Abstract
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 (leave the queue without getting service). Due to fluctuating modes of service delivery, the system may provide service with complete or reduced efficiency. Consequently, we construct the mathematical model and derive the probability generating functions of the steady state probabilities of several states of the system including the steady state queue size distribution. Further, we discuss some particular cases of the proposed queueing model. We present numerical examples in order to demonstrate the effects of server vacation and reneging on the various states of the system. The study revealed that an increase in reneging and a decrease in server vacation results in a decrease in server utilization and an increase in server’s idle time provided rates of server breakdown and repair completion are constant. In addition, the probability of server vacation, the probability of system is under repair and the probabilities that the server provides service in three fluctuating modes decreases due to an increase in reneging and a decrease in vacation completion rates.
Highlights
A batch arrival single server queueing system is one in which customers arrive for service in groups or in batches and are served individually by one server
∞, adding the results to Equation (10) and utilizing the probability generating functions defined in Equation (7) gives d dy multiply Equation (11) by zn, take the sum of both sides over n from 1 to ∞, add the result to Equation (12) and use the probability generating functions defined in Equation (7) to obtain d dy Multiplying Equation (13) by zn, taking sum of both sides over n from 1 to
We introduced and studied a batch arrival single server queueing model for providing service in three fluctuating modes during reneging, server breakdown and server vacation
Summary
A batch arrival single server queueing system is one in which customers arrive for service in groups or in batches and are served individually by one server. Some examples of batch arrivals include families that go to a restaurant for lunch in a particular period of the day, convoy of a political officer that came to a filling station for refuelling and sets of triplets brought to a hospital for medical treatment. Many batch arrival single server queuing systems assume that the server offers one type of general service at the same average service rate. Climatic conditions as well as other unforeseen circumstances may affect the performance of radio networks and as such, communication companies cannot provide service to their customers at the same average rate
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