On infinite server queues with batch arrivals

Abstract

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 + δ t ) with probability λ( t )δ t + o (δ t ) (λ( t ) > 0) and no arrival takes place during ( t , t + δ t ) with the probability 1 –λ( t )δ t + o (δ t ), (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.