Batch Service Queue Research Articles

Strategic Behavior and Social Welfare Optimization in a Batch-service Queueing System with Threshold Policy

Strategic Behavior and Social Welfare Optimization in a Batch-service Queueing System with Threshold Policy

Analytic computation for stationary distributions of D-BMAP/D -MSP(a,b)/1 queueing system

ABSTRACT This paper analyses single server correlated batch arrival and correlated batch service queue with innumerable waiting space in discrete-time. The arrival and service processes are governed by discrete batch Markovian arrival process and discrete Markovian service process with general bulk service rule, respectively. Observing two consecutive random epochs, we construct general structure transition probability matrix and it is then reblocked to the desired M/G/1 structure. The UL-type RG-factorization approach combined with spectral expansion method is adopted to unravel the queue length distribution at random epoch. The relationships are undertaken to obtain the queue length distributions at pre-arrival, outside observer's, intermediate and post-departure epochs. Our most challenging effort is to derive the probability mass functions of waiting time and service batch size for arbitrary customer in an arriving batch. We address the warehouse management process for finished products from dealer to retailer as a potential practical use of our batch queue with correlation in both arrival and service, and provide the effect of our queueing behaviour in this application based on our analytical and computational investigation. Based on the implementation of parametrized numeric experiments with numerous categories, we impart our computational results to validate the analytical findings reported in this investigation.

Finite population and finite capacity single server batch service queues with compulsory working vacation

Finite population and finite capacity single server batch service queues with compulsory working vacation

Exact and asymptotic analysis of infinite server batch service queues with random batch sizes

Exact and asymptotic analysis of infinite server batch service queues with random batch sizes

Equilibrium Analysis for Batch Service Queueing Systems with Strategic Choice of Batch Size

Various transportation services exist, such as ride-sharing or shared taxis, in which customers receive services in a batch of flexible sizes and share fees. In this study, we conducted an equilibrium analysis of a variable batch service model in which customers who observe no waiting customers in an incomplete batch can strategically select a batch size to maximize the individual utilities. We formulated this model as a three-dimensional Markov chain and created a book-type transition diagram. To consider the joining/balking dilemma of customers for this model, we proposed an effective algorithm to construct a necessary and sufficient size of state space for the Markov chain provided that all customers adopt the threshold-type equilibrium strategy. Moreover, we proved that the best batch size is a non-decreasing function for i if the reward for the completion of batch service with size l is an increasing function of l assuming that a tagged customer observes i complete batches in the system upon arrival; in other words, the fee decreases as the batch becomes larger. We then derive several performance measures, such as throughput, social welfare, and monopolist’s revenue. Throughout the numerical experiment, a comparison between the present variable batch service model and regular batch service model in which customers were served in a constant batch, was discussed. It was demonstrated that the three performance measures can be optimized simultaneously in the variable batch service model, as long as the fee was set relatively high.

A Moment Approach for a Conditional Central Limit Theorem of Infinite-Server Queue: A Case of M/MX/∞ Queue

Several studies have been conducted on scaling limits for Markov-modulated infinite-server queues. To the best of our knowledge, most of these studies adopt an approach to prove the convergence of the moment-generating function (or characteristic function) of the random variable that represents a scaled version of the number of busy servers and show the weak law of large numbers and the central limit theorem (CLT). In these studies, an essential assumption is the finiteness of the phase process and, in most of them, the CLT for the number of busy servers conditional on the phase (or the joint states) has not been considered. This paper proposes a new method called the moment approach to address these two limitations in an infinite-server batch service queue, which is called the M/MX/∞ queue. We derive the conditional weak law of large numbers and a recursive formula that suggests the conditional CLT. We derive series expansion of the conditional raw moments, which are used to confirm the conditional CLT by a symbolic algorithm.

Analysis of non-Markovian batch service queue with second optional service under transient and steady state domain

Analysis of non-Markovian batch service queue with second optional service under transient and steady state domain

Performance analysis of an infinite-buffer batch-size-dependent bulk service queue with server breakdown and multiple vacation

<p style='text-indent:20px;'>In several manufacturing systems, server's breakdown is a common phenomenon for which renovation of the service station is required and has a definite effect on the system. The server's vacation is a significant tool to deal with the systems where a common server is shared by several users. The principal goal of this paper is to analyze a batch-service queueing model with server's breakdown, multiple vacations and batch-size-dependent service time. To be more precise, the center of attention is on the queue length together with server content distribution for which we obtain the bivariate probability generating function using supplementary variable technique. Server content distribution is very worthy for the manufacturing systems. The probabilities have been extracted and used to acquire the arbitrary epoch probabilities. In order to upgrade the qualitative aspects of the concerned model, diverse numerical results and graphical observations are illustrated.</p>

Analysis and evaluation of synchronous and asynchronous FLchain

Motivated by the heterogeneous nature of devices participating in large-scale federated learning (FL) optimization, we focus on an asynchronous server-less FL solution empowered by blockchain technology. In contrast to mostly adopted FL approaches, which assume synchronous operation, we advocate an asynchronous method whereby model aggregation is done as clients submit their local updates. The asynchronous setting fits well with the federated optimization idea in practical large-scale settings with heterogeneous clients. Thus, it potentially leads to higher efficiency in terms of communication overhead and idle periods. To evaluate the learning completion delay of BC-enabled FL, namely FLchain, we provide an analytical model based on batch service queue theory. Furthermore, we provide simulation results to assess the performance of both synchronous and asynchronous mechanisms. Important aspects involved in the BC-enabled FL optimization, such as the network size, link capacity, or user requirements, are put together and analyzed. As our results show, the synchronous setting leads to higher prediction accuracy than the asynchronous case. Nevertheless, asynchronous federated optimization provides much lower latency in many cases, thus becoming an appealing solution for FL when dealing with large datasets, tough timing constraints (e.g., near-real-time applications), or highly varying training data.

To wait or not to wait: Strategic behaviors in an observable batch-service queueing system

This paper deals with an observable batch service queueing system in which customers rationally choose whether to form a batch with another customer or not, in addition to deciding whether or not to join the queue. When choosing whether to form a batch, a customer in an incomplete batch decides on an optimal waiting time for the next customer to arrive and share the service fee. When choosing whether to join the queue, customers follow a threshold strategy, which depends on the strategy identified in the former game.

Analysis of Non-Markovian Batch Service Queue with Second Optional Service under Transient and Steady State Domain

Analysis of Non-Markovian Batch Service Queue with Second Optional Service under Transient and Steady State Domain

Performance of batch service queue model with second optional service, repairable breakdown and standby server

In this paper, we investigate the performance of batch service queue model with second optional service, repairable breakdown and warm standby server. Both primary operating and warm standby servers provide First Essential Service (FES) and Second Optional Service (SOS) to customers, wherein FES is all arriving customers and only some of them may further request the SOS. The service times, failure times and repair times of both primary operating and warm standby server are assumed to follow exponential distributions. We use Runge-Kutta method to obtain the transient state probabilities and matrix-decomposition method to obtain the steady-state probabilities of the model. Also, a cost model is presented to determine the optimal service rates so that the expected cost is minimised. Finally, the effect of the model parameters on the system behaviour is demonstrated through numerical results and discussions.

On state dependent batch service queue with single and multiple vacation under Markovian arrival process

On state dependent batch service queue with single and multiple vacation under Markovian arrival process

Performance of batch service queue model with second optional service, repairable breakdown and standby server

Performance of batch service queue model with second optional service, repairable breakdown and standby server

Study on Infinite Buffer Batch Size Dependent Bulk Service Queue with Queue Length Dependent Vacation.

Queueing models with vacations have drawn the attention of researchers over several decades as a handy tool for tackling real-life congestion problems. Keeping this in mind, we pay attention to an infinite buffer single server batch-size dependent batch service queue with queue size (queue length) dependent vacation. The arrival pattern of the customers in the system follows the Poisson process where they get the service in packets/group following the general batch service (GBS) rule. An embedded Markov chain technique is used for the mathematical analysis where service (vacation) completion epochs have been taken as an embedded Markov point. We obtain the bivariate generating functions of the queue size and vacation type (queue size at vacation initiation epoch) at vacation termination epoch, and the bivariate generating function of the queue size and batch size with the server at service completion epoch, and then we successfully extract the steady-state joint probabilities of the queue size and batch size with the server and the joint probabilities of the queue size and vacation type at various epochs. Finally, various performance measures are presented. Also, the behavior of the considered model is presented by the graphs and tables.

Transient and steady state analysis of M/M^([b])/1 queue with second optional service

ABSTRACT In this paper, the transient and steady state of a single server batch service queueing system with second optional service is analyzed. All incoming units receive essential service, once the essential service is completed, the units may opt the optional service. The study applies the probability generating function, Rouche’s theorem, and Laplace transform techniques to obtain the transient state probabilities. The stationary probabilities are obtained by using the Tauberian property in the Laplace transform expressions. This study contributes to filling the gap on investigation the batch queue with essential and optional services with interesting practical application in health care systems. Furthermore, we have presented numerical results and cost optimization. The results reveal that a higher service rate in the essential service helps the system manager to run the system effectively and emphasize on optimal service rates in order to have a cost benefit and less congestion in the queue.

Complete analysis of a discrete-time batch service queue with batch-size-dependent service time under correlated arrival process: D-MAP/Gn(a,b)/1

Discrete-time queueing models find a large number of applications as they are used in modeling queueing systems arising in digital platforms like telecommunication systems and computer networks. In this paper, we analyze an infinite-buffer queueing model with discrete Markovian arrival process. The units on arrival are served in batches by a single server according to the general bulk-service rule, and the service time follows general distribution with service rate depending on the size of the batch being served. We mathematically formulate the model using the supplementary variable technique and obtain the vector generating function at the departure epoch. The generating function is in turn used to extract the joint distribution of queue and server content in terms of the roots of the characteristic equation. Further, we develop the relationship between the distribution at the departure epoch and the distribution at arbitrary, pre-arrival and outside observer’s epochs, where the first is used to obtain the latter ones. We evaluate some essential performance measures of the system and also discuss the computing process extensively which is demonstrated by some numerical examples.

One-dimensional service networks and batch service queues

The proliferation of smart devices, computational and storage resources is predicted to continue aggressively in the near future. Such “networked” devices and resources which are distributed in a physical space and provide services are collectively referred to as a distributed service network. Assigning users or applications to available resources is important to sustain high performance of the distributed service network. In this work, we consider a one-dimensional service network where both users and resources are located on a line, and analyze a unidirectional assignment policy Move To Right (MTR), which sequentially assigns users to resources available to their right. We express the communication cost for a user-resource assignment as an increasing function of the distance traveled by the user request (request distance) and analyze the expected communication cost for the service network when locations of users and resources are modeled by different spatial point processes. We use results from the literature that map the request distance of an assigned user in a one-dimensional service network to the sojourn time of a customer in an exceptional service accessible batch queueing system. We compute the Laplace–Stieltjes transform of the sojourn time distribution for this queueing system for Poisson distributed users with general inter-resource distance distributions and in the process also generate new results for batch service queues. Unlike previous work (Panigrahy et al. in Perform Eval 142:102, 2020), our framework not only captures the first-order moment of the request distance, but also the request distance distribution itself, thus allowing us to compute the expected communication cost under different cost models.

On the joint distribution of an infinite-buffer discrete-time batch-size-dependent service queue with single and multiple vacations

ABSTRACT Due to the widespread applicability of discrete-time queues in wireless networks or telecommunication systems, this paper analyzes an infinite-buffer batch-service queue with single and multiple vacations where customers/messages arrive according to the Bernoulli process and service time varies with the batch-size. The foremost focal point of this analysis is to get the complete joint distribution of queue length and server content at the service completion epoch, for which first the bivariate probability generating function has been derived. We also acquire the joint distribution at an arbitrary slot. We also provide several marginal distributions and performance measures for the utilization of the vendor. During the transmission of data through a particular communication channel, high transmission error may take place due to several factors. For this reason, one may skip the transmission through that particular channel. However the discrete phase-type distribution plays a noteworthy role to control this transmission error which eventually motivates us to include a numerical example where service time distribution follows discrete phase-type distribution. A comparison between batch-size-dependent and -independent services has been drawn through the graphical representation of some performance measures and total system cost.

Stationary distribution of an infinite-buffer batch-arrival and batch-service queue with random serving capacity and batch-size-dependent service

In traditional batch-service queueing systems, the mean service time of batches are generally assumed to be constant. However, in numerous applications this assumption may not be appropriate. In telecommunication networks, the transmission rates depend on the number of packets in the batch which can be framed as batch-size-dependent service queue. The objective of this paper is to focus on both queue and server content distribution in an infinite-buffer batch-arrival and batch-service queue with random serving capacity rule and batch-size-dependent service. After deriving a bivariate probability generating function of queue length and server content distribution at departure epoch of a batch, we extract the complete joint distribution in terms of roots of the characteristic equation. We also obtain the system as well as queue length distribution at arbitrary epoch. Finally, a significant number of numerical examples are appended to show the feasibility of the analytic procedure and results where the occurrence of multiple roots have been dealt without facing any difficulty. At the end, a graphical representation of cost of the system shows that batch-size-dependent service is more significant as compared to batch-size-independent service.