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The service facility or server is the key constituent to keep a system operational for desired period of time. As any eventuality with the system necessitates immediate presence of it (server) so the time point of arrival and treatment of server significantly affects the system performance. This paper works out the steady state behavior of a cold standby system equipped with two similar units and a server with elapsed arrival and treatment times following general probability distributions. It practices the theory of semi-Markov processes, regenerative point technique and Laplace transforms to derive the expressions for state transition probabilities, mean sojourn times, mean time to system failure, system availability, server busy period and expected frequencies of repairs and treatments. The profit function is also developed taking different costs and revenue in to account. For tracing wider applicability of the model for different reliability and cost-effective systems, a particular case study is also presented as an illustration.

The reliability analysis is an essential practice for the installations where failure may turn out hazardous either in terms of huge financial loss or threat to human life. These causes inspired the literature to a greater extent [

a) Transition probabilities and mean sojourn times in different states.

b) MTSF and reliability of the system.

c) System availability.

d) Server busy period.

e) Expected number of repairs and treatments.

f) And expected profit.

To provide ease to the computational work, the model is developed using the following set of assumptions:

a) The model consists of two identical units. Initially, one unit is in operation and another as cold-standby.

b) The unit in standby mode can’t fail.

c) Upon failure of the operative unit the standby becomes operative instantly.

d) All the failures are repairable to be repaired by the server but the server takes some time to arrive.

e) The server may fail while working but curable.

f) The server restoration subjects to treatment with some elapsed time.

g) All the repairs, treatments and switching are perfect.

h) The system works as long as at least one unit remains working.

i) All the random variables are assumed to be statistically independent.

j) All the random variables follow general probability distribution with different distribution functions.

The system model comprises of regenerative and non-regenerative states. The states

_{i} up to time ‘t’ without making any transition to any other regenerative state or returning to the same state via one or more non-regenerative states.

_{i}) in state S_{i} when system transit directly to state j.

_{i}) in state S_{i} when system transit to state j via k and n times between

(s)/(c): Stieltjes convolution/Laplace convolution.

Taking account of all possible transitions and the re-generative points a system schematic state transition diagram is constructed as given in

Simple probabilistic considerations, yields the following expressions for the non-zero elements

For these Transition Probabilities, it can be verified that

The Mean sojourn time µ_{i }in state S_{i }are given by:

The unconditional mean time taken by the system to transit from any state S_{i} when time is counted from epoch at entrance into state S_{j} is stated as:

Let _{i} to a failed state. Regarding the failed state as absorbing state, we have the following recursive relations for

where S_{j} is an un-failed regenerative state to which the given regenerative state S_{i} can transit and S_{k} is failed state to which the state S_{i} can transit directly.

Taking LST of Equation (3) and solving for

The reliability R(t) is given by

Let the system entered the regenerative state S_{i} at t = 0. Considering S_{j} as a regenerative state to which the given regenerative state S_{i} transits, the recursive relations for various profit measures in (0, t] are given as follow:

Here,

And

Using LT/ LST, of Equations (4)-(7) and solving we get the results in steady state as below:

Further, using the values of above performance measures, the profit incurred to the system model in steady state is given as below.

For the sake of convenience, let us suppose all the random variables follow the exponential distribution with the probability density functions given below.

We assume some particular values for various time rates and costs i.e.

Failure rate of server (g) = 0.02 per unit time, Failure rate of unit (λ) = 0.008 per unit time.

Repair rate of unit (α) = 0.3 per unit time, Treatment rate of server (β) = 0.05 per unit time.

Server arrival rate (y) = 0.08 per unit time, Waiting treatment time (x) = 0.08 per unit time.

And

For this, we obtained the values for different measures of system performance as follows:

MTSF = 1110.909 unit time, Availability = 0.977238, Busy period of server = 0.023716,

Expected number of repairs = 0.000918, Expected number of treatments =0.000364 and

System profit = 19532.31.

The detailed results are given in tabular form. Here Tables 1-3 respectively, illustrate the effect of server treatment rate for various combinations of parameters on Mean Time to System Failure (MTSF), availability and profit assuming

MTSF | |||||
---|---|---|---|---|---|

Treatment Rate β | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 |

1004.696 | 1056.204 | 1083.138 | 1099.698 | 1110.909 | |

689.7959 | 717.9856 | 733.3333 | 742.9864 | 749.6183 | |

1004.696 | 1056.204 | 1083.138 | 1099.698 | 1110.909 | |

774.7201 | 855.3967 | 902.9509 | 934.3066 | 956.5359 | |

1004.696 | 1056.204 | 1083.138 | 1099.698 | 1110.909 | |

1146.843 | 1189.648 | 1211.383 | 1224.532 | 1233.344 | |

1004.696 | 1056.204 | 1083.138 | 1099.698 | 1110.909 | |

1007.834 | 1060.815 | 1088.586 | 1105.682 | 1117.266 | |

1004.696 | 1056.204 | 1083.138 | 1099.698 | 1110.909 | |

1115.433 | 1182.51 | 1218.052 | 1240.063 | 1255.036 |

Availability | |||||
---|---|---|---|---|---|

Treatment rate β | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 |

0.936585 | 0.961641 | 0.970256 | 0.974611 | 0.977238 | |

0.91928 | 0.949691 | 0.960247 | 0.965604 | 0.968842 | |

0.936585 | 0.961641 | 0.970256 | 0.974611 | 0.977238 | |

0.840416 | 0.908284 | 0.932934 | 0.94563 | 0.953361 | |

0.936585 | 0.961641 | 0.970256 | 0.974611 | 0.977238 | |

0.958533 | 0.973861 | 0.979065 | 0.981685 | 0.983261 | |

0.936585 | 0.961641 | 0.970256 | 0.974611 | 0.977238 | |

0.936889 | 0.961972 | 0.970594 | 0.974952 | 0.977581 | |

0.936585 | 0.961641 | 0.970256 | 0.974611 | 0.977238 | |

0.939355 | 0.964577 | 0.97325 | 0.977634 | 0.98028 |

Profit | |||||
---|---|---|---|---|---|

Treatment Rate β | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 |

18719.42 | 19220.45 | 19392.7 | 19479.77 | 19532.31 | |

18370.65 | 18978.74 | 19189.81 | 19296.9 | 19361.65 | |

18719.42 | 19220.45 | 19392.7 | 19479.77 | 19532.31 | |

16796.16 | 18153.13 | 18645.96 | 18899.78 | 19054.33 | |

18719.42 | 19220.45 | 19392.7 | 19479.77 | 19532.31 | |

19163.11 | 19469.62 | 19573.7 | 19626.07 | 19657.6 | |

18719.42 | 19220.45 | 19392.7 | 19479.77 | 19532.31 | |

18725.48 | 19227.04 | 19399.44 | 19486.57 | 19539.14 | |

18719.42 | 19220.45 | 19392.7 | 19479.77 | 19532.31 | |

18774.62 | 19278.98 | 19452.39 | 19540.05 | 19592.94 |

A stochastic model for a repairable cold standby system, with waiting arrival and treatment times of server, is discussed in this paper. The theory of semi-Markov process and regenerative point technique is used to derive expressions for measures of reliability and profit. An example is given under the setup of exponential distribution by assigning distinct values to various parameters and costs considered for the system model. The further numerical results (as given in Tables 1-3) indicate that MTSF, availability and the profit rise with increasing server treatment rate (β), repair rate of units (α) and server arrival time (ψ) but the trend reverts with increasing the failure rates of server (g) and unit (λ). The persisting trend reveals that the waiting arrival and treatment times impact a lot on the system performance. Therefore, the study re-iterates the practicalities that a cold standby system served by a repairable server can be kept reliable and profitable by:

1) Using standard units with low failure rates.

2) Deploying efficient server with high repair rates.

3) Planning for higher arrival rate of server and,

4) Arranging rapid after failure treatment of the server.

The re-iteration of the true facts evidently proves the acceptability of the probabilistic model developed in this paper. The study may be inspiring and useful for system planners and reliability engineers for developing highly reliable and profitable systems to earn users’ satisfaction.

The study finds its application in diverse areas such as power generating systems with standby reservoirs, communication systems with redundant channels, remote sensing systems with alternate power backups etc.

This work is a part of Major Research Project F. N. 42-34/2013(SR) financially supported by UGC under MHRD, Govt. of India. The authors are grateful to anonymous referee for their valuable comments and suggestions.

Rohtash K. Bhardwaj,Ravinder Singh, (2016) Stochastic Model of a Cold-Stand by System with Waiting for Arrival & Treatment of Server. American Journal of Operations Research,06,334-342. doi: 10.4236/ajor.2016.64031