On infinite server queues with batch arrivals

The queueing system studied in this paper is the one in which (i) there are an infinite number of servers, (ii) initially (at t = 0) all the servers are idle, (iii) one server serves only one customer at a time and the service times are independent and identically distributed with distribution function B(t) (t > 0) and mean β(< ∞), (iv) the arrivals are in batches such that a batch arrives during (t, t + δt) with probability λ(t)δt + o(δt) (λ(t) > 0) and no arrival takes place during (t, t + δt) with the probability 1 –λ(t)δt + o(δt), (v) the batch sizes are independent and identically distributed with mean α(< ∞), and the probability that a batch size equals r is given by a r(r ≧ 1), (vi) the batch sizes, the service times and the arrivals are independent.