ОПЕРАТОРНЫЙ АНАЛИЗ УРАВНЕНИЙ ТИПА КОЛМОГОРОВА – ЧЕПМЕНА С ДИФФЕРЕНЦИАЛЬНЫМ ОПЕРАТОРОМ

In this paper, we are interested in solving queueing systems having Poisson batch arrivals, exponential servers and negative customers. Preconditioned Conjugate Gradient (PCG) method is applied to solving the steady-state probability distribution of the queueing system. Preconditioners are constructed by exploiting near-Toeplitz structure of the generator matrix and the Gohberg-Semumcul formula. We proved that the preconditioned system has singular values clustered around one. Therefore Conjugate Gradient (CG) methods when applied to solving the preconditioned system, we expect fast convergence rate. Numerical examples are given to demonstrate our claim.

Some optimality conditions for M b/M/C queueing systems

In this paper, the waiting time and response time processes in continuous-time two parallel queueing systems are analyzed as well as the system size process in discrete-time two parallel queueing systems. Upon arrival, each job splits into two tasks, denoted by task 1 and task 2. Task k, $k = 1,2$, is assigned to the kth queueing system to be served on a FCFS (first-come-first-served) basis. Each queueing system consists of a single server and an infinite-capacity queue. For the continuous-time queueing system, we assume that jobs arrive in a state-independent Poisson fashion with rate $\lambda $; the servers have exponential service times with rates $\mu _1 $ and $\mu _2 $, respectively. A closed-form expression for the Laplace transform of the ergodic bivariate waiting time distribution is derived. The distribution of the sum of the two waiting times is then obtained. The distribution of the response time when $\mu _1 = \mu _2 $ is derived analytically. For the discrete-time queueing systems, it is assu...

This paper considers a Poisson arrival queuing system with an exponential server (server-1) and a general server (server-2) operating under dual control.We extend our analysis of the same queuing system with an infinite buffer to the finite case.The case presented here has many applications in manufacturing, computer centers and telecommunications systems.

Two different kinds of heavy-traffic limit theorems have been proved for s-server queues. The first kind involves a sequence of queueing systems having a fixed number of servers with an associated sequence of traffic intensities that converges to the critical value of one from below. The second kind, which is often not thought of as heavy traffic, involves a sequence of queueing systems in which the associated sequences of arrival rates and numbers of servers go to infinity while the service time distributions and the traffic intensities remain fixed, with the traffic intensities being less than the critical value of one. In each case the sequence of random variables depicting the steady-state number of customers waiting or being served diverges to infinity but converges to a nondegenerate limit after appropriate normalization. However, in an important respect neither procedure adequately represents a typical queueing system in practice because in the (heavy-traffic) limit an arriving customer is either almost certain to be delayed (first procedure) or almost certain not to be delayed (second procedure). Hence, we consider a sequence of (GI/M/S) systems in which the traffic intensities converge to one from below, the arrival rates and the numbers of servers go to infinity, but the steady-state probabilities that all servers are busy are held fixed. The limits in this case are hybrids of the limits in the other two cases. Numerical comparisons indicate that the resulting approximation is better than the earlier ones for many-server systems operating at typically encountered loads.

This paper studies the optimal policy for joint control of admission, routing, service, and jockeying in a queueing system consisting of two exponential servers in parallel. Jobs arrive according to a Poisson process. Upon each arrival, an admission/routing decision is made, and the accepted job is routed to one of the two servers with each being associated with a queue. After each service completion, the servers have an option of serving a job from its own queue, serving a jockeying job from another queue, or staying idle. The system performance is inclusive of the revenues from accepted jobs, the costs of holding jobs in queues, the service costs and the job jockeying costs. To maximize the total expected discounted return, we formulate a Markov decision process (MDP) model for this system. The value iteration method is employed to characterize the optimal policy as a hedging point policy. Numerical studies verify the structure of the hedging point policy which is convenient for implementing control actions in practice.

In this paper, we consider a queueing system in which there are two exponential servers, each having his own queue, and arriving customers will join the shorter queue. Based on the results given in Flatto and McKean, we rewrite the formula for the probability that there are exactlyk customers in each queue, wherek = 0, 1,…. This enables us to present an algorithm for computing these probabilities and then to find the joint distribution of the queue lengths in the system. A program and numerical examples are given.

Reduction of everywhere convergent power series with respect to Gröbner bases

With platelets, as with T lymphocytes, the blood cell system behaves as a neuron

The paper considers the possibility of using a queuing system to determine the required number of repair and maintenance personnel in the event of mass failures in rural distribution electric networks. In this case, the separation of requirements for restoring failures (requests) into priority and secondary is used. The multi-channel queuing system is used with two priority levels, with the delay of both priority requests and secondary requests. The algorithm has been developed for composing a system of differential equations describing a given queuing system. The system of differential equations is solved by numerical methods, and, using a specially developed algorithm, based on the results of the solution for a given point in time, the required number of repair personnel is estimated based on such parameters as the probability of queuing and the average waiting time of requests in the queue. The developed algorithms have been implemented in software; the necessary number of repair personnel has been calculated with their use, as well as on the basis of statistical data on failures in rural distribution electric networks. Assessment of the intensity of the flow of priority and secondary requests has been carried out by two methods. In the first method the data from the dispatcher’s record book or acts of investigation of technological disturbances have been used. Failures were divided by such parameters as reliability categories of power supply for cut-off consumers, volume of cut-off load and social significance of cut-off objects. In the second method, those requests that could be fixed by the forces of the field service teams were priority, the remaining requests were considered secondary.

Normal matrices over hermitian discrete valuation rings

In the analysis of metastable diffusion processes, Transition Path Theory (TPT) provides a way to quantify the probability of observing a given transition between two disjoint metastable subsets of state space. However, many TPT-based methods for diffusion processes compute the primary objects from TPT, such as the committor and probability current, by solving partial differential equations. The computational performance of these methods is limited by the need for mesh-based computations, the need to estimate the coefficients of the stochastic differential equation that defines the diffusion process, and the use of Markovian processes to approximate the diffusion process. We propose a Monte Carlo method for approximating the primary objects from TPT from sample trajectory data of the diffusion process, without estimating drift or diffusion coefficients. We discretise the state space of the diffusion process using Voronoi tessellations and construct a non-Markovian jump process on the dual Delaunay graph. For the jump process, we define committors, probability currents, and streamlines, and use these to define piecewise constant approximations of the corresponding objects from TPT for diffusion processes. Rigorous error bounds and convergence theorems establish the validity of our approach. A comparison of our method with TPT for Markov chains (Metzner et al., Multiscale Model Simul. 2009) on a triple-well 2D potential provides proof of principle.

The direct and inverse scattering problems in two-dimensional acoustics at a fixed frequency are considered. It is shown how the null-field approach can be modified so that the Q matrix (which in a straightforward manner gives the transition matrix) is obtained as a series instead of an integral. For an obstacle which is a perturbation of a circle this series form gives an approximate, very explicit, expression for the transition matrix. A few numerical examples are given to show the utility and limitations of this approximation. For the inverse problem, the series form of Q gives a system of nonlinear polynomial equations which are solved by the imbedding method. Some numerical examples show that quite accurate results are obtained by this method in cases where the system of equations can be kept small. The linear approximation of the system of polynomial equations yields a method that works surprisingly well and which is also promising for the more difficult three-dimensional and vector problems.

The article considers the components of the target function of optimization of HUNDRED losses associated with the introduction of the post (posts) of acceptance-diagnosis with the increase of the set of diagnostic equipment. In particular, the target function of optimizing losses at the service station is presented in the form of an economic criterion that takes into account the maximum possible number of components on which depends the technological efficiency of the service station and the level of "satisfaction" of car owners with services. The object of research is a car service station. The purpose of the work is to conduct a study of the target function of optimization of HUNDRED losses associated with the type of organization of acceptance-diagnosis of cars and to find out algorithms for finding the components of the target function based on the representation of HUNDRED as a queuing system. Research method - analysis and formalization: quantitative indicators of the objective function, methods of forming a group of events, alternative solutions and calculation of the main components of optimization of HUNDRED losses as a queuing system. It is established that the target function of optimization of HUNDRED losses has economic essence and can be presented in the form of complex criterion. It is found that based on the theory of queuing for the area of reception-diagnosis it becomes possible to find solutions of the system of equations that describe its current state, namely: 1) the probability that the system has no requirements, ie free all "workers »Posts and no applications in the queue; 2) the probability of finding the system in a certain state; 3) the probability of denial of the received application for maintenance, ie the probability of finding the system in the last valid state; 4) the average number of occupied reception posts. Similar solutions were found for the TO-R zone. The formulas for determining the components of the duration of the application are considered and given. The formulas for determining the average value of the duration of work in the area of maintenance by the flow of faults and their share in the total flow of applications. The necessity of the 5th - 6th step of building the equipment in the reception-diagnostic area for detection of additional malfunctions (hidden defects) is substantiated. It is established that the average processing time of the application at the posts of acceptance-diagnosis includes three components: 1) the average time of acceptance-diagnosis of the declared ambiguous manifestation of the fault; 2) the average time of additional preliminary diagnosis on the application, which contained ambiguous manifestations of malfunctions. Depends on the number of diagnostic operations performed and the available equipment; 3) the average time of additional diagnosis on the application, which either did not contain ambiguous manifestations of malfunctions, or they did not fall within the capabilities of the existing diagnostic complex at the reception. The results of the article can be used to model the operation of the service station as a queuing system in computer software products and will reveal the low efficiency of diagnosing the main systems of cars in a particular service station. KEY WORDS: DIAGNOSTIC PARAMETER, APPLICATION, PROBABILITY, FAULT, EQUIPMENT, POST, PROFIT, ACCEPTANCE-DIAGNOSIS, SYSTEM.

In the paper, a nonlocal initial-boundary value problem for a non-homogeneous one-dimensional heat equation is considered. The domain under consideration is a rectangle. The classical initial condition with respect to t is put. A nonlocal periodic boundary condition by a spatial variable x is put. It is well-known that a solution of problem can be constructed in the form of convergent orthonormal series according to eigenfunctions of a spectral problem for an operator of multiple differentiation with periodic boundary conditions. Therefore Green's function can be also written in the form of an infinite series with respect to trigonometric functions (Fourier series). For classical first and second initial-boundary value problems there also exists a second representation of the Green's function by Jacobi function. In this paper we find the representation of the Green's function of the nonlocal initial-boundary value problem with periodic boundary conditions in the form of series according to exponents.