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The limited scheme with excess distribution (LS-ED), being a superior grant-sizing protocol in terms of average delay minimization while providing fairness and quality of service (QoS) satisfaction in Ethernet passive optical networks (EPONs), has already been deployed in diverse technologies. An analysis of the protocol is of paramount importance for obtaining deeper insight into several network design and deployment aspects. Thus, the primary objective of this paper is to perform a rigorous mathematical analysis of LS-ED, for the first time to the best of our knowledge, with the help of a discrete time Markov chain (DTMC). Owing to the dependency of the grant size of a particular optical network unit (ONU) on the queue length of other ONUs, the exact queuing analysis of this scheme necessitates solving a Markov model of the same dimension as the number of ONUs, which makes the analysis complicated and mathematically intractable. A lower-dimensional valid approximation model may allow us to perform such an analysis. In this paper, for the first time to the best of our knowledge, we present an analysis of the scheme, wherein the system dynamics are captured from the perspective of the scheduler leading to an approximate model. The model aids in obtaining a closed-form expression for the average cycle time and throughput along with closed-form expressions for average buffer occupancy and average delay at low load. Finally, we provide useful design insights for EPONs employing LS-ED.
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