Dynamic Multiple-Message Broadcast: Bounding Throughput in the Affectance Model

Abstract

We study a dynamic version of the Multiple-Message Broadcast problem, where packets are continuously injected in network nodes for dissemination throughout the network. Our performance metric is the ratio of the throughput of such protocol against the optimal one, for any sufficiently long period of time since startup. We present and analyze a dynamic Multiple-Message Broadcast protocol that works under an affectance model, which parameterizes the interference that other nodes introduce in the communication between a given pair of nodes. As an algorithmic tool, we develop an efficient algorithm to schedule a broadcast along a BFS tree under the affectance model. To provide a rigorous and accurate analysis, we define two novel network characteristics based on the network topology and the affectance function. The combination of these characteristics influence the performance of broadcasting with affectance (modulo a logarithmic function). We also carry out simulations of our protocol under affectance. To the best of our knowledge, this is the first dynamic Multiple-Message Broadcast protocol that provides throughput guarantees for continuous injection of messages and works under the affectance model.

Appendix

1.1 A Notes

In this section, we highlight the differences between this paper and our preliminary work appeared in [35].

In [35], we studied a model of affectance that subsumes only some SINR models, by combining the effect of Radio Network collisions with affectance from nodes at more than one hop. Here, we generalize our model to subsume any arbitrary interference model. For instance, in the present model it is possible to receive a transmission even when more than one neighboring node transmits, as in some SINR models.

Also, in [35] our maximum path affectance metric was based on fast links only, which yields possibly tighter bounds. However, the definition was based on a specific BFS tree (a GBST [20]) which related the network characterization to our specific algorithmic solution. In the present work the characterization is related only to topology, since it is based on arbitrary BFS trees.

We also notice here that the proof of the maximum rank in [35] has an error, introduced while bounding the maximum number of ranks needed for updating the rank according to affectance. Lemma 1 here provides the correct bound.

1.2 B Instantiation of Affectance in SINR

Claim

The affectance matrix For $u,v,w\in V$, where $u\ne w$, the affectance matrix

$$\begin{aligned} A(u,(v,w))&= \frac{P/d_{uw}^\alpha }{P/(\beta ' d_{vw}^\alpha )-N}, \end{aligned}$$

with transmission power level P, background noise N, SINR lower bound for reception $\beta '$, path-loss exponent $\alpha$, and Euclidean distance between nodes u and v denoted as $d_{uv}$, corresponds to the SINR model.

Proof

To prove this claim, we show that there is a successful transmission in the SINR model if and only if there is a successful transmission in the affectance model with matrix A. Consider a successful transmission in the SINR model. We have

$$\begin{aligned} \frac{P/d_{uv}^\alpha }{N+\sum _{w\ne u} P/d_{wv}^\alpha }&> \beta '\\ P/(\beta ' d_{uv}^\alpha )&> N+\sum _{w\ne u} P/d_{wv}^\alpha \\ P/(\beta ' d_{uv}^\alpha ) - N&> \sum _{w\ne u} P/d_{wv}^\alpha . \end{aligned}$$

If $\sum _{w\ne u} P/d_{wv}^\alpha =0$ then $\sum _{w\ne u}A(w,(u,v))=0\Rightarrow$ success in affectance model. Otherwise, it is $\sum _{w\ne u} P/d_{wv}^\alpha >0$ and we have

$$\begin{aligned} \frac{P/(\beta ' d_{uv}^\alpha ) - N}{\sum _{w\ne u} P/d_{wv}^\alpha }&> 1. \end{aligned}$$

Thus, it is $P/(\beta ' d_{uv}^\alpha ) - N>0$ and, hence, we have

$$\begin{aligned} \frac{\sum _{w\ne u} P/d_{wv}^\alpha }{P/(\beta ' d_{uv}^\alpha )-N}&< 1. \end{aligned}$$

Therefore, it is $\sum _{w\ne u} A(w,(u,v)) < 1 \Rightarrow$ success in affectance model.

Consider now a non-successful transmission in the SINR model. We have

$$\begin{aligned} \frac{P/d_{uv}^\alpha }{N+\sum _{w\ne u} P/d_{wv}^\alpha }&\le \beta '\\ P/(\beta ' d_{uv}^\alpha )&\le N+\sum _{w\ne u} P/d_{wv}^\alpha \\ P/(\beta ' d_{uv}^\alpha ) - N&\le \sum _{w\ne u} P/d_{wv}^\alpha . \end{aligned}$$

If $P/(\beta ' d_{uv}^\alpha ) \le N$, it would mean that P is not large enough to overcome the noise, even if no other node transmits. Then, rather than being produced by interference, the failure would be due to consider a link that is not even feasible. That is, $(u,v)\notin E$. Thus, it must be $P/(\beta ' d_{uv}^\alpha ) > N$ and we have

$$\begin{aligned} \frac{\sum _{w\ne u} P/d_{wv}^\alpha }{P/(\beta ' d_{uv}^\alpha )-N}&\ge 1\\ \end{aligned}$$

Therefore, it is $\sum _{w\ne u}A(w,(u,v)) \ge 1 \Rightarrow$ failure in affectance model.