Batch Service Queue Research Articles

Transient Analysis of Batch Service Queue with Second Optional Service and Reneging

This paper investigates a single server batch service queueing system with second optional service and reneging. The transient state probabilities of the queueing model are computed by using fourth order Runge Kutta method. Cost analysis is performed to determine the optimal values of the service rates of First Essential Service (FES) and Second Optional Service (SOS) simultaneously at the minimum total expected cost per unit time. Some important performance measures and numerical illustrations are provided in order to show the managerial intuitions of the model.

Stationary queue and server content distribution of a batch-size-dependent service queue with batch Markovian arrival process:

Queueing systems with batch Markovian arrival process (BMAP) have paramount applications in the domain of wireless communication. The BMAP has been used to model the superposition of video sources and to approximate the super-position of data, voice and video traffic. This article analyzes an infinite-buffer generally distributed batch-service queue with BMAP, general bulk service (a, b) rule and batch-size-dependent service time. In this proposed analysis, we mainly focus on deriving the bivariate vector generating function of queue and server content distribution together at departure epoch using supplementary variable technique. The mathematical procedure for the complete extraction of distribution at departure epoch has been discussed and using those extracted probabilities, we achieve the queue and server content distribution at arbitrary epoch. Finally, numerical illustrations have been carried out in order to make a deep insight to the readers which contains deterministic as well as phase-type service time distributions.

A $ {BMAP/BMSP/1} $ queue with Markov dependent arrival and Markov dependent service batches

Batch arrival and batch service queueing systems are of importance in the context of telecommunication networks. None of the work reported so far consider the dependence of consecutive arrival and service batches. Batch Markovian Arrival Process( \begin{document}$ BMAP $\end{document} ) and Batch Markovian Service Process ( \begin{document}$ BMSP $\end{document} ) take care of the dependence between successive inter-arrival and service times, respectively. However in real life situations dependence between consecutive arrival and service batch sizes also play an important role. This is to regulate the workload of the server in the context of service and to restrict the arrival batch size when the flow is from the same source. In this paper we study a queueing system with Markov dependent arrival and service batch sizes. The arrival and service batch sizes are assumed to be finite. Further, successive inter-arrival and service time durations are also assumed to be correlated. Specifically, we consider a \begin{document}$ BMAP/BMSP/1 $\end{document} queue with Markov dependent arrival and Markov dependent service batch sizes. The stability of the system is investigated. The steady state probability vectors of the system state and some important performance measures are computed. The Laplace-Stieltjes transform of waiting time and idle time of the server are obtained. Some numerical examples are provided.

A stochastic model and social optimization of a blockchain system based on a general limited batch service queue

Blockchain is well known as a database technology supporting digital currencies such as Bitcoin, Ether and Ripple. For the purpose of maximizing the overall revenue of the blockchain system, we propose a pricing policy to impose on transactions. Regarding the mining process as a vacation, and the block-verification process as a service, we establish a type of non-exhaustive queueing model with a limited batch service and a possible zero-transaction service. By selecting the beginning instant of a block-verification process as a Markov point and using the method of a generating function, we obtain the stationary probability distribution for the number of transactions in the system at the Markov points and analyze the elapsed time for the mining cycle. Based on the model analysis results, we derive the average latency of transactions and demonstrate how the average latency of transactions changes in relation to the arrival rate of transactions. With a reward-cost structure, we construct an individual benefit function and a social benefit function. By improving the Grasshopper Optimization Algorithm (GOA), we search for the Nash equilibrium and the socially optimal arrival rates of transactions. Numerical results show that the Nash equilibrium arrival rate of transactions is always higher than the socially optimal arrival rate of transactions for a given mining parameter and a specific block capacity. For this, we propose a pricing policy that forces the transactions to accept the socially optimal arrival rate and maximize the overall revenue of the blockchain system, including all transactions and miners.

Managerial decision analysis of bulk arrival queuing system with state dependent breakdown and vacation

The aim of this paper is to analyse state dependent breakdown in bulk arrival and batch service queueing system with vacation. Server affords service in two service modes as single and batch service, depending upon the queue length. In this model occurrence of breakdown during single service and batch service are different. If the server fails during single service, service interrupted and sent to repair station immediately. Also, when the server got failure during batch service, service will not be interrupted, it continues for current batch by doing some technical arrangements. After service completion, if the queue length is less than the minimum server capacity then the server leaves for a vacation. For the designed queueing system probability generating function of the queue size at an arbitrary time is obtained. Various performance measures are derived with numerical illustration. Additionally optimum cost analysis is also carried out for the model.

System occupancy in a multiclass batch-service queueing system with limited variable service capacity

In multi-class telecommunications or manufacturing systems, customers belonging to the same class can often be processed together. This results in a service capacity that depends on the classes of the customers in the queue. In this paper, we analyse a discrete-time batch-service queue with two customer classes. The single batch server can group all same-class customers at the head of the queue up to a constant class-dependent maximum service capacity. We focus on the analysis of the system occupancy at service initiation opportunities, and also compute both a light- and heavy traffic approximation in order to reduce the numerical complexity introduced by the maximum service capacities. Additionally, we propose a method for interpolating between these approximations in order to study the behaviour in the intermediate region. We also deduce the system occupancy and its approximations at random slot boundaries. In the numerical experiments, we examine the conditions under which these proposed approximations are accurate.

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.

A Discrete-Time $$GI^{X}/Geo^{Y}/1$$ Queue with Early Arrival System

A discrete-time batch service queue with batch renewal input and random serving capacity rule under the late arrival delayed access system, has recently appeared in the literature (Barbhuiya and Gupta in Queueing Syst 91(3):347–365, 2019b). In this paper, we consider the same model under the early arrival system, since it is more applicable in telecommunication systems where an arriving batch of packets needs to be transmitted in the same slot in which it has arrived. In doing so, we derive the steady-state queue length distributions at various epochs, and show that in limiting case the result gets converted to the continuous-time queue (Barbhuiya and Gupta in J Differ Equ Appl 25(2):1–10, 2019a). We discuss few numerical results as well.

Analysis of a batch-service queue with variable service capacity, correlated customer types and generally distributed class-dependent service times

Queueing models with batch service have been studied frequently, for instance in the domain of telecommunications or manufacturing. Although the batch server’s capacity may be variable in practice, only a few authors have included variable capacity in their models. We analyse a batch server with multiple customer classes and a variable service capacity that depends on both the number of waiting customers and their classes. The service times are generally distributed and class-dependent. These features complicate the analysis in a non-trivial way. We tackle it by examining the system state at embedded points, and studying the resulting Markov Chain.We first establish the joint probability generating function (pgf) of the service capacity and the number of customers left behind in the queue immediately after service initiation epochs. From this joint pgf, we extract the pgf for the number of customers in the queue and in the system respectively at service initiation epochs and departure epochs, and the pgf of the actual server capacity. Combined with additional techniques, we also obtain the pgf of the queue and system content at customer arrival epochs and random slot boundaries, and the pgf of the delay of a random customer. In the numerical experiments, we focus on the impact of correlation between the classes of consecutive customers, and on the influence of different service time distributions on the system performance.

Analysis of renewal input batch service queue with impatient customers and multiple working vacations

ABSTRACTThis paper presents a renewal input single server batch service queue with impatient customers and multiple working vacations. An arriving customer may join the system with probability or balk with probability . The service is carried out in batches with a maximum threshold limit . The server takes a working vacation when there are no customers in the system, and continues taking multiple working vacations as long as there are no customers at working vacation completion epochs. Applying the embedded Markov chain approach and the displacement operator method, we find the probability distribution of the queue-length at pre-arrival epoch. With the help of the Markov renewal theory and semi-Markov processes, we compute the probability distribution of the queue-length at an arbitrary epoch. We also obtain the system-length distributions at both pre-arrival and arbitrary epochs and present a potential application of the model in cloud systems. Some corresponding results underneath special cases are found by putting suitable values. We present various performance measures and waiting-time distributions in the queue. Numerical results are demonstrated to emphasize the qualitative features of the model.

Markov processes in blockchain systems

In this paper, we develop a more general framework of block-structured Markov processes in the queueing study of blockchain systems, which can provide analysis both for the stationary performance measures and for the sojourn time of any transaction or block. In addition, an original aim of this paper is to generalize the two-stage batch-service queueing model studied in Li et al. (Blockchain queue theory. In: International conference on computational social networks. Springer: New York; 2018. p. 25–40) both “from exponential to phase-type” service times and “from Poisson to MAP” transaction arrivals. Note that the MAP transaction arrivals and the two stages of PH service times make our blockchain queue more suitable to various practical conditions of blockchain systems with crucial factors, for example, the mining processes, the block generations, the blockchain building and so forth. For such a more general blockchain queueing model, we focus on two basic research aspects: (1) using the matrix-geometric solution, we first obtain a sufficient stable condition of the blockchain system. Then, we provide simple expressions for the average stationary number of transactions in the queueing waiting room and the average stationary number of transactions in the block. (2) However, on comparing with Li et al. (2018), analysis of the transaction–confirmation time becomes very difficult and challenging due to the complicated blockchain structure. To overcome the difficulties, we develop a computational technique of the first passage times by means of both the PH distributions of infinite sizes and the RG factorizations. Finally, we hope that the methodology and results given in this paper will open a new avenue to queueing analysis of more general blockchain systems in practice and can motivate a series of promising future research on development of blockchain technologies.

A difference equation approach for analysing a batch service queue with the batch renewal arrival process

ABSTRACTWe consider an infinite buffer single server queue wherein customers arrive according to the batch renewal arrival process and are served in batches following the random serving capacity rule. The service-batch times follow exponential distribution. This model has been studied in the past using the embedded Markov chain technique and probability generating function. In this paper we provide an alternative yet simple methodology to carry out the whole analysis which is based on the supplementary variable technique and the theory of difference equations. The procedure used here is simple in the sense that it does not require the complicated task of constructing the transition probability matrix. We obtain explicit expressions of the steady-state system-content distribution at pre-arrival and arbitrary epochs in terms of roots of the associated characteristic equation. We also present few numerical results in order to illustrate the computational procedure.

On the distribution of an infinite-buffer queueing system with versatile bulk-service rule under batch-size-dependent service policy: M/G^(a,Y)_n/1

Batch-service queues have a wide range of noteworthy applications in wireless telecommunication to deal with the multimedia type of data, manufacturing systems, group testing procedure, etc. The knowledge of both the queue and server content distributions helps the system designer to evaluate the efficiency of the queueing system in a better way. We analyse a single server infinite-buffer batch-size-dependent service queue with Poisson arrival and versatile batch-service rule. Based on supplementary variable technique, a bivariate probability generating function, the entire spectrum of new contributions, of queue content and number in a served batch at departure epoch is derived. Moreover, we perceive the complete queue and server content distribution at departure as well as arbitrary epochs. The utility of analytical results is illustrated by the inclusion of some numerical examples, which also includes the investigation of multiple zeroes.

Modeling data backups as a batch-service queue with vacations and exhaustive policy

For data backup processes to cloud infrastructure, there is a clean trade off between backing up frequently (improving data safety) and reducing resource usage (power consumption and communication cost). With rapid growth of data storage requirements in recent years, we need to find the right balance between both objectives. To explicitly address this trade off, we model a wide set of exhaustive data backup processes as a general batch service queueing model with multiple vacations and probabilistic restarts.We study this queueing model and establish expressions for its performance measures such as system content and queue content distributions. This analysis aids in computing Quality of Service (QoS) measures of the data backup process such as the fraction of time the backup server is busy, the frequency of new connections and the age of the data at the beginning of a backup period. This enables us to quickly examine the dependence of QoS on the model parameters as well as to compute the optimal parameters in the backup process. We illustrate the latter by defining a particular cost function of a user and by framing an optimization problem.

Analysis of multistage M[X]/GK/1 queue with different server’s interruptions and its application in energy consumption

ABSTRACTIn this paper, we consider multistage batch arrival, batch service (fixed batch size) queue having multiple vacation policy subject to server breakdown with the delayed repair. In addition, the server may extend the vacation at each vacation completion points and the server may renege during vacation and breakdown periods. Using the supplementary variable technique, transient and its corresponding steady-state solutions are derived. The results obtained are comprehended by using a numerical example.

Performance analysis of container yard based on batch service queueing system

China Railway Express has been developed rapidly since “the Belt and Road Initiative” in China was proposed. Many cities in the Midwest of China have developed into the transport hub from the transport terminal. The key question is how to reasonably formulate the size of the container station and the frequency of the shift for China Railway Express. This paper takes the container as the research object. Regarding the containers as customers in the queuing system, customers’ services ending with the loading of containers into China Railway Express, based on a finite-capacity queueing system with batch service, this paper provides an optimal layout method for the container yard of China Railway Express.

Performance characteristics of a batch service queueing system with functioning server failure and multiple vacations

This paper examines bulk arrival and batch service queueing system with functioning server failure and multiple vacations. Customers are arriving into the system in bulk according to Poisson process with rate λ. Arriving customers are served in batches with minimum of ‘a’ and maximum of ‘b’ number of customers according to general bulk service rule. In the service completion epoch if the queue length is less than ‘a’ then the server leaves for vacation (secondary job) of random length. After a vacation completion, if the queue length is still less than ‘a’ then the server leaves for another vacation. The server keeps on going vacation until the queue length reaches the value ‘a’. The server is not stable at all the times. Sometimes it may fails during functioning of customers. Though the server fails service process will not be interrupted.It will be continued for the current batch of customers with lower service rate than the regular service rate. The server will be repaired after the service completion with lower service rate. The probability generating function of the queue size at an arbitrary time epoch will be obtained for the modelled queueing system by using supplementary variable technique. Moreover various performance characteristics will also be derived with suitable numerical illustrations.

Delay analysis of a two-class batch-service queue with class-dependent variable server capacity

In this paper, we analyse the delay of a random customer in a two-class batch-service queueing model with variable server capacity, where all customers are accommodated in a common single-server first-come-first-served queue. The server can only process customers that belong to the same class, so that the size of a batch is determined by the length of a sequence of same-class customers. This type of batch server can be found in telecommunications systems and production environments. We first determine the steady state partial probability generating function of the queue occupancy at customer arrival epochs. Using a spectral decomposition technique, we obtain the steady state probability generating function of the delay of a random customer. We also show that the distribution of the delay of a random customer corresponds to a phase-type distribution. Finally, some numerical examples are given that provide further insight in the impact of asymmetry and variance in the arrival process on the number of customers in the system and the delay of a random customer.

Probabilistic selfish routing in parallel batch and single-server queues

We consider a network of parallel queues, operating under probabilistic routing, where users can choose to join either a batch service queue, or one of several FIFO single-server queues. Afimeimounga et al. (Queueing Syst 49:321–334, 2005) considered the 2-queue case, which is known to exhibit the Downs–Thomson paradox, where delays may increase as capacity is increased. We show that in larger parallel systems, with multiple single-server queues, the user equilibrium is always unique when the batch size is sufficiently large relative to the number of queues; no more than three equilibria exist; Braess paradox may appear when adding extra queues.

Equilibrium threshold joining strategies in partially observable batch service queueing systems

We study the strategic customer behavior in queueing systems with batch services under incomplete information. In particular, we assume that arriving customers have the opportunity to observe only the number of waiting batches upon arrival and, afterwards, they make their join/balk decisions. We prove that equilibrium strategies always exist within the legitimate class of threshold strategies, but they may not be unique. We also provide an algorithmic scheme for their computation. Moreover, we compare the strategic behavior under this information level with the corresponding behavior in the complete information case.