In this paper a unified approach is used for proving relationships between customer-stationary and time-stationary characteristics of service systems with varying service rate and point processes. This approach is based on an intensity conservation principle for general stationary continuous-time processes with imbedded stationary marked point processes. It enables us to work under weaker independence assumptions than usual in queueing theory.

We consider the problem of conditioning a continuous-time Markov chain (on a countably infinite state space) not to hit an absorbing barrier before time T; and the weak convergence of this conditional process as T → ∞. We prove a characterization of convergence in terms of the distribution of the process at some arbitrary positive time, t, introduce a decay parameter for the time to absorption, give an example where weak convergence fails, and give sufficient conditions for weak convergence in terms of the existence of a quasi-stationary limit, and a recurrence property of the original process.

In the spirit of Albrecher and Hipp (2007), and Albrecher, Renaud, and Zhou (2008) we consider a Lévy insurance risk model with tax payments of a more general structure than in the aforementioned papers, which was also considered in Albrecher, Borst, Boxma, and Resing (2009). In terms of scale functions, we establish three fundamental identities of interest which have stimulated a large volume of actuarial research in recent years. That is to say, the two-sided exit problem, the net present value of tax paid until ruin, as well as a generalized version of the Gerber–Shiu function. The method we appeal to differs from Albrecher and Hipp (2007), and Albrecher, Renaud, and Zhou (2008) in that we appeal predominantly to excursion theory.

Let F be a probability distribution function with density f. We assume that (a) F has finite moments of any integer positive order and (b) the classical problem of moments for F has a nonunique solution (F is M-indeterminate). Our goal is to describe a , where h is a ‘small' perturbation function. Such a class S consists of different distributions Fε (fε is the density of Fε) all sharing the same moments as those of F, thus illustrating the nonuniqueness of F, and of any Fε, in terms of the moments. Power transformations of distributions such as the normal, log-normal and exponential are considered and for them Stieltjes classes written explicitly. We define a characteristic of S called an index of dissimilarity and calculate its value in some cases. A new Stieltjes class involving a power of the normal distribution is presented. An open question about the inverse Gaussian distribution is formulated. Related topics are briefly discussed.

Let Xn, n ≥ 1 be a sequence of trials taking values in a given set A, let ∊ be a pattern (simple or compound), and let Xr,∊ be a random variable denoting the waiting time for the rth occurrence of ∊. In the present article a finite Markov chain imbedding method is developed for the study of Xr,∊ in the case of the non-overlapping and overlapping way of counting runs and patterns. Several extensions and generalizations are also discussed.

A queue with Poisson arrivals and two different exponential servers is considered. It is assumed that customers are allowed to stall, i.e., to wait for a busy fast server at times when the slow server is free. A stochastic analysis of the queue is given, steady-state probabilities are computed, and policies for overall optimization are characterized and computed. The issue of individual customer's optimization versus overall optimization is also discussed.

The asymptotic final size distribution of a multitype Reed–Frost process, a chain-binomial model for the spread of an infectious disease in a finite, closed multitype population, is derived, as the total population size grows large. When all subgroups are of comparable size, the infection pattern irreducible and the epidemic started by a small number of initial infectives, the classical threshold behaviour is obtained, depending on the basic reproduction rate of the disease in the population, and the asymptotic distributions for small and large outbreaks can be found. The same techniques can then be used to study other asymptotic situations, e.g. small groups in an otherwise large population, large numbers of initial infectives and reducible infection patterns.

We consider a class of secretary problems in which the order of arrival of candidates is no longer uniformly distributed. By a suitable embedding in a time-transformed extremal process it is shown that the asymptotic winning probability is again 1/e as in the classical situation. Extensions of the problem to more than one choice are also considered.

Small sample and large sample properties of the bundle strength of parallel filaments studied earlier by Daniels (1945) and Sen, Bhattacharyya, Suh (1969) have been developed here by probabilistic argument. The statistics belong to a family or class of statistics, each of which forms a reverse semi-martingale sequence. Certain moment properties are also discussed.

Some minor corrections to Heyde and Seneta (1972) are made, and new convergence rate results given. Estimation by recurrence methods is discussed, as announced earlier.

We consider the problem of approximating the stationary distribution of a positive-recurrent Markov chain with infinite transition matrix P, by stationary distributions computed from (n × n) stochastic matrices formed by augmenting the entries of the (n × n) northwest corner truncations of P, as n →∞.

This paper arose from interest in assessing the quality of random number generators. The problem of testing randomness of a string of binary bits produced by such a generator gained importance with the wide use of public key cryptography and the need for secure encryption algorithms. All such algorithms are based on a generator of (pseudo) random numbers; the testing of such generators for randomness became crucial for the communications industry where digital signatures and key management are vital for information processing.

The concept of approximate entropy has been introduced in a series of papers by S. Pincus and co-authors. The corresponding statistic is designed to measure the degree of randomness of observed sequences. It is based on incremental contrasts of empirical entropies based on the frequencies of different patterns in the sequence. Sequences with large approximate entropy must have substantial fluctuation or irregularity. Alternatively, small values of this characteristic imply strong regularity, or lack of randomness, in a sequence. Pincus and Kalman (1997) evaluated approximate entropies for binary and decimal expansions of e, π, √2 and √3 with the surprising conclusion that the expansion of √3 demonstrated much less irregularity than that of π. Tractable small sample distributions are hardly available, and testing randomness is based, as a rule, on fairly long strings. Therefore, to have rigorous statistical tests of randomness based on this approximate entropy statistic, one needs the limiting distribution of this characteristic under the randomness assumption. Until now this distribution remained unknown and was thought to be difficult to obtain. To derive the limiting distribution of approximate entropy we modify its definition. It is shown that the approximate entropy as well as its modified version converges in distribution to a χ2-random variable. The P-values of approximate entropy test statistics for binary expansions of e, π and √3 are plotted. Although some of these values for √3 digits are small, they do not provide enough statistical significance against the randomness hypothesis.

In this paper we investigate sufficient conditions that ensure the optimality of threshold strategies for optimal stopping problems with finite or perpetual maturities. Our result is based on a local-time argument that enables us to give an alternative proof of the smooth-fit principle. Moreover, we present a class of optimal stopping problems for which the propagation of convexity fails.

In this paper we show that the problem of existence and uniqueness of stationary distributions for piecewise-deterministic Markov processes (PDPs) is equivalent to the same problem for the associated Markov chain, so long as some mild conditions on the parameters of the PDP are satisfied. Our main result is the construction of an invertible mapping from the set of stationary distributions for the PDP to the set of stationary distributions for the Markov chain. Some sufficient conditions for existence are presented and an application to capacity expansion is given.

This paper presents the theory of a multivariate birth-and-death process and its representation as a branching process. The bivariate linear birth-and-death process may be used as a model for various epidemic situations involving two types of infective. Various properties of the transient process are discussed and the distribution of epidemic size is investigated. For the case of a disease spread solely by carriers when the two types of infective are carriers and clinical infectives the large population version of a model proposed by Downton (1968) is further developed and shown under appropriate circumstances to closely approximate Downton's model.

A class of multivariate exponential distributions is suggested which includes those presented by Marshall and Olkin, Downton, and Hawkes. It is based on a concept of hierarchical successive damage. Recursive expressions for the Laplace transforms and for first and second moments are derived in the bivariate case. Related multivariate geometric distributions are described.

Probability generation functions of waiting time distributions of runs and patterns have been used successfully in various areas of statistics and applied probability. In this paper, we provide a simple way to obtain the probability generating functions for waiting time distributions of compound patterns by using the finite Markov chain imbedding method. We also study the characters of waiting time distributions for compound patterns. A computer algorithm based on Markov chain imbedding technique has been developed for automatically computing the distribution, probability generating function, and mean of waiting time for a compound pattern.

We give a unified presentation of stability results for stochastic vector difference equations based on various choices of binary operations and , assuming that are stationary and ergodic. In the scalar case, under standard addition and multiplication, the key condition for stability is E[log |A0|] < 0. In the generalizations, the condition takes the form γ< 0, where γis the limit of a subadditive process associated with . Under this and mild additional conditions, the process has a unique finite stationary distribution to which it converges from all initial conditions.

The variants of standard matrix algebra we consider replace the operations + and × with (max, +), (max,×), (min, +), or (min,×). In each case, the appropriate stability condition parallels that for the standard recursions, involving certain subadditive limits. Since these limits are difficult to evaluate, we provide bounds, thus giving alternative, computable conditions for stability.

Dependence structures for bivariate extremal events are analyzed using particular types of copula. Weak convergence results for copulas along the lines of the Pickands-Balkema-de Haan theorem provide limiting dependence structures for bivariate tail events. A characterization of these limiting copulas is also provided by means of invariance properties. The results obtained are applied to the credit risk area, where, for intensity-based default models, stress scenario dependence structures for widely traded products such as credit default swap baskets or first-to-default contract types are proposed.

We prove that if the renewal function M(t) corresponding to a life distribution F is convex (concave) then F is NBU (NWU), and hence answer two questions posed by Shaked and Zhu (1992). Moreover, based-on the renewal function, some characterizations of the exponential distribution within certain classes of life distributions are given.