In this paper we consider a general Lévy process X reflected at a downward periodic barrier At and a constant upper barrier K, giving a process VKt=Xt +LAt−LKt. We find the expression for a loss rate defined by lK=ELK1 and identify its asymptotics as K→∞ when X has light-tailed jumps and EX1<0.

A continuum of stochastic dominance rules, also referred to as fractional stochastic dominance (SD), was introduced by Müller, Scarsini, Tsetlin, and Winkler (2017) to cover preferences from first- to second-order SD. Fractional SD can be used to explain many individual behaviors in economics. In this paper we introduce the concept of fractional pure SD, a special case of fractional SD. We investigate further properties of fractional SD, for example the generating processes of fractional pure SD via $\gamma$ -transfers of probability, Yaari’s dual characterization by utilizing the special class of distortion functions, the separation theorem in terms of first-order SD and fractional pure SD, Strassen’s representation, and bivariate characterization. We also establish several closure properties of fractional SD under quantile truncation, under comonotonic sums, and under distortion, as well as its equivalence characterization. Examples of distributions ordered in the sense of fractional SD are provided.

This paper is motivated by relations between association and independence of random variables. It is well known that, for real random variables, independence implies association in the sense of Esary, Proschan and Walkup (1967), while, for random vectors, this simple relationship breaks. We modify the notion of association in such a way that any vector-valued process with independent increments also has associated increments in the new sense - association between blocks. The new notion is quite natural and admits nice characterization for some classes of processes. In particular, using the covariance interpolation formula due to Houdré, Pérez-Abreu and Surgailis (1998), we show that within the class of multidimensional Gaussian processes, block association of increments is equivalent to supermodularity (in time) of the covariance functions. We also define corresponding versions of weak association, positive association, and negative association. It turns out that the central limit theorem for weakly associated random vectors due to Burton, Dabrowski and Dehling (1986) remains valid, if the weak association is relaxed to the weak association between blocks.

A linear list is a collection of items that can be accessed sequentially. The cost of a request is the number of items that need to be examined before the desired item is located, i.e. the distance of the requested item from the beginning of the list. The transposition rule is one of the algorithms designed to reduce the search cost by organizing the list. In particular, upon a request for a given item, the item is transposed with the preceding one. We develop a new approach for analyzing the algorithm, based on a coupling to a certain constrained asymmetric exclusion process. This allows us to establish an asymptotic optimality of the rule for two families of request distributions.

Customers arriving at a queue do not usually see its time-average behavior unless arrivals occur according to a Poisson process. In this article we study how nearly customers see time-average behavior. We give total variation error bounds for comparing the distance between the time- and customer-average distributions of a queueing system in terms of properties of the interarrival distribution. Some refinements are given for special cases and numerical computations are used to demonstrate the performance of the inequalities.

The limiting behavior of the number of runs up to time n of a regenerative phenomenon on the positive integers is related to the one of the number of regenerations. Applications are considered in fluctuation theory, and for the GI/G/1 queue.

The classical technique of uniformization (or randomization) for bounded continuous-time Markov chains and Markov reward structures is extended to dynamic systems generated by arbitrary non-negative generators. Most notably, these include so-called input-output models in economic analysis. The results are of practical interest for both computational and theoretical purposes. Particularly, the recursive computation and the limiting behaviour of cumulative reward structures for non-negative dynamic systems is concluded as a special application. Two numerical examples are included to illustrate the conditions and the results.

We define a class of anticipative flows on Poisson space and compute its Radon-Nikodym derivative. This result is applied to statistical testing in an anticipative queueing problem.

Miyazawa and Taylor (1997) introduced a class of assemble-transfer batch service queueing networks which do not have tractable stationary distribution. However by assuming a certain additional arrival process at each node when it is empty, they obtain a geometric product-form stationary distribution which is a stochastic upper bound for the stationary distribution of the original network. In this paper we develop a stochastic lower bound for the original network by introducing an additional departure process at each node which tends to remove all the customers present in it. This model in combination with the aforementioned upper bound model gives a better sense for the properties of the original network.

A general two-dimensional random walk is considered with a barrier along the y-axis. Absorption probabilities are derived when the barrier is absorbing, and when it is semi-reflecting.

A carrier-borne epidemic model, in which the carrier removal process is influenced by the number of detected infected individuals, is considered. The distribution of the total size of the epidemic is found and expressions for the expected ultimate size of the epidemic and the expected area under the trajectory of carriers are given.

The stochastic version of the simple epidemic is considered in this paper. A method which has computational advantages is developed for computing the probability distribution of the process. The method is based on the Jordan canonical form of the matrix representing the system of differential-difference equations for the simple epidemic. It is also shown that the method can be applied to the general epidemic as well as any other right-shift process.

Consider situations where the treatment may cause an initial effect and may also cause a long-range effect. We want to evaluate the treatment, or to compare two treatments, when the effect of treatment may result from the two distinct mechanisms, M1 and M2. We may wish to evaluate M1 and M2 separately, but we may also want to evaluate their combined effect M3. Examples are given and the general results are applied to the special case arising in weather modification studies and elsewhere: the possible effects are multiplicative and the distribution of non-zero variables is gamma with at most the scale parameter affected by treatment. An example demonstrates that the two components may be too weak to be judged significant while their sum is large and significant. The locally optimum C(α) test is used.

There is a brief discussion of the power function of the tests. The asymptotic power agrees well, in general, by the results of the Monte Carlo simulation for the test Z3 of the combined effect. If the zero values are discarded and then Z2 employed, there is large bias in the power. The bias is more pronounced if the Wilcoxon, Mann–Whitney test is employed. Notice that the two effects under study may be acting in the same direction or they may be in opposition.

Draw two lines, L1 and L2, in a plane. Along L1 place the points of a Poisson process, and through each point draw a line, the angles of intersection with L1 being distributed independently and uniformly on (0, π) . The intercepts of these random lines with L2 form a new process, which is Poisson if and only if L1 and L2 are parallel. Curiously, if L1 and L2 are inclined then, with probability 1, the new process forms a dense subset of L2. It is not really even a point process. In the present paper we shall investigate this anomaly and some of its generalizations.

We consider non-homogeneous Markov chains generated by the simulated annealing algorithm. We classify states according to asymptotic properties of trajectories. We identify recurrent and transient states. The set of recurrent states is partitioned into disjoint classes of asymptotically communicating states. These classes correspond to atoms of the tail sigma-field. The results are valid under the weak reversibility assumption of Hajek.

Large stochastic systems of marked particles are considered. These ‘populations' grow according to pairwise mutually catalytic reactions and in addition particles may exchange their type (mark) by a sampling procedure. We are interested in the explosive behavior of the model and its high density properties (law of large numbers).

The Itô formula is the fundamental theorem of stochastic calculus. This short note presents a new proof of Itô's formula for the case of continuous semimartingales. The new proof is more geometric than previous approaches, and has the particular advantage of generalizing immediately to the multivariate case without extra notational complexity.

Let ℰ be an event governed by a homogeneous second-order two-state Markov chain. Assume the steady state has been achieved. Consider Nm the number of times that ℰ occurs in m successive trials. The limiting distribution of P(Nm = k) is obtained under the condition that mP(ℰ)=λ.

A problem of regrinding and recycling worn train wheels leads to a Markov population process with distinctive properties, including a product-form equilibrium distribution. A convenient framework for analyzing this process is via the notion of dynamic reversal, a natural extension of ordinary (time) reversal. The dynamically reversed process is of the same type as the original process, which allows a simple derivation of some important properties. The process seems not to belong to any class of Markov processes for which stationary distributions are known.

A number of items are to be distributed among n individuals, arranged in a line and labelled by the integers 1, 2, ···, n, in such a way that the distance between any two members presented should be at least d, a given number, but otherwise any subset of the population should have the same probability of each of its members being presented. The number of items thus distributed is a random variable X(d, n) and it turns out that X(d, n) is asymptotically a fixed percentage of the population size n. This fact makes it possible to apply the model in a number of situations. For example, if in a foreign aid program a number of tools are offered and to be distributed in a fair way but, for lack of sufficient quantity, one tool is expected to be used by a number of neighbours, the model is applicable if the amount of aid is not fixed in advance but may vary within certain limits. A similar situation arises in cases of disaster (floods, earthquakes, etc.) when the victims are sent gifts, again in an insufficient quantity, hence for the sake of justice, a family of a smaller size is supposed to receive only one gift. In many other similar cases, the model can be used.