In this paper, several linear two-dimensional consecutive k-type systems are studied, which include the linear connected-(k, r)-out-of-$(m,n)\colon\! F$ system and the linear l-connected-(k, r)-out-of-$(m,n)\colon\! F$ system without/with overlapping. Reliabilities of these systems are studied via the finite Markov chain imbedding approach (FMCIA) in a novel way. Some numerical examples are provided to illustrate the theoretical results established here and also to demonstrate the efficiency of the developed method. Finally, some possible applications and generalizations of the developed results are pointed out.

Let Xi,i ∈ ℕ, be independent and identically distributed random variables with values in ℕ0. We transform (‘prune’) the sequence {X1,…,Xn},n∈ ℕ, of discrete random samples into a sequence {0,1,2,…,Yn}, n∈ ℕ, of contiguous random sets by replacing Xn+1 with Yn +1 if Xn+1 >Yn. We consider the asymptotic behaviour of Yn as n→∞. Applications include path growth in digital search trees and the number of tables in Pitman's Chinese restaurant process if the latter is conditioned on its limit value.

We consider a multivariate distributional recursion of sum type, as arises in the probabilistic analysis of algorithms and random trees. We prove an upper tail bound for the solution using Chernoff's bounding technique by estimating the Laplace transform. The problem is traced back to the corresponding problem for binary search trees by stochastic domination. The result obtained is applied to the internal path length and Wiener index of random b-ary recursive trees with weighted edges and random linear recursive trees. Finally, lower tail bounds for the Wiener index of these trees are given.

Nash's generalization of Gittins’ classic index result to so-called generalized bandit problems (GBPs) in which returns are dependent on the states of all arms (not only the one which is pulled) has proved important for applications. The index theory for special cases of this model in which all indices are positive is straightforward. However, this is not a natural restriction in practice. An earlier proposal for the general case did not yield satisfactory index-based suboptimality bounds for policies — a central feature of classical Gittins index theory. We develop such bounds via a notion of duality for GBPs which is of independent interest. The index which emerges naturally from this analysis is the reciprocal of the one proposed by Nash.

It is known [8] that a certain class of bond-decorated graphs exhibits multiple AB percolation phase transitions. Sufficient conditions are given under which the corresponding AB percolation critical probabilities may be identified as points of intersection of the graph of a certain polynomial with the boundary of the percolative region of an associated two-parameter bond-site percolation model on the underlying undecorated graph. The main result of the article is used to prove that the graphs in [8] exhibit multiple AB percolation critical probabilities. The possibility of identifying AB percolation critical exponents with corresponding limits for the bond-site model is discussed.

It is shown here that questions about the probability distributions of the partial sums of a sequence of geometric distributions, all with different parameters, can be answered by considering the transition probabilities of a homogeneous Markov chain. The result is applied to the embedded random walk of an epidemic process.

Continuous-time threshold autoregressive (CTAR) processes have been developed in the past few years for modelling non-linear time series observed at irregular intervals. Several approximating processes are given here which are useful for simulation and inference. Each of the approximating processes implicitly defines conditions on the thresholds, thus providing greater understanding of the way in which boundary conditions arise.

Five distributions on the non-negative integers are given which use the orthogonal polynomials of Hermite, Legendre, Laguerre and Chebyshev.

Consider a device which begins operating at time t = 0 and then operates continuously in time. The operation of the device is satisfactory so long as a physical parameter X(t) of the device remains less than a critical value c > 0 (X(0) = 0). The device does not operate satisfactorily (i.e., it is “failed”) at a time t if X(t) ≧ c. Suppose that X(t) is a random variable with mean EX(t) = ta (a> 0), i.e., the device parameter X drifts with a linear rate a.

A general epidemic process is said to be completed whenever the number of susceptibles or the number of infectives reduces to zero. The mean duration time for this process to be completed is derived.

Distributions of some random variables have been characterized by independence of certain functions of these random variables. For example, let X and Y be two independent and identically distributed random variables having the gamma distribution. Laha showed that U = X + Y and V = X | Y are also independent random variables. Lukacs showed that U and V are independently distributed if, and only if, X and Y have the gamma distribution. Ferguson characterized the exponential distribution in terms of the independence of X – Y and min (X, Y). The best-known of these characterizations is that first proved by Kac which states that if random variables X and Y are independent, then X + Y and X – Y are independent if, and only if, X and Y are jointly Gaussian with the same variance. In this paper, Kac's hypotheses have been somewhat modified. In so doing, we obtain a larger class of distributions which we shall call class λ1. A subclass λ0 of λ1 enjoys many nice properties of the Gaussian distribution, in particular, in non-linear filtering.

A simple birth-death model of particle fluctuations is studied where at each discrete time a birth and/or death may occur. We show that if the probability of a birth does not depend on the number of particles present and if births and deaths are independent, then the times between successive deaths are independent geometrically distributed random variables, which is false in the general case. Since the above properties of the times between successive neuron firings have been observed in nerve cells, the model proposed in [2] obtains added credence.

This paper is a generalization to Markov chains of the work of Shepp [6] in the i.i.d case. Shepp studies the limiting values of the averages Tn = (Sn+ f(n) – Sn)/f(n) where Sn = X0 + X1+ · ·· + Xn, X0 = 0, n = 1, 2, ···, is a sum of mutually independent and identically distributed random variables. The function f takes positive integer values and non-decreasingly tends to infinity. Here we take a class of functions f in central position f(n) = [c log n], c > 0, n = 1, 2, ···. There are many refinements of the function f in the i.i.d case [1], [2]. Here we consider the more general case where X1, · ··, Xn is an irreducible and recurrent Markov chain. The state space of the chain is either compact or countable.

It is apparent that much of the congestion in road traffic is due to interference between vehicles at intersections, and this has led to the appearance of several mathematical discussions of the problems of delay at intersections of various types (Cohen and Stam (1963), Cohen and de Lange (1965), Hawkes (1965), (1966), Tanner (1962), Weiss and Maradudin (1962)). Comparatively little work has appeared on the problem of delay caused by vehicles turning right* across an opposing stream of vehicles, although Hawkes (1966) and Newell (1959) discuss this problem explicitly, and the models considered by all of these authors may be regarded as having some application to the right-turning problem.

An iterated sequence of Bernoulli trials is carried out and the success probability estimated at each point on the sequence by the current success ratio. We find the probability P1 that this estimate always lies above some pre-selected rational fraction p′, and its complement P2, the probability that it will reach p′ or below at least once. In the region p′ ≧ p, P1 = 0. In the region p′ < p, P1 ≠ 0 and is furthermore a discontinuous function of p′ at every rational p′.

Two models of a system with a single activated unit which can be in a finite number of states are considered. The unit is subject to Markovian deterioration, and it is possible to replace it before its failure. Inspections of the system are carried out at discrete time instants. The only difference between the two models is when the replacements take effect — immediately at the instant when the corresponding decision is made, or with the next inspection. The paper shows that this difference is much more essential than one might expect, and proves a relation between the optimal replacement strategies in the models concerned.

In this paper, N particles el, · ··, eN occupying positions in finite state spaces S1, · ··, 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 {Xt: 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.

For every hth member of a two-state Markov chain the value of a random variable Y is observed where the distribution of Y is conditional on the state of the corresponding member of the chain. A recursive set of equations is derived giving the posterior probabilities for both the observed and unobserved members. The use of this recursive solution to investigate the optimality of certain simple classification rules is discussed, and a “classification by runs” is also presented.

Signatures are useful in analyzing and evaluating coherent systems. However, their computation is a challenging problem, especially for complex coherent structures. In most cases the reliability of a binary coherent system can be linked to a tail probability associated with a properly defined waiting time random variable in a sequence of binary trials. In this paper we present a method for computing the minimal signature of a binary coherent system. Our method is based on matrix-geometric distributions. First, a proper matrix-geometric random variable corresponding to the system structure is found. Second, its probability generating function is obtained. Finally, the companion representation for the distribution of matrix-geometric distribution is used to obtain a matrix-based expression for the minimal signature of the coherent system. The results are also extended to a system with two types of components.

In this paper we study a large system of N servers, each with capacity to process at most C simultaneous jobs; an incoming job is routed to a server if it has the lowest occupancy amongst d (out of N) randomly selected servers. A job that is routed to a server with no vacancy is assumed to be blocked and lost. Such randomized policies are referred to JSQ(d) (Join the Shortest Queue out of d) policies. Under the assumption that jobs arrive according to a Poisson process with rate $N\lambda^{(N)}$ where $\lambda^{(N)}=\sigma-\frac{\beta}{\sqrt{N}\,}$ , $\sigma\in\mathbb{R}_+$ and $\beta\in\mathbb{R}$ , we establish functional central limit theorems for the fluctuation process in both the transient and stationary regimes when service time distributions are exponential. In particular, we show that the limit is an Ornstein–Uhlenbeck process whose mean and variance depend on the mean field of the considered model. Using this, we obtain approximations to the blocking probabilities for large N, where we can precisely estimate the accuracy of first-order approximations.