Availability of 1-for-2 shared protection systems with general repair-time distributions

Abstract

In this paper, we investigate the availability of 1-for-2 shared protection systems. We assume that there are 2 working units each serving a single user and one shared protection (spare) unit in the system. We also assume that the time to failure is subject to an exponential distribution and that the time to repair is subject to a general distribution. Under these assumptions, we derive the availability for each user by combining the state transition analysis and the supplementary variable method. We also show the effect of the repair time distribution, the failure rates and the repair rates of the units through the case study of small-sized 2 enterprises that share one spare device for backup.

Appendices

Appendix 1

Equation (3) is derived from the following expression:

$$\begin{aligned}&p_0 \left( {t+\Delta t} \right) =p_0 \left( t \right) \left\{ {1-3\lambda \Delta t} \right\} \nonumber \\&\quad +\int _{0}^{t} {p_1 \left( {t,x} \right) \left( {1-2\lambda \Delta t} \right) \frac{B\left( {x+\Delta t} \right) -B\left( x \right) }{{\overline{B}} \left( x \right) }dx}. \end{aligned}$$

(36)

In the above equation, the left hand side means the system is in state $0$ at time $t+\Delta t$ , while the right hand side means the sum of two events i.e. (1) the system stays in state $0$ during the interval $(t,t+\Delta t)$, (2) the system moves from state $1$ to state $0$ as a broken unit gets repaired at some point in the interval $(t,t+\Delta t)$.

Expression (6) is derived from the following expression and (4):

$$\begin{aligned} p_1 \left( {t+\Delta t,x+\Delta t} \right) \!=\!p_1 \left( {t,x} \right) \left( {1\!-\!2\lambda \Delta t} \right) \frac{1-B\left( {x+\Delta t} \right) }{{\overline{B}} \left( x \right) }.\nonumber \\ \end{aligned}$$

(37)

In the above equation, the left hand side means the user stays in state $1$ during the interval $(t,t+\Delta t)$, while the right hand side means there is neither failure nor repair during the same interval. By substituting (4) to (37), we obtain the following (38).

$$\begin{aligned} \begin{array}{l} \exp \left\{ {-2\lambda \left( {x+\Delta t} \right) } \right\} g_1 \left( {t+\Delta t,x+\Delta t} \right) {\overline{B}} \left( x \right) \\ \quad =\exp \left\{ {-2\lambda x} \right\} g_1 \left( {t,x} \right) {\overline{B}} \left( x \right) \left( {1-2\lambda \Delta t} \right) \frac{1-B\left( {x+\Delta t} \right) }{{\overline{B}} \left( x \right) } \\ \end{array} \end{aligned}$$

(38)

(38) is equivalent to the following (39) and (6):

$$\begin{aligned} g_1 \left( {t+\Delta t,x+\Delta t} \right)&= g_1 \left( {t,x} \right) +\frac{\partial g_1 }{\partial t}\Delta t\nonumber \\&+\frac{\partial g_1 }{\partial x}\Delta t=g_1 \left( {t,x} \right) \end{aligned}$$

(39)

Equation (10) is derived from the following expression and (4), (8):

$$\begin{aligned} p_2 \left( {t+\Delta t,x+\Delta t} \right)&= p_2 \left( {t,x} \right) \left( {1-\lambda \Delta t} \right) \frac{1-B\left( {x+\Delta t} \right) }{{\overline{B}} \left( x \right) }\nonumber \\&+p_1 \left( {t,x} \right) \frac{1-B\left( {x+\Delta t} \right) }{{\overline{B}} \left( x \right) }2\lambda \Delta t.\nonumber \\ \end{aligned}$$

(40)

In the above equation, the left hand side means the user stays in state $2$ during the interval $(t,t+\Delta t)$, while the right hand side means the sum of two events i.e. (1) there is neither failure nor repair during the same interval, (2) the system moves from state $1$ to state $2$ as a unit was broken at some point in the interval $(t,t+\Delta t)$.By substituting (4) and (8) to (40) and letting $\Delta t\rightarrow 0$, we obtain (10).

Equation (11) is derived from the following expression and (8), (9):

$$\begin{aligned} p_3 \left( {t+\Delta t,x+\Delta t} \right)&= p_3 \left( {t,x} \right) \frac{1-B\left( {x+\Delta t} \right) }{{\overline{B}} \left( x \right) }\nonumber \\&+p_2 \left( {t,x} \right) \frac{1-B\left( {x+\Delta t} \right) }{{\overline{B}} \left( x \right) }\lambda \Delta t.\nonumber \\ \end{aligned}$$

(41)

In the above equation, the left hand side means the system stays in state $3$ during the interval $(t,t+\Delta t)$, while the right hand side means the sum of two events i.e. (1) there is no repair during the same interval, (2) the system moves from state $2$ to state $3$ as a unit is broken at some point in the interval $(t,t+\Delta t)$.By substituting (8) and (9) to (41) and letting $\Delta t\rightarrow 0$, we obtain (11).

Equations (14) through (16) are derived from the following expressions and (4), (8), (9), (12), (13):

$$\begin{aligned}&p_1 \left( {t+\Delta t,0} \right) \Delta t=p_0 \left( t \right) 3\lambda \Delta t\nonumber \\&\quad +\int _{0}^{t} {p_2 } \left( {t,x} \right) \left( {1-\lambda \Delta t} \right) \frac{B\left( {x+\Delta t} \right) -B\left( x \right) }{{\overline{B}} \left( x \right) }dx\end{aligned}$$

(42)

$$\begin{aligned}&p_2 \left( {t+\Delta t,0} \right) \Delta t=\int _{0}^{t} {p_3 } \left( {t,x} \right) \frac{B\left( {x+\Delta t} \right) -B\left( x \right) }{{\overline{B}} \left( x \right) }dx,\nonumber \\ \end{aligned}$$

(43)

and

$$\begin{aligned} p_3 \left( {t+\Delta t,0} \right) \Delta t=0. \end{aligned}$$

(44)

In (42), the left hand side means the system arrived in state $1$ just at the time $t+\Delta t$ , while the right hand side means the sum of two events i.e. (1) the system moves from state $0$ to state $1$ by a failure of a unit within $(t,t+\Delta t)$, (2) the system moves from state $2$ to state $1$ as a broken unit gets repaired at some point in the interval $(t,t+\Delta t)$.By substituting (8) to (42) and letting $\Delta t\rightarrow 0$, we obtain (14).

In (43), the left hand side means the system arrived in state $2$ just at the time $t+\Delta t$ , while the right hand side means the system moves from state $3$ to state $2$ as a broken unit gets repaired at some point in the interval $(t,t+\Delta t)$. By substituting (9) to (43) and letting $\Delta t\rightarrow 0$, we obtain (15).

(44) means the system is never in state $3$ without the elapsed repair time. By letting $\Delta t\rightarrow 0$, we obtain (16).

Appendix 2

$A_{1:2}$ (Expression (24) for the case of the Erlang type-$k$ repair-time distribution (25)) is expressed as follows:

$$\begin{aligned} A_{1:2}&=\frac{\left( {k\mu } \right) ^{k}}{\Delta _{Ek} }\left[ \left( {k\mu } \right) ^{2k}+\frac{3}{2}\left( {\lambda \!+\!k\mu } \right) ^{k}\left\{ {\left( {2\lambda \!+\!k\mu } \right) ^{k}\!+\!\left( {k\mu } \right) ^{k}} \right\} \right. \nonumber \\&\quad \ \left. \times \left\{ {1-\left( {\frac{k\mu }{\lambda +k\mu }} \right) ^{k}} \right\} \right] , \end{aligned}$$

(45a)

where

$$\begin{aligned} \Delta _{Ek}&= 3k\lambda \left\{ {\left( {2\lambda +k\mu } \right) ^{k}+\left( {k\mu } \right) ^{k}} \right\} \\&\left[ \left( {k\mu } \right) ^{k-1}\left( {\lambda +2k\mu } \right) \left( {\lambda +k\mu } \right) ^{k-1}\right. \nonumber \\&\left. +\left( {k\mu } \right) ^{k}\left\{ {\left( {k\mu } \right) ^{k-1}-\left( {\lambda +k\mu } \right) ^{k-1}} \right\} \right] \\&+\left( {k\mu } \right) ^{2k}\!\left\{ \! {\left( {2\lambda +k\mu } \right) ^{k}\!+\!3\lambda k\left( {2\lambda \!+\!k\mu } \right) ^{k-1}\!+\!2\left( {k\mu } \right) ^{k}} \!\right\} \\&-\left( {k\mu } \right) ^{2k}\left\{ {\left( {2\lambda +k\mu } \right) ^{k}+\left( {k\mu } \right) ^{k}} \right\} \\&-3k\lambda \left( {k\mu } \right) ^{2k-1}\left\{ k\mu \left( {2\lambda +k\mu } \right) ^{k-1}\right. \\&\left. +2\left( {2\lambda +k\mu } \right) ^{k}+\left( {k\mu } \right) ^{k} \right\} . \end{aligned}$$

If $k$ is equal to one, we obtain the following availability in the case of the exponential distribution of repair time.

$$\begin{aligned} A_{1:2} =\frac{\mu ^{3}+3\lambda \mu ^{2}+3\lambda \mu ^{2}}{6\lambda ^{3}+6\lambda ^{2}\mu +3\lambda \mu ^{2}+\mu ^{3}} \end{aligned}$$

(45b)

$A_{1:2}$ (Expression (24) for the case of the fixed repair-time (27))is expressed as follows:

$$\begin{aligned} A_{1:2}&= \frac{1}{\Delta _F }\left[ \hbox {exp}\left( {-\frac{3\lambda }{\mu }} \right) +\frac{3}{2}\left\{ {1+\hbox {exp}\left( {-\frac{2\lambda }{\mu }} \right) } \right\} \right. \nonumber \\&\times \left. \left\{ {1-\hbox {exp}\left( {-\frac{\lambda }{\mu }} \right) } \right\} \right] , \end{aligned}$$

(45)

where

$$\begin{aligned} \Delta _F \!=\!\hbox {exp}\left( \! {-\frac{3\lambda }{\mu }} \right) +\frac{3\lambda }{\mu }\left\{ {1+\hbox {exp}\left( \! {-\frac{2\lambda }{\mu }} \right) -\hbox {exp}\left( {-\frac{\lambda }{\mu }} \right) } \right\} . \end{aligned}$$

$A_{1:2}$ (Expression (24) for the case of the sharp-peaked and long-tailed repair-time distribution (29) ($k=1))$ is expressed as follows:

$$\begin{aligned} A_{1:2}\! =\!\frac{1}{\Delta _{F+E1} }\!\left[ \!\! {\begin{array}{l} \left( {a\mu } \right) ^{3}\exp \!\left( \! {-\frac{3\lambda }{a\mu }} \!\right) \!+\! \frac{3}{2}\left( {a\mu } \right) \left\{ {\left( {a\!-\!1} \right) \lambda \!+\!a\mu } \right\} \\ \quad \times \left\{ {2\left( {a-1} \right) \lambda +a\mu +a\mu \exp \left( {-\frac{2\lambda }{a\mu }} \right) } \right\} \\ \quad \times \left\{ {1-\exp \left( {-\frac{\lambda }{a\mu }} \right) \frac{a\mu }{\left( {a-1} \right) \lambda +a\mu }} \right\} \\ \end{array}} \!\!\right] \!,\nonumber \\ \end{aligned}$$

(46)

where

$$\begin{aligned}&\Delta _{F+E1} =3\lambda \left\{ {2\left( {a-1} \right) \lambda +a\mu +a\mu \exp \left( {-\frac{2\lambda }{a\mu }} \right) } \right\} \\&\quad \times \left\{ {a\left( {a-1} \right) \lambda +a^{2}\mu +a\left( {a-2} \right) \mu \exp \left( {-\frac{\lambda }{a\mu }} \right) } \right\} \\&\quad +\left( {a\mu } \right) ^{2}\exp \left( {-\frac{\lambda }{a\mu }} \right) \left\{ {3\left( {a-1} \right) \lambda -\left( {3\lambda -a\mu } \right) \exp \left( {-\frac{2\lambda }{a\mu }} \right) } \right\} \\&\quad +3a\lambda \mu \exp \left( {-\frac{\lambda }{a\mu }} \right) \left\{ {2\left( {a-1} \right) \lambda +a\mu +a\mu \exp \left( {-\frac{2\lambda }{a\mu }} \right) } \right\} \\&\quad -3a\lambda \mu \exp \left( {-\frac{\lambda }{a\mu }} \right) \left[ 4\left( {a-1} \right) \left\{ {\left( {a-1} \right) \lambda +a\mu } \right\} \right. \\&\quad \left. +a\left( {a-2} \right) \mu \exp \left( {-\frac{2\lambda }{a\mu }} \right) +2\left( {a-1} \right) \lambda -a\left( {a-2} \right) \mu \right] . \end{aligned}$$

$A_{1:2}$ (Expression (24) for the case of the sharp-peaked and long-tailed repair-time distribution (29) ($k=2))$ is expressed as follows:

$$\begin{aligned} A_{1:2} \!=\!\frac{1}{\Delta _{F+E2} }\!\left[ \!\! {\begin{array}{l} 4\left( {a\mu } \right) ^{6}\exp \!\left( \! {-\frac{3\lambda }{a\mu }} \!\right) \!+\!\frac{3}{2}\left( {a\mu } \right) ^{2}\left\{ {\left( {a\!-\!1} \right) \lambda \!+\!2a\mu } \right\} ^{2} \\ \quad \times \left[ {\left\{ {\left( {a\!-\!1} \right) \lambda \!+\!a\mu } \right\} ^{2}\!+\!\left( {a\mu } \right) ^{2}\exp \left( {-\frac{2\lambda }{a\mu }} \right) } \right] \\ \quad \times \left[ {1-\exp \left( {-\frac{\lambda }{a\mu }} \right) \left\{ {\frac{2a\mu }{\left( {a-1} \right) \lambda +2a\mu }} \right\} ^{2}} \right] \\ \end{array}} \!\!\right] \!,\nonumber \\ \end{aligned}$$

(47)

where

$$\begin{aligned}&\Delta _{F+E2} =3\lambda \left[ a^{2}\mu \left\{ {\left( {a-1} \right) \lambda +2a\mu } \right\} ^{2}+4\left( {a-2} \right) \left( {a\mu } \right) ^{3}\right. \\&\left. \quad \times \exp \left( {-\frac{\lambda }{a\mu }} \right) \right] \times \left[ {\left\{ {\left( {a-1} \right) \lambda +a\mu } \right\} ^{2}\!+\!\left( {a\mu } \right) ^{2}\exp \left( {-\frac{2\lambda }{a\mu }} \right) } \right] \\&\quad +4\left( {a\mu } \right) ^{4}\exp \left( {-\frac{\lambda }{a\mu }} \right) \Big [ \left\{ {\left( {a-1} \right) \lambda +a\mu } \right\} \left\{ {4\left( {a-1} \right) \lambda +a\mu } \right\} \\&\quad +a\mu \exp \left( {-\frac{2\lambda }{a\mu }} \right) \left( {2a\mu -3\lambda } \right) \Big ] \\&\quad +4\left( {a\mu } \right) ^{3}\left( {3\lambda -a\mu } \right) \exp \left( {-\frac{\lambda }{a\mu }} \right) \left[ \left\{ {\left( {a-1} \right) \lambda +a\mu } \right\} ^{2}\right. \\&\quad \left. +\left( {a\mu } \right) ^{2}\exp \left( {-\frac{2\lambda }{a\mu }} \right) \right] -12\lambda \left( {a\mu } \right) ^{3}\exp \left( {-\frac{\lambda }{a\mu }} \right) \\&\qquad \times \left[ {\begin{array}{l} 2\left( {a-1} \right) \left\{ {\left( {a-1} \right) \lambda +a\mu } \right\} \left\{ {\left( {a-1} \right) \lambda +2a\mu } \right\} \\ \quad +\left( {a-2} \right) \left( {a\mu } \right) ^{2}\exp \left( {-\frac{2\lambda }{a\mu }} \right) +\left\{ {\left( {a-1} \right) \lambda +a\mu } \right\} \\ \quad \left\{ {\left( {a-1} \right) \lambda -a\left( {a-2} \right) \mu } \right\} \\ \end{array}} \right] . \end{aligned}$$