Control Chart Analysis for a Bulk Arrival, Non-Markovian Queue, Random Breakdown, Delay Time, Different Vacation Policies and a Stand by Server with State Dependent Restricted Admissibility of Customers

Abstract

The content of the article is a single server bulk arrival non-Markovian queue. The arrival process is a compound Poisson process with arrival size is a random variable follows a discrete distribution. The customers wait in a queue of infinite capacity if the service is not immediate. In queue the first in first out queue discipline is applied. At each service completion point the server apply two different vacation policies. Based on these two models are defined and analysed. The vacation period follows general distribution. The customers are given service singly; the service time follows general distribution. When the server is busy, the server may breakdown, the number of breakdowns follows exponential distribution. After breakdown, there is a potential delay before the repair process starts. The delay period and the repair period are generally distributed. After vacation ends, the server may continue the vacation, called extended vacation follows general distribution. The server is on vacation or repair there is a standby server, the server is service time follows exponential distribution. In addition, the arriving group will be restricted to join the system with state dependent parameter. The model is analysed by obtaining the probability generating function of the number of customers in the queue at various server’s states. Numerical illustrations are provided. Control chart analysis is carried out.