A Batch Arrival Two Types of Bulk Service Queue with Server Breakdown and Modified M-Vacation Research (original) (raw)
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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.
We study the behavior of a batch arrival queuing system equipped with a single server providing general arbitrary service to customers with different service rates in two fluctuating modes of service. In addition, the server is subject to random breakdown. As soon as the server faces breakdown, the customer whose service is interrupted comes back to the head of the queue. As soon as repair process of the server is complete, the server immediately starts providing service in mode 1. Also customers waiting for service may renege (leave the queue) when there is breakdown or when server takes vacation. The system provides service with complete or reduced efficiency due to the fluctuating rates of service. We derive the steady state queue size distribution. Some special cases are discussed and numerical illustration is provided to see the effect and validity of the results.
2019
This paper investigates an infinite buffer batch arrival single server queueing system with variant working vacations and server breakdowns wherein customers arrive according to a Poisson process. The server may breakdown during regular busy period. During working vacations, the customer may renege due to impatience and retention policy is adopted to retain the reneging customers. It is assumed that service times, vacation times, repair times and reneging times are all exponentially distributed and are all mutually independent. We derive the probability generating function of steady state probabilities and obtain performance measures of the system. In addition, cost model is formulated to determine the optimal service rate during busy period using quadratic fit search method. KeywordsBatch Arrival Queue, Working Vacations, Server Breakdowns, Probability Generating Function, Reneging, Retention.
2013
Consider a single server fixed batch service queueing system under multiple vacation with a possibility of catastrophe in which the arrival rate ? follows a Poisson process and the service time follows an exponential distribution with parameter ?. Further we assume that the catastrophe occur at the rate of ? which follows a Poisson process and the length of time the server in vacation follows an exponential distribution with parameter ?. Assume that the system initially contains k customers when the server enters in to the system and starts the service immediately in a 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 ?. 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. We are analyzing the possibility of catastrophe that is whenever a catastrophe occurs in the system, all the customers wh...
2018
This paper is concerned with batch arrival queue with an additional second optional service to a batch of customers with dissimilar service rate where the idea of restricted admissibility of arriving batch of customers is also introduced. The server may take two different vacations (i) Emergency vacation-during service the server may go for vacation to an emergency call and after completion of the vacation, the server continues the remaining service to a batch of customers. (ii) Bernoulli vacation-after completion of first essential or second optional service, the server may take a vacation or may remain in the system to serve the next unit, if any. While the server is functioning with first essential or second optional service, it may break off for a short period of time. As a result of breakdown, a batch of customers, either in first essential or second optional service is interrupted. The service channel will be sent to repair process immediately. The repair process presumed to b...
Mathematical theory and modeling, 2021
This paper investigates a batch arrival queueing system in which customers arrives at the system in a Poisson stream following a compound Poisson process and the system has a single server providing three types of general heterogeneous services. At the beginning of each service, a customer is allowed to choose any one of the three services and as soon as a service of any type gets completed, the server may take a vacation or may continue staying in the system. The vacation time is assumed to follow a general (arbitrary) distribution and the server vacation is based on Bernoulli schedule under a single vacation policy. During the server vacation period, impatient customers are assumed to balk. This paper described the model as a bivariate Markov chain and employed the supplementary variable technique to find closed-form solutions of the steady state probability generating function of number of customers, the steady state probabilities of various states of the system, the average queue size, the average system size, and the average waiting time in the queue as well as the average waiting time in the system. Further, some interesting special cases of the model are also derived.
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.
Open Journal of Optimization, 2020
This article examines the effects of reneging, server breakdown and server vacation on the various states of the batch arrivals queueing system with single server providing service to customers in three fluctuating modes. In this queueing system, any batch arrival joins the queue if the server is busy or on vacation or under repair. However, if the server is free, one customer from the arriving batch joins the service immediately while others join the queue. In case of server breakdown, the customer whose service is interrupted returns back to the head of the queue. As soon as the server has is repaired, the server attends to the customer in mode 1. For this queueing system, customers that are impatient due to breakdown and server vacation may renege
International Journal of Internet and Enterprise Management, 2012
This paper studies a general retrial M X /G/1 queue with an additional phase of second optional service and Bernoulli vacation where breakdowns occur randomly at any instant while servicing the customers. If an arriving batch finds that the server is busy in providing either first essential service (FES)/second optional service (SOS) or on vacation then arriving batch enters an orbit called retrial queue. Otherwise, one customer from arriving batch starts to be served by the server while the rest join the orbit. The vacation times and service times of both first essential and second optional services are assumed to be general distributed while the retrial times are exponential distributed. Introducing supplementary variables and by employing embedded Markov chain technique, we derive some important performance measures of the system such as average orbit size, average queue size, mean waiting time, expected lengths of busy period, etc. Numerical results have been facilitated to illustrate the effect of different parameters on several performance measures.