Probability generating function, mean and variance of the service time distribution of an M/Gc/1 queuing model

The advent of multimedia applications over wireless networks has necessitated the need for high data rates and a reliable communication over dynamic wireless links. Adaptive Modulation and coding (AMC), Automatic Repeat re-Quest (ARQ) and Forward Error Correction (FEC) have been widely used to overcome the time varying nature of these links and consequently enhancing their performance. The number of packets served per Coherence Time Interval (CTI) when AMC is employed is a function of the quality of the received Signal-to-Noise Ratio (SNR) in that CTI. Since the SNR is variable, the packet/message service time for multi-rate systems is variable. Consequently, when realistic traffic arrivals (random length messages) are considered in multi-rate systems, coupled with the assumption that the transitions from one CTI to the next are Markovian, the queue service process non-Markovian because the message may take more than one CTI to transmit/serve. In this paper, we derive the first and second moments of/and the service time distribution of the modified M/G/1 queuing model that employs AMC herein referred to as M/G^c/1 for TDMA systems over Nakagami-m fading channel. Geometrically distributed message lengths with fixed packet lengths are considered, whose arrivals are Markovian. However, individual packet service times are correlated and affects overall message service time which subsequently determines the delay experienced by a message in a queue awaiting transmission/service.