STOCHASTIC MODELING OF THE FUNCTIONING OF WIRELESS SENSOR NETWORKS

 Introduction 

At present, there is an urgent need to control and measure almost all physical quantities and in all spheres of human activity. The use of sensors and associated communication nodes gives an idea of the universality of the problem of the development of wireless sensor networks (wireless network sensors, NSHV), in particular, in homes and buildings; industrial facilities; warehouses; in the natural environment in an environment affected by biological and chemical weapons; in cars and airplanes; on mobile intersections; at the bottom of the ocean; in rivers in combination with water energy, etc.

The development of electronics, information and communication technologies (ICT) provides grounds for implementing the idea of measuring and controlling any necessary physical quantities of the environment, industrial processes, management processes, monitoring, etc. Such a huge volume of applications of measuring technology requires solutions related to the technology of collecting, transmitting and processing information. Many network solutions have been developed and implemented based on previous experience in implementing ICT in the Internet of Things (IoT) concept, which are computer networks of physical objects (i.e. actually, things) that are equipped with technologies to interact with each other. These solutions are dominated by deterministic methods of access to the functioning of the network. The number of solutions is quite large and diverse: LAN, MAN, WAN, WLAN, Wi-Fi, mobile telephony, Bluetooth, ZeegBee, etc. [1].

However, for a wide range of applications in modern monitoring information systems, deterministic solutions are of little use in view of large equipment costs, complexity, high energy requirements, complexity of algorithms, and a wide occupied radio band. All this significantly limits the possibility of their use. At the same time, the search for stochastic solutions opens up wide opportunities for additions that were previously little-used network solutions in some applications (for implementation that has not been possible so far). They spread the category of solutions for modern applications, such as environmental monitoring, medical monitoring, etc. In this regard, the development of information technologies for environmental monitoring in the IoT concept is an urgent scientific and technical task [2].

The analysis of the principles of construction, technological solutions and the direction of development of monitoring systems in the concept of the Internet of Things (IoT), which consists of physical devices equipped with built-in technologies for interacting with each other or with the external environment using standard communication protocols, indicates that these devices can be automatically read, connected and carried into operation using highly intelligent interface without human involvement. Due to the intensive development of information and communication technologies, in particular, the spread of NSHV systems of wireless networks, the emergence of cloud technologies and computing, the development of technologies for interaction between machines of practical solutions. It is necessary to emphasize that the IoT concept encompasses three interrelated basic problems: ensuring information security (Internet of Things security); scaling the growing volume of technical devices and data (Scalability of the Internet of Things), and also taking into account the requirements for reducing energy consumption (Technical solutions of the Internet of Things and low-power design) [3].

Modeling of wireless sensor networks

Let's turn to stochastic models of WSN functioning to estimate the probability of signal collisions in the system [3]. Let's take a closer look at these models. Let As' be an event that will mean that there is no collision in the interval [0, s] (s>0). We take as P( As ') the probability of no collision in the interval [0, s]. Consider the interval [0, s], where s>tp.

Suppose that N(s)=j, that is, the number of transmissions in the interval [0, s] is equal to j j≥1. The random vector (U1,…,Uj) of the time between transmissions is uniformly distributed on the set

Ωt*={(u1,….,uj): u1+….+uj≤s}

with conditional density

f(u1,….,uj) / N(s) = j)=j!/ sj

for (u1,….,uj) ∈ Ωt*, as well as 0 beyond that.

Then the conditional density of the absence of a collision in the interval [0, s], assuming N(s) = j, is equal to

PAS'/Ns=j=PU1>tp,…,Uj>tp=1-jtpS +j,

where the expression x+ is defined as follows: x+=x for x≥0and x+=0 for x<0. The conditional probability of a collision in the interval of length s, where s>tp, provided Ns= j, is given by the expression so:

                        PAS/Ns= j=1-1-jtps+j.                                  (1)

The probability of a collision in the interval of length s ( s>tp) is determined by the expression:

   As=j=2∞e-ntpT(nST)jj![1-(1-jtpS)+j],                   (2)

where n is the number of nodes, T is the average time between node transfers, tp is the protocol transfer time. Let's analyze the issues of the number of nodes that remain in collision in the interval s for the case s>tp. We investigate the probability of a collision in the interval s for the case.

Below are the models characterizing the lower and upper estimates of the conditional probability of the number of gears remaining in collision in the interval s, assuming that the number of gears in the transmission interval in the interval s ( s>tp) is equal to j.  Let Ys  be the number of gears remaining in collision in the interval of length s.

Then

jtp Sк-11-jtp Sj-к≤PYs =к/Ns=j  ≤jtp Sк+121-jtp Sj-к+12   

  (3)

j=2∞e-nST∙nSTjj!jtpSк-11-jtpS +j-к≤PYS=к≤j=2∞e-nST∙nSTjj!jtpSк+121-jtpS +j-к+12.

Models characterizing the lower and upper estimates of the expected number of gears in collision and the variance of the number of gears in collision in the interval of length D(Ys )=D2(Ys ) are obtained. Let s>tp.

к=2∞кj=2∞e-nST(nST)jj!(jtpS)K-11-jtpS +j-K≤EYS≤к=2∞кj=2∞e-nST(nST)jj!(jtpS)K+121-tpS +j-K+12

к=2∞кj=2∞e-nSTnSTjj!jtpSK-11-jtpS +j-K+к=2∞кj=2∞e-nSTnSTjj!jtpSK+121-tpS +j-K+12≤

D2YS≤к=2∞кj=2∞e-nSTnSTjj!jtpSK+121-tpS +j-K+12+к=2∞кj=2∞e-nST(nST)jj!(jtpS)K-11-jtpS +j-K

Two dependences for the probability of collision are obtained (Fig.). Expression (1) describes the probability of a collision in a short continuation time tp of the provision of the protocol, determining the probability of undisturbed provision of the protocol. Expression (2) is derived using other properties of the Poisson process regarding the probability of a collision in a sufficiently long transmission continuation time. The graphs illustrate the possibility of a collision depending on the number of nodes (sensors) for the set average time between message transmissions, and also shows the dependence on the average protocol transmission time if the number of nodes is set (Fig.). For the average time between node transmissions equal to 10 seconds, the maximum number of nodes at which quality is ensured at a probability level of no more than  10-2 is 10, and for the average time between node transmissions equal to 30 seconds, the maximum number of nodes is 50 [5,6].

Conclusion

The work is devoted to solving an important and in-demand task-the development of information technology for monitoring environmental parameters in the modern concept of the Internet of Things (IoT) - taking into account the uncertainty of information sources and the possibility of crisis situations. 

The paper analyzes the principles of construction, development of technological solutions and the direction of development of monitoring systems in the concept of the Internet of Things (IoT), consisting of physical devices, as a result of which the shortcomings of known approaches are revealed and the expediency and necessity of creating mathematical models, methods, communication protocols of WSN networks with random access and corresponding information technologies for monitoring environmental parameters are proved. among them, to ensure high performance, quality and survivability by the creators of the systems [6,7]. 

Stochastic models of the functioning of wireless sensor networks using randomized network parameters (with a variable number of nodes and random participation of nodes in separate groups of network nodes) have been improved, which made it possible to assess the possibility of signal collisions and design IoT communication protocols more efficiently. These models allowed us to estimate the probability of a signal collision: the maximum number of nodes providing transmission quality at the level of the probability of a collision not higher than 10-2 is 50 pcs., and the number of nodes involved in the collision was negligible compared to the average number of transmissions, in particular, the ratio of the average number of nodes involved in the collision to the average The number of gears is 10-7.

devices and data (Scalability of the Internet of Things), and also taking into
account the requirements for reducing energy consumption (Technical solutions
of the Internet of Things and low-power design) [3].  

1. F., Baccelli, The Role of PASTA In Network Measurement Computer Communication Review, Proceedings of ACM Sigcomm, 2006, Vol. 36, № 4-pp 231-242.
2. A.A.A. Alkhatib, G.S. Baichier, An Overiview of Wirelees Sensor Networks, Computer Networks and Communication System (CNCS-2012): 2012 International Conference, IPCSIT, Singapore: IACSIT Press, 2012, V. 35-pp 11-15.
3. D. Nadig, S.S. Lyengur, A new architecture for distributed sensor integration, Procudings of IEEE Sout heastcon ¢ 93, Carlone, NC, April, 1991: thesis, 1993-pp 1-8.
4. K. Sohrabi, J.Gao, V.Ailawadhi, G.J. Pottie, Protocols for SelfOrganization of a Wireless Sensor Network, Personal Communications, IEEE, October 2000, Vol. 7, N 5, -pp. 16-27.
5. Adrian Perrig, John Stankovic, David Wagner, “Security in Wireles Sensor Networks” Communications of the ACM, 2004-pp. 53-57.
6. Z. Hu, A. Gizun, V.Gnatyuk, T. Zhyrova, Method for rules set forming of cyber incidents extrapolation in network-centric monitoring, Proceedings of 2017 4th International Scientific-Practical Conference Problems of Infocommunications Science and Technology (PIC S&T 2017), 2017. -pp. 121-132.
7. G. Hoblos, M. Staroswiecki, A. Aitouche, Optimal design of fault tolerant Sensor networks, Control Applications: IEEE International Conference, Anchorage, AK, September 2000-pp. 467-472.