Stationary Characteristics of an Unreliable Single-Server Queueing System with Losses and Preventive Maintenance

Abstract

This paper considers an unreliable restorable single-server queueing system with losses in which server failures may occur during operation. The random variables describing the system have general distributions. For increasing the efficiency of this system, it is proposed to carry out preventive maintenance of the server as soon as the accumulated operating time exceeds an upper permissible threshold. A semi-Markov model of the system’s evolution over time is constructed. The explicit-form expressions for the final probabilities and mean sojourn times of the system in different physical states are derived using the stationary distribution of the embedded Markov chain. The frequency of preventive maintenance of the server is optimized via maximizing the average specific profit and minimizing the average specific costs of the system.

Appendix

Proof of Theorem 1. The system of equations

$$\rho (B)=\int_{E}\rho (dx)P(x,B)$$

for the stationary distribution ρ( ⋅ ) of the EMC $\left\{{S}_{n},\,n\ge 0\right\}$ with the transition probabilities P(x, B) has the form

$$\begin{array}{cl}\rho (0xuv)=\mathop{\int}\limits_{0}^{v}f(t){v}_{g}(t,x)\rho (1,t+u,v-t)dt+{\rho }_{1}f(v){v}_{g}(v,x)\varphi (v+u),\\ \rho (1uv)=\mathop{\int}\limits_{0}^{\infty }\rho (0xuv)dx,\\ \rho (3x)=\mathop{\int}\limits_{0}^{\tau }{v}_{g}(t,x)\overline{F}(t)dt\mathop{\int}\limits_{t}^{\infty }\rho (1u,\tau -t)du+{\rho }_{1}\overline{F}(\tau ){v}_{g}(\tau ,x)\overline{\Phi }(\tau ),\\ \rho (2x)=\mathop{\int}\limits_{0}^{\tau }{v}_{g}(t,x)\overline{F}(t)dt\mathop{\int}\limits_{t}^{\tau }\rho (1t,v-t)dv+{\rho }_{1}\mathop{\int}\limits_{0}^{\tau }\overline{F}(t){v}_{g}(t,x)\varphi (t)dt,\\ \rho (0x)=\mathop{\int}\limits_{0}^{\infty }{\psi }_{{\rm{pr}}}(t)\rho (3,t+x)dt+\mathop{\int}\limits_{0}^{\infty }{\psi }_{{\rm{pr}}}(t)dt\mathop{\int}\limits_{0}^{t}{v}_{g}(t-s,x)\rho (3s)ds\\ +\mathop{\int}\limits_{0}^{\infty }{\psi }_{{\rm{em}}}(t)\rho (2,t+x)dt+\mathop{\int}\limits_{0}^{\infty }{\psi }_{{\rm{em}}}(t)dt\mathop{\int}\limits_{0}^{t}{v}_{g}(t-s,x)\rho (2s)ds,\\ {\rho }_{1}=\mathop{\int}\limits_{0}^{\infty }\rho (0x)dx.\end{array}$$

Eliminate the function ρ(0xuv) from the first two equations of the system to obtain the following restoration equation for the density ρ(1uv) in the variable v:

$$\rho (1uv)={\rho }_{1}f(v)\varphi (u+v)+\mathop{\int}\limits_{0}^{v}f(t)\rho (1,t+u,v-t)dt.$$

The solution of this equation is ρ(1uv) = ρ₁h_f(v)φ(u + v). The expressions for the other density functions are derived using the properties of the density functions of the restoration function and the direct residual restoration time. The stationary probability ρ₁ is determined from the normalization condition. The proof of Theorem 1 is complete.

Proof of Theorem 2. The limiting values ${p}_{i}^{* }$ of the transition probabilities of the semi-Markov process S(t), t ≥ 0, are given by the relations [14, 15]

$${p}_{i}^{* }=\mathop{{\rm{lim}}}\limits_{t\to \infty }\Phi (t,x,{E}_{i})=\int_{{E}_{i}}m(x)\rho (dx){\left[\int_{E}m(x)\rho (dx)\right]}^{-1},\quad i=\overline{0,3}.$$

In view of the stationary distribution of the EMC (1), the mean sojourn times

$$\begin{array}{rcl}{\rm{M}}{\theta }_{1}&=&\mathop{\int}\limits_{0}^{\tau }\overline{F}(t)\overline{\Phi }(t)dt,\quad {\rm{M}}{\theta }_{1uv}=\mathop{\int}\limits_{0}^{u\wedge (\tau -v)}\overline{F}(t)dt,\\ {\rm{M}}{\theta }_{2x}&=&{\rm{M}}{\sigma }_{{\rm{em}}};\quad {\rm{M}}{\theta }_{3x}={\rm{M}}{\sigma }_{{\rm{pr}}};\quad M{\theta }_{oxuv}=M{\theta }_{0x}=x\end{array}$$

in different states of the system, and the identity Mβ_t = Mβ(1 + H_g(t)) − t, finally write

$$\begin{array}{cl}\int_{{E}_{1}}m(x)\rho (dx)={\rho }_{1}\mathop{\int}\limits_{0}^{\tau }\overline{F}(t)\overline{\Phi }(t)dt+{\rho }_{1}\mathop{\int}\limits_{0}^{\tau }{h}_{f}(v)dv\mathop{\int}\limits_{0}^{\infty }\varphi (u+v)du\mathop{\int}\limits_{0}^{u\wedge (\tau -v)}\overline{F}(t)dt\\ ={\rho }_{1}\mathop{\int}\limits_{0}^{\tau }\overline{F}(t)\overline{\Phi }(t)dt+{\rho }_{1}\mathop{\int}\limits_{0}^{\tau }\overline{\Phi }(s)ds\mathop{\int}\limits_{0}^{s}{h}_{f}(t)\overline{F}(s-t)dt={\rho }_{1}\mathop{\int}\limits_{0}^{\tau }\overline{\Phi }(t)dt={\rho }_{1}{\rm{M}}(\gamma \wedge \tau );\\ \int_{{E}_{2}}m(x)\rho (dx)={\rm{M}}{\sigma }_{{\rm{em}}}\mathop{\int}\limits_{0}^{\infty }\rho (2x)dx={\rho }_{1}{\rm{M}}{\sigma }_{{\rm{em}}}\Phi (\tau )={\rho }_{1}{\rm{M}}{\sigma }_{{\rm{em}}}P(\gamma <\tau );\\ \int_{{E}_{3}}m(x)\rho (dx)={\rm{M}}{\sigma }_{{\rm{pr}}}\mathop{\int}\limits_{0}^{\infty }\rho (3x)dx={\rho }_{1}{\rm{M}}{\sigma }_{{\rm{pr}}}\overline{\Phi }(\tau )={\rho }_{1}{\rm{M}}{\sigma }_{{\rm{pr}}}P(\gamma >\tau );\\ \int_{{E}_{0}}m(x)\rho (dx)={\rho }_{1}{\rm{M}}\beta \left[{\overline{N}}_{{\rm{ser}}}(\tau )+{\overline{N}}_{{\rm{los}}}^{{\rm{ser}}}(\tau )+{\overline{N}}_{{\rm{los}}}^{p}(\tau )+{\overline{N}}_{{\rm{los}}}^{a}(\tau )\right]\\ -{\rho }_{1}\mathop{\int}\limits_{0}^{\tau }\overline{\Phi }(t)dt-{\rho }_{1}\overline{\Phi }(\tau ){\rm{M}}{\sigma }_{{\rm{pr}}}-{\rho }_{1}\Phi (\tau ){\rm{M}}{\sigma }_{{\rm{em}}};\\ \int_{E}m(x)\rho (dx)={\rho }_{1}{\rm{M}}\beta \left[{\overline{N}}_{{\rm{ser}}}(\tau )+{\overline{N}}_{{\rm{los}}}^{{\rm{ser}}}(\tau )+{\overline{N}}_{{\rm{los}}}^{p}(\tau )+{\overline{N}}_{{\rm{los}}}^{a}(\tau )\right].\end{array}$$

The proof of Theorem 2 is complete.

Proof of Theorem 3. The mean sojourn times T(E_i) in the states E_i are given by the relations [14, 15]

$$T({E}_{i})=\int_{{E}_{i}}m(x)\rho (dx){\left[\int_{E\backslash {E}_{i}}\rho (dx)P(x,{E}_{i})\right]}^{-1},\quad i=\overline{0,3}.$$

(A.1)

Taking into account the transition probabilities and the stationary distribution of the EMC (1), the integrals in the denominators of (A.1) can be transformed to

$$\begin{array}{cl}\int_{E\ \backslash {E}_{1}}\rho (dx)P(x,{E}_{1})=\mathop{\int}\limits_{0}^{\infty }\rho (0x)dx+\mathop{\int}\limits_{0}^{\tau }dv\mathop{\int}\limits_{0}^{\infty }du\mathop{\int}\limits_{0}^{\infty }\rho (0xuv)dx\\ ={\rho }_{1}+{\rho }_{1}\mathop{\int}\limits_{0}^{\tau }{h}_{f}(x)\ \overline{\Phi }(x)\ dx={\rho }_{1}\left[1+{\overline{N}}_{{\rm{ser}}}(\tau )\right];\\ \int_{E\ \backslash {E}_{2}}\rho (dx)P(x,{E}_{2})=\int_{{E}_{2}}\rho (dx)P(x,E\ \backslash {E}_{2})=\mathop{\int}\limits_{0}^{\infty }\rho (2x)dx={\rho }_{1}\Phi (\tau );\\ \int_{E\ \backslash {E}_{3}}\rho (dx)P(x,{E}_{3})=\int_{{E}_{3}}\rho (dx)P(x,E\ \backslash {E}_{3})=\mathop{\int}\limits_{0}^{\infty }\rho (3x)dx={\rho }_{1}\overline{\Phi }(\tau );\\ \int_{E\ \backslash {E}_{0}}\rho (dx)P(x,{E}_{0})=\int_{{E}_{0}}\rho (dx)P(x,E\ \backslash {E}_{0})\\ =\mathop{\int}\limits_{0}^{\infty }\rho (0x)dx+\mathop{\int}\limits_{0}^{\tau }dv\mathop{\int}\limits_{0}^{\infty }du\mathop{\int}\limits_{0}^{\infty }\rho (0xuv)dx={\rho }_{1}\left[1+{\overline{N}}_{{\rm{ser}}}(\tau )\right].\end{array}$$

The proof of Theorem 3 is complete.