The main contribution of this chapter is to design an effective MAC protocol for multihop WSNs that can achieve three concurrent goals: increasing the through of network, reducing the end-to-end delay packets and minimising the energy consumed during service a packet independently of topology configurations, traffic patterns or routing policies. The fundamental logic of the proposed protocol is to enable each node to generate its back-off intervals according to a Gamma distribution whose parameters are inferred for the status of the channel. In particular, a node infers the
M. Baz, PhD Thesis, University of York 2014
rate with which its channel is busy and the type of collision (either due to hidden or non-hidden nodes). A node then uses the inferred values as the rate and shape parameters for the Gamma distribution.
Justification for using the Gamma distribution to generate the back-off interval can be acquired by considering the versatility aspects of the Gamma distribution and the fact that the Gamma distribution is the conjugate prior for most of the probability distributions that are used to model the traffic over communication networks. Hence using the Gamma distribution whose rate parameter is set according to the rate with which the channel is found busy enables a node to back-off with high probability when the channel is potentially busy and to assess the channel when it is presumably idle. This in turn saves the energy consumed in assessing busy channels and more importantly allows a node to deliver its packets without deferring them for random periods. Another key advantage of enabling a node to generate its back-off intervals according to the rate with which the channel is found busy is that such adjustment can reduce the end-to-end delay of packets and thereby increases the throughput of the network. Furthermore, a justification for using the shape parameter to mitigate the collision probability is that the shape parameter controls the skewness of the Gamma distribution which in turn enables colliding nodes to manage their channel assessment activities without spending long back-off intervals. Justification for using the inference process is that this process enables a node to cater for its distinct traffic demands and to adapt its contention parameters in accordance with the traffic varying of network adequately. More importantly, the inference process prolongs the lifetime of nodes by enabling them to gather the required information without a need to listen for the channel for long periods or overwhelming the network with control packets.
Besides the aforementioned advantages, using the Gamma distribution to generate the back-off intervals yields a Gamma process. The Gamma process is a pure-jump stochastic process with stationary and independent increment [200-203]. A Gamma process is a mathematically tractable process whose sample paths increase monotonically, hence construction of its likelihood is straightforward. Moreover, the Gamma process features a much lower variation compared to the power law process used in BEB. Reducing the variation of the back-off process has the advantage that it improves the traffic flow over the network. Due to the infinite divisibility property of the Gamma distribution, the value of the Gamma process at any instance follows a Gamma process. This property is particularly important in designing the proposed back-off scheme as it ensures that the back-off intervals generated from the proposed model will always have the appealing statistical characterises of Gamma distributions. An illustration for the underlying approach of the proposed protocol is shown in Figure 6.2.
M. Baz, PhD Thesis, University of York 2014 Figure 6.2 Underlying approach of the proposed protocol
Figure 6.2 shows the inter-departure distributions of a number of nodes that contend to send their packet to a common receiver where 𝐿 refers to the length of packets in time unit. It is worth noting that all of these inter-departure distributes are truncated by 𝐿 since the departure time between two consequence packets from the same nodes has to be separated by at least 𝐿, hence the probability of the event that inter- departure is less than equal 𝐿 is zero. Using the independent assumptions [163], the inter-arrival distribution at the common receiver is the convolution of all inter- departure distributions of all contenders. As the convolution is a smooth operator, the inter-arrival distribution at the common receiver will be extended over larger intervals including [0, 𝐿]. These probabilities in the interval [0, 𝐿] represent the probability of overlapping at the common receiver, i.e., the collisions. Hence the obvious solution is control the inter-departure distributions of the contenders in such a way that reduces the probabilities of overlapping which is the fundamental logic of the proposed protocol. Recall that equation (5.11) shows that the inter-departure distribution is a function of inter-arrival distribution and service distribution; as the inter- arrival distribution is application-specific then the proposed protocol uses the Gamma distribution to control the contending behaviour of nodes by allows each node to adjust the shape and rate parameter based on the states of the channel. A detailed description for such adjustment is given in the pseudo-code in the next section.