Batch Arrival Queue Research Articles

The Markovian Batch Arrival Queue with Differentiated Vacation

The Markovian Batch Arrival Queue with Differentiated Vacation

A Markovian batch arrival queueing system with disasters, working breakdowns, and impatience: Mathematical modeling and economic analysis

A Markovian batch arrival queueing system with disasters, working breakdowns, and impatience: Mathematical modeling and economic analysis

Non-stationary departure process in a batch-arrival queue with finite buffer capacity and threshold-type control mechanism

Non-stationary departure process in a batch-arrival queue with finite buffer capacity and threshold-type control mechanism

Analysis of batch arrival single server queue with random vacation policy and two types of general heterogeneous repeated service

Analysis of batch arrival single server queue with random vacation policy and two types of general heterogeneous repeated service

Analysis of batch arrival single server queue with random vacation policy and two types of general heterogeneous repeated service

Analysis of batch arrival single server queue with random vacation policy and two types of general heterogeneous repeated service

Analysis of a model of batch arrival single server queue with random vacation policy

This article deals with batch arrival single server queue with random vacation policy, in which the server takes the maximum number of random vacations till it finds some customers waiting in a queue at a vacation completion epoch. If no arrival occurs after completing maximum number of random vacations, the server stays dormant in the system and waits for the upcoming arrival. Here, it is assumed that the customer’s arrival in batches conforms to compound Poisson process, although service time and vacation time are generally distributed. Explicit expressions are obtained for steady state queue size distribution at service completion point and steady state system size probabilities. Some significant measures such as a mean number of batches served during the busy period, Laplace-Stieltjes transform of waiting time, unfinished work and its corresponding mean values are also obtained. A cost optimal policy is developed in terms of the average cost function to determine a locally optimal random vacation policy at a lower cost. Finally, various numerical results are presented for the above system performance measures.

Analysis of an unreliable single server batch arrival queue with two types of services under Bernoulli vacation policy

This article considers a single server batch arrival queue providing two types of general heterogeneous parallel services under Bernoulli Vacation Policy. The server assumed here is unreliable and incorporates the concept of delayed repair period. An extensive study of this model is carried out and the queue size distribution at random as well as departure epoch is obtained. Further, Laplace Stieltjes Transform of a busy period and waiting time distribution is also being derived. Finally, some important performance measures along with the reliability indices are accomplished in the study.

An M^[X]/G(a,b)/1 queue with unreliable server, second optional service, closedown, setup with N-policy and multiple vacation

Batch arrival bulk service queueing system has been effectively used to model many real life systems like production, manufacturing, transportation as well as telecommunication. Characteristic like second optional service, service interruption, closedown time, multiple vacation, setup time with N policy, etc. have been respectively considered in such models. So far, no comprehensive combination of these characteristics have been reported in the literature. This paper deals with the analysis of a non-Markovian batch arrival bulk service queue with a general combination of all the above mentioned characteristics.

Analysis of batch arrival bulk service queue with additional optional service multiple vacation and setup time

This paper studies an M[x]/G(a, b)/1 queuing system with additional optional service, multiple vacation and setup time. After completing the first service, the customers may opt for the second service with probability α or leave the system with probability 1 ‒ α. After completing a bulk service, if the queue size is less than 'α', then the server leaves for a vacation of random length. When he returns from the vacation, if the queue length is still less than 'α', he leaves for another vacation and so on. This process continues until he finds at least 'α' customer in the queue. After a vacation, if the server finds at least 'α' customer waiting for service, he requires a setup time 'G' to start the service. After this setup, he serves a batch of ξ customers (a ≤ ξ ≤ b). Using supplementary variable technique, the probability generating function (PGF) of the queue size, expected queue length, expected waiting time, expected busy period and expected idle period are derived. Numerical illustrations are presented to visualise the effect of system parameters.

An <i>M</i><SUP align="right">[<i>X</i>]</SUP>/<i>G</i>(<i>a</i>, <i>b</i>)/1 queueing system with server breakdown and repair, stand-by server and single vacation

In this paper, we discuss a non-Markovian batch arrival general bulk service single server queueing system with server breakdown and repair, single vacation and stand-by server. There is a stand-by server which is employed during the period for which the regular server remains under repair. Both the servers serve the customer under general bulk service rule. Suppose at the instant main servers service completion if the queue length is less than 'a' then the main server goes for a vacation. Suppose at the instant of repair completion or vacation completion if the system size is less than 'a' then the main server stays in the system and waits for the next batch of arriving customers. The probability generating function of queue size at an arbitrary time and some performance measures of the system are derived. An extensive numerical result for a particular case of the model is illustrated.

Analysis of a non-Markovian single server batch arrival queueing system of compulsory three stages of services with fourth optional stage service, service interruptions and deterministic server vacations

This paper deals with the steady state analysis of a single server batch arrival queueing system with three stages of compulsory service. An additional supposition of a fourth stage optional service is well thought-out. The server may take a vacation after completion of service. In this model, the vacation is of predetermined duration. A busy server may break down at any moment. It is tracked by a repair process. Service time, vacation time and repair time follow general distribution. The steady state probability generating function for the system and also system recital measures is obtained by using a supplementary variable technique. Some special cases of the model are also discussed. The model is justified by means of numerical illustrations followed by graphical representation.

Analysis of a non-Markovian single server batch arrival queueing system of compulsory three stages of services with fourth optional stage service, service interruptions and deterministic server vacations

Analysis of a non-Markovian single server batch arrival queueing system of compulsory three stages of services with fourth optional stage service, service interruptions and deterministic server vacations

Batch Arrival Queueing Model with Unreliable Server

The unreliable server with provision of temporary server in the context of application has been investigated. A temporary server is installed when the primary server is over loaded i.e., a fixed queue length of K-policy customers including the customer with the primary server has been build up. The primary server may breakdown while rendering service to the customers; it is sent for the repair. This type of queuing system has been investigated using Matrix Geometric Method to obtain the probabilities of the system steady state.AMS subject classification number— 60K25, 60K30 and 90B22.

A Note on the Waiting-Time Distribution in an Infinite-Buffer GI[X]/C-MSP/1 Queueing System

This paper deals with a batch arrival infinite-buffer single server queue. The interbatch arrival times are generally distributed and arrivals are occurring in batches of random size. The service process is correlated and its structure is presented through a continuous-time Markovian service process (C-MSP). We obtain the probability density function (p.d.f.) of actual waiting time for the first and an arbitrary customer of an arrival batch. The proposed analysis is based on the roots of the characteristic equations involved in the Laplace-Stieltjes transform (LST) of waiting times in the system for the first, an arbitrary, and the last customer of an arrival batch. The corresponding mean sojourn times in the system may be obtained using these probability density functions or the above LSTs. Numerical results for some variants of the interbatch arrival distribution (Pareto and phase-type) have been presented to show the influence of model parameters on the waiting-time distribution. Finally, a simple computational procedure (through solving a set of simultaneous linear equations) is proposed to obtain the “R” matrix of the corresponding GI/M/1-type Markov chain embedded at a prearrival epoch of a batch.

Performance measures of variant working vacations on batch arrival queue with server breakdowns

ABSTRACTThis paper analyzes an variant working vacation queue with server breakdowns and repair. The server takes working vacations as soon as the system becomes empty. The service times during regular busy period, working vacation period, vacation times, breakdown times and repair times are all assumed to be exponentially distributed and are mutually independent. We have used the probability generating functions to derive the steady-state probabilities and obtain the closed form expressions of the system size. In addition, we obtain some other performance measures and their monotonicity with respect to variant working vacations.

M[x]/G/1 Multistage Queue with Stand-by Server During Main Server’s Interruptions

<p>This paper investigates a multistage batch arrival queue with different server interruptions and a second server replaces the main server during the interruptions. The different server interruptions are assumed to be: multiple vacation, extended vacation, breakdown with delay time and server under two phases of repair. Customers are assumed to arrive in batches according to Poisson process and a single server provides service to the customers. When the main server is inactive due to the interruptions, stand-by server provide service to the arrivals. In addition, customers may renege during server breakdown or during server vacation due to impatience. Transient solution and the corresponding steady state solution is derived using supplementary variable technique.</p>

A batch arrival bulk service queue with multiple vacations, optional re-service, closedown time and setup time under restricted admissibility of arriving batches

A batch arrival bulk service queue with multiple vacations, optional re-service, closedown time and setup time under restricted admissibility of arriving batches

Analysis of bulk queueing system with single service and single vacation

A batch arrival queueing system with two types of service pattern with single vacation is considered. If the queue length is more than or equal to the minimum of batch size ‘a’, the bulk service can be provided otherwise the single service or single vacation is provided. After completing the bulk service, if the queue length is less than ‘a’ then the server leaves for single vacation. After completing the single vacation, if the queue length is greater than or equal to ‘a’ then the server will provide the bulk service, otherwise, the server goes to single service. The queue size distribution of the developed model is obtained and the performance measures of the developed queueing model are established. The particular cases of the proposed queuing model are also discussed.

Batch arrival single-server queue with variable service speed and setup time

In this paper, we consider an M $${}^X$$ /M/1/SET-VARI queue which has batch arrivals, variable service speed and setup time. Our model is motivated by power-aware servers in data centers where dynamic scaling techniques are used. The service speed of the server is proportional to the number of jobs in the system. The contribution of our paper is threefold. First, we obtain the necessary and sufficient condition for the stability of the system. Second, we derive an expression for the probability generating function of the number of jobs in the system. Third, our main contribution is the derivation of the Laplace–Stieltjes transform (LST) of the sojourn time distribution, which is obtained in series form involving infinite-dimensional matrices. In this model, since the service speed varies upon arrivals and departures of jobs, the sojourn time of a tagged job is affected by the batches that arrive after it. This makes the derivation of the LST of the sojourn time complex and challenging. In addition, we present some numerical examples to show the trade-off between the mean sojourn time (response time) and the energy consumption. Using the numerical inverse Laplace–Stieltjes transform, we also obtain the sojourn time distribution, which can be used for setting the service-level agreement in data centers.

Analysis of non-Markovian batch arrival queueing model with multi stages of service of restricted admissibility, feedback service and three optional vacations in production and manufacturing

In this paper, a multi stage of heterogeneous service in a batch arrival queue is investigated. Each customer undergoes multi stages of heterogeneous service. As soon as the completion of nth stage of service, if the customer is dissatisfied with his service, he can immediately join the tail of the original queue to get a feedback service. According to vacation policy, it has three heterogeneous phases. After service completion of a customer the server may take a phase one Bernoulli vacation. Further, after completion of phase one Bernoulli vacation the server may take phase two and phase three optional vacations. The vacation times are assumed to be general. Also we assume restricted admissibility of arriving batches. By employing generating functions and supplementary variable techniques, the transient solution and the corresponding steady state results have been obtained explicitly. Also the mean number of customers in the queue and the system are also derived. Practical rationalisation of the model is described. Special cases are discussed. In order to validate the model, numerical illustration is specified.