The paper considers some properties of the reversible counters system of Lampard [1]. Lampard considers the properties of the process imbedded at the epochs when the counter receiving pulses from the ‘down’ process reaches zero. We consider here the time-dependent and stationary distributions of the original process. Their derivation gives rise to some interesting functional equations, besides being useful for studying queues with Markov-dependent inter-arrival or service times.

Consider two queues with k servers. Inequalities between their sample paths and thence between the distribution functions of their waiting times and queue sizes are deduced from inequalities between the conditional distribution functions, given previous interarrival and service times, for the two queues.

We consider an additive shock process where shocks occur according to a Poisson point process and they are accumulated in an appropriate way to the damage. It is shown that suitably normalized shock processes converge weakly to a process which is represented as a sum of a stable process and a deterministic process.

For fixed accuracy functions, let Gnm be a discrete-fire (silent, noisy, or silent-noisy) duel in which the amounts of ammunition available to the two players are n and m, respectively. As n and m both become large, a class of ‘ε -optimal' pure strategies (with support (anm, 1)) is derived. If vnm denotes the value of Gnm, the asymptotic relationships among the quantities n, m, vnm, and anm are investigated.

We study statistical properties of the range (= number of distinct sites visited) of a lattice random walk in discrete time constrained to visit a given site at a given time. In particular, we calculate the mean and obtain a bound on the variance of the range in the large time limit. The results are applied to a problem involving an unconstrained random walk in the presence of randomly distributed traps. A key role is played by the associated random walk that is obtained from the original random walk via a Cramer transform.

This paper deals with the preservation of strong forms of mixing of point processes under the clustering operation. Strong, ϕ -, and *-mixing possess an increasing degree of uniformity of their asymptotic independence, and this turns out to be a significant factor in their preservation. In particular, it is indicated that ϕ-mixing may be maintained only under very stringent conditions (bounded clusters), whereas strong mixing is maintained under considerably milder conditions.

In this paper we use the Siegert formula to derive alternative expressions for the moments of the first passage time of the Ornstein-Uhlenbeck process through a constant threshold. The expression for the nth moment is recursively linked to the lower-order moments and consists of only n terms. These compact expressions can substantially facilitate (numerical) applications also for higher-order moments.

This short note offers a new proof of the following fact: in a discrete-time arbitrage-free market model, a contingent claim is attainable if and only if its expected value is the same under all equivalent martingale measures. The proof is based on Rogers' (1994) proof of the Dalang-Morton-Willinger (1990) theorem.

Recurrence equations for the number of types and the frequency of each type in a random sample drawn from a finite population undergoing discrete, nonoverlapping generations and reproducing according to the Cannings exchangeable model are deduced under the assumption of a mutation scheme with infinitely many types. The case of overlapping generations in discrete time is also considered. The equations are developed for the Wright-Fisher model and the Moran model, and extended to the case of the limit coalescent with nonrecurrent mutation as the population size goes to ∞ and the mutation rate to 0. Computations of the total variation distance for the distribution of the number of types in the sample suggest that the exact Moran model provides a better approximation for the sampling formula under the exact Wright-Fisher model than the Ewens sampling formula in the limit of the Kingman coalescent with nonrecurrent mutation. On the other hand, this model seems to provide a good approximation for a Λ-coalescent with nonrecurrent mutation as long as the probability of multiple mergers and the mutation rate are small enough.

We consider the queueing system denoted by M/MN/1/N where customers are served in batches of maximum size N. The model is motivated by a traffic application. The time-dependent probability distribution for the number of customers in the system is obtained in closed form. The solution is used to predict the optimal service rates during a finite time horizon.

In this paper, we show that the time-average distributions of excess, age, and spread are given by the solution of first-order differential equations. These differential equations can be directly derived in a simple, unified way using a general level-crossing formula based on the balance of up and down crossings on sample paths, which may be helpful for the intuitive interpretation.

Most research in first-passage percolation has been done under the assumption of a finite mean for the underlying time coordinate distribution. We demonstrate that the basic ergodic results can be derived under a weaker moment assumption, which still permits us to evaluate the time constant in the case where the atom at zero of the time coordinate distribution exceeds one-half. Further almost sure convergence is investigated more closely.

We consider a Cramér–Lundberg insurance risk process with the added feature of reinsurance. If an arriving claim finds the reserve below a certain threshold γ, or if it would bring the reserve below that level, then a reinsurer pays part of the claim. Using fluctuation theory and the theory of scale functions of spectrally negative Lévy processes, we derive expressions for the Laplace transform of the time to ruin and of the joint distribution of the deficit at ruin and the surplus before ruin. We specify these results in much more detail for the threshold set-up in the case of proportional reinsurance.

In this paper, by using the tail asymptotics derived by Dębicki et al. (2016), we prove the Gumbel limit laws for the maximum of a class of nonhomogeneous Gaussian random fields. As an application of the main results, we derive the Gumbel limit law for Shepp statistics of fractional Brownian motion and Gaussian integrated processes.

Consider a sequence X1,…,Xn of independent random variables with the same continuous distribution and the event Xi-r+1 < ⋯ < Xi of the appearance of an increasing sequence with length r, for i=r,…,n. Denote by W the number of overlapping appearances of the above event in the sequence of n trials. In this work, we derive bounds for the total variation and Kolmogorov distances between the distribution of W and a suitable compound Poisson distribution. Via these bounds, an associated theorem concerning the limit distribution of W is obtained. Moreover, using the previous results we study the asymptotic behaviour of the length of the longest increasing sequence. Finally, we suggest a non-parametric test based on W for checking randomness against local increasing trend.

In this paper we propose a strategy that gives an optimal lower bound of the average gain for the two-envelope problem within the McDonnell and Abbott (2009) and McDonnell et al. (2011) framework. We obtain this result with partial information about the probability distribution of the envelope's contents.

The Greenberg–Hastings model (GHM) is a simple cellular automaton which emulates two properties of excitable media: excitation by contact and a refractory period. We study two ways in which external stimulation can make ring dynamics in the GHM recurrent. The first scheme involves the initial placement of excitation centres which gradually lose strength, i.e. each time they become inactive (and then stay inactive forever) with probability 1 − pf. In this case, the density of excited sites must go to 0; however, their long–term connectivity structure undergoes a phase transition as pf increases from 0 to 1. The second proposed rule utilizes continuous nucleation in that new rings are started at every rested site with probability ps. We show that, for small ps, these dynamics make a site excited about every ps−1/3 time units. This result yields some information about the asymptotic shape of a closely related random growth model.

Queueing models have been used to describe the slowing or stopping of a stream of vehicles due to a traffic light, a merging disturbance, or an accident. The buildup of stopped vehicles, and their eventual dispersal after the delaying factor has been removed, has been likened to the busy period of a queue, e.g., Jewell (1964). Some studies have dealt with the total delay to all drivers involved in such a queue, Daley and Jacobs (1969), Gaver (1969); or the combination of this total delay and the distance of the last vehicle from the originating incident, McNeil (1969), Shaw (1970).

The problem of how to compare repairable systems has long been of interest to those who are engaged in reliability analysis. In this article we have introduced a type of comparison of two counting processes. Various properties and applications of our stochastic comparison are also studied.

It is known that when costs are unbounded satisfaction of the appropriate dynamic programming ‘optimality' equation by a policy is not sufficient to guarantee its average optimality. A ‘lowest-order potential' condition is introduced which, along with the dynamic programming equation, is sufficient to establish the optimality of the policy. Also, it is shown that under fairly general conditions, if the lowest-order potential condition is not satisfied there exists a non-memoryless policy with smaller average cost than the policy satisfying the dynamic programming equation.