between various types of policies is made in order to determine when and under which condition one type of policy is better than another.
Abstract: In this paper we introduce the concept of repair replacement. Repair replacement is a maintenance policy in which items are preventively maintained when a certain time has elapsed since their last repair. This differs from age replacement where a certain amount of time has elapsed since the last replacement. If the last repair was a complete repair, repair replacement is essentially the same as age replacement. It is in the case of minimal repair that these two policies differ. We make comparison between various types of policies in order to determine when and under which condition one type of policy is better than another.
56 citations
TL;DR: In this article, a new family of Hougaard processes, called Hougaard Processes, is introduced, which preserve the monotonicity properties of stochastic processes under subordination to Poisson and stable processes.
Abstract: The paper investigates stochastic processes directed by a randomized time process. A new family of directing processes called Hougaard processes is introduced. Monotonicity properties preserved under subordination, and dependence among processes directed by a common randomized time are studied. Results for processes subordinated to Poisson and stable processes are presented. Potential applications to shock models and threshold models are also discussed. Only Markov processes are considered.
TL;DR: In this article, the first-order autoregressive semi-Mittag-Leffler (SMLAR(1)) process is introduced and its properties are studied, as well as the special case of the firstorder autoresignation semi-mittag leffler process.
Abstract: The first-order autoregressive semi-Mittag-Leffler (SMLAR(1)) process is introduced and its properties are studied. As an illustration, we discuss the special case of the first-order autoregressive Mittag-Leffler (MLAR(I)) process. EAR(1) PROCESS; SEMI-MITTAG-LEFFLER DISTRIBUTION
TL;DR: In this paper, the full-information secretary problem is shown to have a value smaller than 7/3 for all n (the number of options) for a simple memoryless threshold rule, and the asymptotic optimal value for the class of such rules is 2.3266.
Abstract: The full-information secretary problem in which the objective is to minimize the expected rank is seen to have a value smaller than 7/3 for all n (the number of options). This can be achieved by a simple memoryless threshold rule. The asymptotically optimal value for the class of such rules is about 2.3266. For a large finite number of options, the optimal stopping rule depends on the whole sequence of observations and seems to be intractable. This raises the question whether the influence of the history of all observations may asymptotically fade. We have not solved this problem, but we show that the values for finite n are non-decreasing in n and exhibit a sequence of lower bounds that converges to the asymptotic value which is not smaller than 1.908.
TL;DR: In this article, a set of stochastic processes that are useful for modeling and analyzing a new genetic mapping method is described, and the central issue is boundary-crossing probabilities, which correspond to pvalues for the existence of genes for particular traits.
Abstract: This paper describes a set of stochastic processes that is useful for modeling and analyzing a new genetic mapping method. Some of the processes are Markov chains, and some are best described as functions of Markov chains. The central issue is boundary-crossing probabilities, which correspond to p-values for the existence of genes for particular traits. The methods elaborated by Aldous (1989) provide very accurate approximate p-values, as spot-checked against simulations. MARKOV CHAIN; BOUNDARY-CROSSING PROBABILITY; POISSON APPROXIMATION AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 60J27 SECONDARY 92 D10
TL;DR: The skeleton of a critical Galton-Watson process with offspring mean 1 + r, r ≧ 0, and finite offspring variance, is considered in this paper, where the time unit is taken as α /r generations (α > 0).
Abstract: The skeleton of a (super-) critical Galton-Watson process with offspring mean 1 + r, r ≧ 0, and finite offspring variance, is considered. When r = 0 it is trivial. If r > 0 is small and the time unit is taken as α /r generations (α > 0) then the skeleton can be approximated by a Yule (linear pure birth) process of rate α. This approximation can be used to study the evolution of genetic types over a long period of time in an exponentially growing population.
TL;DR: In this article, the authors examined the main properties of the Markov chain X t = T(X t-1 )+σ (X t -1 ) +σ(x t −1 ǫ t ) under general and tractable assumptions, and derived bounds for the tails of the stationary density of the process {X t } in terms of the common density.
Abstract: We examine the main properties of the Markov chain X t = T(X t-1 )+σ(X t-1 )ɛ t . Under general and tractable assumptions, we derive bounds for the tails of the stationary density of the process {X t } in terms of the common density of the ɛ t 's.
TL;DR: In this paper, a new series formula of Sharma and its connection with traditional Bessel function series is established, and an alternative new series is developed which isolates the steady-state component for all values of traffic intensity and which turns out to be computationally superior.
Abstract: Past work relating to the computation of time-dependent state probabilities in M/M/1 queueing systems is reviewed, with emphasis on methods that avoid Bessel functions. A new series formula of Sharma [13] is discussed and its connection with traditional Bessel function series is established. An alternative new series is developed which isolates the steady-state component for all values of traffic intensity and which turns out to be computationally superior. A brief comparison of our formula, Sharma's formula, and a classical Bessel function formula is given for the computation time of the probability that an initially empty system is empty at time t later.
TL;DR: A reference probability is explicitly constructed under which the signal and observation processes are independent, and a simple, explicit recursive form is obtained for the conditional density of the signal given the observations.
Abstract: 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.
TL;DR: In this article, the authors considered the problem of routing jobs to parallel queues with identical exponential servers and unequal finite buffer capacities, and established the extremal properties of the shortest non-full queue (SNQ) and the longest NNQ policies, in systems with concave/convex service rates.
Abstract: We consider the problem of routing jobs to parallel queues with identical exponential servers and unequal finite buffer capacities. Service rates are statedependent and non-decreasing with respect to queue lengths. We establish the extremal properties of the shortest non-full queue (SNQ) and the longest non-full queue (LNQ) policies, in systems with concave/convex service rates. Our analysis is based on the weak majorization of joint queue lengths which leads to stochastic orderings of critical performance indices. Moreover, we solve the buffer allocation problem, i.e. the problem of how to distribute a number of buffers among the queues. The two optimal allocation schemes are also 'extreme', in the sense of capacity balancing. Some extensions are also discussed. OPTIMAL ROUTING; BUFFER ALLOCATION; WEAK MAJORIZATION; STATE-DEPENDENT SERVICE RATES AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 60K25
TL;DR: The class of new better than used in convex ordering (NBUC) is shown to be closed under formation of parallel systems with independent and identically distributed components as discussed by the authors, where the class of NBUC is closed under the formation of a parallel system.
Abstract: The class of new better than used in convex ordering (NBUC) is shown to be closed under formation of parallel systems with independent and identically distributed components.
TL;DR: In this paper, a two-parameter Ehrenfest urn model is derived according to the approach taken by Karlin and McGregor [7] where Krawtchouk polynomials are used.
Abstract: A two-parameter Ehrenfest urn model is derived according to the approach taken by Karlin and McGregor [7] where Krawtchouk polynomials are used. Furthermore, formulas for the mean passage times of finite homogeneous Markov chains with general tridiagonal transition matrices are given. In the special case of the Ehrenfest model they have quite a different structure as compared with those of Blom [2] or Kemperman [9].
TL;DR: In this paper, the distribution of maxima of independent stochastic processes is characterized in terms of spectral functions acting on a Poisson point process, where the spectral function is defined as a function of the maximum of a point process.
Abstract: Limits in distribution of maxima of independent stochastic processes are characterized in terms of spectral functions acting on a Poisson point process.
TL;DR: In this article, the concept of a non-homogeneous semi Markov system in a stochastic environment (S-NHSMS) was introduced and defined for the first time, and the problem of finding the expected population structure as a function of the basic parameters of the system was studied.
Abstract: In the present we introduce and define for the first time the concept of a non-homogeneous semi Markov system in a stochastic environment (S-NHSMS). We study the problem of finding the expected population structure as a function of the basic parameters of the system. Important properties are established among the basic parameters of a non-homogeneous semi Markov system in a stochastic environment.
TL;DR: In this paper, the Palm theory of point processes is transformed into the Palm inversion formula and Miyazawa's rate conservation law, and a single formula which, when its components are adequately chosen, transforms itself into the main Palm formulas.
Abstract: We obtain a single formula which, when its components are adequately chosen, transforms itself into the main formulas of the Palm theory of point processes: Little's L = λW formula [10], Brumelle's H = λG formula [5], Neveu's exchange formula [14], Palm inversion formula and Miyazawa's rate conservation law [12]. It also contains various extensions of the above formulas and some new ones.
TL;DR: In this article, the authors considered the absorption of a non-decreasing compound Poisson process of finite order in a general upper boundary and derived the probability of absorption and first-passage times.
Abstract: This paper considers the absorption of a non-decreasing compound Poisson process of finite order in a general upper boundary. The problem is relevant in fields such as risk theory, Kolmogorov-Smirnov statistics and sequential analysis. The probability of absorption and first-passage times are given in terms of a generating function which depends on the boundary only and can be computed readily. Absorption is certain or not as the asymptotic slope of the boundary is greater or less than the expected increase of the process in unit time. The case of the linear
TL;DR: In this paper, the authors derive a simple procedure to obtain the moments of on-hand inventory in (s, S) inventory systems with periodic reviews and immediate deliveries, under mild assumptions on the demand distribution.
Abstract: By applying a method proposed by W. B. Gong and J. Q. Hu (1992), the authors derive a simple procedure to obtain the moments of on-hand inventory in (s, S) inventory systems with periodic reviews and immediate deliveries. Under mild assumptions on the demand distribution, it is possible to express the moments of on-hand inventory in terms of two power series, without involving any renewal functions. Preliminary numerical examples show that the method is very promising. >