Performance Analysis of a Discrete-Time Queue with Versatile Batch Transmission Rule Under Batch Size Sensitive Policy

Abstract

This paper considers a single channel, infinite buffer, batch transmission system in a slotted time set up. Packets arrive according to the Bernoulli process and the transmission time is arbitrarily distributed and depends on the number of packets undergoing transmission. The service is provided according to versatile service policy i.e., the server can decide the number of packets (threshold bound) to be transmitted on beforehand with certain probability. Present study may greatly help in understanding the related performances of batch transmission channels which allows packets of variable size with the transmission time depending on the size of the packets undergoing transmission. We obtain the distribution of the number of packets in the queue and also with the server at post transmission epochs in a tractable and presentable form. We use probability generating function approach for our analysis. The joint distribution of the state probabilities essentially provides the necessary information about the busy states of the system. Furthermore, we establish the relationships between arbitrary and post transmission epoch state probabilities. Next, we also obtain some essential marginal distributions, performance characteristics and develop a related cost structure for the present model, which may be very useful to the vendors for optimal utilization of the present system by possibly controlling only the sensitive parameters in preimplementation stage. We also present some illustrative numerical examples to accomplish our study.