General Bulk Service Rule Research Articles

Transient analysis of interdependent M/M(<SUP align="right">a,d,b</SUP>)/1 queue with discouragement and controllable arrival rate

This paper deals with the state dependent single server queue having a general batch service rule where the arrival and service are assumed to be correlated and follow the bivariate Poisson processes. According to general bulk service rule, the service starts only when a specified number of customers, say 'a', is present in the queue and the maximum service capacity of a batch service is 'b'. Due to accessible batch, the late arriving customers are allowed to join the batch during ongoing service, if the size of the batch served is less than some fixed integer 'd' (a < d < b). When the batch size reaches the threshold limit d, no further customers are allowed to join the batch which is called non-accessible batch. To demonstrate the validity of the analytical results, numerical illustration has been taken into consideration. The sensitivity has been analysed by examining the effect of system parameters on the average queue length.

A simple analysis of system characteristics in the batch service queue with infinite-buffer and Markovian service process using the roots method:GI/C-MSP(a,b)/1/∞

We consider an infinite-buffer single-server queue with renewal input and Markovian service process where customers are served in batches according to a general bulk service rule. Queue-length distributions at epochs of pre-arrival, arbitrary and post-departure have been obtained along with some important performance measures such as mean queue lengths and mean waiting times in both the system as well as the queue. We also obtain the steady-state service batch size distributions as well as system-length distributions. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of queue-length distribution at a pre-arrival epoch. Also, we provide analytical and numerical comparison between the roots method used in this paper and the matrix geometric method in terms of computational complexities and required computation time to evaluate pre-arrival epoch probabilities for both the methods. Later, we have established heavy- and light-traffic approximations as well as an approximation for the tail probabilities at pre-arrival epoch based on one root of the characteristic equation. Numerical results for some cases have been presented to show the effect of model parameters on the performance measures.

Analysis of a finite-buffer bulk-service queue under Markovian arrival process with batch-size-dependent service

We consider a finite capacity single server queue in which the customers arrive according to a Markovian arrival process. The customers are served in batches following a ‘general bulk service rule’. The service times, which depend on the size of the batch, are generally distributed. We obtain, in steady-state, the joint distribution of the random variables of interest at various epochs. Efficient computational procedures in the case of phase type services are presented. An illustrative numerical example to bring out the qualitative nature of the model is presented.

Cost Analysis of MAP/G(a, b)/1/N Queue with Multiple Vacations and Closedown Times

This paper gives the cost analysis of a finite capacity single server bulk queueing model with closedown times. The server serves the customers in batches of maximum size ‘b’ with a minimum threshold value ‘a’. Customers arrive according to a Markovian Arrival Process (MAP). On completion of a service, if the queue length is less than ‘a’, then the server performs a closedown work and then leaves for a vacation of random length. When the server returns from vacation and if the queue length is still less than ‘a’ he avails another vacation and so on until the server finds ‘a’ customers waiting in the queue. After the completion of a service, if the number of customers in the queue is greater than a specified value ‘a’ then the server will continue the batch service with general bulk service rule. On the other hand, if the server finds at least ‘a’ customers during closedown period, he immediately starts serving the batch of ‘a’ customers. Using supplementary variable and imbedded Markov chain technique, queue length distribution at arbitrary epoch is obtained. Some key performance measures are also obtained. Cost model is discussed with Numerical illustration.

A single server finite source loss model with general bulk service rule

A single server general bulk service finite source loss model has been studied. For this model the system steady state probabilities and waiting time distributions are obtained. Some performance measures are also calculated. Particular model is deduced and some numerical examples are also given.

Single server queueing model with server delayed vacation and switch over state

This paper focus on a server delayed vacation of M/M (a, b) / 1queueing system switch over state. In this model it is assumed that the arrival pattern is Poisson fashion with parameter λ and service is done in batches which is exponentially distributed with parameter µ according to the general bulk service rule introduced by Neuts (11).The batches are served according to FCFS discipline. The length of the vacations can be controlled by means of the number of customers ‘kb’ arriving to the system during vacation, and the level ‘k’ may be chosen according to the arrival rate, the service rate, the cost per unit of waiting time and the cost of the server being transferred from vacation to work. Secondly a server allowed for a delay time before a vacation begins. During the delay time, the server is situated in warm standby state and the service starts immediately if the batches of ‘a’ customers are present. If server finds (a-1) customers, then server will stay idle in the system called delay time before he goes for vacation. If the server finds (a-2) customers in the system the server switch over the system. So in this system, sever can take only one vacation between two successive service times.

Cost analysis of a bulk service retrial queue

This paper studies a batch arrival general bulk service retrial queueing model with constant retrial rate. The primary customers arrive in bulk according to Poisson process and they get service under general bulk service rule with minimum of one customer and maximum of ‘b’ customers. If the arriving batch of customers, of size ‘ζ ’, 1≤ζ ≤ b , finds the server free, then all of them get service immediately; while, if the size of the arriving batch is more than ‘b’, then, ‘b’ customers enter the service station and the remaining ζ - b customers join the orbit. However, if an arriving batch of customers finds the server busy, then the entire batch joins the orbit in order to seek service again. The customers in the orbit will try for service one by one with a constant retrial rate ‘v’ when the server is idle. For the proposed model, the probability generating function of the steady-state queue size distribution at an arbitrary time, expected number of customers in the orbit, expected waiting time, expected length of busy period and expected length of busy cycle are obtained. The cost analysis of the queueing system is discussed. The effects of several parameters on the system are analysed numerically.

New results on bulk service queue with finite-buffer: M/G (a,b)/1/N

We consider a single-server finite-buffer queue with general bulk service rule where customers arrive according to a Poisson process and service times of the batches are arbitrarily distributed. Using the embedded Markov chain technique we first obtain the steady-state joint distribution of the number of customers in the queue and number with the departing batch at departure-epoch. Further in order to obtain the joint distribution of the number of customers in the queue and number with the server at arbitrary-epoch, we develop relations between arbitrary- and departure-epoch probabilities. Finally, we also obtain the distributions of number of customers in the queue (system), and number with the server. Various performance measures of interest such as average number of customers in the queue (system) and average number of customers with the server etc. are obtained. The model was earlier discussed by Gold and Tran-Gia (Queueing Syst 14:413–426, 1993) and Chaudhry and Gupta (Queueing Syst 31:95–100, 1999) wherein they first obtained the distribution of the number of customers in the queue at departure-epoch without taking into the consideration of the size of the departing batch. Then they obtained the distribution of the number of customers in the queue at arbitrary-epoch when server is idle/busy.

A Non-Markovian Bulk Queue with Multiple Vacations and Control Policy on Request for Re-Service

The steady state behaviour of a single server non-Markovian queue with multiple vacations and controlled optional re-service is analysed. A batch of customers arrive according to Poisson with rate A, whereas the bulk service is rendered by a single server with a minimum batch size of ‘a and maximum of ‘b’. The service times follow a general distribution. At the completion of an essential service, the leaving batch of customers may request for a re-service with probability n. However, the re-service is rendered only when the number of customers waiting in the queue is less than ‘a’. If no request for re-service is made after the completion of an essential service and numbers of customers in the queue is less than ‘a, then the server will avail a vacation of a random length. When the server returns from vacation and if the queue length is still less than ‘a’ he avails another vacation so on until the server finds ‘a’ customers in the queue. After the completion of an essential service and numbers of customers in the queue is greater than ‘a then the server will continue the batch service with general bulk service rule. The probability generating function of queue size at a random epoch is obtained. Some important performance measures such as expected queue size, expected busy period and idle period are derived. Cost model is discussed with numerical illustration.

On the Distribution of the Number of Customers in the D-BMAP/Ga,b/1/M Queue—A Simple Approach to a Complex Problem

In this paper, we consider a single-server finite-capacity queue with general bulk service rule where customers arrive according to a discrete-time batch Markovian arrival process (D-BMAP). The model is denoted by D-BMAP/Ga,b/1/M which includes a wide class of queueing models as special cases. We give a relation between the steady-state probabilities of the number of customers in the queue at departure- and arbitrary-epochs using the concept of the mean sojourn time in the phase of the underlying Markov chain of D-BMAP before the next arrival. We use the embedded Markov chain technique to obtain the departure-epoch probability of the number of customers in the queue. The pre-arrival probability of the number of customers in the queue is also obtained. Finally, a complete solution to the distribution of the number of customers in the D-BMAP/Ga,b/1/M queue, some computational results, and performance measures such as loss probability and mean queue length are presented.

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.

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.

The truncated Hyper-Poisson queues: Hk/Ma,b/C/N with balking, reneging and general bulk service rule

The aim of this paper is to derive the analytical solution of the queue: Hk/Ma,b/C/N with balking and reneging in which (I) units arrive according to a hyper-Poisson distribution with k independent branches, (II) the queue discipline is FIFO; and (III) the units are served in batches according to a general bulk service rule. The steady-state probabilities, recurrence relations connecting various probabilities introduced are found and the expected number of units in the queue is derived in an explicit form. Also, some special cases are obtained. .

Steady state analysis of a M&lt;SUP align=right&gt;x&lt;/SUP&gt;/G/1 queue with two service modes and multiple vacations

This paper deals with a single server Poisson queue with state dependent service. The service is rendered with different service modes as singly or in bulk according to the queue length. The service starts only if at least one customer is waiting for service. The server switches over from single service to bulk service only at service initiation epoch according to the queue length. If the number of customers waiting for service is ξ (1 ≤ ξ < a), then the server serves one by one. At the service initiation epoch, if the number of waiting customers ξ is at least 'a' (a < b), then the service is rendered according to Neut's general bulk service rule. At the service completion epoch, if there are no customers in the queue, the server avails multiple vacations, until he finds at least one customer in the queue. The Probability Generating Function (PGF) of the queue size and performance measures are obtained. Numerical results are presented in order to illustrate the effectiveness of performance of the state dependent queue.

Discrete-time [formula omitted] queues with single and multiple vacations

In this paper, we consider a discrete-time single-server finite-buffer batch arrival queue in which customers are served in batches according to a general bulk-service rule, that is, at least ‘ a ’ customers are needed to start a service with a maximum serving capacity of ‘ b ’ customers. The server takes a (single and multiple) vacation as soon as the queue length falls below ‘ a ’ at the completion of service. The interarrival times of batches are assumed to be independent and geometrically distributed. The service times of the batches and vacation times of the server are generally distributed and their durations are integral multiples of slot duration. We obtain queue length distributions at service completion, vacation termination, arbitrary, and prearrival epochs. Finally, we discuss various performance measures and numerical results.

The finite-buffer M/ G/1 queue with general bulk-service rule and single vacation

This paper studies a single server finite-buffer bulk-service queue in which the inter-arrival and service times are exponentially and arbitrarily distributed, respectively. The service is performed in batches of maximum size ‘ b’ and minimum size ‘ a’. Server takes a single vacation when he finds less than ‘ a’ customers after the service completion. The distributions of the number of customers in the queue at arbitrary, service completion and vacation termination epochs have been obtained. Finally, some key performance measures such as average queue length, probability of blocking etc. are discussed.

Performance analysis of finite buffer discrete-time queue with bulk service

Discrete-time queues are extensively used in modelling the ATM environment at the cell level. In this paper, we model a single-server queue with finite buffer space in a discrete-time environment where the services are performed in batches of maximum size ‘b’ with a minimum threshold value ‘a’. The analytic analysis of the model is carried out for the both arrival first (AF) and departure first (DF) management policies and distributions of buffer content at various epochs have been obtained. The system performance measures such as average buffer content, average delay and probability of blocking are discussed. A comparative study of AF and DF management policies on certain performance measures is also presented. Scope and purpose B-ISDN and in particular ATM networks are designed in order to transfer and switch all type of information services such as voice, data and video communications. As these systems operate on a discrete-time basis, many classical results of queuing theory which are used in performance analysis cannot be applied and hence require a fresh investigation. The purpose of this paper is to analyse a discrete-time finite buffer queue with general bulk service rule which may be useful to system designers and other practitioners involved in investigating the performance of communication systems based on B-ISDN.

Analysis of the MAP/G a , b /1/N Queue

The Markovian arrival process (MAP) is used to represent the bursty and correlated traffic arising in modern telecommunication network. In this paper, we consider a single server finite capacity queue with general bulk service rule in which arrivals are governed by MAP and service times are arbitrarily distributed. The distributions of the number of customers in the queue at arbitrary, post-departure and pre-arrival epochs have been obtained using the supplementary variable and the embedded Markov chain techniques. Computational procedure has been given when the service time distribution is of phase type.

Modelling and analysis of M/G^{a,b}/1/N queue – A simple alternative approach

In this paper, we consider a single-server finite-capacity queue with general bulk service rule where customers arrive according to a Poisson process and service times of the batches are arbitrarily distributed. The queue is analyzed using both the supplementary variable and imbedded Markov chain techniques. The relations between state probabilities at departure and arbitrary epochs have been presented in explicit forms.

Two units connected in series with general bulk service of accessible batches in unit 2

We study a queueing model consisting of two units connected in series with a finite intermediate waiting room. The customers in the buffer are served according to a new general bulk service rule with exponential times which admits each batch served to have not less than a and not more than b customers such that the arriving customers can enter service, without affecting the service time, if the size of the batch under service is less than b. For this model we obtain the steady-state probability vector for the number of customers in the queue using a matrix-geometric algorithmic approach.