Probabilistic Analysis of a Ceramic Tile Production System Considering Various Redundant Subsystems and Inspection

The present paper gives analysis of a stochastic model for a ceramic tile production system considering its five main subsystems viz. Ball Mill, Spray Dryer, Hydraulic Press, Glaze Line and Kiln and a storage system, Silo. It was considered that the failure of Ball Mill and Spray Dryer subsystems depends upon Silo while failure of other subsystems are independent of Silo. Also the storage in Silo is consumed during the repair/replacement of the failed subsystems then the system goes to failure otherwise it is operative. During the occurrence of failure in subsystems Hydraulic Press, Glaze Line and Kiln, leads to the failure in other subsequent subsystem and the repairman first inspects in which redundant unit of the subsystem has a fault and then accordingly repair/replacement is carried out. The system has been analysed using Markov process and regenerative point techniques. Various conclusions for the system regarding its reliability and availability are drawn by plotting several graphs for a particular case.


Introduction
Just from the beginning of the inception of technological systems, over a long period of time anticipation of the public of the society has been that the systems to perform its work satisfactorily and accurately. To meet the challenge and expectation of the public, various reliability models have been elaborated since long by sevaral researchers taking into account different circumstances/aspects and acquiring system performance influencing measures. Kulshreshtha [1] investigated a multi-component system, EL-Sherbeny [2] analysed two-stage repair time system, Yusuf and Yusuf [3] determined the problem of three types of failures, Taneja and Sachdeva [4] discussed on optical communication process and Rajesh et al. [5] studied about the reliability of a gas turbine. All the above studies achieved different types of failures, which optimize the reliability of the systems.
Proschan [6] introduced redundancy of the system, Arora [7] investigated reliability of many standby redundant systems with priority, Dhillon [8] studied a multistate component redundant system having common cause failures. Wang et al. [9] determined the comparison of availability between four systems having warm standby subsystems with standby switching failures. Recently, Reena and Kumar [10] obtained reliability analysis of a ceramic tile manufacture system having various subsystem failures. So, lot of studies has been done by several researchers taking redundant systems. In this paper we deal with a stochastic models developed for a ceramic tile production system considering its five main subsystems viz. Ball Mill, Spray Dryer, Hydraulic Press, Glaze Line and Kiln and a storage system, Silo that is used for storing raw material on the basis of information made while visiting the system. It was observed that the failure of Ball Mill and Spray Dryer subsystems depends upon Silo while failures of other subsystems are independent of Silo. If the storage in Silo is consumed during the repair/replacement of the failed subsystems then the system goes to failure otherwise it is operative. On failure of the system, subsystems Hydraulic Press, Glaze Line and Kiln are inspected by the repairman to judge in which subsystems Hydraulic Press, Glaze Line and Kiln fault has occurred and accordingly repair the subsystem. During the occurrence of fault in the subsystems Hydraulic Press, Glaze Line and Kiln leads to the failure in other subsequent subsystem.

Notations
λ 1 /λ 2 /λ 3 / λ 4 /λ 5 constant failure rate of A/B/C/D/E subsystems of the system X dust storage level at a time x 0 required level of dust for the operation of the system t* time duration in which dust storage reduced to required level x0 p 1 probability that the dust storage (capacity) is more than the required level = P(X ≥ x0) q 1 probability that the dust storage (capacity) is less than the required level = P(X < x0) p 2 probability that the repair is done before the dust storage (capacity) goes less than required level x0 q 2 probability that the repair is done after the dust storage (capacity) goes less than required level x0. c i probability that the fault is in the ith press, i=1,2,3,4 d i probability that the fault is in the ith glaze line, i=1,2,3,4 e i probability that the fault is in the ith kiln, i=1,2,3,4 I 3 (t)/i 3 Figure 1 shows possible states of transitions of the system. The points of entry into states 0, 1, 2, 3, 4, 5, 6, 7, 8,9,10,11,12,13,14,15,16,17,18 and 19 are regeneration points and hence these are regenerative states. The states 2 and 4 are failed states and 5, 6,7,8,9,10,11,12,13,14,15,16,17,18

Mean Sojourn Time
Mean sojourn time ( i µ ) is the mean first passage time taken by the ith state before transiting to any other state. The unconditional mean time obtained by the system to transit for any state j when it is calculated from the point of entrance into the state i, is mathematically expressed as:

Mean Time to System Failure
MTSF is determined regarding the failed states as absorbing states of the system. Using probabilistic arguments, following are the recurrence relations for φ i (t), c.d.f of the first passage time from regenerative state 'i' to failed state: Taking L.S.T. of these equations and after solving for

Availability of the System with Full Capacity
By the assertions of the theory of regenerative processes, the availability of the system with full capacity (AF i (t)), the probability that the system is working at instant 't' with full capacity given that it entered regenerative state 'i' at t = 0, is seen to satisfy the following recurrence relations.  AF t q t AF t q t AF t q t AF t q t AF t