## 3 Model  
The topology of the network is modeled by a graph \(\mathcal{G}=(V,E)\) where \(V\) is set of node indices and \(E\) is the set of wireless links. We consider a _node exclusive interference_ model for the wireless sensors. The primary constraint is that a node cannot simultaneously receive and transmit data. A slight modification is made to the node exclusive model in this paper. We assume that a node can _overhear_ the data transmitted by a one-hop neighbor unless the node is either a transmitter or receiver. In the original model it is assumed that only one node can receive from a transmitter, which is mainly for the ease of modeling, however our modified version better captures the broadcast nature of wireless networks. We consider a time-slotted system. The overall time is broken into _frames_ where each frame has a constant duration denoted by \(t_{f}\). A frame is divided into two phases similar to the model used in [18, 20]. The first phase, called _contention phase_, is used for contention-based channel access and contention resolution. A contention phase consists of \(m\) time slots which are called _minislots_ of duration \(\delta\). The minislots are indexed by \(\{1,2,\ldots,m\}\). The duration of the contention phase is denoted by \(t_{c}\). The second phase is called _data transmission phase_, and is used for data transmission. The data transmission phase has a fixed duration of \(t_{d}\), e.g., see Figure 1. Nodes are allowed to contend for channel access and transmit data only at the beginning of each frame. We will assume that fine-grained synchronization schemes, such as [27], are available. Thus the nodes are assumed to be synchronized to the timeslots with negligible errors. The nodes are allowed to begin transmission at only one of the minislots. If there is only one sensor that transmits the earliest in the contention phase, we say the sensor wins the frame. The winner of the frame will transmit during the frame. The transmission from the winning node is assumed to be detectable by the contending nodes. When the contending nodes senses the channel busy, they stop further attempts for contending and wait for the next frame in which they retry transmission. If more than one nodes attempt to transmit in the same minislot, a collision is declared. We assume that such collision can be sensed by contending nodes, _i.e._, both collision and successful transmissions are 'channel busy' events. When a collision occurs, all the contending nodes cease the attempts to transmit on the current frame, and resume conten  
Figure 1: A time diagram for frame structures. We assume the data gathering is globally initiated, and the initiation signal is synchronized to a frame boundary. At the signaling, the measured data is assumed to be available at every node, thus all the nodes initiate transmissions simultaneously. The initiation signaling is assumed to be instantaneously done through a network-wide broadcast by certain'supernode' e.g., an aircraft transmitting a strong signal that can be received by all the nodes in a sensing field. From the initiation signal a deadline \(d\), measured in terms of the number of frames, is imposed on the network. Let us denote the number of frames it takes for the sink to retrieve the maximum data value by a RV \(T\). Our goal is to maximize \(\mathbb{P}(T\leq d)\). From the definition of the latency, the objective of'maximizing the probability of retrieval subject to deadlines' is equivalent to'maximizing the cumulative distribution function (CDF) of the latency'. In Section 5, using simulations, we will demonstrate that the CDF of the latency of the proposed scheme exceeds that of the baseline scheme. The set of node indices is denoted by \(I=\{1,2,\ldots,n\}\). We assume the reading from sensor node \(i\in I\) is a continuous RV \(X_{i}\) which has a smooth cumulative distribution function (CDF) \(F_{X_{i}}(\cdot)\) and is independent of other sensors' data. We begin by providing a precise definition of the 'extreme' or'maximum' data value.