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Iran Journal of Computer Science

, Volume 1, Issue 3, pp 147–153 | Cite as

Profit analysis of a computer system with preventive maintenance and priority subject to maximum operation and repair times

  • Ashish Kumar
  • Monika Saini
Original Article
  • 249 Downloads

Abstract

The aim of the present study is to carry out the profit analysis of a computer system by considering the concept of priority to preventive maintenance over different hardware and software repair activities. For this purpose, three stochastic models have been developed under different priority policies. A single repair facility is available to perform all repair activities. The hardware unit undergoes replacement after a pre-specific maximum repair time. The repairman performs the preventive maintenance of the whole system after a pre-fixed maximum operation time. All time-dependent failure rates follow exponential distribution, while the repair rates follow arbitrary distribution. Switch devices, repairs and preventive maintenance are perfect. All random variables are statistically independent. Semi-Markov process and regenerative point technique were used to derive the necessary system effectiveness measure to evaluate the profit function of all the three models. To highlight the importance of the study, graphs for profit difference between models have been depicted for a particular set of values of various parameters with respect to preventive maintenance.

Keywords

Computer system Profit function Semi-Markov process Regenerative point technique Priority 

1 Introduction

The outstanding progress in the field of computer technology has resulted in the widespread usage of computer applications in almost all academic, medical, manufacturing, business and industrial sectors. Technology demands high-performance hardware and high-quality software for making improvements and breakthrough. The size and complexity of computer-intensive systems has grown dramatically during the last few decades. The demand for integrated hardware and software systems has increased rapidly than the ability to design, implement, test and maintain them. When the requirements for and dependencies on computers increase, the possibility of their failures also increases. The impact of these failures ranges from economic loss to human causalties. Therefore, it becomes necessary to operate such systems with high importance and reliability. Thus, engineers and scientists stress on the development of reliable computer systems considering various operational and design policies. Several researchers have time to time tried to design the reliability models for computer systems under various sets of assumptions.

By considering the above facts in mind, in the present study an effort has been made to analyze the effect of priority to preventive maintenance over different hardware and software repairs and upgradation activities on the profit function of computer systems. For this purpose, three stochastic models have been developed using the concepts of maximum repair time, maximum operation time, preventive maintenance and priority. In the first model, no provision is made, while in the second and third models priority to preventive maintenance over software upgradation and h/w repair activities has been given, respectively. A single repair facility is available with a system which performs all repair activities such as software upgradation, preventive maintenance, hardware repair and hardware replacement. The unit undergoes replacement after maximum operation time and under preventive maintenance after maximum operation time. All time-dependent failure rates follow exponential distribution, while repair rates follow arbitrary distribution. Switch, repairs and preventive maintenance are perfect. All random variables are statistically independent. Semi-Markov process and regenerative point technique have been used to derive the necessary system effectiveness measure to evaluate the profit function of all the three measures. To highlight the importance of the study, graphs for profit and profit difference between models have been depicted for a particular set of values of various parameters with respect to preventive maintenance.

The manuscript is organized into the following sections. The present section is introductory in nature. Section 2 is concerned with the literature review. Notations, models description and transition probabilities are given in Sect. 3. Development of recurrence relations and analysis of results are given in Sect. 4. Section 5 deals with the profit analysis, along with the graphical results. Finally, the comparative analysis of results have been appended in Sect. 6.

2 Literature review

Friedman and Tran [1] developed some reliability techniques for combined hardware/software systems. Welke et al. [2] tried to develop reliability models for hardware/software systems. Lai et al. [3] established a stochastic model for availability analysis of distributed software/hardware systems. The work carried out by these authors has been limited to the consideration of either hardware or software components. First of all, Malik and Anand [4] studied a computer system considered as a single entity with independent hardware and software failures. Koutras and Platis [5] formulated a semi-Markov performance model of a redundant system with partial, full and failed rejuvenation. Jain et al. [6] carried out the availability analysis of the software–hardware system with common cause shock failure, spare and switching failure. The concept of maximum operation and repair time in the field of computer systems reliability was introduced for the first time by Kumar et al. [7]. Kumar and Malik [8] developed a stochastic model for a computer system with priority to PM over software replacement. Kumar and Malik [9] carried out the cost–benefit analysis of a computer system with maximum operation and repair time. In the paper, priority to software replacement over hardware repair activities has been given. Malik and Munday [10] designed a stochastic model of computer systems with hardware redundancy. Kumar et al. [11] carried out the performance analysis of a computer system with imperfect fault detection of hardware components. Jain and Preeti [12] evaluated the availability of software rejuvenation in an active/standby cluster system. Kumar and Saini [13] analyzed comparatively various reliability measures of a computer system under the concept of priority for preventive maintenance over hardware repair. Kumar et al. [14] studied the profit function of a computing machine with priority and s/w rejuvenation. Kumar et al. [15] probabilistically analyzed various performance measures of a redundant system using Weibull failure and repair laws, along with the concept of preventive maintenance. But, the effect of priority to maintenance policies over repair activities has not been extensively studied in detail and comparatively for profit analysis of computer systems. So, in the present study an effort has been made in this direction and, for comparative analysis, graphs of profit function have been depicted. The necessary data for profit analysis were collected with the help of IT personnel of a private university.

3 Notations, model state description and transition probabilities

3.1 Notations

The following notations have been used in the present study:

\(N_{o}/ Cs\)

Denotes operative/cold standby units

a / b

Denotes probability of hardware/software failure

\(\lambda _{1}\)/\(\lambda _{2}\)/ \(\alpha _0 / \beta _0 \)

Constant rate of hardware failure/software failure/ maximum operation time/maximum repair time

\(h(t)/g(t)/ m(t)/f(t)\)

Probability density function of upgradation time of the software/repair/replacement of hardware/preventive maintenance of system

Pm/WPm

Denotes unit under preventive maintenance/waiting for preventive Maintenance

PM/WPM

Denotes unit continuously under preventive maintenance/waiting for preventive Maintenance from previous state

HFur/HFurp /HFwr

Denotes failed hardware unit under repair/under replacement/waiting for repair

HFUR/HFURP/HFWR

Denotes failed hardware unit continuously under repair/under replacement/waiting for repair from preceding state

SFurp/SFwrp

Denotes failed software unit under upgradation/waiting for upgradation

SFURP/SFWRP

Denotes failed software unit continuously under upgradation/waiting for upgradation from previous state

©/\({\circledS }\)

Laplace convolution/Laplace Stieltjes convolution

\(K_{0}\)

Revenue generated by system per unit up time

\(K_i, i=1,\ldots 7\)

Expenditure per unit time during repairman is engaged in doing preventive maintenance, hardware repair, software upgradation, hardware replacement, per unit hardware replacement, software upgradation and visits by server

LT/LST

Laplace transformation/Laplace Stieltjes transformation

\(q_{ij}(t)/ Q_{ij}(t)\)

pdf /cdf of passage time from regenerative state i to a regenerative state j or to a failed state j without visiting any other regenerative state in (0, t]

\(q_{ij.kr} (t)/Q_{ij.kr}(t)\)

pdf/cdf of direct transition time from regenerative state i to a regenerative state j or to a failed state j visiting state k, r once in (0, t]

pdf/cdf

Probability density function/cumulative density function

3.2 Model state description

In this subsection, three stochastic models have been developed for a two-unit cold standby system by considering the computer system as a single unit. The computer system comprises hardware and software components which fail independently. The unit undergoes preventive maintenance after a maximum operation time and failed hardware is replaced by the new one after a maximum repair time. Initially, one unit is operative and other is taken as a cold standby. The state description of all models is as follows:

Model I total states

Model II total states

Model III total states

\(S_{0}\) = (N\(_{o}\), Cs),

\(S_{0}\) = (N\(_{o}\), Cs),

\(S_{0}\) = (N\(_{o}\), Cs),

\(S_{1}\) = (N\(_{o}\), Pm),

\(S_{1}\) = (N\(_{o}\), Pm),

\(S_{1}\) = (N\(_{o}\), Pm),

\(S_{2}\) = (N\(_{o}\), HFur),

\(S_{2}\) = (N\(_{o}\), HFur),

\(S_{2}\) = (N\(_{o}\), HFur),

\(S_{3}\) = (N\(_{o}\),SFurp),

\(S_{3}\) = (N\(_{o}\),SFurp),

\(S_{3}\) = (N\(_{o}\),SFurp),

\(S_{4}\) = (N\(_{o}\),HFurp),

\(S_{4}\) = (N\(_{o}\),HFurp),

\(S_{4}\) = (N\(_{o}\),HFurp),

\(S_{5}\) = (HFUR,Wpm),

\(S_{5}\) = (HFUR,Wpm),

\(S_{5}\) = (HFUR,Wpm),

\(S_{6}\) = (HFwr, PM),

\(S_{6}\) = (HFwr, PM),

\(S_{6}\) = (HFwr, PM),

\(S_{7}\) = (SFURP, HFwr),

\(S_{7}\) = (SFURP, HFwr),

\(S_{7}\) = (SFURP, HFwr),

\(S_{8}\) = (PM, SFwrp),

\(S_{8}\) = (PM, SFwrp),

\(S_{8}\) = (PM, SFwrp),

\(S_{9}\) = (SFURP, WPm),

\(S_{9}\) = (SFwrp, Pm),

\(S_{9}\) = (SFURP, WPm),

\(S_{10}\) = (SFURP, SFwrp),

\(S_{10}\) = (SFURP, SFwrp),

\(S_{10}\) = (SFURP, SFwrp),

\(S_{11}\) = (HFUR,SFwrp)

\(S_{11}\) = (HFUR,SFwrp)

\(S_{11}\) = (HFUR,SFwrp)

\(S_{12}\) = (HFUR, HFwr)

\(S_{12}\) = (HFUR, HFwr)

\(S_{12}\) = (HFUR, HFwr)

\(S_{13}\) = (WPm, PM),

\(S_{13}\) = (WPm, PM),

\(S_{13}\) = (WPm, PM),

\(S_{14}\) = (SFWRP,HFurp),

\(S_{14}\) = (SFWRP,HFurp),

\(S_{14}\) = (SFWRP,HFurp),

\(S_{15}\) = (HFurp, HFWR),

\(S_{15}\) = (HFurp, HFWR),

\(S_{15}\) = (HFurp, HFWR),

\(S_{16}\) = (HFurp,WPM),

\(S_{16}\) = (HFurp,WPM),

\(S_{16}\) = (HFWRP,Pm),

\(S_{17}\) = (HFURP, Wpm)

\(S_{17}\) = (HFURP, Wpm)

\(S_{17}\) = (HFURP, SFwrp)

\(S_{18}\) = (HFURP, SFwrp)

\(S_{18}\) = (HFURP, SFwrp)

\(S_{18}\) = (HFURP, HFwr)

\(S_{19}\) = (HFURP, HFwr)

\(S_{19}\) = (HFURP, HFwr)

 

Operative and regenerative states

Operative and regenerative states

Operative and regenerative states

\(S_{0}\) = (N\(_{o}\), Cs),

\(S_{0}\) = (N\(_{o}\), Cs),

\(S_{0}\) = (N\(_{o}\), Cs),

\(S_{1}\) = (N\(_{o}\), Pm),

\(S_{1}\) = (N\(_{o}\), Pm),

\(S_{1}\) = (N\(_{o}\), Pm),

\(S_{2}\) = (N\(_{o}\), HFur),

\(S_{2}\) = (N\(_{o}\), HFur),

\(S_{2}\) = (N\(_{o}\), HFur),

\(S_{3}\) = (N\(_{o}\),SFurp),

\(S_{3}\) = (N\(_{o}\),SFurp),

\(S_{3}\) = (N\(_{o}\),SFurp),

\(S_{4}\) = (N\(_{o}\),HFurp),

\(S_{4}\) = (N\(_{o}\),HFurp),

\(S_{16}\) = (HFWRP,Pm)

 

\(S_{9}\) = (SFwrp, Pm)

\(S_{4}\) = (N\(_{o}\),HFurp),

  

\(S_{5}\) = (HFUR,Wpm),

3.3 Transition probabilities and mean sojourn times

By considering simple probabilistic arguments, the below mentioned process given in Eq. (1) yields the transition probabilities at all stages of model I.
$$\begin{aligned} \varepsilon _{01}= & {} \int \limits _0^\infty \alpha _0 e^{-\alpha _0 t}e^{-a\lambda _1 t}e^{-b\lambda _2 t}e^{-st}\mathrm{d}t\nonumber \\= & {} \alpha _0 \int \limits _0^\infty {e^{-(a\lambda _1 +b\lambda _2 +\alpha _0 +s)t}\mathrm{d}t}\nonumber \\= & {} \frac{\alpha _0 }{a\lambda _1 +b\lambda _2 +\alpha _0 +s}. \end{aligned}$$
(1)
Taking the limit \(s\rightarrow 0\) in Eq. (1), we get
$$\begin{aligned} p_{01}= & {} \lim \frac{\alpha _0 }{a\lambda _1 +b\lambda _2 +\alpha _0 +s}\\= & {} \frac{\alpha _0 }{a\lambda _1 +b\lambda _2 +\alpha _0 }. \end{aligned}$$
The transition probabilities and mean sojourn times for model II and model III are obtained by applying the same procedure.
For Model I
$$\begin{aligned} p_{01}= & {} \frac{ \alpha _0 }{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 },\nonumber \\ p_{02}= & {} \frac{ {a\lambda }_1 }{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 },\nonumber \\ p_{03}= & {} \frac{ {b\lambda }_2 }{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 },\nonumber \\ p_{10}= & {} f^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0}), \nonumber \\ p_{16}= & {} \frac{ {a\lambda }_1 }{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 }[ 1- f^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0})]\nonumber \\= & {} p_{12.6},\nonumber \\ p_{18}= & {} \frac{ {b\lambda }_2 }{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 }[ 1- f^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0})]\nonumber \\= & {} p_{13.8},\nonumber \\ p_{20}= & {} g^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0}),\nonumber \\ p_{1.13}= & {} \frac{ \alpha _0 }{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 }[ 1- f^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0})]\nonumber \\= & {} p_{11.13}, \nonumber \\ p_{30}= & {} h^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0}),\nonumber \\ p_{24}= & {} \frac{ \beta _0 }{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 + \beta _0 }[ 1- g^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0}\nonumber \\&\quad +\beta _{0})], \nonumber \\ p_{40}= & {} m^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0}),\nonumber \\ p_{25}= & {} \frac{ \alpha _0 }{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 + \beta _0 }[ 1- g^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0}\nonumber \\&\quad +\beta _{0})], \nonumber \\ p_{51}= & {} g^{*}(\beta _{0}),p_{5,16}=1- g^{*}(\beta _{0}),\nonumber \end{aligned}$$
$$\begin{aligned} p_{2.11}= & {} \frac{ {b\lambda }_2 }{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 + \beta _0 }[ 1- g^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0}\nonumber \\&\quad +\beta _{0})], \nonumber \\ p_{62}= & {} f^{*}(0), p_{72}=h^{*}(0),\nonumber \\ p_{2.12}= & {} \frac{ {a\lambda }_1 }{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 + \beta _0 }[ 1- g^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0}\nonumber \\&\quad +\beta _{0})], \nonumber \\ p_{83}= & {} f ^{*}(0), p_{91}=h ^{*}(0),\nonumber \\ p_{37}= & {} \frac{ {a\lambda }_1 }{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 }[ 1- h^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0})]\nonumber \\= & {} p_{32.7},\nonumber \\ p_{10.3}= & {} h^{*}(0),p_{11.3}=g^{*}(\beta _{0}),\nonumber \\ p_{39}= & {} \frac{ \alpha _0 }{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 }[ 1- h^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0})],\nonumber \\= & {} p_{31.9}\nonumber \\ p_{11.14}= & {} 1- g^{*}(\beta _{0}),p_{12.2}=g^{*}(\beta _{0}),\nonumber \\ p_{3,10}= & {} \frac{ {b\lambda }_2 }{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 }[ 1- h^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0})]\nonumber \\= & {} p_{33.10},\nonumber \\ p_{12.15}= & {} 1- g^{*}(\beta _{0}),p_{13.1}=f^{*}(0),\nonumber \\ p_{4.17}= & {} \frac{ \alpha _0 }{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 }[ 1- m^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0})]\nonumber \\= & {} p_{41.17,} \nonumber \\ p_{14.3}= & {} m^{*}(0),p_{15.2}=m^{*}(0),\nonumber \\ p_{4,18}= & {} \frac{ {b\lambda }_2 }{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 }[ 1- m^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0})]\nonumber \\= & {} p_{43.18},\nonumber \\ p_{16.1}= & {} m^{*}(0),p_{17.1}=m^{*}(0),\nonumber \\ p_{4.19}= & {} \frac{ {a\lambda }_1 }{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 }[ 1- m^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0})]\nonumber \\= & {} p_{42.19}, \nonumber \\ p_{18.3}= & {} m^{*}(0),p_{19.2}=m^{*}(0),\nonumber \\ p_{21.5}= & {} \frac{ \alpha _0 }{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 + \beta _0 }[ 1- g^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0}\nonumber \\&\quad +\beta _{0})]g^{*}(\beta _{0}),\nonumber \\ p_{21.5,16}= & {} \frac{ \alpha _0 }{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 + \beta _0 }[ 1- g^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0}\nonumber \\&\quad +\beta _{0})][1-g^{*}(\beta _{0})],\nonumber \\ p_{23.11}= & {} \frac{ {b\lambda }_2 }{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 + \beta _0 }[ 1- g^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0}\nonumber \\&\quad +\beta _{0})][g^{*}(\beta _{0})],\nonumber \\ p_{23.11,14}= & {} \frac{ {b\lambda }_2 }{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 + \beta _0 }[ 1- g^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0}\nonumber \\&\quad +\beta _{0})][1-g^{*}(\beta _{0})],\nonumber \\ p_{22.12}= & {} \frac{ {a\lambda }_1 }{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 + \beta _0 }[ 1- g^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0}\nonumber \\&\quad +\beta _{0})]g^{*}(\beta _{0}),\nonumber \\ p_{22.12,15}= & {} \frac{ {a\lambda }_1 }{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 + \beta _0 }[ 1- g^{*}({a}\lambda _{1}+b\lambda _{2}+\alpha _{0}\nonumber \\&\quad +\beta _{0})][1-g^{*}(\beta _{0})]. \end{aligned}$$
(2)
The mean sojourn times (\(\mu _{i})\) is the state \(S_{i}\) are
$$\begin{aligned} \mu _{0}= & {} \frac{1}{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 },\nonumber \\ \mu _{1}= & {} \frac{1}{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 +\alpha },\nonumber \\ \mu _{2}= & {} \frac{1}{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 +\theta + \beta _0 },\nonumber \\ \mu _{3 }= & {} \frac{1}{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 + \beta },\nonumber \\ \mu _{4 }= & {} \frac{1}{ {a\lambda }_1 + {b\lambda }_2 + \alpha _0 + \gamma }, \end{aligned}$$
(3)
$$\begin{aligned}&\mu _2^{\prime } =\beta 0+\frac{\gamma (\theta +\beta 0)}{(a\lambda _1 +b\lambda _2 +\theta +\alpha 0+\beta 0)}\\&\quad +\frac{{\begin{array}{l} (a\lambda _1 +b\lambda _2 +\alpha 0)\{- \theta ^2 {(\theta +\beta 0)}^2 +\gamma \theta (a\lambda _1 \\ +b\lambda _2 +\theta +\alpha 0+\beta 0) \\ +\beta 0(\theta +\beta 0)(a\lambda _1 +b\lambda _2 +\theta +\alpha 0\\ +\beta 0)(a\lambda _1 +b\lambda _2 +\alpha 0+\beta 0) \\ -\beta 0(\theta +\beta 0)\gamma \theta (a\lambda _1 +b\lambda _2 +\theta +\alpha 0+\beta 0) \\ +(\theta +\beta 0)\gamma (a\lambda _1 +b\lambda _2 +\alpha 0+2\beta 0)(a\lambda _1 +b\lambda _2 +\alpha 0+\beta 0)\} \end{array}}}{ {(\theta {+}\beta 0)}^2 (a\lambda _1 {+}b\lambda _2 {+}\theta {+}\alpha 0+\beta 0)(a\lambda _1 {+}b\lambda _2 +\alpha 0+\beta 0)}. \end{aligned}$$

4 Development and analysis of recurrence relations

4.1 Availability analysis

By simple probabilistic arguments, semi-Markov process and regenerative point technique, the recurrence relation for systems availability are as follows:

For Models I, II and III:
$$\begin{aligned}&A_i \left( t \right) =M_i \left( t \right) +\sum _j {q_{i,j}^{(n)} \left( t \right) \copyright A_j \left( t \right) }; i,j\nonumber \\&\quad \text {denotes the regenerative states,} \end{aligned}$$
(4)
where \(A_i \left( t \right) \) is the probability that the system is in up-state at instant ’t’ given that the system entered the regenerative state \(S_i \) at \(t~=~0\) and \(M_i \left( t \right) \) is the probability of remaining in up-state at the regenerative state without visiting any other state.
For Models I, II and III:
$$\begin{aligned} M_0 (t)= & {} e^{-(a\lambda _1 +b\lambda _2 +\alpha _0 )t},\nonumber \\ M_1 (t)= & {} e^{-(a\lambda _1 +b\lambda _2 +\alpha _0 )t}\overline{F(t)} ,\nonumber \\ M_2 (t)= & {} e^{-(a\lambda _1 +b\lambda _2 +\alpha _0 +\beta _0 )t}\overline{G(t)},\nonumber \\ M_3 (t)= & {} e^{-(a\lambda _1 +b\lambda _2 +\alpha _0 )t}\overline{H(t)} ,\nonumber \\ M_4 (t)= & {} e^{-(a\lambda _1 +b\lambda _2 +\alpha _0 )t}\overline{M(t)}. \end{aligned}$$
(5)
Using Laplace transformation on Eqs. (4) and (5) and solving for \(A_0^*(s)\), the steady state availability is given by \(A_0 (\infty )=\mathop {\lim }\limits _{s\rightarrow 0} sA_0^*(s).\)

4.2 Busy period analysis

By simple probabilistic arguments, semi-Markov process and regenerative point technique, the recurrence relation for repairman’s busy period are as follows:

For Models I, II and III
$$\begin{aligned} B_i^p \left( t \right)= & {} W_i \left( t \right) +\sum _j {q_{i,j}^{(n)} \left( t \right) \copyright } B_j^p \left( t \right) ;\nonumber \\ B_i^R \left( t \right)= & {} W_i \left( t \right) +\sum _j {q_{i,j}^{(n)} \left( t \right) \copyright } B_j^R \left( t \right) ;\nonumber \\ B_i^S \left( t \right)= & {} W_i \left( t \right) +\sum _j {q_{i,j}^{(n)} \left( t \right) \copyright } B_j^S \left( t \right) ;\nonumber \\ B_i^{HRp} \left( t \right)= & {} W_i \left( t \right) +\sum _j {q_{i,j}^{(n)} \left( t \right) \copyright } B_j^{HRp} \left( t \right) .\nonumber \\&\quad i, j \text { denote the regenerative states,} \end{aligned}$$
(6)
where \(B_i^P (t), B_i^R (t), B_i^S (t) and B_i^{HRp} (t)\) are the busy period probabilities due to preventive maintenance, hardware repair, software upgradation and hardware replacement at an instant ‘t’, given that the system entered state \(S_i \)at t = 0 and \(W_i \left( t \right) \) is the probability of remaining busy due to any repair activity. The expression of \(W_i \left( t \right) \) for I & III model was derived using the same procedure as that used in model II.
For Model II:
$$\begin{aligned} W_1= & {} e^{-(a\lambda _1 +b\lambda _2 +\alpha _0 )t}\overline{F}(t)\nonumber \\&+(\alpha _0 e^{-(a\lambda _1 +b\lambda _2 +\alpha _0)t}\copyright \hbox {1)}\overline{\hbox {F}} (t)\nonumber \\&+(a\lambda _1 e^{-(a\lambda _1 +b\lambda _2 +\alpha _0 )t}\copyright \hbox { 1)}\overline{F} (t) \nonumber \\&+(b\lambda _2 e^{-(a\lambda _1 +b\lambda _2 +\alpha _0 )t}\copyright \hbox { 1)}\overline{F} (t), W_9^ =\overline{F} (t), \nonumber \\ W_2= & {} e^{-(a\lambda _1 +b\lambda _2 +\alpha _0 +\beta _0 )t}\overline{G} (t)\nonumber \\&+(\alpha _0 e^{-(a\lambda _1 +b\lambda _2 +\alpha _0 +\beta _0 )t}\copyright \hbox { 1)}\overline{\hbox {G}} (t) \nonumber \\&+(a\lambda _1 e^{-(a\lambda _1 +b\lambda _2 +\alpha _0 +\beta _0 )t}\copyright \hbox { 1)}\overline{G} (t)\nonumber \\&+(b\lambda _2 e^{-(a\lambda _1 +b\lambda _2 +\alpha _0 +\beta _0 )t}\copyright \hbox { 1)}\overline{G} (t), \nonumber \\ W_3= & {} e^{-(a\lambda _1 +b\lambda _2 +\alpha _0 )t}\overline{H} (t)\nonumber \\&+(a\lambda _1 e^{-(a\lambda _1 +b\lambda _2 +\alpha _0 )t}\copyright \hbox { 1)}\overline{H} (t) \nonumber \\&+(b\lambda _2 e^{-(a\lambda _1+b\lambda _2 +\alpha _0 )t}\copyright \hbox { 1)}\overline{H} (t),\nonumber \\ W_4= & {} e^{-(a\lambda _1 +b\lambda _2 +\alpha _0 )t}\overline{M} (t)\nonumber \\&+(\alpha _0 e^{-(a\lambda _1 +b\lambda _2 +\alpha _0)t}\copyright \hbox { 1)}\overline{\hbox {M}} (t)\nonumber \\&+(a\lambda _1 e^{-(a\lambda _1 +b\lambda _2 +\alpha _0 )t}\copyright \hbox {1)}\overline{M} (t) \nonumber \\&+(b\lambda _2 e^{-(a\lambda _1 +b\lambda _2 +\alpha _0)t}\copyright \hbox { 1)}\overline{M} (t). \end{aligned}$$
(7)
Using Laplace transformation on equation (6) & (7) and solving for \(B_i^{*P} (s)B_i^{*R} (s)B_i^{*S} (s) and B_i^{*HRp} (s)\), the time for which the server is busy due to preventive maintenance, h/w repair and h/w and s/w upgradations, respectively, is given by:
$$\begin{aligned} B_0^H= & {} \mathop {\lim }\limits _{s\rightarrow 0} sB_0^{*H} (s)=\frac{N_3^H }{D_2 },\\ B_0^S= & {} \mathop {\lim }\limits _{s\rightarrow 0} sB_0^{*S} (s)=\frac{N_3^S }{D_2 },\\ B_0^R= & {} \mathop {\lim }\limits _{s\rightarrow 0} sB_0^{*R} (s)=\frac{N_3^R }{D_2 }\\ \hbox { and }B_0^{HRp}= & {} \mathop {\lim }\limits _{s\rightarrow 0} sB_0^{*HRp}\\ (s)= & {} \frac{N_3^{HRp} }{D_2 }. \end{aligned}$$
Fig. 1

Profit difference of model I and model II vs preventive maintenance rate \(\alpha \)

Fig. 2

Profit difference of model I and model III vs preventive maintenance rate \(\alpha \)

4.3 Expected number of hardware repairs, software upgradation and visits of repairman

By simple probabilistic arguments, semi-Markov process and regenerative point technique, the recurrence relation for the expected number of hardware repairs, software upgradation and visits of repairman are as follows:

For Models I, II and III:
$$\begin{aligned} R_i^H \left( t \right)= & {} \sum _j {Q_{i,j}^{(n)} \left( t \right) \left[ {\delta _j +R_j^H \left( t \right) } \right] } ;\nonumber \\ R_i^S \left( t \right)= & {} \sum _j {Q_{i,j}^{(n)} \left( t \right) \left[ {\delta _j +R_j^S \left( t \right) } \right] };\nonumber \\ N_i \left( t \right)= & {} \sum _j {Q_{i,j}^{(n)} \left( t \right) \left[ {\delta _j +N_j \left( t \right) } \right] } ;\nonumber \\&\quad i, j \text { denotes the regenerative states}, \end{aligned}$$
(8)
where \(R_i^H \left( t \right) \), \(R_i^S \left( t \right) \)and \(N_i \left( t \right) \) is the expected number of hardware repair, software upgradation and visits by repairman at an instant ‘t’ given that the system entered state \(S_i \) at \(t~=~0\) and \(\delta _j =\left\{ {\begin{array}{l} 1; \mathrm{if} j \text { is regenerative state} \\ 0; \mathrm{otherwise} \\ \end{array}} \right. \).
Using LST in Eq. (10) and solving for \(\tilde{R}_0^H (s)\), \(\tilde{R}_0^S (s)\) and \(\tilde{N}_0 (s)\). We obtain the expected number of hardware repairs, software upgradations and visits by the server using the following expressions:
$$\begin{aligned} R_0^H (\infty )= & {} \mathop {\lim }\limits _{s\rightarrow 0} s\tilde{R}_0^H (s);\nonumber \\ R_0^S (\infty )= & {} \mathop {\lim }\limits _{s\rightarrow 0} s\tilde{R}_0^S (s);\nonumber \\ N_0 (\infty )= & {} \mathop {\lim }\limits _{s\rightarrow 0} s\tilde{N}_0 (s). \end{aligned}$$

5 Profit functions

The profit gained by the system models has been obtained by using the formula:
$$\begin{aligned} P= & {} K_0A_0-K_{1B_{0}^{P}} -K_{2{B_{0}^{R}}} -K{3_ {B_{0}^{S}}} -K_{4{B_{0}^{HRp}}}\nonumber \\&\quad -K_{5{R_{0}^{H}}} -K_{6{R_{0}^{S}}} -K_{7{N_{0}}}. \end{aligned}$$
(9)

6 Conclusion

The graphs of profit difference of all the models with respect to preventive maintenance rate (\(\alpha )\) for fixed values of other parameters are drawn for a particular case \(g(t)=\theta e^{-\theta t}\),\(h(t)=\beta e^{-\beta t}\), \(f(t)=\alpha e^{-\alpha t}\)and\(m(t)=\gamma e^{-\gamma t}\) as shown in Figs. 1 and 2. The figures reveal that the profit increases with the increase of PM rate (\(\alpha )\) and hardware repair rate (\(\theta )\). However, the value of these measures increases with the increase of maximum operation time (\(\alpha _{0})\). Again, if we increase the value of the maximum constant rate of repair time (\(\beta _{0})\), the value of MTSF, availability and profit also increase. From Figs. 1 and 2, it is concluded that model II and model III are more profitable than model I. Hence, the concept of priority to preventive maintenance over s/w upgradation and hardware repair activities is more economical to use, as compared to the system where no provision of priority is made.

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsManipal University JaipurJaipurIndia

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