Length distributions for random secants through a convex region K are derived for three types of randomness. The results are formulated in terms of geometric properties of K, e.g. the overlap surface content of K with its translated self. The distribution of distance between two random points in K, expressed in terms of the overlap volume, is shown to extend to non-convex (including disjoint) regions.

We consider a discrete-time risk process driven by proportional reinsurance and an interest rate process. We assume that the interest rate process behaves as a Markov chain. To reduce the risk of ruin, we may reinsure a part or even all of the reserve. Recursive and integral equations for ruin probabilities are given. Generalized Lundberg inequalities for the ruin probabilities are derived given a stationary policy. To illustrate these results, a numerical example is included.

In this paper we consider an M/M/c queue modified to allow both mass arrivals when the system is empty and the workload to be removed. Properties of queues which terminate when the server becomes idle are firstly developed. Recurrence properties, equilibrium distribution, and equilibrium queue-size structure are studied for the case of resurrection and no mass exodus. All of these results are then generalized to allow for the removal of the entire workload. In particular, we obtain the Laplace transformation of the transition probability for the absorptive M/M/c queue.

Customers initially enter a service unit via a waiting room. The customers to be served are stored in a service room which is replenished by the transfer of all those in the waiting room at the points in time where the service room becomes empty. At those epochs of transfer, positive random numbers of ‘overhead customers' are also added to the service room. Algorithmically tractable expressions for the stationary distributions of queue lengths and waiting times at various embedded random epochs are derived. The discussion generalizes an earlier treatment by Takács in several directions.

It is known that for a Bienaymé– Galton–Watson process {Zn} whose mean m satisfies 1 < m < ∞, the limiting random variable in the strong limit theorem can be represented as a random sum of i.i.d. random variables and hence that convergence rate results follow from a random sum central limit theorem.

This paper develops an analogous theory for the case m = ∞ which replaces ‘sum' by ‘maximum'. In particular we obtain convergence rate results involving a limiting extreme value distribution. An associated estimation problem is considered.

Let {ξk : k ≧ 1} be a sequence of independent, identically distributed random variables with E{ξ1} = μ ≠ 0. Form the random walk {Sn : n ≧ 0} by setting S0, Sn = ξ1 + ξ2 + ··· + ξn, n ≧ 1. Define the random function Xn by setting where α is a norming constant. Let N denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of ξ1) that the finite-dimensional distributions of Xn, conditioned on n < N < ∞ converge to those of the Brownian excursion process.

Based on a Wiener process approximation, a sequential test for the bundle strength of filaments is proposed and studied here. Asymptotic expressions for the OC and ASN functions are derived, and it is shown that asymptotically the test is more efficient than the usual fixed sample size procedure based on the asymptotic normality of the standarized form of the bundle strength of filaments, studied earlier by Daniels (1945), and Sen, Bhattacharyya and Suh (1973).

Limit processes for sequences of partial sums of non-linear regression residuals are obtained under assumptions on regressor functions imposed by Jennrich (1969). The limit process and covariance kernel are calculated explicitly for functions of exponential type.

We investigate some preservation properties of two nonparametric classes of survival distributions and their duals, under appropriate reliability operations. The aging properties defining these nonparametric classes are based on comparing the mean life of a new unit to the mean residual life function of the asymptotic remaining survival time of the unit under repeated perfect repairs. They are motivated from a point of view that realistic notions of degradation, applicable to repairable systems, should be based on contrasting some aspect of the remaining life of a repairable unit (under a given repair strategy, such as renewals) to the life of a new unit.

In this paper we consider the operation of the move-to-front scheme where the requests form a Markov chain of N states with transition probability matrix P. It is shown that the configurations of items at successive requests form a Markov chain, and its transition probability matrix has eigenvalues that are the eigenvalues of all the principal submatrices of P except those of order N—1. We also show that the multiplicity of the eigenvalues of submatrices of order m is the number of derangements of N — m objects. The last result is shown to be true even if P is not a stochastic matrix.

In this paper we study the excursion time of a Brownian motion with drift outside a corridor by using a four-state semi-Markov model. In mathematical finance, these results have an important application in the valuation of double-barrier Parisian options. We subsequently obtain an explicit expression for the Laplace transform of its price.

Various authors have derived the necessary and sufficient conditions for optimality in semi-Markov decision processes in which the state remains constant between jumps. In this paper similar results are presented for a generalized semi-Markov decision process in which the state varies between jumps according to a Markov process with continuous sample paths. These results are specialized to a general storage model and an application to the service rate control in a GI/G/1 queue is indicated.

This note concerns the distribution of the equilibrium M/G/1 queue length. A representation for the probability generating function is given which allows for an explicit finite sum representation of the associated probabilities. The radius of convergence of the probability generating function and an asymptotic formula for the right tail of the distribution also follow from this representation, as well as infinite divisibility of the queue-length distribution when the service distribution is infinitely divisible. Extension of these results to the bulk arrival case is straightforward.

It is shown that the limiting random variable W(si) of an irregular branching process with infinite mean, defined in [5], has a continuous and positive distribution on {0 < W(si) < ∞}. This implies that for all branching processes (Zn) with infinite mean there exists a function U such that the distribution of V = limnU(Zn)e–n a.s. is continuous, positive and finite on the set of non-extinction. A kind of law of large numbers for sequences of independent copies of W(si) is derived.

In his paper ‘Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test’ (J. Appl. Prob.8, 431–453), Durbin derives an integral equation whose kernel has a singularity. Since direct solution of an approximating set of simultaneous equations would be very inaccurate, he uses probability arguments to approximate to integrals of sub-intervals. In this note, two alternative procedures are discussed. One makes a linear transformation of the original integral equation to eliminate the singularity; the other, due to Weiss and Anderssen, integrates the singular factor in the kernel over the sub-interval. Computation of a special case indicates this latter method to be the most effective of the three.

We consider a generalised Vervaat perpetuity of the form X = Y1W1 +Y2W1W2 + · · ·, where $W_i \sim {\cal U}^{1/t}$ and (Yi)i≥0 is an independent and identically distributed sequence of random variables independent from (Wi)i≥0. Based on a distributional decomposition technique, we propose a novel method for exactly simulating the generalised Vervaat perpetuity. The general framework relies on the exact simulation of the truncated gamma process, which we develop using a marked renewal representation for its paths. Furthermore, a special case arises when Yi = 1, and X has the generalised Dickman distribution, for which we present an exact simulation algorithm using the marked renewal approach. In particular, this new algorithm is much faster than existing algorithms illustrated in Chi (2012), Cloud and Huber (2017), Devroye and Fawzi (2010), and Fill and Huber (2010), as well as being applicable to the general payments case. Examples and numerical analysis are provided to demonstrate the accuracy and effectiveness of our method.

A continuum structure function γ is a non-decreasing mapping from the unit hypercube to the unit interval. Block and Savits (1984) use the sets and to determine bounds on the distribution of γ (X) when X is a vector of associated random variables. It is shown that, if γ admits of a modular decomposition, improved bounds may be obtained.

This paper is devoted to a study of the integral of the workload process of the single server queue, in particular during one busy period. Firstly, we find asymptotics of the area 𝒜 swept under the workload process W(t) during the busy period when the service time distribution has a regularly varying tail. We also investigate the case of a light-tailed service time distribution. Secondly, we consider the problem of obtaining an explicit expression for the distribution of 𝒜. In the general GI/G/1 case, we use a sequential approximation to find the Laplace—Stieltjes transform of 𝒜. In the M/M/1 case, this transform is obtained explicitly in terms of Whittaker functions. Thirdly, we consider moments of 𝒜 in the GI/G/1 queue. Finally, we show asymptotic normality of .

An object is hidden in one of two boxes and occasionally moves between the boxes in accordance with some specified continuous-time Markov process. The objective is to find the object with a minimal expected cost. In this paper it is assumed that search efforts are unlimited. In addition to the search costs, the ‘real time' until the object is found is also taken into account in the cost structure. Our main results are that the optimal policy may consist of five regions and that the controls applied should be of the extreme 0 or ∞ type. The resulting expected cost compares favorably with that of the expected cost with bounded controls studied previously in the search literature.