# End to End Delay and Energy Consumption in a Two Tier Cluster Hierarchical Wireless Sensor Networks
## 1 Introduction  
A Wireless Sensor Network (WSN) consists of hundreds of small and low cost nodes or motes, that are spatially dispersed in a wide area to monitor and capture physical parameters of a target area. Nodes or sensors are powered with small and low cost batteries, and offers several capabilities such as sensing information (temperature, humidity, speed of the wind, etc.), data processing (compression, aggregation, ciphering, etc.) and transferring the data packets towards a gateway or central node, named as sink. For those purposes sensors are organized in a hierarchical way, using the clustering technique the most common practical solution. By clustering, a given number of nodes form a closed set, where the group leader (CH) [1] has been chosen and automatically the remaining nodes become members of the group.  
One type of random access protocol (RAP) [2] is usually chosen as basic communication scheme between CH members and their CH. On the other hand, CHs communicate between themselves with the idea of transferring the sensed information to the sink, typically using a deterministic protocol such as Time Division Multiple Access (TDMA) [3]. This topology has been recognized in the open literature as a two tier cluster hierarchical WSN [4]. Two main approaches have been considered for communication between CHs. The first one is a multi-hop routing scheme, in which the information travels hop-by-hop, from one CH to another CH located in a position closer to the sink, i.e., CHs also acts as relay nodes. In this first approach, the CHs near the sink process a large traffic load, a situation that leads to a rapid depletion of the energy of those CHs, which results in the so-called _problem of the energy hole_. These phenomena have been widely discussed quite frequently in the literature, [4; 5]. In the second approach, each CH forwards the information directly to the sink (sensor-to-sink) i.e., CH do not act as relay (intermediate) nodes. In this direct routing, we have the opposite effect: CHs away from the sink consume more energy per bit than CHs near to the sink, therefore their batteries run out faster than the batteries of the CHs near the sink [6; 7].  
In this work, sensors communicate with their CH by means of a simple RAP like Frame Slotted ALOHA (FSA) [8], and the communication between CHs is implemented by means of a deterministic protocol such as the TDMA adopting the hop-by-hop routing scheme. Our contribution is twofold; on one hand we provide a Markovian model to evaluate the end-to-end (E2E) transfer delay of an arbitrary data packet from the time it is originated at a given sensor until it reaches the sink. On the other hand we have addressed the evaluation of the energy consumption of the WSN as a whole. E2E delay and energy cost are key parameters that we have used as indicators of the performance of several WSN topologies.  
The paper offers the following structure. In Section 2 we give a basic description of related work. In Section 3 we describe the WSN scenario. In Section 4 we deal with the communication model where the Combi-Frame is defined, keeping in mind the concept of slot-reuse and routing strategy for load balancing. Sections 5 and 6 is concerned with intra-cluster or intra-cell and inter-cluster or inter-cell communications, respectively. In Sections 7 and 8 we formulate the end-to-end delay model and the energy consumption model, respectively. Section 9 provides some illustrative results. Finally, the conclusions and future work are reported in Section 10.