Novel Mittag-Leffler stability of linear fractional delay difference equations with impulse

In this letter we propose a class of linear fractional difference equations with discrete-time delay and impulse effects. The exact solutions are obtained by use of a discrete Mittag-Leffler function with delay and impulse. Besides, we provide comparison principle, stability results and numerical illustration.     

An alternative approach to Privault's discrete-time chaotic calculus

In this paper, we present an alternative approach to Privault's discrete-time chaotic calculus. Let Z be an appropriate stochastic process indexed by N (the set of nonnegative integers) and l²(Γ) the space of square summable functions defined on Γ (the finite power set of N). First we introduce a stochastic integral operator J with respect to Z, which, unlike discrete multiple Wiener integral operators, acts on l²(Γ). And then we show how to define the gradient and divergence by means of the operator J and the combinatorial properties of l²(Γ). We also prove in our setting the main results of the discrete-time chaotic calculus like the Clark formula, the integration by parts formula, etc. Finally we show an application of the gradient and divergence operators to quantum probability.     

On the discrete-time Geo/G/1 queue with randomized vacations and at most J vacations

This paper examines a discrete-time Geo/G/1 queue, where the server may take at most J − 1 vacations after the essential vacation. In this system, messages arrive according to Bernoulli process and receive corresponding service immediately if the server is available upon arrival. When the server is busy or on vacation, arriving messages have to wait in the queue. After the messages in the queue are served exhaustively, the server leaves for the essential vacation. At the end of essential vacation, the server activates immediately to serve if there are messages waiting in the queue. Alternatively, the server may take another vacation with probability p or go into idle state with probability (1 − p) until the next message arrives. Such pattern continues until the number of vacations taken reaches J. This queueing system has potential applications in the packet-switched networks. By applying the generating function technique, some important performance measures are derived, which may be useful for network and software system engineers. A cost model, developed to determine the optimum values of p and J at a minimum cost, is also studied.     

Generalized Markov Models of Infectious Disease Spread: A Novel Framework for Developing Dynamic Health Policies

We propose a class of mathematical models for the transmission of infectious diseases in large populations. This class of models, which generalizes the existing discrete-time Markov chain models of infectious diseases, is compatible with efficient dynamic optimization techniques to assist real-time selection and modification of public health interventions in response to evolving epidemiological situations and changing availability of information and medical resources. While retaining the strength of existing classes of mathematical models in their ability to represent the within-host natural history of disease and between-host transmission dynamics, the proposed models possess two advantages over previous models: (1) these models can be used to generate optimal dynamic health policies for controlling spreads of infectious diseases, and (2) these models are able to approximate the spread of the disease in relatively large populations with a limited state space size and computation time.     

A single-server discrete-time retrial <Emphasis Type="Italic">G</Emphasis>-queue with server breakdowns and repairs

This paper concerns a discrete-time Geo/Geo/1 retrial queue with both positive and negative customers where the server is subject to breakdowns and repairs due to negative arrivals. The arrival of a negative customer causes one positive customer to be killed if any is present, and simultaneously breaks the server down. The server is sent to repair immediately and after repair it is as good as new. The negative customer also causes the server breakdown if the server is found idle, but has no effect on the system if the server is under repair. We analyze the Markov chain underlying the queueing system and obtain its ergodicity condition. The generating function of the number of customers in the orbit and in the system are also obtained, along with the marginal distributions of the orbit size when the server is idle, busy or down. Finally, we present some numerical examples to illustrate the influence of the parameters on several performance characteristics of the system.     

The randomized threshold for the discrete-time Geo/G/1 queue

This paper discusses a discrete-time Geo/G/1 queue, in which the server operates a random threshold policy, namely 〈p, N〉 policy, at the end of each service period. After all the messages are served in the queue exhaustively, the server is immediately deactivated until N messages are accumulated in the queue. If the number of messages in the queue is accumulated to N, the server is activated for services with probability p and deactivated with probability (1 − p). Using the generating functions technique, the system state evolution is analyzed. The generating functions of the system size distributions in various states are obtained. Some system characteristics of interest are derived. The long-run average cost function per unit time is analytically developed to determine the joint optimal values of p and N at a minimum cost.     

A multi-parametric programming approach for multilevel hierarchical and decentralised optimisation problems

In this paper, we outline the foundations of a general global optimisation strategy for the solution of multilevel hierarchical and general decentralised multilevel problems, based on our recent developments on multi-parametric programming and control theory. The core idea is to recast each optimisation subproblem, present in the hierarchy, as a multi-parametric programming problem, with parameters being the optimisation variables belonging to the remaining subproblems. This then transforms the multilevel problem into single-level linear/convex optimisation problems. For decentralised systems, where more than one optimisation problem is present at each level of the hierarchy, Nash equilibrium is considered. A three person dynamic optimisation problem is presented to illustrate the mathematical developments.