Batch Of Customers Research Articles

A single-server batch arrival queue with returning customers

We consider a new class of batch arrival retrial queues. By contrast to standard batch arrival retrial queues we assume if a batch of primary customers arrives into the system and the server is free then one of the customers starts to be served and the others join the queue and then are served according to some discipline. With the help of Lyapunov functions we have obtained a necessary and sufficient condition for ergodicity of embedded Markov chain and the joint distribution of the number of customers in the queue and the number of customers in the orbit in steady state. We also have suggested an approximate method of analysis based on the corresponding model with losses.

Performance analysis of discrete-time GI/D-MSP<SUP align=right>a;b</SUP>/1/N queue

This paper presents a discrete-time single-server finite buffer queue with renewal input and discrete Markovian service process (D-MSP) where server serves customers in batches according to general batch service rule. Using the supplementary variable and the imbedded Markov chain techniques, we obtain the queue-length distributions at pre-arrival and arbitrary epochs. Some performance measures such as loss probability, mean queue length and mean waiting time in the queue along with some numerical results have also been discussed.

Stationary distribution of the closed queuing network with batch transitions of customers

Consideration was given to the closed queuing network with batch jockeying customers. This network extends the Gordon-Newell model to the case of batch customer servicing. The serviced batch discharges a node without changing its size and arrives to another node with a probability depending on its size. Some constraints must be imposed on the servicing process. The form of the stationary distribution was established, and an efficient algorithm to determine it was proposed.

Lattice path approach for busy period density of [formula omitted] queues using [formula omitted] Coxian distributions

In this paper busy period analysis of non-Markovian queueing system GI a / G b / 1 , starting initially with i 0 batches of customers, is carried out via lattice path approach. Both interarrival and service time distributions are approximated by 2-phase Cox distributions, C 2 , that have Markovian property, amenable to the application of lattice paths combinatorial analysis. Arrivals occur in batches of size a and services occur in batches of size b , a and b are co-prime. Distributions having rational Laplace–Stieltjes transform and square coefficient of variation lying in [ 1 / 2 , ∞ ) form a very wide class of distributions. As any distribution of this class can be approximated by a C 2 , the use of C 2 , therefore, has led us to achieve results applicable to almost any real life queueing system GI a / G b / 1 occurring in computer systems, communication systems, manufacturing systems, etc. Numerical computations have been performed for different sets of values of the parameters involved using software Mathematica and presented graphically.

The MAP + MAP/PH/1/N queuing system with single and batch arrivals of customers

The single-server queuing system with finite buffer was considered. The customers may arrive one-by-one or in batches. Arrivals of single customers and their batches obey the Markov input processes. The customers from a batch taken for servicing come one at a time at the exponentially distributed time intervals. The numbers of customers in batches are distributed geometrically. The time of customer servicing has a phase-type distribution. The numbers of batches and single customers that may be simultaneously accepted by the system are controllable parameters. The joint distribution of the number of batches and the number of customers in system, loss probabilities, distribution of the time of batch sojourn, and problems of optimization were analyzed.

Finite-Buffer Bulk Service Queue Under Markovian Service Process: GI/MSP (a,b)/1/N

We consider a single-server queue with renewal input and Markovian service process where server serves customers in batches according to a general bulk service rule. Queue length distributions at pre-arrival and arbitrary epochs have been obtained along with some important performance measures such as probability of blocking, mean queue lengths and mean waiting times. The analysis has been carried out assuming finite-buffer space for the arriving customers. The model has potential applications in areas such as computer networks, telecommunication systems and manufacturing systems.

Approximation Algorithms for the Supplier's Supply Chain Scheduling Problem to Minimize Delivery and Inventory Holding Costs

We study the upstream supplier's batch scheduling problem in a supply chain, which was defined by Hall and Potts [Hall, N. G., C. N. Potts. 2003. Supply chain scheduling: Batching and delivery. Oper. Res. 51(4) 566–584]. The supplier has to manufacture multiple products and deliver them to customers in batches. There is an associated delivery cost with each batch. The objective of the supplier is to minimize the total inventory holding and delivery costs. We present simple approximation algorithms for this strongly NP-hard problem, which find a solution that is guaranteed to have a cost at most 3/2 times the minimum. We also prove that the approximation algorithms have worst-case bounds that vary parametrically with the data and that for realistic parameter values are much better than 3/2. The theoretical results are also supported by the findings of a computational study.

Approximation algorithms for minimizing the total weighted number of late jobs with late deliveries in two-level supply chains

We study a supply chain scheduling problem in which n jobs have to be scheduled on a single machine and delivered to m customers in batches. Each job has a due date, a processing time and a lateness penalty (weight). To save batch-delivery costs, several jobs for the same customer can be delivered together in a batch, including late jobs. The completion time of each job in the same batch coincides with the batch completion time. A batch setup time has to be added before processing the first job in each batch. The objective is to find a schedule which minimizes the sum of the weighted number of late jobs and the delivery costs. We present a pseudo-polynomial algorithm for a restricted case, where late jobs are delivered separately, and show that it becomes polynomial for the special cases when jobs have equal weights and equal delivery costs or equal processing times and equal setup times. We convert the algorithm into an FPTAS and prove that the solution produced by it is near-optimal for the original general problem by performing a parametric analysis of its performance ratio.

Queueing analysis and optimal control of [formula omitted] and [formula omitted] systems

We first consider a finite-buffer single server queue where arrivals occur according to batch Markovian arrival process ( BMAP ) . The server serves customers in batches of maximum size ‘ b’ with a minimum threshold size ‘ a’. The service time of each batch follows general distribution independent of each other as well as the arrival process. We obtain queue length distributions at various epochs such as, pre-arrival, arbitrary, departure, etc. Some important performance measures, like mean queue length, mean waiting time, probability of blocking, etc. have been obtained. Total expected cost function per unit time is also derived to determine the optimal value N ∗ of N at a minimum cost for given values of a and b. Secondly, we consider a finite-buffer single server queue where arrivals occur according to BMAP and service process in this case follows a non-renewal one, namely, Markovian service process ( MSP ) . Server serves customers according to general bulk service rule as described above. We derive queue length distributions and important performance measures as above. Such queueing systems find applications in the performance analysis of communication, manufacturing and transportation systems.

HWCVD of polymers: Commercialization and scale-up

GVD Corporation specializes in process development and equipment design for the production of ultra-thin polymer coatings using hot wire chemical vapor deposition (HWCVD, also known as initiated chemical vapor deposition, iCVD). HWCVD allows many coating compositions to be produced, including fluorocarbon and silicone polymers, copolymers, and vinyl hydrocarbon polymers. It is especially valuable for creating ultra-thin layers of insoluble, infusible polymers which are hard to process by conventional means, such as polytetrafluoroethylene (PTFE, Teflon ®). HWCVD PTFE coatings are chemically robust, comprised of essentially 100% CF 2, resistant to solvents, conformal to complex surface geometry, and have excellent adhesion to a wide range of substrates. Since the part to be coated remains at room temperature, fragile materials like plastics and fabrics can be coated with ease. GVD has focused on scale-up of the process equipment and has developed several standard coating systems, which will be discussed in this paper. These include laboratory-scale batch coating systems, a medium sized production batch coating system, a large scale custom batch coater, and a pilot scale roll-to-roll web coater. All of GVD's systems are complete with fully automated, computer based control systems and include options for effluent monitors and an exhaust scrubber.

Maintenance in Single‐Server Queues: A Game‐Theoretic Approach

We use antagonistic stochastic games and fluctuation analysis to examine a single‐server queue with bulk input and secondary work during server′s multiple vacations. When the buffer contents become exhausted the server leaves the system to perform some diagnostic service of a minimum of L jobs clustered in packets of random sizes (event A). The server is not supposed to stay longer than T units of time (event B). The server returns to the system when A or B occurs, whichever comes first. On the other hand, he may not break service of a packet in a middle even if A or B occurs. Furthermore, the server waits for batches of customers to arrive if upon his return the queue is still empty. We obtain a compact and explicit form functional for the queueing process in equilibrium.

ColorTree: a batch customization tool for phylogenic trees

BackgroundGenome sequencing projects and comparative genomics studies typically aim to trace the evolutionary history of large gene sets, often requiring human inspection of hundreds of phylogenetic trees. If trees are checked for compatibility with an explicit null hypothesis (e.g., the monophyly of certain groups), this daunting task is greatly facilitated by an appropriate coloring scheme.FindingsIn this note, we introduce ColorTree, a simple yet powerful batch customization tool for phylogenic trees. Based on pattern matching rules, ColorTree applies a set of customizations to an input tree file, e.g., coloring labels or branches. The customized trees are saved to an output file, which can then be viewed and further edited by Dendroscope (a freely available tree viewer). ColorTree runs on any Perl installation as a stand-alone command line tool, and its application can thus be easily automated. This way, hundreds of phylogenic trees can be customized for easy visual inspection in a matter of minutes.ConclusionColorTree allows efficient and flexible visual customization of large tree sets through the application of a user-supplied configuration file to multiple tree files.

Analysis of GI X /M(n)//N systems with stochastic customer acceptance policy

We investigate GI X /M(n)//N systems with stochastic customer acceptance policy, function of the customer batch size and the number of customers in the system at its arrival. We address the time-dependent and long-run analysis of the number of customers in the system at prearrivals and postarrivals of batches and seen by customers at their arrival to the system, as well as customer blocking probabilities. These results are then used to derive the continuous-time long-run distribution of the number of customers in the system. Our analysis combines Markov chain embedding with uniformization and uses stochastic ordering as a way to bound the errors of the computed performance measures.

A Learning Approach for Interactive Marketing to a Customer Segment

When a marketer in an interactive environment decides which messages to send to her customers, she may send messages currently thought to be most promising (exploitation) or use poorly understood messages for the purpose of information gathering (exploration). We assume that customers are already clustered into homogeneous segments, and we consider the adaptive learning of message effectiveness within a customer segment. We present a Bayesian formulation of the problem in which decisions are made for batches of customers simultaneously, although decisions may vary within a batch. This extends the classical multiarmed bandit problem for sampling one-by-one from a set of reward populations. Our solution methods include a Lagrangian decomposition-based approximate dynamic programming approach and a heuristic based on a known asymptotic approximation to the multiarmed bandit solution. Computational results show that our methods clearly outperform approaches that ignore the effects of information gain.

Product Numbering System Design for Single Piece or Small Batch Customization Manufacture and its Process Organization

In this paper, a kind of product numbering system for single piece/small batch customization manufacturing and its process organization is put forward based on the current status of Shenyang Heavy Machine Manufacture Group Co. Ltd. and the trends of advanced manufacturing technology such as PDM/ERP etc. The numbering system developed includes all of relative public information data such as product order, design, process plan, manufacturing, marketing, consumer servicing and so on. A more favorable condition is created to realize the share of relative information in the different departments of the company and to improve the response to the market.

An M [X]/G/1 retrial queue with server breakdowns and constant rate of repeated attempts

We consider an M[X]/G/1 retrial queue subject to breakdowns where the retrial time is exponential and independent of the number of customers applying for service. If a coming batch of customers finds the server idle, one of the arriving customers begins his service immediately and the rest joins a retrial group (called orbit) to repeat his request later; otherwise, if the server is busy or down, all customers of the coming batch enter the orbit. It is assumed that the server has a constant failure rate and arbitrary repair time distribution. We study the ergodicity of the embedded Markov chain, its stationary distribution and the joint distribution of the server state and the orbit size in steady-state. The orbit and system size distributions are obtained as well as some performance measures of the system. The stochastic decomposition property and the asymptotic behavior under high rate of retrials are discussed. We also analyse some reliability problems, the k-busy period and the ordinary busy period of our retrial queue. Besides, we give a recursive scheme to compute the distribution of the number of served customers during the k-busy period and the ordinary busy period. The effects of several parameters on the system are analysed numerically.

Lattice Path Approach for Busy Period Density ofGIb/G/1 Queues UsingC2Coxian Distributions

In this paper busy period analysis of non-Markovian queuing system GIb/G/1, starting initially with i0 batches of customers, is carried out via lattice path approach. Both interarrival and service time distributions are approximated by 2-phase Cox distributions, C2, that have Markovian property, amenable to the application of lattice paths combinatorial analysis. Arrivals occur in batches of size b. Distributions having rational Laplace-Stieltjes transform and square coefficient of variation lying in [1/2, ∞] form a very wide class of distributions. As any distribution of this class can be approximated by a C2, the use of C2, therefore, has led us to achieve results applicable to almost any real life queuing system GIb/G/1 occurring in computer systems, communication systems, manufacturing systems, etc. Numerical computations have been performed for different sets of values of the parameters involved using software Mathematica and presented graphically.

A simple approximation method for workload analyses in some queueing systems with control policies

We consider two systems: M/G/1 with a finite workload capacity and an infinite dam model under P λ M policy. It is necessary to analyze the workload processes for these systems, which is mostly difficult and complex. An efficient and easy approximation procedure for these systems is provided. Our method is based on the idea that the service time of a customer can be approximated as the sum of small deterministic service times of a batch of customers. This enables us to work with system size processes instead of workload processes. The proposed method turns out to be in excellent agreement with simulation results and exact solution.

Batch scheduling in a two-level supply chain— a focus on the supplier

This paper studies the batch scheduling problem in a supply chain from the supplier’s point of view. The supplier has to manufacture multiple products in large quantities and deliver them to customers in batches. There is an associated delivery cost with each delivered batch. The objective of the supplier is to minimize the inventory holding and delivery costs. The supplier’s system is modeled as a one-stage system in which processing a batch requires a setup when the system switches to a different product. A polynomial-time algorithm is presented to minimize the sum of the total inventory holding cost and the batch delivery cost of the supplier.

Age Process, Workload Process, Sojourn Times, and Waiting Times in a Discrete Time SM[K]/PH[K]/1/FCFS Queue

In this paper, we study a discrete time queueing system with multiple types of customers and a first-come-first-served (FCFS) service discipline. Customers arrive according to a semi-Markov arrival process and the service times of individual customers have PH-distributions. A GI/M/1 type Markov chain for a generalized age process of batches of customers is introduced. The steady state distribution of the GI/M/1 type Markov chain is found explicitly and, consequently, the steady state distributions of the age of the batch in service, the total workload in the system, waiting times, and sojourn times of different batches and different types of customers are obtained. We show that the generalized age process and a generalized total workload process have the same steady state distribution. We prove that the waiting times and sojourn times have PH-distributions and find matrix representations of those PH-distributions. When the arrival process is a Markov arrival process with marked transitions, we construct a QBD process for the age process and the total workload process. The steady state distributions of the waiting times and the sojourn times, both at the batch level and the customer level, are obtained from the steady state distribution of the QBD process. A number of numerical examples are presented to gain insight into the waiting processes of different types of customers.