Batch arrival Queue with dual-stage service with an additional service and multi-phase Bernoulli governed vacations under N-Policy

Abstract

We consider a batch arrival queueing system where the server provides dual stages of heterogeneous service with an optional stage under a modified Bernoulli schedule and N-policy. The server stays idle until the queue size reaches N (≥1). As soon as the queue size becomes at least N, it begins serving each customer through primary and secondary stages of service. After the dual stages of service, the customer leaves with probability (1–α) or receives optional service with probability α. Upon completing all service stages, the server enters a vacation mode governed by a Bernoulli schedule: with probability (1–β), it serves the next customer; with probability β, it takes a k-phases of vacation. After this, the server may take an optional K+1 phase vacation with probability or resume service with (1–θ). We derive queue size distribution and performance measures using generating functions, and propose an optimal threshold N to minimize total cost.