Using a characterizing equation for the beta distribution, Stein's method is applied to obtain bounds of the optimal order for the Wasserstein distance between the distribution of the scaled number of white balls drawn from a Pólya-Eggenberger urn and its limiting beta distribution. The bound is computed by making a direct comparison between characterizing operators of the target and the beta distribution, the former derived by extending Stein's density approach to discrete distributions. In addition, refinements are given to Döbler's (2012) result for the arcsine approximation for the fraction of time a simple random walk of even length spends positive, and so also to the distributions of its last return time to 0 and its first visit to its terminal point, by supplying explicit constants to the present Wasserstein bound and also demonstrating that its rate is of the optimal order.

This paper is an investigation into the reliability and stochastic properties of three-state networks. We consider a single-step network consisting of n links and we assume that the links are subject to failure. We assume that the network can be in three states, up (K = 2), partial performance (K = 1), and down (K = 0). Using the concept of the two-dimensional signature, we study the residual lifetimes of the networks under different scenarios on the states and the number of failed links of the network. In the process of doing so, we define variants of the concept of the dynamic signature in a bivariate setting. Then, we obtain signature based mixture representations of the reliability of the residual lifetimes of the network states under the condition that the network is in state K = 2 (or K = 1) and exactly k links in the network have failed. We prove preservation theorems showing that stochastic orderings and dependence between the elements of the dynamic signatures (which relies on the network structure) are preserved by the residual lifetimes of the states of the network (which relies on the network ageing). Various illustrative examples are also provided.

We consider a generalized stochastic epidemic on a Bernoulli random graph. By constructing the epidemic and graph in unison, the epidemic is shown to be a randomized Reed–Frost epidemic. Hence, the exact final-size distribution and extensive asymptotic results can be derived.

Using Slepian processes as a standard of comparison, estimates are given for the probability that a centered multiparameter stationary Gaussian process reaches a constant barrier u on a subset of the parameter domain.

The general cost structure of a unit on line is assumed to be governed by a stochastic process , where R(t) denotes the operating cost on [0, t)and ζ denotes the time of an unscheduled (or unplanned) replacement by a new unit at a cost c1. For an age replacement maintenance policy, scheduled (or planned) replacements occur whenever an operating unit reaches age T, whereas in the block replacement case, scheduled replacements occur every T units of time. Such scheduled replacements cost c2. The expected long-run cost per unit time can then be expressed in the form A(T)/Ε [min(ζ, T)] and B(T)/T respectively. Our main result shows that where U is the associated renewal function generated by ζ.

In the paper the problem of the extinction of non-homogeneous Galton-Watson processes with one type of particle is studied. It is proved that the process becomes extinct if the expectations of the process do not converge to a limit (finite or infinite). If the expectations have a finite limit, then simple necessary and sufficient conditions for the extinction are proved. The general case remains open; however two more sufficient conditions which are also necessary under some restrictions are given.

The concept of cost has not, until recently, been investigated in any detail within the framework of epidemic theory. In the case of minor illnesses a pertinent estimate of cost may be the man-hours lost in industry during an outbreak. For the general epidemic, in which the removals are immune recoveries (or deaths), this is conveniently represented by the area generated by the trajectory in time t of infectives I(t) until the epidemic ceases. A small element of area under the trajectory is I(t)δt and represents I(t) infectives withdrawn from the labour force for a small time δt; the total work-time lost is thus represented by where T is the duration of the epidemic.

We provide some results for the comparison of the failure times and interfailure times of two systems based on a replacement policy proposed by Kapodistria and Psarrakos (2012). In particular, we show that when the first failure times are ordered in terms of the dispersive order (or, the excess wealth order), then the successive interfailure times are ordered in terms of the usual stochastic order (respectively, the increasing convex order). As a consequence, we provide comparison results for the cumulative residual entropies of the systems and their dynamic versions.

A number of items are arranged in a line. At each unit of time one of the items is requested, the i th being requested with probability Pi. We consider rules which reorder the items between successive requests in a fashion which depends only on the position in which the most recently requested item was found. It has been conjectured that the rule which always moves the requested item one closer to the front of the line minimizes the average position of the requested item. An example with six items shows that the conjecture is false.

Cohen (1969) has studied the transient and stationary queue length distributions for the M/G/1, K queue, with a fixed maximum number of customers, K, in the system at any time. The present note applies Cohen's method to generalize his results to the MX/GY/1, K queue.

We consider discrete-time dynamic scheduling problems of the following three types of G/G/1 queue with K different customer classes: (i) a G/DFR/1 queue with K classes under preemptive resume service discipline, (ii) a G/IFR/1 queue with two classes under preemptive resume service discipline, and (iii) a G/G/1 queue with two classes under non-preemptive service discipline. Interchange arguments are used to show that simple index policies of different type minimize the total holding cost of customers in a finite-horizon scheduling period for the three cases. Our results extend the result for a G/M/1 queue by Buyukkoc et al. (1985) to general queues.

This paper considers a modified block replacement with two variables and general random minimal repair cost. Under such a policy, an operating system is preventively replaced by new ones at times kT (k= 1, 2, ···) independently of its failure history. If the system fails in [(k − 1)T, (k − 1)T+ T0) it is either replaced by a new one or minimally repaired, and if in [(k − 1) T + T0, kT) it is either minimally repaired or remains inactive until the next planned replacement. The choice of these two possible actions is based on some random mechanism which is age-dependent. The cost of the ith minimal repair of the system at age y depends on the random part C(y) and the deterministic part ci (y). The expected cost rate is obtained, using the results of renewal reward theory. The model with two variables is transformed into a model with one variable and the optimum policy is discussed.

In a recent paper Rényi (1964) has obtained the following characterization of the Poisson process.

Consider a sequence of exchangeable or independent binary trials ordered on a line or on a circle. The statistics denoting the number of times an F-S string of length (at least) k1 + k2, that is, (at least) k1 failures followed by (at least) k2 successes in n such trials, are studied. The associated waiting time for the rth occurrence of an F-S string of length (at least) k1 + k2 in linearly ordered trials is also examined. Exact formulae, lower/upper bounds and approximations are derived for their distributions. Mean values and variances of the number of occurrences of F-S strings are given in exact formulae too. Particular exchangeable and independent sequences of binary random variables, used in applied research, combined with numerical examples clarify further the theoretical results.

In this paper, some ordering properties of convolutions of heterogeneous Bernoulli random variables are discussed. It is shown that, under some suitable conditions, the likelihood ratio order and the reversed hazard rate order hold between convolutions of two heterogeneous Bernoulli sequences. The results established here extend and strengthen the previous results of Pledger and Proschan (1971) and Boland, Singh and Cukic (2002).

Expressions are derived for the third and fourth moments of the distribution of the numbers of sides of the polygons formed by random lines in a plane. These are evaluated numerically; it seems unlikely that they can be expressed simply in terms of elementary functions or constants. Using these results, the third and fourth moments of the distribution of the perimeters of the polygons are derived.

Starting from the theory of point processes the concept of a process with an embedded marked point process is defined. It is shown that the known formula expressing the relation between the stationary and synchronous version of a regenerative process remains valid without the assumption of independence of cycles. General formulae for stationary availability and interval reliability of complex systems with repair are also obtained. In this way generalizations of Keilson's results for Markovian systems and Ross's results for systems with separately maintained elements are presented. The formulae are applied to a two-unit parallel system with a single repair facility.

The study of high-dimensional distributions is of interest in probability theory, statistics, and asymptotic convex geometry, where the object of interest is the uniform distribution on a convex set in high dimensions. The ℓp-spaces and norms are of particular interest in this setting. In this paper we establish a limit theorem for distributions on ℓp-spheres, conditioned on a rare event, in a high-dimensional geometric setting. As part of our proof, we establish a certain large deviation principle that is also relevant to the study of the tail behavior of random projections of ℓp-balls in a high-dimensional Euclidean space.

In this paper we study polynomial and geometric (exponential) ergodicity for M/G/1-type Markov chains and Markov processes. First, practical criteria for M/G/1-type Markov chains are obtained by analyzing the generating function of the first return probability to level 0. Then the corresponding criteria for M/G/1-type Markov processes are given, using their h-approximation chains. Our method yields the radius of convergence of the generating function of the first return probability, which is very important in obtaining explicit bounds on geometric (exponential) convergence rates. Our results are illustrated, in the final section, in some examples.

In this paper we consider a class of quasi-birth-and-death processes for which explicit solutions can be obtained for the rate matrix R and the associated matrix G. The probabilistic interpretations of these matrices allow us to describe their elements in terms of paths on the two-dimensional lattice. Then determining explicit expressions for the matrices becomes equivalent to solving a lattice path counting problem, the solution of which is derived using path decomposition, Bernoulli excursions, and hypergeometric functions. A few applications are provided, including classical models for which we obtain some new results.