Tripod Turnstile Machines Performance Analysis for the System Safety and Security without Considering Simultaneous Failures using Reliability Approach

As the crime is increasing day by day around the world. To stop the crime many security gadgets and machines have been developed by various security agencies. The tripod turnstile machine also helps in stopping the crime and restricting the entry of unauthorized person on the premises. In this paper, the aim is to analyze the reliability measures of the two tripod turnstile machines which work in a parallel configuration. In the case of non-functionality of the machine, unauthorized persons may enter into the premises. Due to security and safety reason, organizations install these machines. Hence these machines must be highly reliable in the operation to avoid any unwanted event. In this paper, for the considered system, the transition state diagram has been drawn with the help of the Markov model. The ChapmanKolmogorov differential equations are developed from the transition state diagram and solved using Laplace transformation. The failure and repair rates are assumed to be constant and follow negative exponential distribution. The system upstate and downstate probabilities are determined. The explicit expression of the system Availability, Reliability and MTTF are also obtained. The sensitivity analysis is also performed to determine which machine affects the system reliability the most. KeywordsTripod turnstile machine, Markov model, Laplace transformation, MTTF, Reliability, Availability.


Introduction
In the whole world, crime is increasing day by day. Therefore, human safety and system safety are the main concerns of the society. With the advancement in the field of science and technology, many useful gadgets and machines have been developed to stop the crime. Though the crime has not been fully eradicated, these gadgets and machines help in the reduction of the rate of crime. The turnstile machine also plays a very crucial role to restrict the entry of the unauthorized person. For this reason, turnstile machines are generally installed at metro railway stations, airports and stadium etc. Turnstile machine is also known as baffle gate. It allows only one person to pass when a person punches a card or insert a coin in the machine. From a security point of view, this is very important to install a turnstile machine as it also stops thefts and crime in the organization. When a person punches his card on the card reader this machine stores the data like name, UID, time of entry, time of exit. Hence, it helps to reduce the security concern of the people. For this reason, failure-free operation of the turnstile machines is required. These turnstile machines come in different sizes like waist height, full-height turnstile, tripod, flap gates etc. Per-minute 15-20 people can pass from this gate using their card or by inserting a coin in the machine. Mosman et al. (2019) introduced the concept of automated stadium management system for managing the crowd of a football stadium. With this system, the data of football match viewer, present in the stadium can be easily saved. With this information, if any unwanted event happens, then the investigation can be done easily with the help of the data available.
On the failure of the turnstile machine, it is either repaired or replaced with a new machine, hence, it is a repairable system. A repairable system is one which has two states, perfect working and complete failure. The repairable system always oscillates between these two states. When the system fails, out of repair or replacement, repair of the system is always preferred. Replacement of the system or component is the best option when the repair is very costly. There are basically two types of maintenance policies, Corrective maintenance and preventive maintenance. Corrective maintenance is performed on the failure of the system and preventive maintenance is performed to delay the failure of the system. Literature related to maintenance policies can be found in McCall (1965), Pierskalla and Voelker (1976), Cho and Parlar (1991), Sherif and Smith (1981), Kumar and Ram (2016). Maintenance action is performed on the system or on system components. Basically, a system is a collection of components or machines that are connected to perform a specific task. A system may be in a series configuration, parallel configuration or in a mixed configuration. Description of all these can be easily found in Srinath (1994), Balagurusamy (1984), Ebeling (2004), Ram et al. (2013), Ram and Kumar (2015), Kumar et al. (2017). Besides, these configurations, redundancy also plays a very important role in increasing system reliability and availability. There are basically two types of redundancies active redundancy and standby redundancy. Li (2016) presented that standby redundancy is better than the active redundancy. Zheng et al. (2018) investigates a real time computing system by the aid of continuous time Markov chain and a comparison was done in between effect of CCFs on the system in warm and hot standby configurations.
To determine the reliability measures of the system, Markov modelling is being used extensively these days by the researchers. The Markov-model is a state-based model. The main advantage of the Markov model is the transition of the future states depend only on the present state but not on the past states. Kumar and Kumar (2019a) used the Markov model for the modelling of the wireless communication system and various reliability measures were calculated in their paper. Authors Kumar and Kumar (2019b), Niwas and Garg (2018) investigated an industrial system which works under cost-free warranty policy. They used the Markov model to determine the reliability indices of the system. Sharma and Vishwakarma (2014) further extended the work and analyzed the performance of the feeding system of the sugar industry. Using Genetic algorithm steady state availability of the system was optimized. Yusuf et al. (2018) analyzed the performance of single host with three types of software. Initially, one software is used but when it fails other software of the same type is used. In this way failure of software doesn't cause system to fail on the failure of the software. This helps in extending the availability of the system and increasing the revenue of the system. Numerical method techniques have also been used in the literature to determine the reliability of the industrial system. Shakuntla et al. (2011) determined the reliability of the polytube industry using by using Runge-Kutta fourth order method. Sensitivity analysis was also carried out to improve the performance of availability. This discussion gives us the idea of the wide application of the Markov model in industries.
To the best of the authors' knowledge, there is not much literature available for the turnstile machines and no one has ever tried to determine the reliability measures of the turnstile machine. In this paper, authors aim is to determine the reliability measure of the turnstile machines working in a series configuration.

Problem Statement
It has been observed, these turnstile machines go out of order many a time when their failure-free operation is required. This allows an unauthorized person to enter into the premises. The entry of these unauthorized persons may affect human security and system safety sometimes. Hence keeping this problem in the mind, reliability measures of the turnstile machine are determined in this paper. This paper is organized as follows: In section (2), system description, assumptions and nomenclature are given. In section (3), Chapman-Kolmogorov differential equations are developed from the transition state diagram. In section (4), for the considered system Availability, Reliability and MTTF are calculated. Besides this, Sensitivity analysis is also performed. In section (5) results are presented. In section (6), Conclusion is given. In section (7), future scope of the work is given. In section (8), conflict of interest is also given.

System Description
In this paper, authors consider two turnstile machine, these machines are working in a parallel configuration. Initially, both the machines are in good working condition. When one turnstile machine fails, then other turnstile machine continues to work and meanwhile, the failed machine is repaired. After repair the turnstile machine is as good as the new machine. On the failure of both machines the entire security system fails. As only one repairman is available with the organization, therefore, the repair of the machine is done on first come and first-served basis. Repair and the failure rates are assumed to be constant. Later in the paper, reliability measures have been determined by substituting the particular value of the repair and failure rates in the system upstate probability.

Nomenclature
The nomenclature of the work are found in Table 1.

Mathematical Formulation of the Turnstile System
In order to determine the reliability measures of the system, as it is presented in Figure 1, the transition state diagram of the system is drawn as shown in Figure 2, Authors has been developed the following Chapman-Kolmogorov differential equations from the transition state diagram of the system given in  On solving equations from (6) -(9), we get;

Numerical Computation
The following values of failure and repair rates will be used for the calculation in this paper.

Availability of the System
Availability of a repairable system is the probability that the system is performing its task at a specified period of time under given operating condition. For calculating the system availability of International Journal of Mathematical, Engineering and Management Sciences Vol. 6, No. 1, 383-395, 2021 https://doi.org/10.33889/IJMEMS.2021.6.1.024 389 the proposed system, take inverse Laplace of equation (14) and substitute the values of failure rates and repair rates given in Table 2, one can easily get the explicit expression of the system availability. On varying time unit from 0 to 100000 with the step size of 10000, one can easily analyze the behavior of system availability from Table 3 and Figure 3.

Reliability of the System
Reliability is the quantitate attribute of the system. Generally, reliability is the probability that the system will perform its task as expected without failure for a specified period of time under specified conditions. If the system does not perform in the specified conditions then it is said to be the failure of the system. For the proposed system, one can easily get the explicit expression for reliability. For this, take Inverse Laplace of the equation (14). After this, substitute the values of failure rates as given in Table 2 and also set the repair rates equal to zero in the obtained expression. One can easily get the explicit expression of the reliability of the system.

MTTF (Mean Time to Failure)
Mean time to failure is the expected time for the system failure. To find the mean time to failure set all repair rates equal to zero in (14), and then taking the limit 0 → s , one can easily get the MTTF of the system. The formula for obtaining the MTTF is given below The explicit expression of the MTTF is Setting values of failure rates as given in Table 2 and varying each failure rate one by one from 0.00001 to 0.00010 in (19), one can get the variation in the MTTF w.r.t variation in the failure rates. This can be easily analyzed from Table 5 and Figure 5.

Sensitivity of Reliability
Sensitivity analysis is performed to determine the most critical component/ components of the system. It actually determines, how actually the system performance measure is affected by changing the failure rate of the system's component. Here, authors perform the sensitivity analysis of the reliability of the system. For this first of all take the inverse Laplace of (14) and differentiate the obtained expression w.r.t all the failure rates one by one. In these derivatives set failure rates values as given in Table 2 and also set all repair rate equal to zero. On varying the time unit from 0 to 100000 with the step size of 10000, one can easily obtain Table 6 and Figure 6.