217 results
Search Results
152. Stationary joint distributions arising in the analysis of the M/G/1 queue by the method of the imbedded markov chain
- Author
-
J. H. Jenkins
- Subjects
Statistics and Probability ,M/G/k queue ,General Mathematics ,M/M/1 queue ,G/G/1 queue ,Combinatorics ,Matrix analytic method ,Burke's theorem ,Fluid queue ,M/G/1 queue ,M/M/c queue ,Statistical physics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
SummaryProbability generating functions are used to relate the joint distribution of the numbers of customers left behind by two successive departing customers to the marginal distribution of the number left behind by each departing customer. A probability generating function is then found for the joint distribution of the numbers of customers arriving in two successive departure intervals using the joint distribution of the numbers of customers left behind by three successive departing customers. The results could be obtained from general Markov chain theory but the method used in this paper is quicker.
- Published
- 1966
153. The quasi-stationary distributions of queues in heavy traffic
- Author
-
E K Kyprianou
- Subjects
Discrete mathematics ,Statistics and Probability ,Queueing theory ,M/G/k queue ,General Mathematics ,010102 general mathematics ,M/M/1 queue ,G/G/1 queue ,M/M/∞ queue ,01 natural sciences ,010104 statistics & probability ,Burke's theorem ,M/G/1 queue ,M/M/c queue ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
This paper demonstrates that, when in heavy traffic, the quasi-stationary distribution of the virtual waiting time process of both the M/G/1 and GI/M/1 queues as well as the quasi-stationary distribution of the waiting times {Wn } of the M/G/1 queue can be approximated by the same gamma distribution. What characterises this approximating gamma distribution are the first two moments of the service time and inter-arrival time distributions only. A similar approximating behaviour is demonstrated for the queue size process.
- Published
- 1972
154. Selective interaction of a poisson and renewal process: the dependency structure of the intervals between responses
- Author
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A. J. Lawrance
- Subjects
Statistics and Probability ,General Mathematics ,010102 general mathematics ,Process (computing) ,Type (model theory) ,Poisson distribution ,Stationary point ,01 natural sciences ,symbols.namesake ,010104 statistics & probability ,Joint probability distribution ,Asynchronous communication ,Statistics ,symbols ,Applied mathematics ,Renewal theory ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics ,Event (probability theory) - Abstract
This paper studies the dependency structure of the intervals between responses in the renewal inhibited Poisson process, and continues the author's earlier work on this type of process ((1970a), (1970b)). A new approach to the intervals between events in a stationary point process, based on the idea of an average event, is introduced. Average event initial conditions (as opposed to equilibrium initial conditions previously determined) for the renewal inhibited Poisson process are obtained and event stationarity of the resulting response process is established. The joint distribution and correlation between pairs of contiguous synchronous intervals is obtained; further, the joint distribution of non-contiguous pairs of synchronous intervals is derived. Finally, the joint distributions of pairs of contiguous synchronous and asynchronous intervals are related, and a similar but more general stationary point result is conjectured.
- Published
- 1971
155. A single server queue in discrete time with customers served in random order
- Author
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D. W. Balmer
- Subjects
Statistics and Probability ,Queueing theory ,Queue management system ,business.industry ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Traffic intensity ,010104 statistics & probability ,Multilevel queue ,Weighted random early detection ,M/G/1 queue ,0101 mathematics ,Statistics, Probability and Uncertainty ,business ,Priority queue ,Bulk queue ,Computer network ,Mathematics - Abstract
This paper aims at showing that for the discrete time analogue of the M/G/l queueing model with service in random order and with a traffic intensity ρ > 0, the condition ρ < ∞ is sufficient in order that every customer joining the queue be served eventually, with probability one (Theorem 2).
- Published
- 1972
156. Negative binomial processes
- Author
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G. F. Yeo and Ole E. Barndorff-Nielsen
- Subjects
Statistics and Probability ,Independent and identically distributed random variables ,Distribution (number theory) ,General Mathematics ,Explained sum of squares ,Negative binomial distribution ,Poisson distribution ,Negative multinomial distribution ,symbols.namesake ,symbols ,Applied mathematics ,Statistics, Probability and Uncertainty ,Gaussian process ,Finite set ,Mathematics - Abstract
SummaryThis paper is concerned with negative binomial processes which are essentially mixed Poisson processes whose intensity parameter is given by the sum of squares of a finite number of independently and identically distributed Gaussian processes. A study is made of the distribution of the number of points of a k-dimensional negative binomial process in a compact subset of Rk, and in particular in the case where the underlying Gaussian processes are independent Ornstein-Uhlenbeck processes when more detailed results may be obtained.
- Published
- 1969
157. A continuous time inventory model
- Author
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J. A. Bather
- Subjects
Statistics and Probability ,Inventory control ,Inventory level ,Mathematical optimization ,Optimization problem ,Character (mathematics) ,Discrete time and continuous time ,General Mathematics ,Statistics, Probability and Uncertainty ,Decision problem ,Integral equation ,Field (computer science) ,Mathematics - Abstract
This paper discusses an optimization problem arising in the theory of inventory control. Much of the previous work in this field has been focused on the Arrow-Harris-Marschak model, [1], [2], in which the inventory level can be modified only at the instants of discrete time. Here, we shall be concerned with a continuous time analogue of the model, in an attempt to avoid the difficulties experienced in solving the basic integral equations. The approach was suggested by recent investigations of a statistical decision problem, [3], [5], which exploited the advantages of a continuous treatment. Although the ideas discussed here are relatively straightforward and involve strong assumptions as to the behavior of the inventory, the explicit character of the optimal policy is encouraging and particular solutions might nevertheless provide useful restocking procedures.
- Published
- 1966
158. A process with chain dependent growth rate
- Author
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Julian Keilson and S. Subba Rao
- Subjects
Wishart distribution ,Statistics and Probability ,Markov chain ,General Mathematics ,010102 general mathematics ,State (functional analysis) ,Space (mathematics) ,01 natural sciences ,Combinatorics ,010104 statistics & probability ,Distribution (mathematics) ,Chain (algebraic topology) ,Discrete time and continuous time ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics ,Central limit theorem - Abstract
Additive processes on finite Markov chains have been investigated by Miller ([8], [9]), Keilson and Wishart ([2], [3], [4]) and by Fukushima and Hitsuda [1]. These papers study a two-dimensional Markov Process {X(t), R(t)} whose state space is R1 × {1, 2, ···, R} characterized by the following properties: (i)R(t) is an irreducible Markov chain on states 1,2, …,R governed by atransition probability matrix Bo = {brs}.(ii)X(t) is a sum of random increments dependent on the chain, i.e., if the ith transition takes the chain from state r to state s, then the increment has the distribution (iii)Nt, is t in discrete time while in the continuous time case Nt, might be an independent Poisson process.
- Published
- 1970
159. Selective interaction of a Poisson and renewal process: the spectrum of the intervals between responses
- Author
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A. J. Lawrance
- Subjects
Statistics and Probability ,General Mathematics ,Spectrum (functional analysis) ,Generating function ,Function (mathematics) ,Interval (mathematics) ,Poisson distribution ,Stationary point ,symbols.namesake ,Joint probability distribution ,Statistics ,symbols ,Statistical physics ,Renewal theory ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
This paper is concerned with the spectrum of the intervals between responses in the renewal inhibited Poisson process, and continues the author's earlier work on this type of process ((1970a), (1970b), (1971)). A generating function for all the pairwise joint distributions of the synchronous intervals following an average response is obtained and leads directly to the associated serial correlations. It is shown that these correlations are equivalent to those predicted on different assumptions by the general stationary point theory. The results are then used to obtain the interval spectrum, and to exhibit a relationship between the sum of the serial correlations and the variance-time function. Explicit results for the spectrum of the renewal inhibited Poisson process are given for gamma inhibitory distributions, and the qualitative behavior is determined. Possible further developments are briefly discussed.
- Published
- 1971
160. A limit theorem for a class of supercritical branching processes
- Author
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R. A. Doney
- Subjects
Statistics and Probability ,Discrete mathematics ,General Mathematics ,Generating function ,Mode (statistics) ,Conditional probability ,Supercritical fluid ,Combinatorics ,Convergence of random variables ,Interval (graph theory) ,Limit (mathematics) ,Statistics, Probability and Uncertainty ,Branching process ,Mathematics - Abstract
In the Bellman-Harris (B-H) age-dependent branching process, the birth of a child can occur only at the time of its parent's death. A general class of branching process in which births can occur throughout the lifetime of a parent has been introduced by Crump and Mode. This class shares with the B-H process the property that the generation sizes {ξn } form a Galton-Watson process, and so may be classified into subcritical, critical or supercritical according to the value of m = E{ξ 1}. Crump and Mode showed that, as regards extinction probability, asymptotic behaviour, and for the supercritical case, convergence in mean square of Z(t)/E[Z(t)], as t → ∞, where Z(t) is the population size at time t given one ancestor at t = 0, properties of the B-H process can be extended to this general class. In this paper conditions are found for the convergence in distribution of Z(t)/E{Z(t)} in the supercritical case to a non-degenerate limit distribution. In contrast to the B-H process, these conditions are not the same as those for ξn /mn to have a non-degenerate limit. An integral equation is established for the generating function of Z(t), which is more complicated than the corresponding one for the B-H process and involves the conditional probability generating functional of N(x), x 0, ≧ the number of children born to an individual in the age interval [0, x].
- Published
- 1972
161. An immigration super-critical branching diffusion process
- Author
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J. Radcliffe
- Subjects
Statistics and Probability ,General Mathematics ,010102 general mathematics ,Extension (predicate logic) ,01 natural sciences ,Mean square convergence ,Branching (linguistics) ,010104 statistics & probability ,Diffusion process ,Statistical physics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Diffusion (business) ,Random variable ,Mathematics - Abstract
This paper is an extension of Davis (1965) by allowing immigration. Mean square convergence is proved for a random variable in a branching diffusion process allowing immigration.
- Published
- 1972
162. Sequential decisions in the control of a space-ship (terminal cost proportional to magnitude of miss distance)
- Author
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Thomas Stroud and Paul I. Feder
- Subjects
Statistics and Probability ,Mathematical optimization ,Spacecraft ,business.industry ,General Mathematics ,Control (management) ,Magnitude (mathematics) ,Terminal cost ,Statistics, Probability and Uncertainty ,business ,Mathematics - Abstract
This is an extension of the paper [1] by Bather and Chernoff, where approximations are obtained to the solution of a sequential control problem involving the decision of when to use fuel to alter course during a space-ship's approach to a target. It is desired to come as close as possible to the target. If fuel is used too early, it may be used incorrectly due to imprecise information about the location of the target; however if the fuel is used too late it may be ineffective in reducing the miss distance.
- Published
- 1971
163. A theorem on Markov branching processes
- Author
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John F. Reynolds
- Subjects
Discrete mathematics ,Statistics and Probability ,Markov chain ,General Mathematics ,Zero (complex analysis) ,Markov process ,Function (mathematics) ,Expression (computer science) ,Correlation function (astronomy) ,symbols.namesake ,Markov renewal process ,symbols ,Applied mathematics ,Statistics, Probability and Uncertainty ,Mathematics ,Branching process - Abstract
This paper derives an expression for the correlation function of a continuous parameter branching process, and obtains conditions for the convergence of this function to a zero or positive value. MARKOV; BRANCHING PROCESS; CORRELATION FUNCTION
- Published
- 1972
164. Characterization of distributions by the identical distribution of linear forms
- Author
-
Morris L. Eaton
- Subjects
Statistics and Probability ,Pure mathematics ,010104 statistics & probability ,Distribution (number theory) ,Similar distribution ,General Mathematics ,010102 general mathematics ,Characterization (mathematics) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Random variable ,01 natural sciences ,Mathematics - Abstract
Throughout this paper, we shall write ℒ(W) = ℒ(Z) to mean the random variables W and Z have the same distribution. The relation “ℒ(W) = ℒ(;Z)” reads “the law of W equals the law of Z”.
- Published
- 1966
165. On the quasi-stationary distributions of the GI/M/1 queue
- Author
-
E. K. Kyprianou
- Subjects
Statistics and Probability ,Discrete mathematics ,M/G/k queue ,General Mathematics ,010102 general mathematics ,M/M/1 queue ,G/G/1 queue ,01 natural sciences ,M/M/∞ queue ,Combinatorics ,010104 statistics & probability ,Burke's theorem ,M/G/1 queue ,M/M/c queue ,0101 mathematics ,Statistics, Probability and Uncertainty ,Bulk queue ,Mathematics - Abstract
This paper studies the existence, in a stable GI/M/1 queue, of the limit as t → ∞ of the distribution of the virtual waiting time process at time t conditioned on the event that at no time in the interval [0, t] the queue has become empty. The conditional limit distribution obtained when the traffic intensity is strictly less than one is the weighted sum of an exponential and a gamma distribution. Similar conditional limit distributions are obtained for the queue size process and the waiting time process as defined by Prabhu (1964).
- Published
- 1972
166. Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test
- Author
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J. Durbin
- Subjects
Statistics and Probability ,Exponential distribution ,General Mathematics ,Mathematical analysis ,010102 general mathematics ,Boundary (topology) ,Poisson distribution ,Kolmogorov–Smirnov test ,01 natural sciences ,Separable space ,symbols.namesake ,010104 statistics & probability ,Distribution function ,Sampling distribution ,symbols ,0101 mathematics ,Statistics, Probability and Uncertainty ,Brownian motion ,Mathematics - Abstract
Let w(t), 0 ≦ t ≦ ∞, be a Brownian motion process, i.e., a zero-mean separable normal process with Pr{w(0) = 0} = 1, E{w(t 1)w(t 2)}= min (t 1, t 2), and let a, b denote the boundaries defined by y = a(t), y = b(t), where b(0) < 0 < a(0) and b(t) < a(t), 0 ≦ t ≦ T ≦ ∞. A basic problem in many fields such as diffusion theory, gambler's ruin, collective risk, Kolmogorov-Smirnov statistics, cumulative-sum methods, sequential analysis and optional stopping is that of calculating the probability that a sample path of w(t) crosses a or b before t = T. This paper shows how this probability may be computed for sufficiently smooth boundaries by numerical solution of integral equations for the first-passage distribution functions. The technique used is to approximate the integral equations by linear recursions whose coefficients are estimated by linearising the boundaries within subintervals. The results are extended to cover the tied-down process subject to the condition w(1) = 0. Some related results for the Poisson process and the sample distribution function are given. The procedures suggested are exemplified numerically, first by computing the probability that the tied-down Brownian motion process crosses a particular curved boundary for which the true probability is known, and secondly by computing the finite-sample and asymptotic powers of the Kolmogorov-Smirnov test against a shift in mean of the exponential distribution.
- Published
- 1971
167. A method for studying the integral functionals of stochastic processes with applications: I. Markov chain case
- Author
-
Prem S. Puri
- Subjects
Statistics and Probability ,Discrete mathematics ,Stratonovich integral ,Stochastic process ,Stochastic modelling ,General Mathematics ,010102 general mathematics ,Type (model theory) ,01 natural sciences ,Measure (mathematics) ,Time reversibility ,Harris chain ,Combinatorics ,010104 statistics & probability ,Markov property ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
The subject of this paper is the study of the distribution of integrals of the type where {X(t); t ≧ 0} is some appropriately defined continuous-time parameter stochastic process, and f is a suitable non-negative function of its arguments. This subject has also sometimes been labelled as “the occupation time or the sojourn time problem” in literature. These integrals arise in several domains of applications such as in the theory of inventories and storage (see Moran [14], Naddor [15]), in the study of the cost of the flow-stopping incident involved in the automobile traffic jams (see Gaver [8], Daley [3], Daley and Jacobs [4]). The author encountered such integrals while studying certain stochastic models suitable for the study of response time distributions arising in various live situations. In fact in [19], it was shown that such a distribution is equivalent to the study of an integral of the type (1). Again, in the study of response of host to injection of virulent bacteria, Y(t) with f(X(t), t) = bX(t), with b > 0, could be regarded as a measure of the total amount of toxins produced by the bacteria during (0, t), assuming a constant toxin-excretion rate per bacterium. Here X(t) denotes the number of live bacteria at time t, the growth of which is governed by a birth and death process (see Puri [16], [17] and [18]).
- Published
- 1971
168. Physical nearest-neighbour models and non-linear time-series. II Further discussion of approximate solutions and exact equations
- Author
-
M. S. Bartlett
- Subjects
Statistics and Probability ,010104 statistics & probability ,Nonlinear system ,Series (mathematics) ,General Mathematics ,010102 general mathematics ,Nearest neighbour ,Applied mathematics ,Exact differential equation ,0101 mathematics ,Statistics, Probability and Uncertainty ,01 natural sciences ,Mathematics - Abstract
The approximate two- and three-dimensional solutions for spatial correlations, using the non-linear time-series approach for nearest-neighbour systems developed in my previous paper, are further discussed. Orthogonal expansions for the correlation functions are also developed which determine with this approach, though so far only in principle, the exact solutions.
- Published
- 1972
169. A stochastic model for two interacting populations
- Author
-
Niels G. Becker
- Subjects
Statistics and Probability ,Birth and death process ,education.field_of_study ,Component (thermodynamics) ,Stochastic modelling ,Differential equation ,General Mathematics ,010102 general mathematics ,Population ,Type (model theory) ,01 natural sciences ,Birth–death process ,010104 statistics & probability ,Nonlinear system ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,education ,Mathematics - Abstract
To explain the growth of interacting populations, non-linear models need to be proposed and it is this non-linearity which proves to be most awkward in attempts at solving the resulting differential equations. A model with a particular nonlinear component, initially proposed by Weiss (1965) for the spread of a carrierborne epidemic, was solved completely by different methods by Dietz (1966) and Downton (1967). Immigration parameters were added to the model of Weiss and the resulting model was made the subject of a paper by Dietz and Downton (1968). It is the aim here to further generalize the model by introducing birth and death parameters so that the result is a linear birth and death process with immigration for each population plus the non-linear interaction component. Consider two populations which we refer to as type 1 and type 2. We suppose that at time t there are Xi(t) individuals of type i present in the habitat and X;(0) = min is the initial number. Further let the birth, death and immigration rates for population i be i, ~i and vi respectively. It is then supposed that the probability of increasing the type i population by one individual during the time
- Published
- 1970
170. Study of a network of serial and non-serial servers with phase type service and finite queueing space
- Author
-
Krishan Lall Arya
- Subjects
Statistics and Probability ,Queueing theory ,Basis (linear algebra) ,Series (mathematics) ,General Mathematics ,Topology ,Poisson distribution ,Space (mathematics) ,Exponential function ,Computer Science::Performance ,symbols.namesake ,Server ,symbols ,Layered queueing network ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
The paper develops the steady-state solution of a finite space queueing system wherein each of the two non-serial servers is separately in series with two non-serial servers. It is assumed that the arriving units of the same type may demand a different number of service phases. Poisson arrivals and exponential service times are assumed at all the four channels of the system. Service of units is completed on a first-come, first-served basis at each channel. The steady-state solution for infinite queueing space is obtained as a special case of finite queueing space.
- Published
- 1972
171. Bounds for moment generating functions and for extinction probabilities
- Author
-
D. Brook
- Subjects
Statistics and Probability ,Moment (mathematics) ,Factorial moment generating function ,Extinction ,General Mathematics ,Statistical physics ,Statistics, Probability and Uncertainty ,Moment-generating function ,Mathematics - Abstract
Suppose that we have a non-negative, real valued random variable x, whose distribution is governed by some unknown moment generating function M(t). Suppose further that we are given certain moments of x, then the question to be discussed in this paper is : can we find a sharp upper bounding function for the m.g.f.? It will be shown that this is usually possible both in the single variate case and in its natural extension to the multivariate case.
- Published
- 1966
172. On random sequential packing in the plane and a conjecture of palasti
- Author
-
Herbert Solomon and B. Edwin Blaisdell
- Subjects
Discrete mathematics ,Statistics and Probability ,Conjecture ,Stochastic process ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Combinatorics ,Combinatorial analysis ,Square packing in a square ,010104 statistics & probability ,Packing dimension ,Sphere packing ,Lattice (order) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Random variable ,Mathematics - Abstract
The random packing of geometric objects in one-, two- or three-dimensions may afford useful insights into the structure of crystals, liquids, absorbates on crystals, and in higher dimensions, into problems of pattern recognition. Random packing has accordingly received increasing attention in recent years. Two principal packing procedures have been formulated and each gives rise to different packing ratios. In one case, all possible configurations of a sphere-packed volume are assumed to be equally likely. In the other and most widely reported case, there is random sequential addition of spheres to the volume until it is packed. This is the situation we study in this paper. Most of the work to date has been limited to the theoretical study of the one-dimensional lattice or to continuous cases particularly in the limit for long lines. The higher dimensional cases have resisted theoretical attack but have been studied by computer simulation by Palasti [12] and Solomon [14] and by physical simulation by Bernal and Scott (see [14]).
- Published
- 1970
173. A correction to 'The area under the infectives trajectory of the general stochastic epidemic'
- Author
-
F. Downton
- Subjects
Statistics and Probability ,General Mathematics ,Trajectory ,Applied mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In a recent paper (Downton (1972)) an attempt was made to use the combinatorial approach of Downton (1967) to obtain an expression for the Laplace transform (and hence the moments) of the area under the infectives trajectory of the general stochastic epidemic. The basic property of that area, which makes this possible, is that it is composed of a random number of rectangles, each of whose area has a distribution depending on the infection rate and the number of susceptibles present at the appropriate stage in the development of the epidemic, but which is independent of the number of infectives. However, Mr A. Abakuks has pointed out that in one respect the argument used was faulty. It was argued that the contribution to the area under the infectives curve of an interval ending in a reduction in the number of infectives from i to i – 1 had an exponential distribution with parameter p (the infection rate); and if the interval ended in a reduction in the number of susceptibles from s to s – 1 the distribution was exponential with parameter s. While this is true unconditionally, the combinatorial situation described was essentially a conditional one, in which each of these contributions to the total area had an exponential distribution, independent of i, but with parameter (ρ + s) regardless of whether the interval ended in a reduction in the number of infectives or of susceptibles.
- Published
- 1972
174. Expenditure patterns for risky R and D projects
- Author
-
Morton I. Kamien and Nancy L. Schwartz
- Subjects
Statistics and Probability ,010104 statistics & probability ,Actuarial science ,General Mathematics ,010102 general mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,01 natural sciences ,Mathematics - Abstract
Summary In this paper we derive the time pattern of optimal planned expenditures for R and D projects in which the total effort required to complete the research satisfactorily is not known. Theorem 1 summarizes our results under alternative assumptions regarding the behavior of the function governing the supposed probability of completing the R and D satisfactorily with incremental effort. In addition, an easily employed necessary condition for the R and D project to be worthwhile is provided in Theorem 2.
- Published
- 1971
175. Linear programming and continuous markovian decision problems
- Author
-
Yoshio Tabata and Hisashi Mine
- Subjects
Statistics and Probability ,Mathematical optimization ,Linear programming ,General Mathematics ,010102 general mathematics ,Markov process ,Decision problem ,01 natural sciences ,symbols.namesake ,010104 statistics & probability ,symbols ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
Summary This paper is concerned with a continuous time parameter Markovian sequential decision process, and presents a method which transforms a given continuous parameter problem into a discrete one. It is proved that the optimal stationary policy for the resulting discrete time parameter Markovian decision process is also the optimal stationary policy for the original continuous one, and vice versa. The resulting discrete parameter problem may be more easily solved than the continuous one by applying the linear programming method. A simple numerical example is presented.
- Published
- 1970
176. On the single-server queue with non-homogeneous Poisson input and general service time
- Author
-
A. M. Hasofer
- Subjects
Statistics and Probability ,General Mathematics ,M/D/1 queue ,010102 general mathematics ,M/M/1 queue ,M/D/c queue ,M/M/∞ queue ,01 natural sciences ,010104 statistics & probability ,M/G/1 queue ,Applied mathematics ,M/M/c queue ,Pollaczek–Khinchine formula ,0101 mathematics ,Statistics, Probability and Uncertainty ,Bulk queue ,Mathematics - Abstract
In this paper, a single-server queue with non-homogeneous Poisson input and general service time is considered. Particular attention is given to the case where the parameter of the Poisson input λ(t) is a periodic function of the time. The approach is an extension of the work of Takács and Reich . The main result of the investigation is that under certain conditions on the distribution of the service time, the form of the function λ(t) and the distribution of the waiting time at t = 0, the probability of a server being idle P 0 and the Laplace transform Ω of the waiting time are both asymptotically periodic in t. Putting where b(t) is a periodic function of time, it is shown that both P o and Ω can be expanded in a power series in z, and a method for calculating explicitly the asymptotic values of the leading terms is obtained. In many practical queueing problems, it is expected that the probability of arrivals will vary periodically. For example, in restaurants or at servicestations arrivals are more probable at rush hours than at slack periods, and rush hours are repeated day after day
- Published
- 1964
177. Equilibrium distributions of polymers on sites which are linearly arrayed
- Author
-
J. L. Verrall and Bev Littlewood
- Subjects
chemistry.chemical_classification ,Statistics and Probability ,General Mathematics ,010102 general mathematics ,Link (geometry) ,Polymer ,Composition (combinatorics) ,01 natural sciences ,Chemical reaction ,Linear array ,010104 statistics & probability ,chemistry ,Chemical physics ,Linear arrays ,Molecule ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
Certain long molecules can be regarded as linear arrays of chemically active sites. During chemical reaction, adjacent groups of sites may link to form dimers, trimers, etc. This paper considers the case when all such groups are unstable, and gives equilibrium distributions for the composition of the molecules as functions of the rates of formation and decay of the linked groups.
- Published
- 1972
178. On two integral equations of queueing theory
- Author
-
J. W. Cohen
- Subjects
Independent and identically distributed random variables ,Statistics and Probability ,Pure mathematics ,Stochastic process ,General Mathematics ,Mathematical analysis ,010102 general mathematics ,Integral equation ,Volterra integral equation ,01 natural sciences ,Traffic equations ,symbols.namesake ,010104 statistics & probability ,symbols ,Functional integration ,G-network ,0101 mathematics ,Statistics, Probability and Uncertainty ,Constant (mathematics) ,Mathematics - Abstract
In the present paper the solutions of two integral equations are derived. One of the integral equations dominates the mathematical description of the stochastic process { v n , n = 1,2, …}, recursively defined by K is a positive constant, τ 1, τ 2, …; Σ 1, Σ 2, …; are independent, non-negative variables, with τ 1, τ 2,…, identically distributed, similarly, the variables Σ 1, Σ 2, …, are identically distributed.
- Published
- 1967
179. Two results in the theory of queues
- Author
-
H. Ali
- Subjects
Statistics and Probability ,Service (business) ,Discrete mathematics ,Distribution (number theory) ,General Mathematics ,010102 general mathematics ,Expected value ,Fork–join queue ,01 natural sciences ,Renewal function ,010104 statistics & probability ,Mean value analysis ,0101 mathematics ,Statistics, Probability and Uncertainty ,Queue ,Instant ,Mathematics - Abstract
Summary In this paper it is shown that the distribution of the instant of service of a customer is symmetric as between the distributions of service and interarrival time. Also U(t), the expected number of departures in (0, t), is a delayed renewal function for the GI/M/1 queue.
- Published
- 1970
180. On the busy periods for the M/G/1 queue with finite and with infinite waiting room
- Author
-
J. W. Cohen
- Subjects
Statistics and Probability ,Discrete mathematics ,Distribution (number theory) ,M/G/k queue ,Joint probability distribution ,General Mathematics ,M/G/1 queue ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
Summary The Laplace-Stieltjes transform of the distribution of the busy period for the M/G/1 system with infinite waiting room can be obtained by using an argument from branching theory. In the present paper it is shown that by applying this argument it is rather easy to derive the expression for the joint distribution of the busy period and the maximum number of customers present simultaneously during this busy period for the M/G/1 system with infinite waiting room as well as the expression for the distribution of the busy period for the M/G/1 system with finite waiting room.
- Published
- 1971
181. Recursive matrix games
- Author
-
Michael Orkin
- Subjects
Statistics and Probability ,Infinite game ,Class (set theory) ,Generality ,General Mathematics ,010102 general mathematics ,ComputingMilieux_PERSONALCOMPUTING ,Vagueness ,Type (model theory) ,01 natural sciences ,Matrix games ,010104 statistics & probability ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematical economics ,Value (mathematics) ,Mathematics - Abstract
“Recursive” games were first defined and studied by Everett. Related results can be found in Gillette, Milnor and Shapley, and Blackwell and Ferguson. In this paper we introduce the notion of a recursive matrix game, which we believe eliminates the vagueness but none of the useful generality of the earlier definition. We then give an inductive proof (different from the proof in [3]) that these games have a value, with ∊-optimal stationary strategies available to each player. We also apply the result and show how a class of games studied in a different framework are games of this type and thus have a value.
- Published
- 1972
182. The definition of a multi-dimensional generalization of shot noise
- Author
-
D. J. Daley
- Subjects
Statistics and Probability ,010104 statistics & probability ,Generalization ,General Mathematics ,010102 general mathematics ,Multi dimensional ,Shot noise ,Image noise ,0101 mathematics ,Statistics, Probability and Uncertainty ,01 natural sciences ,Algorithm ,Mathematics - Abstract
The paper studies the formally defined stochastic process where {tj } is a homogeneous Poisson process in Euclidean n-space En and the a.e. finite Em -valued function f(·) satisfies |f(t)| = g(t) (all |t | = t), g(t) ↓ 0 for all sufficiently large t → ∞, and with either m = 1, or m = n and f(t)/g(t) =t/t. The convergence of the sum at (*) is shown to depend on (i) (ii) (iii) . Specifically, finiteness of (i) for sufficiently large X implies absolute convergence of (*) almost surely (a.s.); finiteness of (ii) and (iii) implies a.s. convergence of the Cauchy principal value of (*) with the limit of this principal value having a probability distribution independent of t when the limit in (iii) is zero; the finiteness of (ii) alone suffices for the existence of this limiting principal value at t = 0.
- Published
- 1971
183. A stochastic process whose successive intervals between events form a first order Markov chain — I
- Author
-
D. G. Lampard
- Subjects
Statistics and Probability ,Markov chain ,Stochastic process ,General Mathematics ,010102 general mathematics ,Discrete phase-type distribution ,Markov process ,01 natural sciences ,Continuous-time Markov chain ,symbols.namesake ,Markov renewal process ,Statistics ,symbols ,Markov property ,Renewal theory ,Statistical physics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this paper we discuss a counter system whose output is a stochastic point process such that the time intervals between pairs of successive events form a first order Markov chain. Such processes may be regarded as next, in order of complexity, in a hierarchy of stochastic point processes, to “renewal” processes, which latter have been studied extensively. The main virtue of the particular system which is studied here is that virtually all its important statistical properties can be obtained in closed form and that it is physically realizable as an electronic device. As such it forms the basis for a laboratory generator whose output may be used for experimental work involving processes of this kind. Such statistical properties as the one and two-dimensional probability densities for the time intervals are considered in both the stationary and nonstationary state and also discussed are corresponding properties of the successive numbers arising in the stores of the counter system. In particular it is shown that the degree of coupling between successive time intervals may be adjusted in practice without altering the one dimensional probability density for the interval lengths. It is pointed out that operation of the counter system may also be regarded as a problem in queueing theory involving one server alternately serving two queues. A generalization of the counter system, whose inputs are normally a pair of statistically independent Poisson processes, to the case where one of the inputs is a renewal process is considered and leads to some interesting functional equations.
- Published
- 1968
184. Some new results in the mathematical theory of phage-reproduction
- Author
-
Prem S. Puri
- Subjects
Statistics and Probability ,Birth and death process ,Distribution (number theory) ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Function (mathematics) ,01 natural sciences ,Combinatorics ,010104 statistics & probability ,Fixed time ,Random event ,0101 mathematics ,Statistics, Probability and Uncertainty ,Reproduction ,media_common ,Mathematics - Abstract
Summary In the theory of phage reproduction, the mathematical models considered thus far (see Gani [5]) assume that the bacterial burst occurs a fixed time after infection, after a fixed number of generations of phage multiplication, or when the number of mature bacteriophages has reached a fixed threshold. In the present paper, a more realistic assumption is considered: given that until time t the bacterial burst has not taken place, its occurence between tand t + Δt is a random event with probability f(· | t)Δt + o(Δt), where f is a non-negative and non-decreasing function of the number X(t) of vegetative phages and of Z(t), the number of mature bacteriophages at time t. More specifically it is assumed that f = b(t)X(t) + c(t)Z(t) with b(t), c(t) ≦ 0. Here X(t) denotes the survivors in a linear birth and death process and Z(t) the number of deaths until time t. The joint distribution of XT and ZT , the respective numbers of vegetative and mature bacteriophages at the burst time is considered. The distribution of ZT is then fitted to some observed data of Delbrück [2].
- Published
- 1969
185. Programming problems and changes in the stable behavior of a class of Markov chains
- Author
-
Richard V. Evans
- Subjects
Statistics and Probability ,Class (set theory) ,Theoretical computer science ,Markov chain ,General Mathematics ,Examples of Markov chains ,Markov decision process ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
This paper develops expressions for the derivatives with respect to a parameter μ of the stable probabilities of a class of Markov chains whose transition matrices are of the form Q + μW. These expressions lead to iterative schemes for calculation which in turn suggest gradient algorithms for finding locally optimal chains.
- Published
- 1971
186. Some theorems on the transient covariance of Markov chains
- Author
-
John F. Reynolds
- Subjects
Statistics and Probability ,Markov chain mixing time ,Markov chain ,Variable-order Markov model ,General Mathematics ,Markov process ,Covariance ,symbols.namesake ,symbols ,Examples of Markov chains ,Transient (computer programming) ,Statistical physics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
Several authors have considered the covariance structure of continuous parameter Markov chains. Most of this work has dealt with particular process ses, notably Morse (1955) who analysed the simple M/M/1 queue and Bene-(1961) who considered a telephone trunking model. Furthermore, the results obtained apply only when the process has attained its limiting (stationary) distribution. A recent paper by Reynolds (1968) gave some general results for finite chains, still assuming stationarity. This note generalises the results obtained therein, and considers the covariance structure during the transient period prior to attaining the stationary distribution where this exists. In the case where no such distribution exists, the results are valid throughout the whole lifetime of the process.
- Published
- 1972
187. On the limiting behaviour of a basic stochastic process
- Author
-
Oussama Achou and İzzet Şahin
- Subjects
Statistics and Probability ,Continuous-time stochastic process ,Pure mathematics ,Stochastic modelling ,Stochastic process ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Integral equation ,010104 statistics & probability ,Simple (abstract algebra) ,Jump ,Bibliography ,Renewal theory ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
Let {W(t), 0 0 for 0 < x < 00; r(0) = 0 and r(x) is non-decreasing (cf. [1]). Under certain assumptions on the renewal process that generates the jump points and/or on the rate of decline, special forms of this process have been studied in the literature as stochastic models for certain storage, queuing, insurance risk and particle counter problems (see [6] and bibliography therein). In this paper, we shall give a simple formulation to determine the limiting behaviour of the general process and obtain some results for the case r(x) = ax by solving an integral equation associated with the process through Fuchs' method. This case was first studied by Takaics [7] to whom we shall refer in due course. The same problem was also investigated in Gaver and Miller [1], Keilson and Mermin [2], Rice [3] and Takics [8] under the assumption that the jump points are generated by a Poisson process.
- Published
- 1972
188. Extreme value theory for a class of discrete distributions with applications to some stochastic processes
- Author
-
C. W. Anderson
- Subjects
Discrete mathematics ,Statistics and Probability ,Class (set theory) ,Stochastic process ,General Mathematics ,010102 general mathematics ,Discrete-time stochastic process ,Function (mathematics) ,01 natural sciences ,Set (abstract data type) ,010104 statistics & probability ,Common distribution ,Statistical physics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Extreme value theory ,Random variable ,Mathematics - Abstract
Let ξn be the maximum of a set of n independent random variables with common distribution function F whose support consists of all sufficiently large positive integers. Some of the classical asymptotic results of extreme value theory fail to apply to ξn for such F and this paper attempts to find weaker ones which give some description of the behaviour of ξn as n → ∞. These are then applied to the extreme value theory of certain regenerative stochastic processes.
- Published
- 1970
189. On a telephone traffic system with several kinds of service distributions
- Author
-
Joseph L. Gastwirth
- Subjects
Statistics and Probability ,Independent and identically distributed random variables ,Service (business) ,Discrete mathematics ,Queueing theory ,Exponential distribution ,General Mathematics ,Information theory ,law.invention ,Traffic equations ,law ,Telephone exchange ,Statistics, Probability and Uncertainty ,Mathematical economics ,Random variable ,Mathematics - Abstract
Let us suppose that telephone calls arrive at a telephone exchange according to a recurrent process. We shall assume that there are infinitely many lines available and that, therefore, no call is ever lost. Ordinarily, it is assumed that the holding times (the durations of the connections) are identically distributed, independent, random variables with an exponential distribution. The American folklore, however, assumes that women talk more than men. In this paper, we shall consider the possibility of calls requiring one of several (s) different exponential holding times.
- Published
- 1964
190. On some limit theorems for the GI/G/1 queue
- Author
-
John A. Hooke
- Subjects
Statistics and Probability ,Discrete mathematics ,Service (business) ,Generality ,Mathematical optimization ,General Mathematics ,010102 general mathematics ,01 natural sciences ,010104 statistics & probability ,Idle ,Work (electrical) ,Burke's theorem ,Limit (mathematics) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Queue ,Mathematics - Abstract
For a GI/G/1 queue a limit theorem is obtained for the total input of work during (0, t]. This result is then used to obtain similar theorems for the waiting and idle times and for busy periods initiated by large service loads. Some of the results contained in the paper have recently been proved by other authors in more general settings. The intent of this work is to show how they may be obtained in lesser generality using simpler techniques.
- Published
- 1970
191. Solutions for some diffusion processes with two barriers
- Author
-
Jay C. Hardin and Arnold L. Sweet
- Subjects
Statistics and Probability ,Diffusion (acoustics) ,Process (engineering) ,Stochastic process ,General Mathematics ,Mathematical analysis ,Separation of variables ,Applied mathematics ,Probability density function ,Statistics, Probability and Uncertainty ,Constant (mathematics) ,Mathematics - Abstract
Use is often made of the Wiener and Ornstein-Uhlenbeck (O.U.) processes in various applications of stochastic processes to problems of engineering interest. These applications frequently involve the presence of barriers. Although mathematical methods for solving Kolmogorov's forward equation for the above processes have previously been discussed ([1], [2]), many solutions for problems with two barriers do not seem to be available in the literature. Instead, one finds solutions for unrestricted processes or simulation used in place of analytical solutions in various applications ([3], [4], [5]). In this paper, solutions of Kolmogorov's forward equations in the presence of constant absorbing and/or reflecting barriers are obtained by means of separation of variables. This enables one to obtain expressions for the probability density functions for first passage times when absorbing barriers are present. The solution for the O.U. process is used to obtain a result of Breiman's [6] concerning first passage times.
- Published
- 1970
192. Stochastic formation of hierarchies
- Author
-
L. L. Helms
- Subjects
Statistics and Probability ,Pure mathematics ,Stochastic process ,General Mathematics ,Markov process ,Asymptotic distribution ,Space (mathematics) ,symbols.namesake ,symbols ,Jump ,Ergodic theory ,State space (physics) ,Statistics, Probability and Uncertainty ,Mathematics ,Probability measure - Abstract
In this paper, N particles el , · ··, eN occupying positions in finite state spaces S 1, · ··, SN , respectively, are considered along with an element of a finite space S N+ 1 of hierarchies which describe the organization of the particles into pairs, pairs of pairs, etc. A stochastic process {X t: t 0} on a probability measure space (Ω, , P) is a hierarchic process if it is a right-continuous Markov jump process taking on values in the state space Sj and having the property that the (N + 1)th component of Xt can jump from a hierarchy to a successor or antecedent of that hierarchy. Asymptotic distributions of perturbed hierarchic processes, bilateral processes, and unilateral processes are determined in terms of an interaction function and the asymptotic distributions of the particles in the absence of any interaction.
- Published
- 1973
193. A Gaussian Markovian process on a square lattice
- Author
-
P. A. P. Moran
- Subjects
Discrete mathematics ,Statistics and Probability ,High Energy Physics::Lattice ,General Mathematics ,010102 general mathematics ,Kinetic scheme ,Integer lattice ,Ornstein–Uhlenbeck process ,Square lattice ,01 natural sciences ,Particle in a one-dimensional lattice ,Reciprocal lattice ,010104 statistics & probability ,Statistical physics ,Markovian arrival process ,0101 mathematics ,Statistics, Probability and Uncertainty ,Lattice model (physics) ,Mathematics - Abstract
A definition of the Markovian property is given for a lattice process and a Gaussian Markovian lattice process is constructed on a torus lattice. From this a Gaussian Markovian process is constructed for a lattice in the plane and its properties are studied. MARKOVIAN PROCESSES; LATTICE PROCESSES 1. A Markov process on a torus In this paper we construct a stationary process on the square lattice formed by all the pairs of positive and negative integers, in which the random variables, Xmn,, say, are identically normally distributed and satisfy a Markovian property. We do this by first constructing a similar such process on a lattice torus and letting the size of the latter tend to infinity.
- Published
- 1973
194. Moments in Markovian systems with lumped states
- Author
-
Greg Taylor
- Subjects
Statistics and Probability ,Class (set theory) ,Markov chain ,General Mathematics ,Markov process ,State (functional analysis) ,Function (mathematics) ,symbols.namesake ,restrict ,symbols ,Statistical physics ,Statistics, Probability and Uncertainty ,Random variable ,Mathematics - Abstract
In dealing with Markov chains, we are sometimes forced by circumstances to make observations on lumped states. We may wish, however, to deal with a random variable which is a function of the numbers in the various (unlumped) states at various times and whose moments, therefore, will involve the probabilities of transition between the unlumped states.This paper demonstrates that, although these probabilities are immeasurable directly, we can, provided that we restrict our attention to a certain well-defined class of functions as our random variables, reduce all moments and product-moments of such random variables to functions of lumped state probabilities.
- Published
- 1971
195. Some characterizations based on the Bhattacharya matrix
- Author
-
D. N. Shanbhag
- Subjects
Statistics and Probability ,Polynomial regression ,Binomial (polynomial) ,Computer Science::Information Retrieval ,General Mathematics ,Diagonal ,Negative binomial distribution ,Probability density function ,Poisson distribution ,Combinatorics ,symbols.namesake ,Matrix (mathematics) ,Quadratic equation ,symbols ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
Laha and Lukacs (1960) have studied distributions with the property that a quadratic statistic has quadratic regression on the sample mean. In doing this, they have arrived at some interesting characterizations for the normal, Poisson, gamma, binomial and negative binomial distributions. Starting with an exponential-type probability density function, the present paper investigates all the distributions for which the 3 × 3 Bhattacharya matrix is diagonal. It is found that the normal, Poisson, gamma, binomial and negative binomial distributions can be characterized by this property. Further, it is observed that for these distributions an s × s Bhattacharya matrix is defined for all s and is also diagonal.
- Published
- 1972
196. Low density inhomogeneous traffic flow
- Author
-
Katsuhisa Ohno and Hisashi Mine
- Subjects
Physics::Fluid Dynamics ,Statistics and Probability ,Flow (mathematics) ,Constant velocity ,General Mathematics ,Low density ,Mechanics ,Statistics, Probability and Uncertainty ,Traffic flow ,Constant (mathematics) ,Road traffic ,Mathematics - Abstract
The purpose of this paper is to show some statistical properties of a one-dimensional inhomogeneous flow of particles with their own constant velocities. The most interesting example of such flows is a low-density inhomogeneous road traffic flow; each vehicle drives at his own constant velocity (desired velocity), since passing is freely allowed. Therefore, to give a definite picture to the one-dimensional inhomogeneous flow of particles, the authors deal with a low-density inhomogeneous traffic flow.
- Published
- 1970
197. Stability Theorems for Solutions to the Optimal Inventory Equation
- Author
-
S. Edward Boylan
- Subjects
Statistics and Probability ,Uniform continuity ,General Mathematics ,Mathematical analysis ,Applied mathematics ,Statistics, Probability and Uncertainty ,Stability (probability) ,Mathematics - Abstract
In a previous paper, [1] it was shown that a solution, f(x) will exist for the optimal inventory equation (where f(y − z) = f(0), y < z) provided: 1. g(x) ≧ 0, x ≧ 0; 2. 0 < a < 1; 3. h(x) is monotonically nondecreasing, h(0) = 0; 4. F is a distribution function on [0, ∞); (In [1], 1–4 were denoted collectively as (A).) and either 5a. g(x) is continuous for all x ≧ 0; 5b. lim x→∞ g(x) = ∞; 5c h(x) is continuous for all x > 0 (Theorem 2 of [1]); or 6. g(x) is uniformly continuous for all x ≧ 0 (Theorem 3 of [1]).
- Published
- 1969
198. Point processes arising in vehicular traffic flow
- Author
-
Edward A. Brill
- Subjects
Statistics and Probability ,M/G/k queue ,General Mathematics ,M/D/1 queue ,010102 general mathematics ,M/M/1 queue ,G/G/1 queue ,M/M/∞ queue ,01 natural sciences ,010104 statistics & probability ,Burke's theorem ,M/G/1 queue ,Applied mathematics ,M/M/c queue ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this paper we investigate the properties of stationary point processes motivated by the following traffic model. Suppose there is a dichotomy of slow and fast points (cars) on a road with limited overtaking. It is assumed that fast points are delayed behind (or are clustered at) a slow point in accordance with the principles of a GI/G /s queue, the order of service being irrelevant. Thus each slow point represents a service station, with the input into each station consisting of a fixed (but random) displacement of the output of the previous queueing station. It is found that tractable results for stationary point processes occur for the cases M/M/s (s = 1,2, ., coc) and M/G/oo. In particular, it is found that for these cases the steady state point processes are compound Poisson and that for the M/M/1 case the successive headways form a two state Markov renewal process. In addition it is shown that the input, output, and queue size processes in a steady state M /G / co queue are independent at any fixed time; this is a result I have been unable to find in the literature. The notion that a car's delay in overtaking a slower car is analogous to the service of a customer in a queueing system is far from new. Tanner [11] and Miller [6] have suggested a model of traffic where platoons (or clusters) are distributed along the road as a Poisson process; also each platoon has its size distributed in accordance with the Borel-Tanner distribution, which happens to be the steady state distribution of queue size in an M /D /1 queue. However, there is no attempt to justify the assumption of independence among successive platoon sizes. In our model, it is seen in fact that independence does not hold for the M/D/1 case. Newell [7] proposed a model for "light traffic" which results in a description of traffic as a superposition of independent point processes; one for single cars and one for pairs of coincident cars. He assumed that overtaking delays behave as do waiting times in an M /G /co queue but neglected multiple car interactions. We shall see that in theory one need not neglect these multiple interactions.
- Published
- 1971
199. Maximal branching processes and ‘long-range percolation’
- Author
-
John Lamperti
- Subjects
Statistics and Probability ,Markov chain ,General Mathematics ,Value (computer science) ,Probability density function ,Combinatorics ,Branching (linguistics) ,Percolation ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Range (statistics) ,Statistics, Probability and Uncertainty ,Random variable ,Branching process ,Mathematics - Abstract
In the first part of this paper, we will consider a class of Markov chains on the non-negative integers which resemble the Galton-Watson branching process, but with one major difference. If there are k individuals in the nth “generation”, and are independent random variables representing their respective numbers of offspring, then the (n + 1)th generation will contain max individuals rather than as in the branching case. Equivalently, the transition matrices Pij of the chains we will study are to be of the form where F(.) is the probability distribution function of a non-negative, integervalued random variable. The right-hand side of (1) is thus the probability that the maximum of i independent random variables distributed by F has the value j. Such a chain will be called a “maximal branching process”.
- Published
- 1970
200. Some asymptotic properties of a two-dimensional periodogram
- Author
-
Marcello Pagano
- Subjects
Statistics and Probability ,Random field ,Welch's method ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Spectral density ,Estimator ,Lipschitz continuity ,01 natural sciences ,010104 statistics & probability ,Bartlett's method ,Orthonormal basis ,0101 mathematics ,Statistics, Probability and Uncertainty ,Random variable ,Mathematics - Abstract
The two-dimensional periodogram has been proposed as an estimator of the spectral density of a real, homogeneous, random field defined over a regular lattice on the plane. In the present paper, results pertaining to the asymptotic distributional properties of such a periodogram are obtained. These results generalize some of the work of Hannan (1960), Walker (1965) and more directly Olshen (1967a,b) concerning asymptotic theory for the periodogram of a stationary time series. Although extension of asymptotic theory for one-dimensional periodograms to a parallel theory for two-dimensional periodograms is not completely straightforward (one runs into problems akin to the problems encountered in extending the theory of one-dimensional trigonometric series to two dimensions), further extensions to asymptotic theory for p-dimensional periodograms (p > 2) is easily accomplished by an obvious mimicry of the definitions, theorems and proofs for the two-dimensional case. Since the notation required for the p-dimensional case is rather cumbersome, we have chosen to give results only for two-dimensional periodograms.
- Published
- 1971
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