Queuing analysis of selective repeat ARQ over a wireless channel

Because of the growing importance of wireless networks and also because of the kind of randomness involved in the transmission of data in there, researchers are attracted to build Queuing models for studying the same. Some such studies are [1-6], where a Discrete Time Markov Chain (DTMC) model has been used for the analysis. In the above, studies [2-5] adopts a generating function approach for finding the performance measures like the average buffer size; whereas [1] and [6] solves for the steady state distribution of a finite dimensional DTMC. The difficulty with the generating function approach is that, when the complexity of the model increases for example by assuming the case of Automatic Repeat Request, it becomes almost impossible to get an explicit expression for the generating functions involved. Hence, getting an explicit expression for the measures like average number of blocks that have been resent, becomes very difficult if not impossible. Having said this, let us point out that the very recent paper by Darabkh et al. [3], considers a discrete model of data transfer with stop and wait ARQ. Assuming that a packet needs at most one time slot for retransmission, they obtain analytical expression for measures like average buffer occupancy using a generating function approach. However, comparing the analysis in papers [2] and [4] with that in [3], one can see that the analysis in [3] is more demanding with the obvious reason that in [3] one has to deal with more generating functions. From [3], it also follows that the generating function approach becomes more complicated if one considers some variations like random number of slots for the retransmission of a corrupted block or a selective repeat ARQ for retransmission. In the other method (see [1] and [6]), that is to solve for the steady state of a finite dimensional DTMC, one cannot expect explicit expressions for the steady state probabilities and hence would have to go for numerical computation. As far as numerical computation of steady state probabilities, especially in the case of infinite dimensional Markov chains (either discrete or continuous) is concerned, the application of Matrix Analytic Methods (MAM) developed by Neuts and his co-researchers provides a unified algorithmic approach for a wide range of problems (see [7, 8]).