This particular three-dimensional random packing limit problem is to determine the mean fraction of a cubic space that would be occupied by aligned, fixed, equalsize cubes, placed at random locations sequentially until no more can be added. No analytical solution has yet been found for this problem. Simulation results for a finite region and finite number of attempts were extrapolated to an infinite number of attempts (N →∞) in an infinite region by multiple linear regression, using volume fraction occupied (F) as a linear combination of the ratio of the length of the small cube sides (S) to the length of the cubic region side (L) and the cube root of the ratio of the region volume to the total volume of cubes tried, (L3/NS3)⅓. These results for random packing in a volume with penetrable walls can be adjusted with a multiplicative correction factor to give the results for impenetrable walls. A total of N = 107 attempts at placement were made for L/S = 20/1 and N = 14 × 106 attempts were made for L/S = 10/1. The results for volume fraction packed are correlated by F = 0.430(±0.008) + 0.966(±0.072)(S/L) – 0.236(±0.029)(L3/NS)⅓. The numbers in parentheses are twice the standard errors of estimate of the coefficients, indicating the 95% confidence intervals due to random errors. This value for the packing density limit, 0.430 ± 0.008, is slightly larger than that given by a conjecture by Palásti [10], 0.4178. Our value is consistent with that obtained by rather different simulation methods by Jodrey and Tory [8], 0.4227 ± 0.0006, and by Blaisdell and Solomon [2], 0.4262.

Asymptotic formulas for means and variances of a multitype decomposable age-dependent supercritical branching process are derived. This process is a generalization of the Kendall–Neyman–Scott two-stage model for tumor growth. Both means and variances have exponential growth rates as in the case of the Markov branching process. But unlike Markov branching, these asymptotic moments depend on the age of the original individual at the start of the process and the life span distribution of the progenies.

The asymptotic behaviour of the Ek/G/1 queue with finite waiting room is studied. Using a combination of the supplementary variable and phase techniques, queue length and waiting time distributions are obtained. Also the idle period distribution and mean length of the idle and busy periods are found.

A characterization of the exponential distribution based on a relevation-type equation and its discrete version are extended to the case of multidimensional spaces via variants of the Choquet–Deny theorem. Comments on some recent results in the literature are made.

Under proportional transaction costs, a price process is said to have a consistent price system, if there is a semimartingale with an equivalent martingale measure that evolves within the bid-ask spread. We show that a continuous, multi-asset price process has a consistent price system, under arbitrarily small proportional transaction costs, if it satisfies a natural multi-dimensional generalization of the stickiness condition introduced by Guasoni (2006).

We show that the solution of a quasi-renewal equation with an exponential distribution of the renewals converges at infinity and we compute explicitly the limit, hence generalizing the classical renewal theorem. We apply this result to a stochastic model of DNA replication introduced by Cowan and Chiu (1994).

A method, making use of characteristic functions, is presented to prove theorems on characterization of distributions by independence of order statistics. Since exact independence in practice is rarely achieved, much attention is being given in the literature to theorems on determining the distributions for which certain statistics are ‘almost’ independent. The technique of this new line of investigation is strongly dependent on the fact that for the case of exact independence there is a solution in terms of characteristic functions. The present work was guided by this fact. In addition to the new method, Theorem 2 appears to be new.

For fixed accuracy functions, let Gnm be a non-discrete duel in which the amounts of ammunition available to the two players are n and m, respectively. Let vnm be the value of Gnm and let the support of the optimal pure strategies for the players be (anm, 1). The asymptotic relationships among the quantities n, m, vnm and anm are investigated.

In a homogeneous Poisson process in Rd, consider an arbitrary point X and let Y be its kth nearest neighbour. Denote by Rk the rank of X in the proximity order defined by Y, i.e., Rk = j if X is the jth nearest neighbour to Y. A representation for Rk in terms of a sum of independent random variables is obtained, and the limiting distribution of Rk, as k →∞, is shown to be normal. This result generalizes to mixtures of Poisson processes.

Let (Xi)i∈ℕ be a sequence of independent and identically distributed random variables with values in the set ℕ0 of nonnegative integers. Motivated by applications in enumerative combinatorics and analysis of algorithms we investigate the number of gaps and the length of the longest gap in the set {X1,…,Xn} of the first n values. We obtain necessary and sufficient conditions in terms of the tail sequence (qk)k∈ℕ0, qk=P(X1≥ k), for the gaps to vanish asymptotically as n→∞: these are ∑k=0∞qk+1/qk <∞ and limk→∞qk+1/qk=0 for convergence almost surely and convergence in probability, respectively. We further show that the length of the longest gap tends to ∞ in probability if qk+1/qk→ 1. For the family of geometric distributions, which can be regarded as the borderline case between the light-tailed and the heavy-tailed situations and which is also of particular interest in applications, we study the distribution of the length of the longest gap, using a construction based on the Sukhatme–Rényi representation of exponential order statistics to resolve the asymptotic distributional periodicities.

Sequence symmetries in DNA and RNA are being discovered at an increasing rate. Conjectures and hypotheses are being proposed for their possible structural and functional role in the nucleic acid. In this paper a probability model is studied which evaluates the probabilities of various repeats occurring by chance alone. Expressions are derived for the mean and variance of the statistics employed. The central limit theorem for dependent trials is used to obtain the asymptotic distributions. An indication is given of how to use the model to search for various gene amplification events in the evolutionary history of the sequences.

Conditions for finiteness of moments of the following quantities have been found: the duration of a busy period of an Μ /G/∞ system; the duration of a partial busy period of an M/G/C loss system, and the duration of a partial busy period of an M/G/C queue.

The chord length distributions of planar convex sets are discussed, particularly the density values at the extremes of the range; there is a qualitative distinction between polygons and sets with smooth boundaries. The distance between convex sets is related to the distance between distribution functions.

In some applications of diffusion theory it is of interest to study the expected time of the first exit from an interval given that a specified boundary will be the first exitpoint. This problem is considered here for a regular Feller process, and applications are given.

We use the properties of the Matuszewska indices to show asymptotic inequalities for hazard rates. We discuss the relation between membership in the classes of dominatedly or extended rapidly varying tail distributions and corresponding hazard rate conditions. Convolution closure is established for the class of distributions with extended rapidly varying tails.

We consider a multiple server queueing loss system where the service times of server i are exponential with rate μi, where μi decreases in i. Arrivals have associated vectors (X1, …, Xn) of binary variables, with Xi = 1 indicating that server i is eligible to serve that arrival. Arrivals finding no idle eligible servers are lost. Letting Ij be the indicator variable for the event that the jth arrival enters service, we show that, for any arrival process, the policy that assigns arrivals to the smallest numbered idle eligible server stochastically maximizes the vector (I1, …, Ir) for every r if the eligibility vector of arrivals is either (a) exchangeable, or (b) a vector of independent variables for which P(Xi = 1) increases in i.

We investigate the effect of a particular mode of approximation by means of four examples of its use; in each case the model approximated is a Markov process with discrete states in continuous time.

We consider the SETAR(l; 1, ···, 1) model: where – ∞= r0 < r1, < · ·· < rl = ∞ and for each j {ε t(j)} forms an i.i.d. zero-mean error sequence independent of {ε t(i)} for i ≠ j and having a density positive on the real line. Chan et al. (1985) obtained the region of the parameter space on which the process is ergodic, and showed the process to be transient on a subset of the remainder. They conjectured that the process was null recurrent everywhere else. In this paper we show that conjecture to be incorrect and under the assumption of finite variance of the error distributions we resolve the remaining questions of transience or null recurrence for this process.

We consider a class of G/G/1 queueing models with independent generalized setup time and exhaustive service. It is shown that a variety of single-server queueing systems with service interruption are special cases of our model. We give a simple computational scheme for the moments of the stationary waiting time and sojourn time. Our numerical investigations indicate that the algorithm is quite accurate and fast in general. For the M/G/1 case, we are able to derive a recursive formula for the moments of the stationary waiting time, which includes the Takács formula as a special case. It immediately results in the stochastic decompòsition property which can be found in the literature.

We consider explicit formulae for equilibrium prices in a continuous-time vertical contracting model. A manufacturer sells goods to a retailer, and the objective of both parties is to maximize expected profits. Demand is an Itô-Lévy process, and to increase realism, information is delayed. We provide complete existence and uniqueness proofs for a series of special cases, including geometric Brownian motion and the Ornstein-Uhlenbeck process, both with time-variable coefficients. Moreover, explicit solution formulae are given, so these results are operational. An interesting finding is that information that is more precise may be a considerable disadvantage for the retailer.