A single server queue with a varying gate mechanism

A Markovian single server queue in a varying environment with gate mechanism is studied. Exact expressions for the duration of the nth batch and the number served are obtained by exploiting the connection between queues, branching processes and continued fractions. Numerical illustrations are presented.     

On a single server queue with two-stage heterogeneous service and deterministic server vacations

We analyse a single server queue with Poisson arrivals, two stages of heterogeneous service with different general (arbitrary) service time distributions and binomial schedule server vacations with deterministic (constant) vacation periods. After first-stage service the server must provide the second stage service. However, after the second stage service, he may take a vacation or may decide to stay on in the system. For convenience, we designate our model as M/G ₁, G ₂/D/1 queue. We obtain steady state probability generating function of the queue length for various states of the server. Results for some particular cases of interest such as M/E_{k 1} , E_{k 2} /D/1, M/M ₁, M ₂/D/1, M/E _k /D/1 and M/G ₁, G ₂/1 have been obtained from the main results and some known results including M/E_k /1 and M/G/1 have been derived as particular cases of our particular cases.     

A single server feedback retrial queue with collisions

A Markovian single server feedback retrial queue with linear retrial rate and collisions of customers is studied. Using generating function technique, the joint distribution of the server state and the orbit length under steady-state is investigated. Some interesting and important performance measures of the system are obtained. Finally, numerical illustrations are provided.     

A single server queue with mixed types of interruptions

Summary The single server M/G/1 queue subject to Poisson interruptions has many useful applications in computer systems modeling. The interruptions are usually characterized by their type of service-preemption discipline. This paper deals with this model in its most general setting, allowing the simultaneous presence of all types of interruptions that may be encountered in real systems. Inspite of the inherent complexity of the analysis, it is possible to derive analytic closed form expressions for interesting performance measures. The results obtained are of theoretical interest as well as of practical significance. In particular, we derive the Laplace Stieltjes transform of the completion time associated with a customer's ervice and obtain the steady-state average number of customers in the system. An application to the modeling of checkpointing and recovery in a transactional system is considered.This work was supported in part by Air Force Office of Scientific Research under grant AFOSR-84-0132, by the Army Research Office under contract DAAG29-84-K0045 and by the National Science Foundation under grant MCS-830200     

A numerical procedure for the selection of the constant interarrival time to a single server queue

We study a single server queue with deterministric arrivals to find the optimal interarrival time. No analytically tractable solution is available, except in the particular D/M/1 case, but under the assumption of a phase type service time distribution, the model has a tractable algorithmic solution.

The purpose of this paper is to present the algorithm for the D/PH/1 queue and to demonstrate its implementation through interactive computation. The algorithm provides enough information about the system to be useful to a variety of problems in engineering design.

Potential applications are in assembly line industrial processes, particularly those in computer-controlled, fully automated factories, and also in the selection of a good appointment system.     

Optimal control of a nonhomogeneous Dirichlet boundary fractional diffusion equation

We study a nonhomogeneous Dirichlet boundary fractional diffusion equation in a bounded domain. The fractional time derivative is considered in the Riemann-Liouville sense. We first prove by transposition the existence and the uniqueness of the solution of the boundary fractional diffusion equation. Then under some appropriate assumptions on the closed convex set of the admissible controls, we obtain a decoupled optimality system.     

A Fokker–Planck equation method predicting Buffer occupancy in a single queue

The VIrtual Predictor BuffER (VIPER) algorithm is a novel algorithm for performing online prediction of the buffer space requirement of each competing traffic stream. It accomplishes its task through the employment of a theoretical, infinite capacity, virtual buffer. Information acquired from the virtual buffer is used to construct a probability distribution function that is based on the Fokker–Planck equation. This distribution function is central to the VIPER algorithm and is used to compute the queue length predictions. The predictions are shown to be promising.     

Optimal allocation of customers in a two server queue withresequencing

The problem of optimal allocation of customers in a two server queue with heterogeneous service rates and resequencing is addressed. The resequencing constraint ensures that the customers leave the system in the order in which they entered it. It is shown that the optimal policy that minimizes the average end-to-end delay of customers in the system is independent of the number of customers in the resequencing buffer. It is also shown that the faster server should be kept busy whenever possible     

A tau method approach for the diffusion equation with nonlocal boundary conditions

An approximate method for solving the diffusion equation with nonlocal boundary conditions is proposed. The method is based upon constructing the double shifted Legendre series to approximate the required solution using Legendre tau method. The differential and integral expressions which arise in the diffusion equation with nonlocal boundary conditions are converted into a system of linear algebraic equations which can be solved for the unknown coefficients. Numerical examples are included to demonstrate the validity and applicability of the method and a comparison is made with existing results.     

Cost-minimization analysis of a working vacation queue with N-policy and server breakdowns

This paper investigates the N-policy M/M/1 queueing system with working vacation and server breakdowns. As soon as the system becomes empty, the server begins a working vacation. The server works at a lower service rate rather than completely stopping service during a vacation period. The server may break down with different breakdown rates during the idle, working vacation, and normal busy periods. It is assumed that service times, vacation times, and repair times are all exponentially distributed. We analyze this queueing model as a quasi-birth–death process. Furthermore, the equilibrium condition of the system is derived for the steady state. Using the matrix-geometric method, we find the matrix-form expressions for the stationary probability distribution of the number of customers in the system and system performance measures. The expected cost function per unit time is constructed to determine the optimal values of the system decision variables, including the threshold N and mean service rates. We employ the particle swarm optimization algorithm to solve the optimization problem. Finally, numerical results are provided, and an application example is given to demonstrate the applicability of the queueing model.     

Single server retrial queue with group admission of customers

We consider a retrial queueing system with a single server and novel customer׳s admission discipline. The input flow is described by a Markov Arrival Process. If an arriving customer meets the server providing the service, it goes to the orbit and repeats attempts to get service in random time intervals whose duration has exponential distribution with parameter dependent on the customers number in orbit. Server operates as follows. After a service completion epoch, customers admission interval starts. Duration of this interval has phase type distribution. During this interval, primary customers and customers from the orbit are accepted to the pool of customers which will get service after the admission interval. Capacity of this pool is limited and after the moment when the pool becomes full before completion of admission interval all arriving customers move to the orbit. After completion of an admission interval, all customers in the pool are served simultaneously by the server during the time having phase type distribution depending on the customers number in the pool. Using results known for Asymptotically Quasi-Toeplitz Markov Chains, we derive stability condition of the system, compute the stationary distribution of the system states, derive formulas for the main performance measures and numerically show advantages of the considered customer׳s admission discipline (higher throughput, smaller average number of customers in the system, higher probability to get a service without visiting the orbit) in case of proper choice of the capacity of the pool and the admission period duration.     

Studying the queue length in a single-server queuing system with unreliable server

The unreliable queuing system was studied in both the nonstationary and stationary modes. It was assumed that a Poisson flow of customers arrives to the system; the times of customer servicing and restoration of the server (servicing system) are random variables with arbitrary distributions; the flow of stable failures makes up the restoration process; the interfailure intervals are distributed hyperexponentially; and the waiting and sojourn times as well as the queue length are not limited.Translated from Avtomatika i Telemekhanika, No. 1, 2005, pp. 72–81.Original Russian Text Copyright © 2005 by Mikadze, Khocholava.This paper was recommended for publication by V.V. Rykov, a member of the Editorial Board     

Stability of the totally implicit finite difference scheme for the one-dimensional diffusion equation with a moving boundary

We analyze the totally implicitO(h ² k) finite difference method applied to a linear unidimensional diffusion equation with a boundary moving with constant velocity. The related physical problem is the solidification of a finite column filled with a binary mixture, in the quasiequilibrium regime. By advancing the boundary along the pathh/k=velocity of the interface, a simple algorithm is obtained which is shown to be consistent and unconditionally stable for any value of the interfacial segregation factor and of the Peclet number.     

Anisotropic diffusion equation with a new diffusion coefficient for image denoising

This paper presents a new anisotropic diffusion model which is based on a new diffusion coefficient for image denoising. In the proposed model, a new diffusion coefficient and a method of automatically set gradient threshold parameter are introduced into an anisotropic diffusion model, which weakens the staircasing effect and preserves fine edges in a processed image. Comparative experiments show that the new model achieves the more satisfied denoising results than the other existing models.     

Cost analysis of a finite capacity queue with server breakdowns and threshold-based recovery policy

This paper analyzes a repairable M/M/1/N queueing system under a threshold-based recovery policy. The threshold-based recovery policy means that the server breaks down only if there is at least one customer in the system, and the recovery can be performed when q (1 ≤ q ≤ N) or more customers are present. For this queueing system, a recursive method is applied to obtain steady-state solutions in neat closed-form expressions. We then develop some important system characteristics, such as the number of customers in the system, the probability that the server is busy, the effective arrival rate and the expected waiting time in the system, etc. A cost model is constructed to determine the optimal threshold value, the optimal system capacity and the optimal service rate at a minimum cost. In order to solve this optimization problem, the direct search method and Newton's method are employed. Sensitivity analysis is also conducted with various system parameters. Finally, we present some managerial insights through an application example.     

Convergence rates of perturbation-analysis-Robbins-Monro-single-runalgorithms for single server queues

In this paper the perturbation-analysis-Robbins-Monro-single-run algorithm is applied to estimating the optimal parameter of a performance measure for the GI/G/1 queueing systems, where the algorithm is updated after every fixed-length observation period. Our aim is to analyze the limiting behavior of the algorithm. The almost sure convergence rate of the algorithm is established. It is shown that the convergence rate depends on the second derivative of the performance measure at the optimal point     

Stabilization of a spatially non‐causal reaction–diffusion equation by boundary control

Stabilization of a reaction–diffusion equation, in which the heat source depends on the temperature of the whole space, is considered by using boundary control. A new backstepping transformation is introduced, in which there are two kernels. Through a series of mathematical tricks, the exact solutions of kernels are obtained, and a control law is obtained specifically. The inverse transformation is derived, and stability of the closed loop system established. Simulation results show that the closed loop system is stable. Copyright © 2012 John Wiley & Sons, Ltd.     

Two-machine flow shop problems with a single server

Scheduling models that allow the handling of pre-operational setup have been a source of major interests because of their practical relevance and theoretical impacts. Two-stage flow-lines have drawn much attention to researchers as they are simple, yet practical and can be easily extended to represent more complex situations. In this paper, two-machine flow-shop problems with a single setup server are surveyed. These problems have been shown to be NP-complete with special cases that are polynomial-time solvable. Several heuristics are proposed to solve the problems in general case, including simulated annealing, Tabu search, genetic algorithms, GRASP, and other hybrids. The results on small inputs are compared with the optimal solutions and results on large inputs are compared to a lower bound. Experiments show that the heuristics developed, obtain nearly optimal solutions.     

Queueing models for a single server LAN with processor sharing disciplines

A local area network (LAN) is a collection of autonomous workstations interconnected by a communication network. A key component of a local network is the file server which stores programs and data and makes them available to the workstations as needed. In practice, a workstation requests a large portion of the file (type 1 request) when a new application is launched. Following this, the workstation requests additional portions of the file as needed (type 2 requests). Clearly, the response time to these requests will depend strongly on the file server scheduling policy to service the two types of incoming requests. In an earlier paper [12] we studied this system when type 2 requests havepreemptive and non-preemptive priority over type 1 requests. From a practical point of view, it is more realistic to describe the file transfer by around-robin scheduling discipline. In this paper we study LANs where the file server followsprocessor sharingdiscipline, which is a limiting case of theround-robin discipline. Assuming that all relevant intervals are exponentially distributed we develop algorithms to analyze the performance of the system. Illustrative examples are presented to study the system behavior under various load conditions. The computational approach presented in this paper helps in resolving some of the analytical difficulties associated with the analysis ofprocessor sharing disciplines.