Transient behaviour of an \(M/ E_k /1\) queue with vacations, balking and control of admission during vacations (Q2247434)

Summary: This paper studies the transient behaviour of an \(M/ E_k /1\) queueing model with single and multiple vacations, balking and control of admission during vacations. An arriving customer undergoes \(k\) exponential phases of service before leaving the system. Whenever the system becomes empty, the server starts vacation (single or multiple). All the arriving customers are not allowed to join the queue during vacations. That is, they are either permitted to join the queue or rejected. During the vacation period, the permitted arrivals may either join the queue or balk. Using the method of generating function, the transient system size probabilities are derived for the proposed model in terms of generalised modified Bessel function of the second kind. The system performance measures such as average and variance of system size, probability of system empty and server idle are also obtained. Numerical illustrations are presented to analyse the influence of the system parameters.