# Development of a generalized mathematical model

### Development of a generalized mathematical model

Abstract - In last year's conference, we presented a system of systems (SoS) approach for the development of a generalized mathematical model for modeling complex systems with integrated WSNs so that necessary benchmark cases can be derived.. Additionally, this will allow for a standardized evaluation of performance measures for the different mobility models of WSNs. The generalized model has been tested mathematically and verified be simulation. Specific mobility models have been tested against this framework and the results have shown that the generalized model provides a powerful framework for the verification of QoS attributes of different models.

Keywords: mobility models; multi-state Markov process; queuing network; packet loss probabilities; end-to-end delay.

### 1 Introduction

Mobility models were proposed to imitate the movements of real mobility nodes of wireless sensor networks where mobile nodes change their speed and direction in reasonable time slots. A variety of mobility models have been proposed to model the mobility of nodes of WSNs [1] and have been classified into four major categories:

- Models with temporal dependency: as an example is the Gauss-Markov model [2]
- Models with spatial dependency: such as Reference point group model [3]
- Models with geographic restrictions: such as Pathway mobility model [4]
- Random models: such as the Random waypoint model [5]

These models have been used extensively to study the performance measures of WSNs for different applications by extensive simulations. Further, mobility models help to identify suitable probability distributions for link connected and disconnected durations between any two moving nodes in a network. These distributions can be used to model the WSNs using suitable mathematical models to obtain appropriate benchmark performance measures.

Recently, researchers have been using queuing networks, as a benchmark case, to analyze the performance measures such as packet loss probability, throughput, optimal buffer size of nodes, end-to-end delay, link and path availability. The nodes of a general open queuing network are static in nature and have a server, a buffer and links to other nodes in the network. The authors have suitably modified queuing networks by introducing intermittent links to capture the mobility of nodes in WSNs [6, 7]. The intermittent durations are governed by suitable statistical distributions depending on the underlying mobility models.

In the literature, many routing protocols have been proposed to improve the QoS of WSNs. However, individual protocols have inherent drawbacks and hence cannot serve as an adequate benchmark. These protocols should give bounds for the best performance measures of the networks under investigation [6] Therefore, by modeling the network of interest using a suitable 'generalized' mathematical model can assist in finding the bounds (upper or lower) for performance measures and thereby can be used as a benchmark model. Mobility models which are then developed for the network can be checked against these bounds.

In this research, the previous proposed System of Systems approach has been used to develop and build a generalized mathematical framework that provides the necessary benchmark cases for modeling complex large-scale systems with integrated WSNs and studying their performance measures [ieee sose 2009].

### 2 Model

The SoS approach treats the system as composed of a producer level, where the nodes are designed and modeled, and the consumer level, where the appropriate performance measures and QoS are determined. This model involves open queuing networks whereby intermittency durations in communication links are captured as mobility parameters. The analytical formulas for the performance measures such as average end-to-end delay, packet loss probability, throughput, and average number of hops are derived using the Queuing Network Analyzer (QNA).

Consider a two-dimensional WSN with mobile sensor nodes. A typical sensor node consists of a sensor, microcontroller, battery, and antenna with fixed transmitting and receiving range (radio range). A sensor node could generate its own packets (sensed data) to be sent to a sink (destination) or it could forward packets from other nodes to be sent to a sink. If a sink is not in the radio range of the node, the node sends the packets to intermediate nodes, which in turn forward the packets to the destined sink. When the battery level of a node goes down, all its activities start going down, including its radio range (weak signal strength). The signal strength between two nodes can also go down when the two nodes move away from each other. Also due to nodes mobility, it may so happen that a node would not be able to forward its own packets if the mobility is too high. This is due to the routing delay which occurs at each sensor node.

In the proposed model, a collection of queuing nodes is considered which have random connections among them according to an intermittent duration probability law (see Figure 1). This collection is referred to as a queuing network whereby at a queuing node, customers can arrive from outside the network (external arrivals) and from other queuing nodes (internal arrivals) according to a routing probability distribution law. Customers who finish service in a node may leave the network according to the routing probability distribution law (departing the network). That is, a customer who finishes service in a queuing node can go to any of the nodes (provided the random connection between the leaving node and the next node is available) or leave the network. Such a model is referred to an intermittent open queuing network. It is an open queuing because external arrivals are allowed in the network.

Note that in the above mapping between an open queuing network and a WSN, details like the nature of the microcontroller and antenna, battery life, collision of packets, and retransmission of packets are not considered. The main focus is on the intermittency in network communication.

To validate the proposed model, its performance measures for specific mobility cases should be validated using the generalized model as a benchmark case. A comparison of selective mobility models performance measures is provided in the next section to validate the generalized framework.

Four models have been used in our experiment. One model from each models classes previously mentioned. The first mobility models that has been compared with the proposed model is the Gauss-Markov mobility, where the nodes in this network has a temporal dependency over time and the distribution used is the Gaussian. The second one is the Random Waypoint model in which exponential distribution has been used. Random Adhoc mobility is modeled with Rayleigh distribution and both Obstacle and Reference Point Group mobility model utilize the exponential distribution. The generalized model (Our mathematical model) has Rayleigh distribution.

### 3 Experiment Setup

### Comparison Measures

The performance measures that are going to be compared for different models are

- End to End Delay () - It is the average waiting time for the packets to go from one network to the other.
- Packet Loss Probability (PL) - Probability of packet loss in a network.
- Throughput - Successfully message delivery in a network.

OMNET++ [8] is an extensible, modular, component-based C++ simulation library and framework, with an Eclipse-based IDE and a graphical runtime environment. To compare the performance measures, the model's author's assumption were taken into consideration and accordingly the performances measures were simulated using OMNeT++. The numbers of nodes were varied from 24 nodes to 225 nodes incrementing by 24 nodes every time and the graph was plotted.

### 4 Results

For these models and the generalized framework, a network with 24 nodes has been considered for the simulation and the performance measures are tabulated in Table 1.

The results from this table show lower end-to-end delay for the Random Waypoint model but the packet loss probability is 0.36 which is considered to be high compared to the generalized model. The results for the Gauss-Markov model show relatively higher end-to-end delay is gained but the packet loss probability is still relatively high which is 0.21. Both Random Waypoint model and Obstacle mobility model gives low network throughput with less end-to-end delay. While the generalized model gives much better results and improves the throughput of the system by guaranteeing low packet loss rate.

Figure 2 shows the packet loss probability and the throughput of the system for the different models. The figure shows that the generalized model gives the best results.

Figure 2. Packet loss probability and throughput for different mobility models comapred with the generalized model

To address the scalability issue, the Random Waypoint model was further investigated to verify its performance measures for different numbers of nodes. Table 2 gives the performance measure for the Random Waypoint and the generalized model with increasing the number of the mobile nodes in the network to 50 nodes.

Figure 3. End-to-end delay for different mobility models comapred with the generalized model

Table 3 gives the performance measure for the Random Waypoint and the generalized model with increasing the number of the mobile nodes in the network to 75 nodes.

Table 3 - Performance measures of different mobility models with 75 sensor nodes

Table 4 gives the performance measure for the Random Waypoint and the generalized model with increasing the number of the mobile nodes in the network to 100 nodes.

Table 4 - Performance measures of different mobility models with 100 sensor nodes

Figure 4. Packet loss probability for veriety number of nodes in the Random Waypoint model and the generalized model

Figure 5. End-to-end delay for veriety number of nodes in the Random Waypoint model and the generalized model

### 5 Conclusion

The generalized model is being used against specific mobility models and applications to illustrate the efficiency of the model by comparing the performance measures computed using simulation with the benchmark case results. Results show that the generalized model give better results for different metrics considered in this research.

Potential research extension may focus on a scalable P2P-based framework that effectively deals with data sharing, heterogeneity and volatility issues. An efficient cooperative, service-based, resource location algorithm that facilitates locating required resources efficiently in dynamic and heterogeneous environments.