Discrete-time queues with single and multiple vacations (original) (raw)
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Performance Evaluation, 2007
This paper treats a discrete-time single-server finite-buffer exhaustive (single-and multiple-) vacation queueing system with discrete-time Markovian arrival process (D-MAP). The service and vacation times are generally distributed random variables and their durations are integral multiples of a slot duration. We obtain the queue-length distributions at departure, service completion, vacation termination, arbitrary and prearrival epochs. Several performance measures such as probability of blocking, average queue-length and the fraction of time the server is busy have been discussed. Finally, the analysis of actual waiting time under the first-come-first-served discipline is also carried out.
The finite-buffer M/G/1 queue with general bulk-service rule and single vacation
Performance Evaluation
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.
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.
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.
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 BMAP∕R∕1 Queues Under Gated-Limited Service with the Server’s Single Vacation Policy
Infosys Science Foundation Series, 2020
This paper deals with the finite-buffer single server vacation queues with batch Markovian arrival process (BMAP). The server follows gated-limited service discipline, i.e., the server can serve a maximum of L customers out of those that are waiting at the start of the busy period or all the waiting customers, whichever is minimum. It has been assumed that the server can take only one vacation, i.e., if no customers are found at the end of a vacation, the server remains idle until a batch of customers arrives. The service time and vacation time distributions are considered to possess rational Laplace-Stieltjes transform. The queue-length distribution at postdeparture, arbitrary, and pre-arrival epochs has been obtained. Various performance measures like mean queue-length, mean waiting time of an arbitrary customer, and mean length of busy and idle periods have been derived for this model. Numerical results have been presented based on the analysis done.
Quality Technology & Quantitative Management, 2021
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 vacation where customers/messages arrive according to the Bernoulii 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 service completion epoch, for which first the bivariate probability generating function has been derived. We also acquire the joint distribution at arbitrary slot. We also provide several marginal distributions and performance measures for the utilization of the vendor. Transmission of data through a particular channel is skipped due to the high transmission error. As the discrete phase type distribution plays a noteworthy role to control this error, we include numerical example where service time distribution follows discrete phase type distribution. A comparison between batch-size dependent and independent service has been drawn through the graphical representation of some performance measures and total system cost.
Discrete Time Batch Arrival Queue with Multiple Vacations
In this paper we consider a discrete time batch arrival queueing system with multiple vacations. It is assume that the service of customers arrived in the system between a fixed intervals of time after which the service goes on vacations after completion of one service of cycle is taken up at the boundaries of the fixed duration of time. This is the case of late arrival. In case of early arrival i.e. arrival before the start of next cycles of service. If the customer finds the system empty, it is served immediately. We prove the Stochastic decomposition property for queue length and waiting time distribution for both the models.