ON CUSTOMERS ACTING AS SERVERS

We consider systems comprised of two interlacing M/M/ • /• type queues, where customers of each queue are the servers of the other queue. Such systems can be found for example in file sharing programs, SETI@home project, and other applications [Arazi, A, E Ben-Jacob and U Yechiali (2005). Controlling an oscillating Jackson-type network having state-dependant service rates. Mathematical Methods of Operations Research, 62, 453–466]. Denoting by Li the number of customers in queue i(Qi), i = 1, 2, we assume that Q1 is a multi-server finite-buffer system with an overall capacity of size N, where the customers there are served by the L2 customers present in Q2. Regarding Q2, we study two different scenarios described as follows: (i) All customers present in Q1 join hands together to form a single server for the customers in Q2, with service time exponentially distributed with an overall intensity μ2L1. That is, the service rate of the customers in Q2 changes dynamically, following the state of Q1. (ii) Each of the customers present in Q1individually acts as a server for the customers in Q2, with service time exponentially distributed with mean 1/μ2. In other words, the number of servers at Q2 changes according to the queue size fluctuations of Q1. We present a probabilistic analysis of such systems, applying both Matrix Geometric method and Probability Generating Functions (PGFs) approach, and derive the stability condition for each model, along with its two-dimensional stationary distribution function. We reveal a relationship between the roots of a given matrix, related to the PGFs, and the stability condition of the systems. In addition, we calculate the means of Li, i = 1, 2, along with their correlation coefficient, and obtain the probability of blocking at Q1. Finally, we present numerical examples and compare between the two models.

PHYSICAL ORIGIN OF THE POWER-LAW TAILED STATISTICAL DISTRIBUTIONS

Starting from the BBGKY hierarchy, describing the kinetics of nonlinear particle system, we obtain the relevant entropy and stationary distribution function. Subsequently, by employing the Lorentz transformations we propose the relativistic generalization of the exponential and logarithmic functions. The related particle distribution and entropy represents the relativistic extension of the classical Maxwell–Boltzmann distribution and of the Boltzmann entropy, respectively, and define the statistical mechanics presented in [Phys. Rev. E66 (2002) 056125] and [Phys. Rev. E72 (2005) 036108]. The achievements of the present effort, support the idea that the experimentally observed power-law tailed statistical distributions in plasma physics, are enforced by the relativistic microscopic particle dynamics.

Kinetic closure conditions for quasi-stationary collisionless axisymmetric magnetoplasmas

A characteristic feature of fluid theories concerns the difficulty of uniquely defining consistent closure conditions for the fluid equations. In fact it is well known that fluid theories cannot generally provide a closed system of equations for the fluid fields. This feature is typical of collisionless plasmas where, in contrast to collisional plasmas, asymptotic closure conditions do not follow as a consequence of an H-theorem This issue is of particular relevance in astrophysics where fluid approaches are usually adopted. On the other hand, it is well known that the determination of the closure conditions is in principle achievable in the context of kinetic theory. In the case of multi-species thermal magnetoplasmas this requires the determination of the species tensor pressure and of the corresponding heat fluxes. In this paper we investigate this problem in the framework of the Vlasov-Maxwell description for collisionless axisymmetric magnetoplasmas arising in astrophysics, with particular reference to accretion discs around compact objects (like black holes and neutron stars). The dynamics of collisionless plasmas in these environments is determined by the simultaneous presence of gravitational and magnetic fields, where the latter may be both externally produced and self-generated by the plasma currents. Our starting point here is the construction of a solution for the stationary distribution function describing slowly-varying gyrokinetic equilibria. The treatment is applicable to non-relativistic axisymmetric systems characterized by temperature anisotropy and differential rotation flows. It is shown that the kinetic formalism allows one to solve the closure problem and to consistently compute the relevant fluid fields with the inclusion of finite Larmor-radius effects. The main features of the theory and relevant applications are discussed.

Fluid queues driven by a discouraged arrivals queue

We consider a fluid queue driven by a discouraged arrivals queue and obtain explicit expressions for the stationary distribution function of the buffer content in terms of confluent hypergeometric functions. We compare it with a fluid queue driven by an infinite server queue. Numerical results are presented to compare the behaviour of the buffer content distributions for both these models.

On the nonlinear damping of a plasma mode

We present a theory of the nonlinear damping of a plasma mode based on a solution of the Vlasov–Poisson system. The formulation is exempt from the objectionable separation of the particles into ‘resonant’ and ‘nonresonant’, and is valid for ion as well as for electron modes. The effect of the damping of the mode on particle motion is taken in account. In particular, we evaluate numerically the damping of an ion mode for a temperature ratio Te/Ti = 16. We also obtain a number of small new shifts in the damping of a plasma mode in general, including contributions from the second derivative of the stationary distribution function.