# Analysing the Performance of  Queuing Model with the Busy Period Breakdown

## Authors

• Lidiya P Department of Mathematics, Nirmala College for Women, Bharathiar University, Coimbatore, Tamil Nadu, India. https://orcid.org/0009-0002-8314-7405
• K Julia Rose Mary Department of Mathematics, Nirmala College for Women, Bharathiar University, Coimbatore, Tamil Nadu, India.

## Keywords:

Breakdown, busy state, Idle, Multiple working vacation, Working state

## Abstract

This paper aims to analyze the M/M(a,b)/1 multiple working vacations queuing model with a breakdown. Instead of the server being fully idle during the vacation period, the server serves at a different rate during multiple working vacations. The system has only one server, and the service rate varies depending on the arrival state. Customers’ enter the system to get service with parameter λ_v following the Poisson distribution. The server provides service for customers’ in regular busy periods with parameter μ and under multiple working vacations, the server provides service with parameter μ_v with the exponential distribution. In this model, batches of customers are served as a group under the general bulk service rule, which was introduced by Neuts. In the batch service process, the service times for each customer within a batch may be independent and random variables. The number of customers’ in each batch can also vary. Thus, each batch of service contains a minimum of ’a’ units and a maximum of ’b’ units of customers’. Suppose that the number of  customers waiting in the queue is less than ’a’ server begins a vacation random variable V with parameter η, the breakdown β_v occurs during the busy state. This paper analyzed the steady-state equation, steady-state solutions, and measures of system performance. Specifically, various performance analyses, namely the mean length and other characteristics like the probability that the server is idle, regular busy and working vacation periods are analyzed. Finally, this paper computed the results with the working vacation and the classical multiple working vacation models.

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