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The water chlorination system of a small town has two separate pipelines, each with a pump which supply chlorine to the water at prescribed rates. The two pumps are denoted A and B respectively. During normal operation, both pumps are functioning, and thus are sharing the load. In this case, each pump is operated on approximately 60% of its capacity (cap% = 0.60). When one of the pumps fails, the corresponding pipeline is closed down, and the other pump has to supply chlorine at a higher rate. In this case, the single pump is operated at full capacity (cap% = 1.00). We assume that the pumps have the following constant failure rates:

cap% = cap% ⋅ 6.3 failures/yr

Assume that the probability of common cause failures is negligible. Repair is initiated as soon as one of the pumps fails. The mean time to repair a pump has been estimated to be eight hours, and the pump is put into operation again as soon as the repair is completed. Repairs are carried out independent of each other (i.e. maintenance crew is thus not a limiting factor). If both pumps are in a failed state at the same time, unchlorinated water will be supplied to the customers. Both pumps are assumed to be functioning at time t = 0. Use Markov analysis to analyze this system. (a) Define the possible system states and establish a state transition diagram for the system. (b) Write down the corresponding state equations on matrix format. (c) Determine the steady state probabilities for each of the system states. (d) Determine the mean number of pump repairs during a period of 3 years. (e) Determine the percentage of time exactly one of the pumps is in a failed state. (f ) Determine the mean time to the first system failure, i.e. the mean time until unchlorinated water is supplied to the customers for the first time after time t = 0. (g) Determine the percentage of time unchlorinated water is supplied to the customers.

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