Two-level control policy of an unreliable queueing system with queue size-dependent vacation and vacation disruption

Abstract

The objective of the paper is to analyse two-level control policy of an M X ∕G(a, b)∕1 queueing system with fast and slow vacation rates and vacation disruption. In the service completion epoch, if the queue length is less than ‘a’, then the server leaves for a vacation. In this model depending upon the queue length, the server is allowed to take two types of vacation called fast vacation and slow vacation. Addressing this in the service completion epoch, if the queue length ψ(say) is less than β where β < a − 1, then the server leaves for slow vacation. On the other hand, if ψ > ζ, where a − 1 ≥ ζ > β during service completion, then the server leaves for fast vacation. During slow vacation if the queue length reaches the value ζ, then the server breaks the slow vacation and switches over to fast vacation. Also if the queue length attains the threshold value ‘a’ during fast vacation, then the server breaks the fast vacation too and moves to tune-up process to start the service. After tune-up process service will be initiated only if ψ ≥ N(N > b). For the designed queueing system probability, generating function of the queue size at an arbitrary time epoch is obtained by using supplementary variable technique. Various performance characteristics will also be derived with suitable numerical illustrations. Cost-effective analysis is also carried out in the paper.