Distribution of total uptime during a given time interval

System availability in a gamma alternating renewal process

“…As stated in the Introduction, all the results on availability remain valid when X and Y are dependent, with h(x) having, however, an expression different from (11). In general, in real-life systems, components do not necessarily fail independently of each other and dependent failures could be classified into three main groups: common cause failures, cascading failures and negative dependencies (Hoyland and Rausand [13]).…”

“…The problem of the off-time distribution has been studied by Muth [26] and the associated on-time by Funaki and Yoshimoto [11]. Let D(t), t > 0, be the total length of time during which the system is off within the time interval [0, t].…”

A method for computing total downtime distributions in repairable systems

“…with strictly positive components. However, our set-up is more general than that in [3], [5], [6], [8], [9], as we allow X i and Y i to be dependent.…”

“…The derivation of Muth [5] is based on consideration of the excess time. Finally, Funaki and Yoshimoto [3] derived the CDF of the total downtime by a conditioning technique. Srinivasan et al [8] derived an expression for the probability density function (PDF) of the total uptime of the system up to time t. They also discussed its covariance structure.…”

A method for computing total downtime distributions in repairable systems

“…For the independent case, Takács [9], Muth [5], Funaki andYoshimoto [3] derived the following formula for the distribution function of the total downtime:…”