Distributed probabilistic medium access with multipacket reception and Markovian traffic

Abstract

Enabling multipacket reception (MPR) at the physical layer is a promising way to achieve higher bandwidth efficiency while reducing the complexity of the medium access control layer in distributed wireless networks. We study distributed probabilistic access where transmitting nodes access the shared wireless medium with a probability based on the node’s information about the aggregate traffic carried by the network. We model bursty traffic by rate-controlled two-state Markov sources and introduce a parameter that describes the “burstiness” level of the offered traffic. A throughput-optimal medium access strategy utilizing limited feedback is then described and its performance is examined for traffic with different levels of burstiness. It is shown that the bursty nature of the traffic in data networks allows for improvement of the bandwidth efficiency. Bounds on the system throughput are proposed and the queuing delay is analyzed.

Appendix

Proof of Proposition 1

First, we show that the following sequence

$$ \xi_N:=Np_K^*(N),\quad N\in\mathbb{N} $$

converges to

$$ \lambda_K^* \triangleq \operatorname*{\arg\max}_\lambda R_K^\mathrm{AL}( \lambda). $$

Define

$$ l_K \triangleq\lim_{n\to\infty} \sup _{N>n} \bigl( \xi_N-\lambda_K^* \bigr). $$

It follows that any subsequence of ξ _N arbitrarily close to $\lambda _{K}^{*}+l_{K}$ is an infinite sequence, i.e., the subsequence indexed by

$$ I_K^{(\varepsilon)} \triangleq \bigl\{N:\big|\xi_N- \bigl(\lambda _K^*+l_K\bigr)\big|<\varepsilon \bigr\} $$

is infinite for any ε>0. Therefore, the subsequence converges to $\lambda_{K}^{*}+l_{K}$ as ε→0⁺. It follows that the distribution of N Bernoulli trials with probability $p_{K}^{*}(N)$ can arbitrarily approach a Poisson distribution with mean $\lambda _{K}^{*}+l_{K}$ with sufficiently small ε>0. This distribution is asymptotically suboptimal unless l _K =0 (following the definition of $\lambda_{K}^{*}$). Since the optimality of $p_{K}^{*}$ must result in the asymptotic optimality of ξ _N and all its subsequences, we have l _K =0.

As a result, we have

$$\begin{aligned} &{\lim_{\lambda\to\infty} \Pr \bigl \{N^\mathrm {tx}=n\bigr \}} \\ &{\quad= \lim_{\lambda\to\infty} \sum_{N=n}^\infty \biggl(\exp(-\lambda )\frac{\lambda^N}{N!} \biggr) f_{\mathrm {bin}}\bigl (n;N,p_K^*(N)\bigr )} \\ &{\quad= \lim_{\lambda\to\infty} \sum_{N=n}^\infty \biggl(\exp(-\lambda )\frac{\lambda^N}{N!} \biggr) \biggl(\exp\bigl(- \lambda_K^*\bigr)\frac{{\lambda _K^*}^n}{n!} \biggr)} \\ &{\quad= \biggl(\exp\bigl(-\lambda_K^*\bigr)\frac{{\lambda_K^*}^n}{n!} \biggr) \lim_{\lambda\to\infty} \sum_{N=n}^\infty \biggl(\exp(-\lambda)\frac{\lambda ^N}{N!} \biggr)} \\ &{\quad= \biggl(\exp\bigl(-\lambda_K^*\bigr)\frac{{\lambda_K^*}^n}{n!} \biggr),} \end{aligned}$$

where the second and the last equality hold because the CDF of the Poisson distribution at any finite value tends to zero as its mean goes to infinity. This means that the asymptotic distribution of the number of transmitted packets with GAPA follows a Poisson distribution with mean $\lambda_{K}^{*}$ as λ→∞. Therefore, the system performs as if the aggregate offered traffic was Poisson with mean $\lambda_{K}^{*}$ and Aloha was used. □