AN $M/G/2$ QUEUE WITH A FINITE BUFFER UNDER DUAL CONTROL: STATIONARY LOST AND COST ANALYSIS

We study the stationary mode of a queuing system (QS) with an infinite storage device, one service device, and exponential service with intensity µ. The input of QS receives a doubly stochastic Poisson flow, the intensity λ(t) of which varies over the interval [α, β] and is a diffusion process with the zero drift coefficient a = 0, diffusion coefficient b > 0, and elastic boundaries α, β. In this paper, two models of a stationary queuing system QS are constructed in the form of systems of differential equations with respect to the characteristics of the applications number. We obtain the necessary and sufficient conditions for the existence and uniqueness of the stationary mode of the QS, the solution of the system of differential equations with respect to the characteristics of the number of applications of the stationary QS, and the non-negativity of the characteristics of the QS. The probability of downtime is estimated. Using the methods of operator analysis, a stationary generating function of the solution in the form of a convergent power series is found.

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 chapter, we study an application of the queuing system discussed in Chapter 3. We apply the queuing system to modeling a manufacturing system of multiple failure prone machines under Hedging Point Production (HPP) policy. In a queuing system, there are servers, customers, and waiting spaces. To model a manufacturing system by a queuing system, one may regard a server as a machine. Depending on the type of manufacturing system, make-to-order or order-to-make,the customers can be regarded as the inventory of product or the jobs to be processed respectively; see Buzacott and Shanthikumar [17]. In our context here, we only discuss the latter case. In a make-to-order system, the machine system produces only when there is an arrived order. But in an order-to-make system, a certain amount of inventory is kept to cope with the fluctuation of demand and therefore production control is necessary. Here we employ the HPP policy as the production control; the definition of the policy will be discussed shortly.KeywordsManufacturing SystemGenerator MatrixInventory LevelMarkov Chain ModelMachine SystemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In recent years manufacturing organisations have been facing increasing changes in their business environment. Those changes are being driven by customers demanding greater choice in products and services and competition from all corners of the globe. Moreover, manufacturing industry has been subject to a number of trends that include outsourcing, time compression, mass customisation and pricing pressures to mention just a few. Information and communication technologies can be used by manufacturing organisations to respond to changes in the business environment.

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.

• Examines scheduling on m identical parallel machines with a common server. • Considers sequence-dependent setup times in parallel machine scheduling. • Introduces an arc-based MILP formulation for makespan minimization. • Presents a GRASP-enhanced Iterated Greedy algorithm with strong performance. • Reports 21 new best-known solutions across benchmark test instances. The identical parallel machine scheduling problem with a single server and sequence-dependent setup times is a challenging optimization problem with important applications in manufacturing and service industries. In such environments, several machines depend on a common server to perform setup operations before production can begin, which creates strong interdependencies and demands more effective scheduling strategies. This characteristic highlights the practical relevance of the problem. The interaction between machine availability and server operations often becomes a critical bottleneck. This study introduces two complementary approaches. The first is an exact method based on a novel arc-based mixed-integer linear programming (ABF) model, which extends the modeling capability of existing formulations by capturing server-related constraints more effectively. The second is an approximation method built on an Iterated Greedy (IG) algorithm. The IG procedure is improved by two evaluation mechanisms: one model-based evaluation derived from the proposed ABF model, and another employing a greedy randomized adaptive search procedure (GRASP)-based strategy that integrates greedy selection, randomization, and reconstruction to enhance solution quality. Computational experiments are conducted on existing benchmark instances. The results show that the proposed ABF model performs well on small and medium-sized instances compared to existing exact methods, while the IG variants, particularly the proposed GRASP-based version, deliver strong performance against state-of-the-art metaheuristics developed for this problem. In addition, 21 new best-known solutions are reported, further demonstrating the effectiveness of the proposed approaches.

Group technology

The Markov-modulated Poisson process (MMPP) is a generalization of the Poisson process and is commonly used in modeling the input process of communication systems such as data traffic systems and ATM networks. In this paper, we give fast algorithms for solving queueing systems and manufacturing systems with MMPP inputs. We consider queueing systems where the input of the queues is a superposition of the MMPP which is still an MMPP. The generator matrices of these processes are tridiagonal block matrices with each diagonal block being a sum of tensor products of matrices. We are interested in finding the steady state probability distributions of these processes which are the normalized null vectors of their generator matrices. Classical iterative methods, such as the block Gauss-Seidel method, are usually employed to solve for the steady state probability distributions. They are easy to implement, but their convergence rates are slow in general. The number of iterations required for convergence increases like O(m). where m is the size of the waiting spaces in the queues. Here, we propose to use the preconditioned conjugate gradient method. We construct our preconditioners by taking circulant approximations of the tensor blocks of the generator matrices. We show that the number of iterations required for convergence increases at most like O(log2 m) for large m. Numerical results are given to illustrate the fast convergence. As an application, we apply the MMPP to model unreliable manufacturing systems. The production process consists of multiple parallel machines which produce one type of product. Each machine has exponentially distributed up time, down time, and processing time for one unit of product. The interarrival of a demand is exponentially distributed and finite backlog is allowed. We consider hedging point policy as the production control. The average running cost of the system can be written in terms of the steady state probability distribution. Our numerical algorithm developed for the queueing systems can be applied to obtain the steady state distribution for the system and hence the optimal hedging point. Furthermore, our method can be generalized to handle the case when the machines have a more general type of repairing process distribution such as the Erlangian distribution.

Machine learning is usually considered a branch of artificial intelligence that enables machines (computers) to learn from data and improve their performance without explicit programming. The paper presents a review of research on meta-learning, also called double and semi-double-loop machine learning, based manufacturing. The paper provides a selected list of meta-learning definitions and generic models and discusses implementation architectures. Following, a set of meta-learning applications in manufacturing is presented, addressing three sub-domains: the basic "classical" processes in manufacturing systems, the manufacturing systems, including some advanced concepts such as smart manufacturing, advanced communication, and Industry 4.0, and manufacturing eco-system as a generic name for a number of application sub-domains that are directly or indirectly related to manufacturing and that could provide virtually easy knowledge transfer to manufacturing. In the last part of the paper, some future research and development directions are given.

A practical architecture for implementing motion control on complex machines is to generate the desired trajectory for all axes on a central computer (PC) at a modest sampling rate, and transmit the desired trajectory to individual axis controllers. This local controller closes an inner feedback loop at a higher sampling rate. If command shaping in the form of Optimal Arbitrary Time-delay (OAT) filtering is to be employed in these architectures to reduce vibration, it can be applied at the central computer or at the axis controller. This paper analyzes these alternatives and how they can be implemented.

During the last few decades, significant progress has been made in solving large-scale finite-dimensional and semi-infinite linear programming problems. In contrast, little progress has been made in solving linear programs in infinite-dimensional spaces despite their importance as models in manufacturing and communication systems. Inspired by the research on separated continuous linear programs, we propose a new class of continuous linear programming problems that has a variety of important applications in communications, manufacturing, and urban traffic control. This class of continuous linear programs contains the separated continuous linear programs as a subclass. Using ideas from quadratic programming, we propose an efficient algorithm for solving large-scale problems in this new class under mild assumptions on the form of the problem data. We prove algorithmically the absence of a duality gap for this class of problems without any boundedness assumptions on the solution set. We show this class of problems admits piecewise constant optimal control when the optimal solution exists. We give conditions for the existence of an optimal solution. We also report computational results which illustrate that the new algorithm is effective in solving large-scale realistic problems (with several hundred continuous variables) arising in manufacturing systems.

This paper is concerned with the analysis of an infinite-capacity batch arrival multi-server queueing system with Bernoulli feedback, synchronous multiple and single working vacation policies, waiting servers, reneging and retention of reneged customers. The steady-state solution of the queueing system is obtained by using probability generating function (PGF). In addition, important performance measures of the queueing system are derived. Then, a cost model is formulated in order to carry out the parameter optimization using genetic algorithm (GA). Finally, numerical study is presented in which various system performance measures are evaluated based on supposed numerical values given to the system parameters.