Batch Of Customers Research Articles

Transient analysis of a bulk stream queue with arbitrarily distributed arrival intervals

We study the classical infinite buffer single server queueing model with renewal input of customers in batches of random size, having arbitrarily distributed arrival intervals and exponentially distributed service times. Using the technique of supplementary variable and shift operator we derive closed form expression of the time dependent system content distribution in terms of its Laplace transform. The analysis is mainly based on the root-finding technique of the non-linear characteristic equation in terms of the Laplace transform variable. Additionally, using asymptotic properties of Laplace transform, we deduce the corresponding steady-state distribution. We discuss some special cases of the model, thus providing an alternative approach in deriving the transient distribution. We further evaluate certain performance measures and present extensive numerical examples in tabular and graphical form to illustrate the applicability of our theoretical work. The effect of system parameters, interarrival time distribution and traffic intensity on the system behavior is also demonstrated.

A math-heuristic approach for scheduling the production and delivery of a mobile additive manufacturing hub

This paper investigates an innovative printing-while-delivering paradigm within a mobile additive manufacturing (MAM) hub. Different from the conventional production-first-delivery-second (PFDS) mode, the MAM hub, comprising a 3D printer and a truck, can seamlessly integrate the production and delivery processes. Throughout the delivery process of the MAM hub, the embedded 3D printer within the truck concurrently prints products for customers in batches. This capability significantly diminishes customer dissatisfaction caused by time delays between order placement and product receipt, which requires coordinating the delivery routing of truck and the production scheduling of 3D printer. To tackle this problem, a mixed integer programming model is firstly established to minimize the required time of the MAM hub returning to depot after completing all deliveries. Then, a math-heuristic approach combining the genetic algorithm and branch-and-bound is devised to solve the mathematical model efficiently. The delivery sequence of truck is represented as a chromosome in the genetic algorithm; while, for each delivery sequence, the optimal production scheduling of 3D printer is obtained by branch-and-bound based on the theoretical analysis about the relationship between production and delivery sequences. In the numerical experiments, the proposed math-heuristic algorithm produces results identical to CPLEX solver outcomes in small-scale cases. For larger instances, it outperforms traditional genetic algorithms by 3.39% to 19.54%. The proposed batch printing mode consistently surpasses the non-batch mode by 10% to 25%. Moreover, compared to PFDS, the printing-while-delivering paradigm reduces trip time by over 40%. Finally, we analyze the effects of customer quantity, distribution radius, and printing time on the objective, providing management insights about optimal production and delivery strategy. The research finding not only aids in scheduling the MAM hub but also holds the potential to address other applications characterized by simultaneous production and delivery requirements.

Performance study of the M/M(a,b)/1/MWV queuing system with types of breakdowns

Abstract This paper analyses the M/M(a,b)/1/MWV queuing model with various types of breakdowns. During peak times, the system has breakdown due to server unavailability. The model considers system with two types of breakdowns. The system may breakdown in two different ways during busy and working vacation stages. Customer arrives to the system with parameter λv which follows Poisson distribution and server provides service in regular busy period with parameter µrb and under the multiple working vacations it provides service with parameter µwv with exponential distribution. In this model batches of customers are served under General Bulk Service Rule. The system may breakdown at any time. The breakdown that occurs during working vacation is denoted as βv 1 and the breakdown during busy state is denoted as βv 2. The steady-state equation, the performance of measures for the system and particular cases of described model are derived. Finally, a real life example demonstrated the validity of the model in a proper way.

ASIP tandem queues with consumption

The Asymmetric Inclusion Process (ASIP) tandem queue is a model of stations in series with a gate after each station. At a gate opening, all customers in that station instantaneously move to the next station unidirectionally. In our study, we enhance the ASIP model by introducing the capability for individual customers to independently move from one station to the next, and by allowing both individual customers and batches of customers from any station to exit the system. The model is inspired by the process by which macromolecules are transported within cells.We present a comprehensive analysis of various aspects of the queue length in the ASIP tandem model. Specifically, we provide an exact analysis of queue length moments and correlations and, under certain circumstances, of the queue length distribution. Furthermore, we propose an approximation for the joint queue length distribution. This approximation is derived using three different approaches, one of which employs the concept of the replica mean-field limit. Among other results, our analysis offers insight into the extent to which nutrients can support the survival of a cell.

Unreliable server with Non-Markovian Retrial Queueing System, Bernoulli Vacation and Fortuitous breakdown

This paper explores the behaviour of a Bulk Arrival Retrial Queue Model (BARQ) with two phases of service under the Bernoulli Vacation schedule and Breakdown (BVSB). Each batch of customers arriving the system finds if the server is available, instantly utilizes the service. If the server is busy, under breakdown, or taking a vacation, then the customers enter into the orbit. After completing both service stages, the server will either take a vacation with probability p or wait until the next customer arrives with probability 1 - p or q. Our approach considers the nature of the customer as balking and also takes into account the breakdown of server, which may occur instantaneously during any stage of service. Significant performance measures have been derived and presented. A numerical study of the proposed model is carried out using MATLAB and results were reported.

Multi-Class, Multi-Server Queueing Inventory System with Batch Service

In this paper, we consider a queueing inventory system with batch arrival and batch service processes. Customers arrive in batches of sizes 1,2,…,k, according to a marked compound Poisson process. We call a batch of customers as belonging to j when there are j individual customers in that batch. The service facility has waiting rooms for each category of customers and also a room to serve them. Except for category 1, all other customers have finite waiting rooms. The service room has only a limited number of seats. These seats are arranged in such a fashion that customers belonging to category j have groups of seats, each with j seats for j=1,2,…,k. Customers are taken for service according to the availability of seats designated to each category. A category j customer can be taken for service only if j items are available in the inventory. The service time of customers of category j is exponentially distributed with parameter depending on j for j=1,2,…,k. The number of seats available in the service room for each category of customers is restricted to a finite number. The replenishment for items follows the (s,S) policy: fill up to S at the time of replenishment. Lead time follows an exponential distribution. We analyze the system in the equilibrium state. Performance characteristics are evaluated and a number of numerical illustrations are provided.

Fast Joint Multi-Robot Trajectory Optimization by GPU Accelerated Batch Solution of Distributed Sub-Problems.

We present a joint multi-robot trajectory optimizer that can compute trajectories for tens of robots in aerial swarms within a small fraction of a second. The computational efficiency of our approach is built on breaking the per-iteration computation of the joint optimization into smaller, decoupled sub-problems and solving them in parallel through a custom batch optimizer. We show that each of the sub-problems can be reformulated to have a special Quadratic Programming structure, wherein the matrices are shared across all the problems and only the associated vector varies. As result, the batch solution update rule reduces to computing just large matrix vector products which can be trivially accelerated using GPUs. We validate our optimizer’s performance in difficult benchmark scenarios and compare it against existing state-of-the-art approaches. We demonstrate remarkable improvements in computation time its scaling with respect to the number of robots. Moreover, we also perform better in trajectory quality as measured by smoothness and arc-length metrics.

Analysis of batch size-dependent bulk service queue with multiple working vacation

ABSTRACT The present article analyses an infinite buffer batch size-dependent bulk service queue with multiple working vacation (MWV). The customer's arrival pattern follows the Poisson process, and they get the service in batches according to the general bulk service (GBS) rule in regular period (RP) as well as in working vacation period (WVP). After one service in RP, if the queue size is greater or equal to the lower threshold of the GBS rule, then the server performs the service in RP, otherwise, it starts the working vacation (WV) following exponential vacation time distribution. During the WVP, the server serves the customers in batches (if any) at a lower service rate than the usual service rate. The service time in the RP as well as in WVP is generally distributed. At an arbitrary epoch, the joint probabilities of the queue size and batch size with the server in RP as well as in WVP have been obtained using the supplementary variable technique (SVT) and the bivariate generating function method. Finally, some numerical observations are presented to enhance the applicability of the analytical results.

Transient and steady-state analysis of hybrid arrivals of single and batch customers queueing systems with switch-off period

In this paper, we analyze the transient behavior of a novel single server queueing system with hybrid arrivals of single and state-dependent batch of customers. Whenever the system becomes empty the server undergoes the switch-off state for random duration of time. Explicit expressions for transient probabilities of the status of the server and system size are obtained. In addition, the first-passage-time probability density function is determined along with its first two moments. We have also derived analytical results for time-dependent performance measures of the system. Further, under stability condition, the corresponding steady-state probabilities and key performance measures of the system are also discussed. Finally, we have provided extensive numerical illustrations to visualize the behaviors of system performance measures graphically by varying the values of system parameters.

Integrated production and distribution scheduling in distributed hybrid flow shops

To improve the operational efficiency and competitive advantage of supply chains, integrated production and distribution has attracted an increasing attention in recent years. This paper focuses on a novel integrated production and distribution scheduling problem (IPDSP) with consideration of factory eligibility and third-party logistics (3PL). In this problem, products are firstly produced in a number of distributed hybrid flow shops (HFS) and then delivered to a customer in batches. To satisfy the production and distribution practice, some products can only be manufactured in a subset of distributed HFSs, and the transportation of some finished products is outsourced to a 3PL provider. Considering the NP-hardness of IPDSP, three fast heuristics (CR-based heuristic, SLACK-based heuristic, and EDD-based heuristic) and an adaptive human-learning-based genetic algorithm (AHLBGA) are proposed to minimize the sum of earliness, tardiness and delivery costs. Motivated by human learning behaviours, AHLBGA integrates an adaptive learning operator with traditional genetic operators to generate candidate solutions. Such learning operator performs social learning, family learning, and individual random learning to improve offspring individuals. The computational experiments on small-sized and large-sized test problems show the superiority of AHLBGA.

Computing stationary distributions of the D-MAP/D-MSP$$^{(a,b) }/1$$ queueing system

The present study investigates a single-server batch service queueing model with arrival process as discrete-time Markovian arrival process and service process as discrete-time Markovian service process. The server serves the customers in batches followed by the general bulk-service rule. We first determine the random epoch probabilities using the matrix-geometric method, where the rate matrix $${\mathbf{R}}$$ is determined by an efficient approach based on the eigenvalues and the corresponding eigenvectors of the associated characteristic equation. Next we obtain the explicit closed-form expressions for the pre-arrival, intermediate, outside observer’s and post-departure epoch probabilities by developing the relations among them in equilibrium state. Further, we provide an analytically simple approach to carry out the waiting-time distribution in the queue measured in slots as well as the distribution of the size of a service batch of an arriving customer. We also demonstrate a cost function to evaluate the optimum value of the minimum service batch size and the corresponding expected cost of the system. Finally, an adequate variety of numerical experiments are performed for validation purpose of our analytical results and they are conferred in the form of tables and graphs.

Analysis of state dependent M[X]/G(a, b)/1 queue with multiple vacation second optional service and optional re-service

The objective of this paper is to analyse an M[X]/G(a, b)/1 queueing model with second optional service, multiple vacation, state dependent arrival and optional re-service. After completing the first essential service, a batch of customers either requests for re-service or leaves the system without re-service. After the completion of first essential service (with or without re-service), the batch of customers either requests for second optional service or leaves the system. At the completion moment of second optional service, the batch of customers either requests for re-service or leaves the system after the second service. Whenever the queue size is less than 'a', the server commences vacation. At the instant of vacation completion, if at least 'a' customers wait for service, the server starts a busy period. Otherwise, the server resumes another vacation. Using supplementary variable technique, the steady-state probability generating function (PGF) of the queue size is obtained.

Analysis of customer's impatience on bulk service queueing system with unreliable server, setup time and two types of multiple vacations

In this investigation, we analyse the customer's impatience on unreliable bulk queueing system with two types of vacation. Depending upon the length of queue, the server is permitted to take two types of vacation called type I vacation and type II vacation. The server is possibly affected by breakdowns and takes considerable time for setup before the repair begins to start. After repair completion, the server will start its remaining service to the batch of customers whose service was interrupted. It could be assumed that the arriving units may be balked from the system when the server is on vacation (type I or type II). By incorporating the supplementary variables technique, the probability generating function of the queue size at an arbitrary and departure time epoch for the designed queueing system are derived. Finally, the stability condition, some performance measures, particular cases are obtained with appropriate numerical illustrations and graphical presentation.

Scheduling on a Single Machine and Parallel Machines with Batch Deliveries and Potential Disruption

In this paper, we study several coordinated production-delivery scheduling problems with potential disruption motivated by a supply chain in the manufacturing industry. Both single-machine environment and identical parallel-machine environment are considered in the production part. The jobs finished on the machines are delivered to the same customer in batches. Each delivery batch has a capacity and incurs a delivery cost. There is a situation that a possible disruption in the production part may occur at some particular time and will last for a period of time with a probability. We consider both resumable case and nonresumable case where a job does not need (needs) to restart if it is disrupted for a resumable (nonresumable) case. The objective is to find a coordinated schedule of production and delivery that minimizes the expected total flow times plus the delivery costs. We first present some properties and analyze the NP-hard complexity for four various problems. For the corresponding single-machine and parallel-machine scheduling problems, pseudo-polynomial-time algorithms and fully polynomial-time approximation schemes (FPTASs) are presented in this paper, respectively.

Entropy Analysis of a Flexible Markovian Queue with Server Breakdowns.

In this paper, a versatile Markovian queueing system is considered. Given a fixed threshold level c, the server serves customers one a time when the queue length is less than c, and in batches of fixed size c when the queue length is greater than or equal to c. The server is subject to failure when serving either a single or a batch of customers. Service rates, failure rates, and repair rates, depend on whether the server is serving a single customer or a batch of customers. While the analytical method provides the initial probability vector, we use the entropy principle to obtain both the initial probability vector (for comparison) and the tail probability vector. The comparison shows the results obtained analytically and approximately are in good agreement, especially when the first two moments are used in the entropy approach.

QUIDDIT - QUantification of infrared active Defects in Diamond and Inferred Temperatures

QUIDDIT is a free Python software-package designed to process Fourier Transform Infrared (FTIR) spectra of diamonds automatically and efficiently. Core capabilities include baseline correction, determination of nitrogen concentration, nitrogen aggregation state and model temperature and fitting of both the 3107 cm-1 and platelet (B’) peaks. These capabilities have allowed the authors to study platelet defects and their relationship to nitrogen aggregation in previous studies. Data visualisation, vital to interpreting and evaluating results, is another key component of the software. QUIDDIT can be applied to single spectra as well as linescan and 2-dimensional map data. Recently, additional features such as manual platelet peak and nitrogen fitting, custom batch peak fitting and two-stage aggregation modelling were made available. QUIDDIT has been used successfully for natural diamonds containing aggregated forms of nitrogen in the past and has since been adapted for the study of diamonds containing C-centres as well.

Extended analysis and computationally efficient results for the queueing system

We consider an infinite waiting space queueing system in which customers arrive according to a renewal process. The server serves the customers in batches under general bulk-service rule. The successive service times of batches are mutually independent and have a common exponential distribution. To obtain the queue-length distributions at prearrival and random epochs, we have used the displacement operator which assists to solve simultaneous non-homogeneous difference equations. An analytically simple and computationally efficient procedures have developed to compute the waiting-time distribution of an arrival customer. Our formula to determine the waiting-time distribution is also useful even when multiple poles occur in the Laplace-Stieltjes transform of the interarrival time distribution. We have determined closed-form analytical expression for the distribution of size of a service batch of an arrived customer. We also have derived the results of some particular well-known queueing models from our model. Some numerical results are provided in the form of tables for a variety of interarrival-time distributions to demonstrate the variation of performance measures of the system.

Analysis of stationary queue-length distributions of the BMAP/R(a,b)/1 queue

This paper investigates an infinite waiting space single-server queueing system in which customers arrive according to a batch Markovian arrival process. The server serves the customers in batches of maximum size ‘b’ with a minimum threshold value ‘a’. The service time of each batch follows R-type distribution, where R-type distribution represents a class of distributions whose Laplace-Stieltjes transform is rational or approximated rational function, which is independent of service batch size and the arrival process. We determine the queue-length distribution at departure epoch in terms of roots of the associated characteristic equation of the vector probability generating function. To show the strength and advantage of roots method, we also provide a comprehensive analysis of the queue-length distribution at departure epoch using the matrix-analytic method (MAM). To determine the queue-length distribution at random epoch, we obtain a relation between the queue-length distributions at departure and random epochs using the supplementary variable technique with remaining service time of a batch in service as the supplementary variable. We also derive the queue-length distributions at pre-arrival epoch of an arrived batch and post-arrival epoch of a random customer of an arrived batch. Some numerical results are demonstrated for different service-time distributions to show the key performance measures of the system.

Analysis of a batch arrival retrial queue with impatient customers subject to the server disasters

<p style='text-indent:20px;'>We consider an <inline-formula><tex-math id="M1">\begin{document}$ M^X/G/1 $\end{document}</tex-math></inline-formula> retrial queue with impatient customers subject to disastrous failures at which times all customers in the system are lost. When the server finishes serving a customer and finds the orbit empty, the server becomes dormant until <inline-formula><tex-math id="M2">\begin{document}$ N $\end{document}</tex-math></inline-formula> or more customers accumulate. If a coming batch of customers finds the server idle, one of the arriving customers begins his service immediately and the rest join the orbit to repeat their request later. Otherwise, if the server is dormant or busy or down, all customers of the coming batch enter the orbit. When the server is under repair, customers in the orbit can become impatient after waiting a random amount of time and leave the system. By using the characteristic method for partial differential equations, the steady-state distributions of the server state and the number of customers in the orbit are obtained along with various performance measures. In addition, the reliability of the system is analyzed detailed. Finally, an application to telecommunication networks is provided and the effects of various parameters on the system performance are demonstrated numerically.

Heavy-traffic limits for Discriminatory Processor Sharing models with joint batch arrivals

We study the performance of Discriminatory Processor Sharing (DPS) systems, with exponential service times and in which batches of customers of different types may arrive simultaneously according to a Poisson process. We show that the stationary joint queue-length distribution exhibits state-space collapse in heavy traffic: as the load ρ tends to 1, the scaled joint queue-length vector (1−ρ)Q converges in distribution to the product of a deterministic vector and an exponentially distributed random variable, with known parameters. The result provides new insights into the behavior of DPS systems. It shows how the queue-length distribution depends on the system parameters, and in particular, on the simultaneity of the arrivals. The result also suggests simple and fast approximations for the tail probabilities and the moments of the queue lengths in stable DPS systems, capturing the impact of the correlation structure in the arrival processes. Numerical experiments indicate that the approximations are accurate for medium and heavily loaded systems.