Consider a stochastic process {Y(t)}t=0, representing the state of a machine at time t alternates between "up" and "deteriorated" states. Denote by u1,d1,u2,d2, the successive sojourn times spent in states up and deteriorated, respectively, and suppose Y(0) is in up state. Assume u1,u2, are i.i.d. random variables with distribution function F(u) and d1,d2, are i.i.d. random variables with distribution function G(d). Denote by Z(t) and W(t) the total sojourn time spent in states up and deteriorated during the time interval (0,t). Clearly Z(t) and W(t) are random variables and Z(t)+W(t)=t. a) Let N(t) be the renewal process generated by u1u2,, i.e., N(t)=max{i:k=0iukt}. Define Argue that (t)=dl+d2++dN(t), P{W(t)x}=P{(tx)x}, and express this in terms of the distribution F and G. b) At time zero it is "on." It then serves before breakdown for a random time Ton with distribution function 1et. It is then off before being repaired for a random time Toff with the same distribution function 1et. It then repeats a statistically independent and identically distributed similar cycle, and so on. Determine the mean of Z(t), the random variable measuring the total time the system is operating during the interval (0,t). c) Assume that a job randomly arrives to the machine when it is on. While the machine is in off state it cannot work on the job, however, once it is repaired it resumes working on the same job until it is finished. Assume that Ton is exponentially distributed with parameter , and Toff is exponentially distributed with parameter . If the time to finish the job is also exponential with parameter , what is the distribution of the total time it takes to complete the job? Consider a stochastic process {Y(t)}t=0, representing the state of a machine at time t alternates between "up" and "deteriorated" states. Denote by u1,d1,u2,d2, the successive sojourn times spent in states up and deteriorated, respectively, and suppose Y(0) is in up state. Assume u1,u2, are i.i.d. random variables with distribution function F(u) and d1,d2, are i.i.d. random variables with distribution function G(d). Denote by Z(t) and W(t) the total sojourn time spent in states up and deteriorated during the time interval (0,t). Clearly Z(t) and W(t) are random variables and Z(t)+W(t)=t. a) Let N(t) be the renewal process generated by u1u2,, i.e., N(t)=max{i:k=0iukt}. Define Argue that (t)=dl+d2++dN(t), P{W(t)x}=P{(tx)x}, and express this in terms of the distribution F and G. b) At time zero it is "on." It then serves before breakdown for a random time Ton with distribution function 1et. It is then off before being repaired for a random time Toff with the same distribution function 1et. It then repeats a statistically independent and identically distributed similar cycle, and so on. Determine the mean of Z(t), the random variable measuring the total time the system is operating during the interval (0,t). c) Assume that a job randomly arrives to the machine when it is on. While the machine is in off state it cannot work on the job, however, once it is repaired it resumes working on the same job until it is finished. Assume that Ton is exponentially distributed with parameter , and Toff is exponentially distributed with parameter . If the time to finish the job is also exponential with parameter , what is the distribution of the total time it takes to complete the job