## Abstract

Consider the problem of a multiple access channel with a large number of users. While several multiuser coding techniques exist, in practical scenarios, not all users can be scheduled simultaneously. Thus, a key problem is which users to schedule in a given time slot. Although the problem has been studied in various time-independent scenarios, capacity scaling laws and algorithms for time dependent channels (e.g., Markov channels) remains relatively unexplored. The basic assumption that users’ capacities are i.i.d. random variables is no longer valid, and a more realistic approach should be taken. The channel observed by a user is time varying, hence its distribution changes in time, i.e., a channel with memory. Since intelligent user selection has significant advantages, analyzing the distribution of the maximal capacity seen by a user in the system can give a good approximations for the capacity scaling law. In this work, we consider a channel with two channel states, Good and Bad, where in each point in time a user is associated with one of them. The process of moving between states is modeled by a Markovian process (Gilbert-Elliott Channel). First, we derive the expected capacity under scheduling for this time dependent environment and show that its scaling law is O(σg √ 2 log K + µg), were σg, µg are the good channel parameters (assuming Gaussian capacity approximation, e.g., under MIMO) and K is the number of users. Analysis uses Extreme Value Theory (EVT) along with finding the suitable normalizing constants. In addition, a distributed algorithm for this scenario is suggested along with it’s channel capacity analysis. The algorithm is threshold-based and the rate for exceeding it is analyzed using Point Process Approximation (PPA). The analysis is done using the properties of the extremes

of chain dependent sequences, which is proved in this study to converge to one of the extreme value distributions. The expected capacity, while imposing the discussed algorithm, scales as O

e−1(σg √ 2 log K + µg) , hence, there is no loss in optimality due to the distributed algorithm. The foundation for extending this study to more channel states is set and can be easily done.

Finally, we turn to performance analysis of such systems while assuming the users are not necessarily fully backlogged, focusing on the queueing problem and, especially, on the strong dependence between the queues. We adopt the celebrated model of Ephremides and Zhu to give new results on the convergence of the probability of collision to its average value (as the number of users grows), and hence for the ensuing system performance metrics, such as throughput and delay. We further utilize this finding to suggest a much simpler approximate model, which accurately describes the system behavior when the number of queues is large. The system performance as predicted by the

approximate models shows excellent agreement with simulation results

of chain dependent sequences, which is proved in this study to converge to one of the extreme value distributions. The expected capacity, while imposing the discussed algorithm, scales as O

e−1(σg √ 2 log K + µg) , hence, there is no loss in optimality due to the distributed algorithm. The foundation for extending this study to more channel states is set and can be easily done.

Finally, we turn to performance analysis of such systems while assuming the users are not necessarily fully backlogged, focusing on the queueing problem and, especially, on the strong dependence between the queues. We adopt the celebrated model of Ephremides and Zhu to give new results on the convergence of the probability of collision to its average value (as the number of users grows), and hence for the ensuing system performance metrics, such as throughput and delay. We further utilize this finding to suggest a much simpler approximate model, which accurately describes the system behavior when the number of queues is large. The system performance as predicted by the

approximate models shows excellent agreement with simulation results

Original language | English GB |
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Volume | abs/1507.03255 |

State | Published - 2015 |

### Publication series

Name | CoRR |
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