As the queue becomes exhausted, different maintenance tasks can be performed according to the fatigue load and wear degree of the service equipment. At the same time, considering the customer's sensitivity to time delay, the service facility will not completely remain inactive during the maintenance period. To describe this objectively existing phenomenon arising in the waiting line system, we consider a hyper-exponential working vacation queue with a batch renewal arrival process. Through the calculation of the well-structured roots of the associated characteristic equation, the shift operator method in the theory of difference equations and the supplementary variable technique for stochastic modeling plays a central role in the queue-length distribution analysis. Comparison with other ways to analyze queueing models, the advantage of our approach is that we can avoid deriving the complex transition probability matrix of the queue-length process embedded at input points. The feasibility of this approach is verified by extensive numerical examples.

In the classical simple random walk the steps are independent, that is, the walker has no memory. In contrast, in the elephant random walk, which was introduced by Schütz and Trimper [19] in 2004, the next step always depends on the whole path so far. Our main aim is to prove analogous results when the elephant has only a restricted memory, for example remembering only the most remote step(s), the most recent step(s), or both. We also extend the models to cover more general step sizes.

In this paper we discuss the Mabinogion urn model introduced by Williams (1991). Therein he describes an optimal control problem where the objective is to maximize the expected final number of objects of one kind in the Mabinogion urn model. Our main contribution is formulae for the expected time to absorption and its asymptotic behaviour in the optimally controlled process. We also present results for the noncontrolled Mabinogion urn process and briefly analyze other strategies that become superior if a certain discount factor is included.

In this paper we present a simple (fixed point) method that yields various results concerning approximate solutions of some difference equations. The results are motivated by the notion of Ulam stability.

In this note, we prove a uniqueness theorem for finite-order meromorphic solutions to a class of difference equations of Malmquist type. Such solutions $f$ are uniquely determined by their poles and the zeros of $f-e_{j}$ (counting multiplicities) for two finite complex numbers $e_{1}\neq e_{2}$.

Some major companies have the policy of annually giving numerical scores to their employees according to their performance, firing those whose performance scores are below a given percentile of the scores of all employees, and then recruiting new employees to replace those who were fired. We introduce a probabilistic model to describe how this practice affects the quality of employee performance as measured over time by the annual scores. Let n be the number of years that the policy has been in effect, and let Fn(x) be the distribution function of the evaluation scores in year n. We show, under certain technical assumptions, that the sequence (Fn(x)) satisfies a particular nonlinear difference equation, and furnish estimates of the solution of the equation and expressions for the quantiles of Fn. The mathematical tools that are used include convex functions, difference equations, and extreme value theory for independent and identically distributed random variables.

Based on the theory of difference equations, we derive necessary and sufficient conditions for the existence of eigenvalues and inverses of Toeplitz matrices with five different diagonals. In the course of derivations, we are also able to derive computational formulas for the eigenvalues, eigenvectors and inverses of these matrices. A number of explicit formulas are computed for illustration and verification.

In this paper, existence criteria for multiple solutions of periodic boundary value problems for the first-order difference equation are established by using the Leggett–Williams multiple fixed point theorem and fixed point theorem of cone expansion and compression. Two examples are also given to illustrate the main results.

The TCP window size process can be modeled as a piecewise-deterministic Markov process that increases linearly and experiences downward jumps at Poisson times. We present a transient analysis of this window size process. Our main result is the Laplace transform of the transient moments. Formulae for the integer and fractional moments are derived, as well as an explicit characterization of the speed of convergence to steady state. Central to our approach are the infinitesimal generator and Dynkin's martingale.

By means of a symbolic calculus for finding solutions of difference equations, we derive explicit eigenvalues, eigenvectors and inverses for tridiagonal Toeplitz matrices with four perturbed corners.

We study the multiserver queue with Poisson arrivals and identical independent servers with exponentially distributed service times. Customers arriving at the system are admitted or rejected according to a fixed threshold policy. Moreover, the system is subject to holding, waiting, and rejection costs. We give a closed-form expression for the average costs and the value function for this multiserver queue. The result will then be used in a single step of policy iteration in the model where a controller has to route to several finite-buffer queues with multiple servers. We numerically show that the improved policy has a close to optimal value.

In this paper one-dimensional correlated random walks (CRW) with various types of barrier such as elastic barriers, absorbing barriers and so on are defined, and explicit expressions are derived for the ultimate absorbing probability and expected duration. Some numerical examples to illustrate the effects of correlation are also presented.

A proof is given of a conjecture in the literature of geometrical probability that the r-content of the r-simplex whose r + 1 vertices are independent random points of which p are uniform in the interior and q uniform on the boundary of a unit n-ball (1 ≦ r ≦ n; 0 ≦ p, q ≦ r + 1, p + q = r + 1) is asymptotically normal (n →∞) with asymptotic mean and variance and , respectively.