A model of periodic acknowledgement

Abstract

We study a problem abstracted from modeling a multicast protocol. In our model, messages generated by a single source are simultaneously forwarded to a set of receivers where they are independently processed. We assume a state-dependent message arrival rate and memoryless service time distributions. The receivers may process messages at different average rates. Messages processed by all receivers are periodically acknowledged and cleared from the system. Due to finite buffer space, the total number of non-acknowledged messages in the system is limited. Our focus in this paper is on the number of messages cleared at acknowledgement time.The problem under consideration bears resemblance to a fork/join process with heterogeneous servers, used in the study of multiprocessing computer systems. Our model includes the additional features of finite buffer space and delayed periodic departure of completed jobs. Even without these features, the resulting type of queuing model has no known closed-form solution in the general case of more than two servers. Because the arrival processes to the servers are correlated, the model is difficult to decompose. We propose a relatively simple decomposition technique and a fixed-point iteration scheme. In our approach, we consider each receiver (server) in isolation, and we account for the influence of other receivers through the probability that a given number of messages can be cleared at acknowledgement time. We derive elementary differential equations for the number of messages processed by a receiver. These equations involve the conditional probability of the number of messages not yet processed given the number of messages waiting to be cleared. We compute an approximate solution using the conditional probability obtained from a model with exponentially distributed acknowledgement periods. Our numerical results for the average number of messages cleared at acknowledgment time are typically within a few percent of simulation midpoints.