On state dependent batch service queue with single and multiple vacation under Markovian arrival process (original) (raw)
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A finite capacity bulk service queue with single vacation and Markovian arrival process
Journal of Applied Mathematics and Stochastic Analysis, 2004
Vacation time queues with Markovian arrival process (MAP) are mainly useful in modeling and performance analysis of telecommunication networks based on asynchronous transfer mode (ATM) environment. This paper analyzes a single-server finite capacity queue wherein service is performed in batches of maximum size “b” with a minimum threshold “a” and arrivals are governed by MAP. The server takes a single vacation when he finds less than “a” customers after service completion. The distributions of buffer contents at various epochs (service completion, vacation termination, departure, arbitrary and pre-arrival) have been obtained. Finally, some performance measures such as loss probability and average queue length are discussed. Numerical results are also presented in some cases.
Analytic and numerical aspects of batch service queues with single vacation
Computers & Operations Research, 2005
This paper deals with an M=G=1 batch service queue where customers are served in batches of maximum size b with a minimum threshold value a. The server takes a single vacation when he ÿnds less than a customers after the service completion. The vacation time of the server is arbitrarily distributed. Using the supplementary variable method we obtain the probability generating functions of the queue length distributions at various epochs. We also obtain relations among queue length distributions at arbitrary, service (vacation) termination epochs. Further their evaluation is also discussed. Finally, some numerical results and graphs are presented.
Analysis of discrete-time queues with batch renewal input and multiple vacations
Journal of Systems Science and Complexity, 2012
This paper analyzes a discrete-time multiple vacations finite-buffer queueing system with batch renewal input in which inter-arrival time of batches are arbitrarily distributed. Service and vacation times are mutually independent and geometrically distributed. The server takes vacations when the system does not have any waiting jobs at a service completion epoch or a vacation completion epoch. The system is analyzed under the assumptions of late arrival system with delayed access and early arrival system. Using the supplementary variable and the imbedded Markov chain techniques, the authors obtain the queue-length distributions at pre-arrival, arbitrary and outside observer's observation epochs for partial-batch rejection policy. The blocking probability of the first-, an arbitraryand the last-job in a batch have been discussed. The analysis of actual waiting-time distributions measured in slots of the first-, an arbitrary-and the last-job in an accepted batch, and other performance measures along with some numerical results have also been investigated.
Bulk Arrival Markovian Queueing System with Two Types of Services and Multiple Vacations
International Journal of Mathematics and Computer Research, 2020
In this paper, we have a bulk arrival queueing Markovian model with two types of services first come first serve and bulk service. This model is also assumed the two servers such as main server and standby server with multiple vacations. If number of customers in the queue is less than 'a' then server will provide the FCFS service or goes for a vacation, if number of customers are more than or equal to 'a' then server will provide the bulk. After coming to the vacation, when the server will find number of customers are less than 'a' then he will provide the FCFS service to the customers. We have obtained the queue size distribution of this considered model and also obtained the performance measures such as idle time for the server, queue length, busy period by using the supplement variable technique.
On the batch arrival batch service queue with finite buffer under server’s vacation: queue
Computers & Mathematics with Applications, 2008
This paper considers a finite-buffer batch arrival and batch service queue with single and multiple vacations. The steady-state distributions of the number of customers in the queue at service completion, vacation termination, departure, arbitrary and pre-arrival epochs have been obtained. Finally, various performance measures such as average queue length, average waiting time, probability that the server is busy, blocking probabilities, etc. are discussed along with some numerical results. The effect of certain model parameters on the key performance measures have also been investigated. The model has potential application in several areas including manufacturing, internet web-server and telecommunication systems.
Analysis of M/G/ 1 queueing model with state dependent arrival and vacation
Journal of Industrial Engineering International, 2012
This investigation deals with single server queueing system wherein the arrival of the units follow Poisson process with varying arrival rates in different states and the service time of the units is arbitrary (general) distributed. The server may take a vacation of a fixed duration or may continue to be available in the system for next service. Using the probability argument, we construct the set of steady state equations by introducing the supplementary variable corresponding to elapsed service time. Then, we obtain the probability generating function of the units present in the system. Various performance indices, such as expected number of units in the queue and in the system, average waiting time, etc., are obtained explicitly. Some special cases are also deduced by setting the appropriate parameter values. The numerical illustrations are provided to carry out the sensitivity analysis in order to explore the effect of different parameters on the system performance measures.
Analysis of a Two Phases Batch Arrival Queueing Model with Bernoulli Vacation Schedule
Investigación Operacional, 2013
We consider a single server bulk arrival queueing system with two phases of heterogeneous service under Bernoulli schedule vacation, where the customers arrive in batches of the random variable 'X'. Using the imbedded Markov chain technique, we first derive the queue size distribution at a stationary point of time. Next, we obtain a recursive solution of the stationary queue size distribution of this model. Finally, we obtain the Laplace Stieltjes Transform of the waiting time distribution and some related performance measures. The method proposed here is not only easily amenable to computation but can be applied to solve more complicated problems of similar nature.
Analysis of a Batch Arrival Poisson Queue Under Single Vacation Policy
Calcutta Statistical Association Bulletin, 2002
We consider a single server Markovian queueing system with compound Poisson arrivals of batches of random size under a single vacation policy, where the server takes exactly one vacation between two successive busy periods. We are concerned with the steady state distributions of the additional queue size, additional delay and queue waiting time distribution.
International Journal of Computer Applications, 2014
Consider a single server fixed batch service queueing system under multiple vacation with gated service in which the arrival rate λ follows a Poisson process and the service time follows an exponential distribution with parameter μ. Assume that the system initially contain k customers when the server enters into the system and starts the service immediately in batch of size k. After completion of a service, if he finds less than k customers in the queue, then the server goes for a multiple vacation of length α. If there are more than k customers in the queue then the first k customers will be selected from the queue and service will be given as a batch. Gated type service policy is adopted in this model that is once the server starts service for a batch of k customers, no customers will be allowed to enter into the queue. Every time a service is finished, and there are less than k customers in the queue, the server leaves for a vacation of length α. This model is completely solved by constructing the generating function and Rouche's theorem is applied and we have derived the closed form solutions for probability of number of customers in the queue during the server busy and in vacation. Further we are providing the closed form solutions for mean number of customers in the queue, variance and various system performance measures of the system. Numerical studies have been done for analysis of system measures for various values of λ, µ, α and k.