A general multi-type branching process where new individuals immigrate according to some point process is considered. An intrinsic submartingale is defined and a convergence result for processes counted with random characteristics is obtained. Some examples are given.

Lenin et al. (2000) recently introduced the idea of similarity in the context of birth-death processes. This paper examines the extent to which their results can be extended to arbitrary Markov chains. It is proved that, under a variety of conditions, similar chains are strongly similar in a sense which is described, and it is shown that minimal chains are strongly similar if and only if the corresponding transition-rate matrices are strongly similar. A general framework is given for constructing families of strongly similar chains; it permits the construction of all such chains in the irreducible case.

A necessary and sufficient condition based on order statistics that a positive random variable having an absolutely continuous probability distribution (with respect to Lebesgue measure) will be exponential is given.

Consider n cells into which balls are thrown at random until all but m cells contain at least l + 1 balls each. Asymptotic results when n →∞, m and l held fixed, are given for the number of cells containing exactly k balls and for related random variables.

The ‘rendezvous time’ of two stochastic processes is the first time at which they cross or hit each other. We consider such times for a Brownian motion with drift, starting at some positive level, and a compound Poisson process or a process with one random jump at some random time. We also ask whether a rendezvous takes place before the Brownian motion hits zero and, if so, at what time. These questions are answered in terms of Laplace transforms for the underlying distributions. The analogous problem for reflected Brownian motion is also studied.

This paper considers a dam of infinite capacity with a discrete-valued stationary input process and a unit release whenever possible. It is shown how, by suitable manipulations of the equation governing the dam content process, the stationary distribution of the dam being empty can be obtained, as also can (with a few additional assumptions) the expected value of the dam content in the stationary case. Results obtained are applied to particular cases of input — independent and identical, Markov, Bivariate Markov and moving-average.

We consider branching processes for which the first three moments of the distribution of offspring exist. Let f(t) andz be, respectively, the generating function of the distribution of offspring and the smallest positive root of the equation f(t) = t. Then if M = f'(z) and fn(t) is equal to the generating function of the distribution of nth generation descendants of a single individual, it is known that, quite generally, (z – fn(0)) / Mn tends toward a constant as n increases. A method is derived for obtaining upper and lower bounds for this constant, which gives an exact solution when there is a geometric distribution of offspring and good bounds when there are Poisson or negative binomial offspring distributions. With some further calculations, one can also obtain finite upper and lower bounds for the mean time to extinction of a line descended from an individual, given there is extinction. These bound apply even ifzis less than 1. Numerical values are given for the Poisson and negative binomial cases.

We consider a stochastic evolutionary model for a phenotype developing amongst n related species with unknown phylogeny. The unknown tree is modelled by a Yule process conditioned on n contemporary nodes. The trait value is assumed to evolve along lineages as an Ornstein-Uhlenbeck process. As a result, the trait values of the n species form a sample with dependent observations. We establish three limit theorems for the sample mean corresponding to three domains for the adaptation rate. In the case of fast adaptation, we show that for large n the normalized sample mean is approximately normally distributed. Using these limit theorems, we develop novel confidence interval formulae for the optimal trait value.

Let X1, X2, …, Xn be independent random variables uniformly distributed on [0,1]. We observe these sequentially and have to stop on exactly one of them. No recall of preceding observations is permitted. What stopping rule minimizes the expected rank of the selected observation? What is the value of the expected rank (as a function of n) and what is the limit of this value when n goes to ∞? This full-information expected selected-rank problem is known as Robbins' problem of minimizing the expected rank, and its general solution is unknown. In this paper we provide an alternative approach to Robbins' problem. Our model is similar to that of Gnedin (2007). For this, we consider a continuous-time version of the problem in which the observations follow a Poisson arrival process on ℝ+ × [0,1] of homogeneous rate 1. Translating the previous optimal selection problem in this setting, we prove that, under reasonable assumptions, the corresponding value function w(t) is bounded and Lipschitz continuous. Our main result is that the limiting value of the Poisson embedded problem exists and is equal to that of Robbins' problem. We prove that w(t) is differentiable and also derive a differential equation for this function. Although we have not succeeded in using this equation to improve on bounds on the optimal limiting value, we argue that it has this potential.

Sequential decision problems can often be modeled as Markov decision processes. Classical solution approaches assume that the parameters of the model are known. However, model parameters are usually estimated and uncertain in practice. As a result, managers are often interested in how estimation errors affect the optimal solution. In this paper we illustrate how sensitivity analysis can be performed directly for a Markov decision process with uncertain reward parameters using the Bellman equations. In particular, we consider problems involving (i) a single stationary parameter, (ii) multiple stationary parameters, and (iii) multiple nonstationary parameters. We illustrate the applicability of this work through a capacitated stochastic lot-sizing problem.

The process of deterioration of repairable systems with each repair is modeled using converging geometric-type processes. It is proved that the expectation of the number of repairs in each interval of time is infinite. A new regularization procedure is suggested and the corresponding optimization problem is discussed.

Mr. G owes $100 000 to a loan shark, and will be killed at dawn if the loan is not repaid in full. Mr. G has $20 000, but partial payments are not accepted, and he has no other source of income or credit. The loan shark owns a primitive casino where one can stake any amount in one's possession, gaining r times the stake with probability w and losing the stake with probability 1 - w (r > 0, 0 < w < 1). Mr. G is permitted to gamble at the casino, but each time he places a bet, the amount of his debt is increased by a factor of 1 + α (α ≥ 0). How should Mr. G gamble to maximize his chance of reaching his (moving) target and thereby surviving? Dubins and Savage showed that an optimal strategy is to stake boldly if the primitive casino is subfair or fair (i.e. w(1 + r) ≤ 1) and the inflation rate α is 0. Intuitively, a positive inflation rate would motivate Mr. G to try to reach his goal as quickly as possible, so it seems plausible that the bold strategy is optimal. However, Chen, Shepp, and Zame found that, surprisingly, the bold strategy is no longer optimal for subfair primitive casinos with inflation if both r > 1 and α satisfies 1/r ≤ α < r. They also conjectured that the bold strategy is optimal for subfair primitive casinos with inflation if r < 1. It is shown in the present paper that this conjecture is true provided that w ≤ ½. Furthermore, by introducing an interesting notion of sharp strategy, additional results are obtained on optimality of the bold strategy.

We present an exact simulation algorithm for the stationary distribution of the customer delay D for first-in–first-out (FIFO) M/G/c queues in which ρ=λ/μ<1. We assume that the service time distribution G(x)=P(S≤x),x≥0 (with mean 0<E(S)=1/μ<∞), and its corresponding equilibrium distribution Ge(x)=μ∫0x P(S>y)dy are such that samples of them can be simulated. We further assume that G has a finite second moment. Our method involves the general method of dominated coupling from the past (DCFTP) and we use the single-server M/G/1 queue operating under the processor sharing discipline as an upper bound. Our algorithm yields the stationary distribution of the entire Kiefer–Wolfowitz workload process, the first coordinate of which is D. Extensions of the method to handle simulating generalized Jackson networks in stationarity are also remarked upon.

By using the alternating projection theorem of J. von Neumann, we obtain explicit formulae for the best linear interpolator and interpolation error of missing values of a stationary process. These are expressed in terms of multistep predictors and autoregressive parameters of the process. The key idea is to approximate the future by a finite-dimensional space.

In this study some of the basic ideas needed for the application of the Wiener-Hopf Technique in solving problems occurring in applied probability theory are discussed; the paper aims to give a short introduction. The method is illustrated by applying it to two problems; one, although basic in probability theory, is rather simple to handle by this method. The second is much more intricate, but shows clearly the power of the method.

We consider Weiss's and Downton's models with parametersπ, αand β depending on i number of susceptibles and j number of carriers. A martingale argument is performed when πand α /β only depend on i or, in Weiss's case, when α /β is the product of a function of i by a function of j. In these cases the martingale approach proves very valuable and gives explicit results quite easily. In particular it shows that well-known relations between moments and integrals along a trajectory are still true for any stopping time and for more general models than the classic ones.

A study is made of the extremal process generated by i.i.d. random variables appearing at the events of a non-homogeneous Poisson process, 𝒫. If 𝒫 has an exponentially increasing rate function, then records eventually occur in a homogeneous Poisson process. The size of the latest record has a classical extreme value distribution.

We consider a controlled M/M/1 queueing system where customers may be subject to two potential rejections. The first occurs upon arrival and is dependent on the number of customers in the queue and the service rate of the customer currently in service. The second, which may or may not occur, occurs immediately prior to the customer receiving service. That is, after each service completion the customer in the front of the queue is assessed and the service rate of that customer is revealed. If the second decision-maker recommends rejection, the customer is denied service with a fixed probability. We show the existence of long-run average optimal monotone switching-curve policies. Further, we show that the average reward is increasing in the probability that the second decision-maker's recommendation of rejection is honored. Applications include call centers with delayed classifications and manufacturing systems when the server is responsible for multiple tasks.

This paper is concerned with a continuous time parameter Markovian sequential decision process, and presents a method which transforms a given continuous parameter problem into a discrete one. It is proved that the optimal stationary policy for the resulting discrete time parameter Markovian decision process is also the optimal stationary policy for the original continuous one, and vice versa. The resulting discrete parameter problem may be more easily solved than the continuous one by applying the linear programming method. A simple numerical example is presented.

The condition of strong mixing for triangular arrays of random variables is a common condition of weak dependence. In this note, it is shown that this condition is not as general as one might believe. In particular, it is shown that there exist triangular arrays of first-order autoregressive random variables which converge almost surely to an independent identically distributed sequence of random variables and for which the central limit theorem holds, but which are not strong mixing triangular arrays. Hence, the strong mixing condition is more restrictive than desired.