Non zero-sum stochastic games in admission,service and routing control in queueing systems

The purpose of this paper is to investigate situations of non-cooperative dynamic control of queueing systems by two agents, having different objectives. The main part of the paper is devoted to analyzing a problem of an admission and a service (vacation) control. The admission controller has to decide whether to allow arrivals to occur. Once the queue empties, the server goes on vacation, and controls the vacations duration (according to the state and past history of the queue). The immediate costs for each controller are increasing in the number of customers, but no convexity assumptions are made. The controllers are shown to have a stationary equilibrium policy pair, for which each controller uses a stationary threshold type policy with randomization in at most one state. We then investigate a problem of a non-zero sum stochastic game between a router into several queues, and a second controller that allocates some extra service capacity to one of the queues. We establish the equilibrium of a policy pair for which the router uses the intuitive Join the shortest queue policy.     

Chaotic system optimal tracking using data-based synchronous method with unknown dynamics and disturbances

We develop an optimal tracking control method for chaotic system with unknown dynamics and disturbances. The method allows the optimal cost function and the corresponding tracking control to update synchronously. According to the tracking error and the reference dynamics, the augmented system is constructed. Then the optimal tracking control problem is defined. The policy iteration(PI) is introduced to solve the min-max optimization problem. The off-policy adaptive dynamic programming(ADP) algorithm is then proposed to find the solution of the tracking Hamilton–Jacobi–Isaacs(HJI) equation online only using measured data and without any knowledge about the system dynamics. Critic neural network(CNN), action neural network(ANN), and disturbance neural network(DNN) are used to approximate the cost function, control, and disturbance. The weights of these networks compose the augmented weight matrix, and the uniformly ultimately bounded(UUB) of which is proven. The convergence of the tracking error system is also proven. Two examples are given to show the effectiveness of the proposed synchronous solution method for the chaotic system tracking problem.     

Continuous Accumulation Games in Continuous Regions

In a continuous accumulation game on a continuous region, a Hider distributes material over a continuous region at each instant of discrete time, and a Seeker examines the region. If the Seeker locates any of the material hidden, the Seeker confiscates it. The goal of the Hider is to accumulate a certain amount of material before a given time, and the goal of the Seeker is to prevent this. In previous works, we have studied accumulation games involving discrete objects and continuous material over discrete locations. The issues raised when the region is continuous are substantially different. In this paper, we study accumulation of continuous material over two types of continuous regions: the interval and the circle.     

Zero-sum delta-systems and multiple copies of graphs

A main result proved in this paper is the following. Theorem. Let G be a noncomplete graph on n vertices with degree sequence d₁ ≥ d₂ ≥ · · · ≥ d_n and t ≥ 2 be a prime. Let m = gcd{t, d_i − d_j: 1 ≤ i < j ≤ n} and set Then R(tG, ℤ_t) = t(n + d) − d, where R is the zero-sum Ramsey number. This settles, almost completely, problems raised in [Bialostocki & Dierker, J Graph Theory, 1994; Y. Caro, J Graph Theory, 1991]. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 207–216, 1999     

Extremal product-one free sequences in Dihedral and Dicyclic Groups

Let $G$ be a finite group, written multiplicatively. The Davenport constant of $G$ is the smallest positive integer $D\left(G\right)$ such that every sequence of $G$ with $D\left(G\right)$ elements has a non-empty subsequence with product $1$. Let $D_{2n}$ be the Dihedral Group of order $2n$ and $Q_{4n}$ be the Dicyclic Group of order $4n$. Zhuang and Gao (2005) showed that $D\left(D_{2n}\right)=n+1$ and Bass (2007) showed that $D\left(Q_{4n}\right)=2n+1$. In this paper, we give explicit characterizations of all sequences $S$ of $G$ such that $\left|S\right|=D\left(G\right)?1$ and $S$ is free of subsequences whose product is 1, where $G$ is equal to $D_{2n}$ or $Q_{4n}$ for some $n$.     

On the existence of the stabilizing solution of generalized Riccati equations arising in zero-sum stochastic difference games: the time-varying case

In this paper, a large class of time-varying Riccati equations arising in stochastic dynamic games is considered. The problem of the existence and uniqueness of some globally defined solution, namely the bounded and stabilizing solution, is investigated. As an application of the obtained existence results, we address in a second step the problem of infinite-horizon zero-sum two players linear quadratic (LQ) dynamic game for a stochastic discrete-time dynamical system subject to both random switching of its coefficients and multiplicative noise. We show that in the solution of such an optimal control problem, a crucial role is played by the unique bounded and stabilizing solution of the considered class of generalized Riccati equations.     

An Iterative Approach to Determining the Length of the Longest Common Subsequence of Two Strings

This paper concerns the longest common subsequence (LCS) shared by two sequences (or strings) of length N, whose elements are chosen at random from a finite alphabet. The exact distribution and the expected value of the length of the LCS, k say, between two random sequences is still an open problem in applied probability. While the expected value E(N) of the length of the LCS of two random strings is known to lie within certain limits, the exact value of E(N) and the exact distribution are unknown. In this paper, we calculate the length of the LCS for all possible pairs of binary sequences from N=1 to 14. The length of the LCS and the Hamming distance are represented in color on two all-against-all arrays. An iterative approach is then introduced in which we determine the pairs of sequences whose LCS lengths increased by one upon the addition of one letter to each sequence. The pairs whose score did increase are shown in black and white on an array, which has an interesting fractal-like structure. As the sequence length increases, R(N) (the proportion of sequences whose score increased) approaches the Chvátal–Sankoff constant a _c (the proportionality constant for the linear growth of the expected length of the LCS with sequence length). We show that R(N) is converging more rapidly to a _c than E(N)/N.