The Behavior of a Finite Queue with Batch Poisson Inputs Resulting from Message Packetization and a Synchronous Server

This paper studies the behavior of a packet switch which provides finite waiting space and receives packetized messages. The arrivals of the messages constitute a Poisson process. Each message consists of a random number of packets. The number of packets contained in a message is assumed to be an integer-valued random variable which may follow any arbitrary probability distribution. All packets residing in the buffer receive service from a single output transmitter operating synchronously at a constant rate. Each packet receives the same fixed service time from the transmitter and then leaves the system. Upon the arrival of a message, if the remaining buffer space is not enough to accommodate all packets of the message, then the entire message is completely rejected. Results such as message blocking probability, packet blocking probability, throughput, and mean delay have been obtained. Two different approaches, minislot approximation and the application of the residue theorem, are used to obtain these results. Especially, this combinatorially very complex problem is successfully solved by the residue theorem in a recursive manner. These results are useful in evaluating the performance of a packet switch. They are also useful for design purposes.