Question

Hello, I need help for this problem.

A system is composed of N identical components. Each independently operates a random length of time until failure. This failure time is exponential with rate λ. When a component fails, it undergoes repair. The repair time is random, exponential with rate µ. The system is said to be in state n at time t if there are exactly n components under repair at time t. This process is a birth and death process.

a- Determine its parameters.

Suppose initially all N components are operative.

b- Find the Laplace transform of the first time that there are two inoperative components.

Hint: Recall the notation τi time to stay in state i. Define T02 and T12 be the first time that there are two inoperative components with X0 = 0 and X0 = 1, respectively. Here Xt is the CTMC which denotes the state of the system at time t. Find their relations.

Thank you for your time.

Answer 1

he depaity function for Solution GiNen that Here the Failure ime is exponential toith rate 2 GANen that if there aTe exactlyo components at Dme ti It meang that if one component is fail,then one component is decreaseal from nsomponen ets Hence the para meter for the death process is p Similarly afailedl com ponent undergoes repair and repaired component is again seplaceal in the $yafem, Hence again one component is inereased io the gystem PoT O.components the paramet er is Hence o Ore sepe nespective paTa met rere for birth t death pTocess Prcbability that there 2 inoperative componente For death process initially Ke 0 & Xa = is CT1C obicb denotes the gtate of X Ahe 9ystem at time t. NBinomial (ne).

PRI C0) 1 Phit) P[N(H)E The differential equations are PRlt)ENA Rt) (nt1) Pox (t) For ( Put 7i in the above equation we get N=2 Plt)-2 Pe (t)+3 P(t)