A batch arrival multi phase queueing system with random feedback in service and single vacation policy

Abstract

We study a batch arrival Poisson input, k phases heterogeneous services with randomly feedback in services. The first phase of service is essential for all customers, but with probability γ1 a tagged customer chose second phase, with probability α1 feedback to tail of original queue or with probability β1 = 1 − γ1 − α1 leave the system. Also, after completion of the second phase, with probability γ2 the customer leads to the third service, or with probability α2 feedback to tail of original queue, or leaves the system with probability β2 = 1 − γ2 − α2. Like that, in kth phase feedback with probability αk or leaves the system with probability 1 − αk. After completion of each phases, the server either goes for a vacation with probability 𝜃(0 ≤ 𝜃 ≤ 1), or may continue to serve the next unit with probability 1 − 𝜃, if any. Otherwise, it remains in the system until a customer arrives, which is single vacation policy. We assume restricted admissibility of arriving batches in which not all batches are allowed to join the system at all times. In this paper we derive the steady- state equations, PGF’s of the system, and measures of sysytem.