The Application of the Residue Theorem to the Study of a Finite Queue with Batch Poisson Arrivals and Synchronous servers

The behavior of a queueing system which accepts batch Poisson arrivals and provides finite waiting space and multiple synchronous servers is studied in this paper. The batch size is assumed to be a positive integer-valued random variable which may follow any arbitrary probability distribution. Upon the arrival of a batch, if the remaining waiting space is not enough to take every customer in the batch, then the entire batch will be completely rejected. The problem, which is combinatorially very complex, has been successfully solved by the residue theorem in complex variables. Closed-form results such as batch blocking probability, test customer blocking probability, system throughput, average waiting time, etc. are derived in this paper.