System Model

We consider a WSN consisting of n nodes. The network is generally configured in the form of a unidirectional tree as shown in Fig. 3.2, for the purpose of data collection from several leaf nodes to the root node.

• A leaf node i performs a set of periodic tasks, Ti = { Si, Ci, Mi}, where Si is the sensing

task, Ci is the computation task (message redundancy determination, data processing

etc.) and Mi is the communication task (message transmission to its parent).

• The parent node aggregates the message received from its child nodes and transmits it to the higher level node. Root node performs data analysis of the collected data.

• At node vij, the computation task is denoted by Cij having CCij computation cycles

and the outgoing message is denoted as Mij with Lij bits.

The scheduling of tasks and messages follows several constraints as enumerated:

• Precedence constraints: Each node performs periodic sensing and computation task, at the completion of which it can begin its communication. Similarly each parent node starts its local computation only after receiving all the messages from its child nodes. A parent node buffers the messages from its child nodes before performing data aggregation. Fig. 3.3 shows the sequence of steps for a cluster with a P arent node and three Child nodes.

• End-to-end latency constraint: Each message that is generated at a leaf node should reach the root node within a specified deadline, D.

• Interference constraints: For channel access, we assume that each node is allotted a time slot for communication with its cluster head. The nodes in a cluster receive exclusive access to the wireless channel since the cluster head can only receive from one transmitter at a time. Nodes in two different clusters can have shared access to the channel if they do not interfere with each other. We follow the protocol model [35] for interference modeling

in wireless networks, according to which, if a node vi transmits to a node vj, then this

transmission is successfully received by node vj if

kvj− vkk > (1 + δ) kvj− vik (3.1)

for any node vk simultaneously transmitting to any node vm at the same time using

the same channel. Using this model, consider two adjacent clusters with cluster heads

denoted as i and j. Let di denote the maximum distance between i and any of its child

nodes and mi denote the minimum distance between i and any of the child nodes in

the adjacent cluster. Now if conditions in Eq. 3.2 are satisfied then the simultaneous transmission by any two nodes in these adjacent clusters do not interfere with each other.

mi> (1 + δ) di, mj > (1 + δ) dj (3.2)

Interference Model: In this chapter, we use the protocol model (a.k.a. disk graph

model) as opposed to the more accurate but highly complex physical model (a.k.a. SINR model) [35]. Link scheduling is shown to be NP-complete in [48]. Given sufficient transmis- sion power, fixed transmission range is considered unattainable due to the interference effect of simultaneously active unintended transmitters. Under the protocol model, the impact of interference from a transmitting node is binary whereas physical model treats interference at a receiving node as an aggregate noise from unwanted transmitters. Due to this simplification, protocol model suffers from the following side effects:

1. Conservative Case: For a transmission to be successful, the receiver should be outside of the interference range of all the non-intended transmitters. This can result in reduced

schedulability.

2. Optimistic Case: When a node falls outside the interference range of each non-intended transmitter, the protocol model assumes that there is no interference. This can result in lossy transmissions, as small interference from different non intended transmitters can aggregate and prevent achieving a minimum threshold necessary for successful receptions.

We emphasize that the following reasons help minimize the negative impact of interference on our proposed schemes:

1. The impact of interference from nodes in neighboring clusters is minimized by following condition in Eq. 3.2.

2. Nodes do not employ any distance based power control. This helps to define a common maximum transmission and interference range throughout the network.

3. We set the ratio of interference to transmission range to be equal to 2. Recent research [49] shows that it is possible to use the protocol model as a good simplification for the physical model when the interference range is set appropriately (the optimal ratio between maximum interference range and maximum transmitter range should be within [1.5, 2]). Our algorithms can be extended by following this work, however this is out of the scope of our current work.

Applications which benefit most from our schemes have strategic deployments with close- knit clusters and relatively large inter-cluster distance (larger than the maximum interference range). Consider application such as transmission line monitoring, where sensors on a trans- mission tower form a cluster and the neighboring cluster is on the neighboring transmission tower; or medical monitoring where a cluster of sensors is monitoring a patient in separate rooms etc. Other factors that can reduce the impact of interference are use of a different fre- quency by each cluster for intra-cluster communication or use of directional antennas instead of omnidirectional antennas.

Scheduling Model: We assume that an initial feasible schedule is constructed using known link scheduling schemes during the initialization phase after sensor deployment. If

static slack prevails after the construction of initial feasible schedule, the centralized offline scheduler can assign frequency levels to tasks and modulation levels to messages using the Gain based Static Scheduling (GSS) algorithm [23]. The resulting schedule is then provided as an input to our schemes. The communication pattern of the tree has a single repeating frame. Each frame consists of time slots for the nodes in a level to send messages to their cluster heads.

Communication model: The modulation scheme considered is Quadrature Amplitude Modulation (QAM). The channel is modeled as frequency-flat Rayleigh fading model. Let

b denote the number of bits per modulation constellation symbol, N0

2 denote the channel

noise power spectral density and BER denote the bit error rate. Etransmit is the necessary

transmitted energy, d is the distance between the transmitter and the receiver and d0 is a

normalizing constant that depends on the wavelength.

We assume that the propagation loss follows a polynomial model and the power decays in the αth order of the distance. As a result, the total energy used in message transmission from node i to node j can be expressed as,

Eijtransmit = ( dij d0 )α.Lij.(2 bij_{− 1)} 6bij . N0 BER (3.3)

In addition to the transmission energy, energy consumed by the transmitter (node i) and receiver (node j) circuitry, is given by,

Eicircuit = Lijβi bij (3.4) Ejcircuit = Lijβj bij (3.5)

where βi and βj are implementation-dependent constants. Thus, the total radio energy spent

for a message transmission at modulation level b is given as,

Eijmessage = Eijtransmit+ Eicircuit+ Ejcircuit (3.6)

W denotes the channel bandwidth in hertz. The corresponding channel transmission rate in bits per second can be calculated as W b. We assume that the BER is low enough to provide

the necessary message reliabilities. The time taken to transmit an L-bit message in the shared wireless medium is calculated as _{W b}L .

In DMS, additional slack allocation to a message results in a lower message modulation thus reducing the communication energy consumption of that node. Slack can be differentiated as static or dynamic.

Static slack: Static slack is the difference between the application specific deadline and the initial schedule makespan. The availability of static slack is known in advance. The existing GSS algorithm [23] computes an energy-aware schedule given the static slack in a centralized manner.

Problem Statement : The static slack allocation problem can be stated as: Given a feasible schedule that satisfies the deadline and precedence constraints, allocate slack to different messages by varying their modulation levels such that the energy consumption is minimized while still preserving the feasibility of the schedule. The problem is NP-hard and the complexity of the problem can be proved by the polynomial reduction of the Multiple Choice Knapsack Problem to the current problem [50].

Dynamic slack: Dynamic slack is generated at runtime due to runtime variations such as absence of messages due to spatial or temporal correlations, channel variations etc. Given the limited connectivity amongst sensor nodes distributed over a large physical area, the huge cost of communication, varying network conditions and varying node capabilities, making centralized scheduling decisions at runtime in order to allocate the dynamic slack is highly energy inefficient. In this chapter, we present three heuristic solutions to address the problem of runtime slack distribution in a distributed manner in order to achieve energy efficiency. Our focus is on effective allocation of slack to relevant messages such that the slack generated at run time is utilized and the energy expenditure is minimized in a best possible manner while satisfying all the constraints, given a feasible static energy-aware schedule that does not violate any deadline or precedence constraints.