Cost benefit analysis of a stochastic model on automated teller machines considering hardware, software, and networking issues

Introduction

ATMs are vital for providing convenient, round-the-clock banking services, enabling transactions like cash withdrawals, deposits, fund transfers, and balance inquiries without bank staff. Advances in technology have expanded their functions to include bill payments, fixed deposits, and loan applications. In today's competitive market, banks must ensure high ATM performance and reliability, as system failures can disrupt services, harm reputation, and lead to financial losses. Minimizing downtime to enhance system dependability is a key priority for both banks and users. ATMs can encounter both minor and major faults. Minor issues are addressed through First Level Maintenance (FLM), which involves basic, non-technical tasks like cash refilling, simple hardware checks, and routine security inspections typically handled by bank staff or security personnel. Major faults require Second Level Maintenance (SLM), performed by trained technicians to resolve complex hardware and software problems. SLM tasks include repairing or replacing components like card readers, cash dispensers, screens, and printers, as well as updating software, fixing network issues, and enhancing security features. ATM faults generally fall into three categories: hardware (e.g., jammed dispensers, card reader or screen failures), software (e.g., system freezes, processing errors), and networkrelated (e.g., server outages, connectivity failures, authentication issues). Many researchers in the field of reliability modeling including Goyal et al (2010) [5] , Kumar and Batra (2012) [8] , and many more analyzed a large number of systems considering various concepts such as different failure modes, repairs, replacements, inspections, different operational stages etc. Kumar and Kumar (2012) [10] introduces a hardware-software system that can encounter failures in the hardware components, software modules, and their mutual interactions. Bhatia and Kumar (2013) [11] present a stochastic model for a single-unit centrifuge system using real data from the Thermal Power Plant in Panipat, Haryana, categorizing faults as minor, major, or negligible. Kumar and Kapoor (2013) [9] introduced a stochastic model for a Base Transceiver System (BTS) comprising multiple hardware and software components. The system is designed to operate in four distinct modes: normal, congested, down, and failed. Two stochastic models are comparatively analyzed in the context of a base transceiver system by authors Kapoor and Kumar (2014) [5] . Yousaf et al (2014) [15] proposes a fuzzy multi-criteria decision-making (MCDM) model using Fuzzy TOPSIS to evaluate the ATM performance, focusing on cost-benefit factors. Barak and Malik (2014) [3] conduct a stochastic analysis of a two-unit parallel system exposed to random shocks. Kapoor and Kumar (2018) [14] present a stochastic model for a base transceiver station accounting for hardware and software faults, traffic congestion, common cause failures, and hardware-induced software failures. Kumari and Kumar (2021) [12] propose a stochastic model for a three-unit system operating over three mission phases with varying configurations: parallel, series, and parallel. Each unit can be in a good, degraded, or failed state. Repairs follow a Last-Come-First-Served (LCFS) policy, prioritizing the most recently failed unit. Apoorva and Taneja (2022) [1] analyze a Markovian reliability model for a self-protecting system that detects security risks and responds by entering a protective mode. Apoorva and Taneja (2023) [2] introduces a stochastic model for a single-unit selfadaptive system, which operates in different modes during its adaptive phase: normal, low-efficiency, and high-efficiency. Malik and Yadav (2024) [13] developed a stochastic model by considering human (operator) redundancy in a cold standby configuration having one unit represents the electronic system, comprising both hardware and software components, while the other unit represents the human operator. Bhardwaj and Kumar (2025) [4] analyze a stochastic model of the human cardiovascular system, taking into account various scenarios of heart damage, failure, and recovery. Kapoor (2018) [14] analyzed the performance of stochastic model on Automated Teller Machine considering major and minor faults, however none of the researcher has carried out the analysis of ATM considering hardware/software faults and networking issues. To bridge this gap, this study considers a stochastic model for Automated Teller Machines (ATM) that includes a number of different software and hardware elements. During operation, an Automated Teller Machine (ATM) can operate in three different modes: active, deterioration, and failure. There could be a variety of major and minor issues in the hardware and software components. A minor fault causes a partial failure, whereas a major fault and networking problems cause the system to fail completely. The bank staff or security people undertake the first level basic maintenance on the ATM whenever it enters a state of deterioration. After determining if the issue is with a software or hardware component, security staff repairs the appropriate part. ATM technicians or engineers are performing second level maintenance if the issue continues or if the ATM enters a failed state. Before recovering the pertinent component, the specialists first check to see if there is a hardware or software component malfunction or a network or connectivity problem. A variety of system performance metrics are derived through the use of Markov processes and the regenerative point technique. The system's profitability is analyzed based on these metrics. Graphical studies are also used to draw a variety of conclusions on the system's reliability, performance, and profitability.

Abbreviation and Notations

Assumptions and State Description

The following assumptions are considered to describe the system 1. Minor or major faults may occur in either hardware or software components. 2. Network-related issues in the system result in a failure state. 3. Minor Faults are promptly addressed by the FLM team and Major faults are promptly addressed by SLM team. 4. Faults occur at a constant rate, while the repair and inspection times follow arbitrary distributions. 5. Following each repair, the system is restored to its original condition. 6. System switching is flawless and occurs instantaneously. 7. All random variables are statistically independent of one another. 8. The technicians are available immediately at the site to handle all types of faults.

Reliability Measures 4.1 Transition Probabilities

A state transition diagram is presented in Fig. 1, illustrating the various transitions between system states. The moments at which the system enters states 0, 1, 2, 3, 4, 5, 6, and 7 are considered regenerative points-meaning all these states are regenerative in nature.

The transition probabilities are q 01 = λ 1 e -(λ 1 +λ 2 )t q 02 = λ 2 e -(λ 1 +λ 2 )t q 13 = p 1 i 1 (t) q 14 = p 2 i 1 (t) q 15 = p 3 i 1 (t) q 26 = q 1 i 2 (t) q 27 = q 2 i 2 (t) q 30 = g h 1 (t) q 40 = g s 1 (t) q 50 = g N 1 (t) q 60 = g h 2 (t) q 70 = g s 2 (t)

The non-zero elements p ij = lim s→0 q ij * (s)

The unconditional mean time taken by the system to transit for any regenerative state j, when it is counted from epoch of entrance into that state i, is mathematically stated as

Mean Time to System Failure (MTSF)

To evaluate the Mean Time to System Failure (MTSF), the failed states are treated as absorbing states. Using probabilistic reasoning, we derive the following recursive relations for χᵢ(t), the cumulative distribution function (c.d.f.) of the first passage time from regenerative state i to a failed state.:

Using Laplace stieltjes transforms, the above recursive relations are solved in terms of χ 0 * * (s), The Mean time to system failure (MTSF) when the system starts from the state 0, is

Estimated Active Time of the System

Based on the principles of regenerative process theory, the estimated active time AC i (t) which represents the probability that the system is active at time t, given it entered regenerative state i at t=0 satisfies the following set of recursive relations:

AC 1 (t) = q 13 ©AC 3 (t) + q 14 ©AC 4 (t) + q 15 ©AC 5 (t) AC 2 (t) = q 26 ©AC 6 (t) + q 27 ©AC 7 (t) AC 3 (t) = q 30 ©AC 0 (t) AC 4 (t) = q 40 ©AC 0 (t) AC 5 (t) = q 50 ©AC 0 (t) AC 6 (t) = q 60 ©AC 0 (t) AC 7 (t) = q 70 ©AC 0 (t)

Where According to the theory of regenerative processes, the estimated deteriorating time of the system DE i (t)defined as the probability that the system is in a failed state at time t, given that it entered regenerative state i at t=0 is governed by the following recursive relations:

Using Laplace transforms, the above recursive relations are solved in terms of * 0

DE (s) ,

The steady state availability of the system is given by

Where

• N 2 = p 02 μ 2 + p 02 p 26 μ 6 + p 02 p 27 μ 7

Estimated Busy Period of Technician (Inspection Time Only)

Applying the principles of regenerative process theory, the following recursive relations have been derived BI 0 (t) = q 01 (t)©BI 1 (t) + q 02 (t)©BI 2 (t) BI 1 (t) = W 1 (t) + q 13 (t)©BI 3 (t) + q 14 (t)©BI 4 (t) + q 15 (t)©BI 5 (t)

Using Laplace transforms, the above recursive relations are solved in terms of 0 BI *(s) , In steady-state, the total fraction of time for which the system is under inspection is given by

Where N 3 = p 01 μ 1 + p 02 μ 2 D 1 =already defined.

Estimated busy period of Technician (First level Maintenance only)

Applying the principles of regenerative process theory, the following recursive relations have been derived

BF 0 (t) = q 01 (t)©BF 1 (t) + q 02 (t)©BF 2 (t) BF 1 (t) = q 13 (t)©BF 3 (t) + q 14 (t)©BF 4 (t) + q 15 (t)©BF 5 (t) BF 2 (t) = q 26 (t)©BF 6 (t) + q 27 (t)©BF 7 (t) BF 3 (t) = q 30 (t)©BF 0 (t) BF 4 (t) = q 40 (t)©BF 0 (t) BF 5 (t) = q 50 (t)©BF 0 (t) BF 6 (t) = W 6 (t) + q 60 (t)©BF 0 (t) BF 7 (t) = W 7 (t) + q 70 (t)©BF 0 (t)

By applying Laplace transforms, the above recursive relations are solved. In the steady-state, the proportion of time the system remains under First level maintenance (FLM) is given by

Where N 4 = p 02 𝑝𝑝 26 μ 6 + p 02 𝑝𝑝 27 μ 7 D 1 =already defined.

Estimated Busy period of Technician (Second level Maintenance only)

Applying the principles of regenerative process theory, the following recursive relations have been derived

BS 0 (t) = q 01 (t)©BS 1 (t) + q 02 (t)©BS 2 (t) BS 1 (t) = q 13 (t)©BS 3 (t) + q 14 (t)©BS 4 (t) + q 15 (t)©BS 5 (t) BS 2 (t) = q 26 (t)©BS 6 (t) + q 27 (t)©BS 7 (t)

By applying Laplace transforms, the above recursive relations are solved. In the steady-state, the proportion of time the system remains under Second level maintenance (SLM) is given by

where N 5 = p 01 𝑝𝑝 13 μ 3 + p 01 𝑝𝑝 14 μ 4 + p 01 𝑝𝑝 15 μ 5 D 1 =already defined. Based on the graphical analysis, the following conclusions are derived in Table 2:

Conclusion

The graphs indicate that both the Mean time to system failure (MTSF) and the Expected Active time of the system decline as major and minor faults occurrence rate increase. Additionally, an increase in the probability of major and minor faults due to hardware and software components, further reduces the MTSF and Expected Active time. It is observed that the Profit rises with an increase in revenue per unit Active time, assuming fixed occurrence rate for both major and minor faults. Conversely, the profit declines as the probability of networking issues increases, given a constant repair rate for those issues.

Acknowledgement

AcknowledgementsFunding: The authors have no financial support for this study.

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