Volume-5, Issue-1, February-2015
International Journal of Engineering and Management Research
Page Number: 57-64
Cost Analysis of A.T.M. (Automated Teller Machine) Employing
Pre-Emptive Repair
Dr Reena Garg
Assistant Professor (Mathematics), YMCA University of Science & Technology, Faridabad – 121006 (Haryana) India
ABSTRACT
In this paper, the author has considered an A.T.M. (automated Teller Machine) system to analysis its cost function. The author has been taken one parallel redundant bank computer to enhance system’s overall performance. The whole system can fail due to either its normal working or human error. All the failures follow exponential time distribution where as all repairs follow general time distribution.
Keywords--- A.T.M., Central Computer, Markovian, Non-Markovian, Laplace transform, Probability of states.
I.
INTRODUCTION
In this chapter, the author has been considered an A.T.M. (automated Teller Machine) system to analysis its cost function. There are four main subsystems namely, generator, A.T.M. station, central computer and bank computer, in the considered system. The author has been taken one parallel redundant bank computer to enhance system’s overall performance. System configuration has been shown in fig-1(a). The first subsystem is generator and it supplies power to rest three subsystems. On failure of generator, the whole system becomes fail. The second subsystem is A.T.M. station and it interacts with human user. It accepts A.T.M. card, dispenses cash and print receipt. On failure of A.T.M. the whole system gets failed. The third subsystem is central computer and it verifies user’s pin, convey message to bank computer, receive reply of bank computer and allow A.T.M. to give cash to user. On failure of central computer, the whole system goes to failed state. The fourth subsystem is bank computer and it receives user’s message from central computer, verify this transaction from user’s account and made entry of this transition in user’s account. In this model, there are two bank computers working in parallel redundancy. Therefore, on failure of any one bank computer, the whole system works in reduced efficiency state. Pre-emptive resume policy has been adopted for repair purpose.
Since the considered system is of Non-Markovian nature, supplementary variables have been used to make it Markovian. State-transition diagram has been shown in fig-1(b). Probability considerations and limiting procedure have been used for mathematical formulation of the system. This mathematical model has been solved with the aid of Laplace transform. Probability of states (depicted in fig-1(b)), has been computed. Availability and cost function have been obtained for the considered system. Steady-state behaviour of the system and a particular case has also been obtained to improve practical utility of the system. Graphical illustration followed by a numerical illustration has also been appended in last to highlight important results of this study.
II.
ASSUMPTIONS
The following assumptions have been associated with this model:
1. Initially, the whole system is good and operable. 2. All failures follow exponential time distribution
and are S-independent.
3. All repairs follow general time distribution and are perfect.
4. Pre-emptive resume policy has been adopted for repair purpose.
5. There are two bank computers working in parallel redundancy.
NOTATIONS USED
III.
FORMULATION OF
MATHEMATICAL MODEL
Probability considerations and limiting procedure yield the following set of difference-differential equations, which is continuous in time, discrete in space and
governing the behaviour of considered system:
SOLUTION OF THE MODEL
PARTICULAR CASE
Fig-2
t
G(t)
5
C
1=
2
2
=
C
G
1(
t
)
5
1
=
C
1
2
=
C
G
2(
t
)
6
1
=
C
1
2
=
C
G
3(
t
)
0
0
0
0
1
2.656669
3.656669
3.588003
2
4.688791
6.688791
6.426549
3
6.1801
9.1801
8.61612
4
7.20311
11.20311
10.24373
5
7.820615
12.82062
11.38474
6
8.086992
14.08699
12.10439
7
8.049327
15.04933
12.45919
8
7.748396
15.7484
12.49808
9
7.219507
16.21951
12.26341
10
6.493235
16.49324
11.79188
Cost function Vs Time
0
5
10
15
20
1 2 3 4 5 6 7 8 9 10 11
t-->
G
(t)-->
G1(t)
G2(t)
G3(t)
Fig-3IV.
RESULTS AND DISCUSSION
We have given the availability of considered system, for various values of time t, in table-1. The graph of this has been shown in fig-2. Critical examination of fig-2 reveals that availability of the system decreases in a constant manner approximately. It should be noted that there are no sudden jumps in the values of availability of considered system.
Table-2 gives the values of cost function at various t
and for three sets of costs
C
1,
C
2. Its graph has been sketched in fig-3. In this graph, we observe that the value of cost function increases constantly. Also, we observe that value ofG
(
t
)
remains better for the second set ofC
1,
C
2(i.e.
G
2(
t
)
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