Reliability and performance analysis of a base transceiver system considering software based hardware and common cause failures

Rajeev Kumar^*, Sunny Kapoor^**

 

Abstract               In this paper a stochastic model for a Base Transceiver System (BTS) is proposed that taking into account that a Base transceiver station (BTS) is the most important networking component of mobile communication system from which all signals are sent and received having both hardware and software components. The hardware and software components may have various types of major and minor faults. The aspect of hardware software interaction failure in the system is also considered. On failure, the repair team first inspects whether there is hardware or software and hardware software interaction failure, then recovery of the relevant component is done. In case of occurrence of major fault, there is complete failure of system whereas in case of minor fault system performance and capacity may decrease The possibility of occurrence of traffic Congestion in the system is also incorporated, wherein traffic congestion is automatic removed by the system. Using Markov processes and regenerative point technique various measures of system performance are obtained. Various conclusions about reliability, performance and profit of the system are made on the basis of the graphical studies. The comparative analysis of the proposed model with the existing model is also incorporated.

Keywords: Base Transceiver System (BTS), hardware software interaction failures, common cause failure, Mean time to system failure, expected uptime, expected degradation time, expected congestion time, Profit, Markov process and regenerative point technique.

 

INTRODUCTION

Mobile communication systems have become part and partial of our day to day lives and are developing regularly to ease life. There is not a single area left, where technology has not transformed life of individuals but on the other part more dependence on systems have raised certain concerns related to their performance and reliability.     A Base transceiver station (BTS) is the most important networking component of mobile communication system from which all signals are sent and received having both hardware and software components. A BTS may fails due to software based hardware fault apart from other reasons as discussed in Kumar and Kapoor (2013). Here hardware based software faults are those software faults which occur due to improper functioning or failure of hardware components, like fault in optical fiber component leads to improper working of DTMU. Software based hardware faults are hardware faults related to software like if a DTRU card hangs then it is reset from BSC, but if there is need of cyclic reset and time interval between these successive resets goes on decreasing then quality of the card degraded and finally we have to replace it. Common cause failures are due to power failure, storms, floods, earthquakes etc. the network may not provide its service continuously to its subscribers. In case of occurrence of major fault, there is complete failure of system whereas in case of minor fault system performance and capacity may decrease. Moreover when there is saturation or traffic congestion in BTS then the services for some subscribers of network is reduced or calls are unattended. Then system operation is such will be restored automatically from congestion.   In the field of reliability modeling several researchers Tuteja et al (1991), Rizwan and Taneja (2000), Taneja et al (2004), Kumar and Bhatia (2011), Kumar and Batra(2013) analyzed a large number of systems considering various aspects. For hardware-software systems, Welke et al (1995), Teng et al (2006), Tumar and Smidts (2011), Kumar and Kumar (2012) analysed hardware-software system considering hardware based software interaction failures and different types of recovery. Bothwell et al (1996), Purohit and Tokekar(2008), Ever et al(2009) and Ashan et al (2010) analyses the reliability and performability of different mobile communication system. Recently Kumar and Kapoor (2013) carried out the economic and performance evaluation of stochastic model on a base transceiver system considering hardware based software faults. However none of the researcher has carried out the analysis of BTS considering both the hardware based software and software based hardware failure. Using Markov processes and regenerative point technique various measures of system performance are obtained. Various conclusions about reliability, performance and profit of the system are made on the basis of the graphical studies. In the present paper, a comparative analysis between this proposed model (say model I), and the model discussed in Kumar and Kapoor (2013) (say model II) is carried out to judge for a base transceiver system which model is better in what situation in terms of reliability and expected uptimes, expected degradation times, expected congestion times and profits. Other assumptions are same as in Kumar and Kapoor (2013).

 

STATES OF THE SYSTEM

	Operative/Congestion state
	Degraded/Failed state under inspection
	Degraded state due to hardware/software/hardware based software fault under repair
	Failed state due to hardware/software/hardware based software fault under repair
	Degraded state/failed state due to software based minor/major hardware fault under replacement
	Failed state due to common cause failure under repair

NOTATIONS

	Rate of occurrence of major/minor faults
	Rate of occurrence of software based major/minor hardware faults
	Rate of occurrence of hardware based major/minor software faults
	Rate of traffic congestion
	Rate of automatic restoration after traffic congestion
	Probability that a major/minor hardware fault occurs in the system
	Probability that a major/minor software fault occurs in the system
	Probability that a hardware based major/minor software fault occurs in the system
	Probability that a common cause failure occurs in the system
	P.d.f/C.d.f of first passage time from state ‘i’ to state ‘j’
	P.d.f. of repair time of major/minor hardware fault
	P.d.f. of repair time of major/minor software fault
	P.d.f. of repair time of hardware based major/minor software fault
	P.d.f./C.d.f of repair time of common cause failure
	P.d.f. of inspection time of major/minor fault
	C.d.f. of inspection time of major/minor fault
	C.d.f. of repair time of major/minor hardware fault
	C.d.f. of repair time of major/minor software fault
	C.d.f. of repair time of hardware based major/minor software fault P.d.f. of replacement time of software based major/minor hardware fault
	C.d.f. of replacement time of software based major/minor hardware fault

THE PROPOSED MODEL

            A transition diagram showing the various states of transition is shown as Figure 1. The epochs of entry in to state 0,1,2,3,4,5,6,7,8,9,10,11,12 are regenerative points, i.e. all the states are regenerative states.

The non-zero elements p_ij =  

             

                   

                  

                            

                                       

              

                               

            

                            

          

By these transition probabilities, it can be verified that

p₀₁+p₀₂ +p₀₃= p₁₄+p₁₅₊ p₁₆ + p₁₇ = p₂₈+p₂₉ + p₂₁₀= p₄₀+p₄₅ = p₆₀+p₆₁₁= p₈₀+p₈₉= 1

 p₁₀₀+p₁₀₁₂ = p₃₀= p₅₆ = p₇₀ = p₉₁₀ = p₁₁₀ = p₁₂₀ = 1

The mean sojourn time (µ_i) in the regenerative state i is defined as the time of stay in that state before transition to any other state. If T denotes the sojourn time in regenerative state i, then

             

                        

                         

                                             

Thus,  

m₀₁ + m₀₂ + m₀₃ = m₀    

m₁₄ + m₁₅ +m₁₆+ m₁₇= m₁         

m₂₈ + m₂₉ + m₂₁₀= m₂               

m₃₀ = m₃           m₄₀ + m₄₅ = m₄                         

m₅₆ = m₅               m₆₀ + m₆₁₁= m₆            

m₇₀ = m₇           m₈₀ +m₈₉= m₈    

m₉₁₀ = m₉            

m₁₀₀ + m₁₀₁₂ = m₁₀                           

m₁₁₀ = m₁₁          

m₁₂₀ = m₁₂

 

Figure 1: State Transition Diagram

MEASURES OF SYSTEM PERFORMANCE

            Using probabilistic arguments for regenerative processes, various recursive relations are obtained and are solved to derive measures of the system performance. In steady state the important measures of system performance obtained are:

These are as given below:

Mean Time to System Failure (T₁)=N₁/D₁

Expected Uptime of the system (UT₁) = N₁₁/ D₁₁

Expected Degradation Time of the System (DT₁)=N₂₁/ D₁₁

Expected Congestion Time of the System (CT₁) = N₃₁/ D₁₁

Busy Period of Repairman (inspection time only) (BI₁) = N₄₁/ D₁₁

Busy period of Repairman (repair time only) (BR₁) =

N₅₁/ D₁₁

Busy Period of Repairman (replacement time only) (BRP₁) = N₆₁/ D₁₁

where  

N₁= µ₀ + p₀₂ µ₂ + p₀₃ µ₃ + p₀₂ p₂₈ µ₈ + (p₀₂ p₂₈ p₈₉ + p₀₂ p₂₉) µ₉ + (p₀₂ p₂₈ p₈₉ + p₀₂ p₂₉+ p₀₂ p₂₁₀) µ₁₀+ (p₀₂ p₂₈ p₈₉ p₁₀₁₂ + p₀₂ p₂₉ p₁₀₁₂ + p₀₂ p₂₁₀ p₁₀₁₂) µ₁₂

D₁= p_01,

N₁₁ =µ₀            ,          

N₂₁ = p₀₂µ₂ + p₀₂ p₂₈ µ₈ + p₀₂ (p₂₈ p₈₉ + p₂₉₎ µ₉ + p₀₂ (p₂₈ p_{89 +} p_{29 +} p₂₁₀) µ₁₀+ p₀₂ (p₂₈ p₈₉ p₁₀₁₂ + p₂₉ p₁₀₁₂ + p₂₁₀ p₁₀₁₂) µ

N₃₁ = p₀₃ µ_3,           

N₄₁= p₀₁µ₁+ p₀₂µ_2,

N₅₁= p₀₁p₁₄ µ₄ + (p₀₁ p₁₄ p₄₅ + p₀₁ p₁₅) µ₅ + (p₀₁ p₁₄ p₄₅ + p₀₁ p₁₅ + p₀₁ p₁₆) µ₆ + p₀₁ p₁₇ µ₇ + p₀₂ p₂₈ µ₈ (p₀₂ p₂₈ p₈₉ + p₀₂ p₂₉) µ₉ + (p₀₂ p₂₈ p₈₉ + p₀₂ p₂₉ + p₀₂ p₂₁₀) µ_10,

N₆₁= p₀₁ (p₁₄ p₄₅ p₆₁₁ +p₁₅ p₆₁₁ +p₁₆ p₆₁₁) μ₁₁+ p₀₂ (p₂₈ p₈₉ p₁₀₁₂ + p₂₉ p₁₀₁₂ + p₂₁₀ p₁₀₁₂) μ₁₂ ,

D₁₁ = µ₀ + p₀₁ µ₁ + p₀₂ µ₂ + p₀₃ µ₃ + p₀₁ p₁₄ µ₄ + p₀₁ (p₁₄ p₄₅ + p₁₅) µ₅ + p₀₁ (p₁₄ p₄₅ + p₁₅ + p₁₆) µ₆+p₀₁ p₁₇µ₇ + p₀₂ p₂₈ µ₈ + p₀₂ (p₂₈ p₈₉ + p₂₉) µ₉ + p₀₂ (p₂₈ p₈₉ + p₂₉ + p₂₁₀) µ₁₀ + p₀₁ (p₁₄ p₄₅ p₆₁₁ + p₁₅ p₆₁₁+ p₁₆ p₆₁₁) µ₁₁ + p₀₂ (p₂₈ p₈₉ p₁₀₁₂ + p₂₉ p₁₀₁₂ + p₂₁₀ p₁₀₁₂) µ₁₂

 

PROFIT ANALYSIS

The expected profit incurred of the system is in steady state given by

P₁ = C₀ UT₁ + C₁DT₁ + C₂ CT₁ - C₃ BI₁ - C₄

 BR₁ - C₅ BRP₁ - C       

where

C₀ = revenue per unit uptime of the system

C₁= revenue per unit degradation time of the system

C₂ = revenue per unit congestion time of the system

C₃ = cost per unit time of inspection

C₄ = cost per unit time of repair

C₅ = cost per unit time of replacement

C = cost of installation of the system

 

GRAPHICAL INTERPRETATION

For graphical analysis purposes, following particular cases are considered:

        

              

        

                                  

             

            Various graphs for measures of system performances viz.MTSF, expected uptime, expected degradation time, expected congestion time and profit are plotted for different values of rates of occurrence of faults , probabilities of occurrence of hardware/software/ hardware based software/common cause failure (a_1,a_2,b_1,b_2,c_1,c_2,d₁) inspection rates , hardware/ software/ hardware based software / common cause failure repair rates ( , , , , ), software based hardware replacement rates traffic congestion and system automatic restoration rates .

Figure 2 gives the graph between MTSF(T₁) and rate of occurrence of software based minor hardware faults (λ₄) for different values of rate of occurence of hardware based minor software faults(λ₆). The graph reveals that MTSF decreases with increase in the values of the rate of occurrence of software based minor hardware faults and has lower values for higher values of rate of hardware based minor software faults.

 

Figure 2: MTSF V/S Rate of software based minor hardware faults for different values of rate of hardware based minor software faults

The curve in Figure 3 depicts the pattern between MTSF(T₁) and rate of occurrence of major faults (λ₁) for different values of probability of occurence of minor hardware faults(a₂). It reveals that MTSF decreases with increase in the values of the rate of occurrence of major faults and has lower for higher values of probability of minor hardware faults.

Figure 3: MTSF V/S Rate of major fault for different values of probability of minor hardware fault

 

Figure 4 shows the graph between expected uptime of the system (UT₁) and rate of occurrence of software based minor hardware faults (λ₄) for different values of rate of occurrence of minor faults (λ₂). It is concluded from the graph that expected uptime of the system decreases with increase in the values of rate of occurrence of software based minor hardware faults and has smaller values for higher values of rate of occurrence of minor faults.

 

Figure 4: Expected uptime v/s Rate of software based minor hardware faults for different values of rate of minor faults

 

            In Figure 5, the curves represent pattern of expected uptime of the system (UT₁) and rate of occurrence of hardware based major software faults (λ₅) for different values of rate of occurrence of major faults (λ₁). It shows that the expected uptime of the system decreases with increase in the values of rate of occurrence of hardware based major software faults and has smaller values for higher values of rate of occurrence of major faults.

Figure 5: Expected uptime v/s Rate of hardware based major software faults for different values of rate of major faults

 

Figure 6 gives the graph between expected congestion time (CT₁) of the system and rate of traffic congestion for different values of automatic restoration rate .The graph indicates that expected congestion time increases with increase in the values of rate of traffic congestion and has lower values for higher values of automatic restoration rate.

 

Figure 6: Expected congestion time v/s Rate of  traffic congestion  for different values of rate of automatic restoration

 

The graph in Figure 7 shows the pattern of profit (P₁) with respect to the rate of occurrence of major faults (λ₁) for different values of rate of hardware based major software faults (λ₅). The curve in the graph reveals that the profit of the system decreases with the increase in the values of the rates of occurrence of major faults as well as with the values of rate of hardware based major software faults. Further from the graph it may also be noticed that for λ₅ =0.0001, the profit is > or = or < 0 according as λ₁ is < or = or > 0.0048. Hence the system is profitable to the company whenever λ₁ ≤ 0.0048. Similarly, for λ₅ = 0.0011 and λ₅ = 0.0021, the profit is > or = or < 0 according as λ₁ is < or = or > 0.0043 and 0.0041, respectively. Hence in these cases the system is profitable to the company whenever λ₁ ≤ 0.0043 and 0.0041 respectively.

Figure 7: Profit v/s Rate of major faults for different values of rate of hardware based major software faults

 

The curves in Figure 8 shows the pattern of profit (P₁) with respect to the rate of occurrence of hardware based minor software faults (λ₆) for different values of rate of software based minor hardware faults (λ₄). The curve in the graph indicates that the profit of the system decreases with the increase in the values of the rates of occurrence of hardware based minor software faults and has smaller values for higher value of rate of software based minor hardware faults. Further from the graph it may also be noticed that for λ₄ =0.0011, the profit is > or = or < 0 according as λ₆ is < or = or > 0.0025. Hence the system is profitable to the company whenever λ₆ ≤ 0.0025. Similarly, for λ₄ = 0.0021 and λ₄ = 0.0031, the profit is > or = or < 0 according as λ₆ is < or = or > 0.0018 and 0.0015, respectively. Hence in these cases the system is profitable to the company whenever λ₆ ≤ 0.0018 and 0.0015 respectively.

 

Figure 8: Profit v/s Rate of hardware based minor software faults for different values of rate of software based minor hardware faults

 

Figure 9 shows the pattern of profit (P₁) with respect to the revenue per unit uptime of the system (C₀) for different values of rate of software based major hardware faults (λ₃). It is observed that the profit of the system increases with the increase in the values of the revenue per unit uptime of the system but decreases with increase in the values of rate of software based major hardware faults. Further from the graph it may also be noticed that for λ₃ =0.0001, the profit is < or = or > 0 according as C₀ is < or = or > 479.30. Hence in this case the system is profitable to the company whenever C₀ ≥ 479.30. Similarly, for λ₃ =0.0009 and λ₃ =0.0017, the profit is < or = or > 0 according as C₀ is < or = or > 637.38 and 727.10, respectively. Hence in these cases the system is profitable to the company whenever C₀ ≥ 637.38 and 727.10 respectively.

Figure 9: Profit v/s Revenue per unit uptime of the system for different values of rate of software based major hardware faults

 

Figure 10: Profit v/s Revenue per unit degradation time of the system for different values of rate of minor  faults

Figure 10 shows the graph of profit (P₁) with respect to the revenue per unit degraded time of the system (C₁) for different values of rate of occurrence of minor faults (λ₂). It is concluded from the graph that the profit of the system increases with the increase in the values of the revenue per unit degradation time of the system and has smaller values for higher values of rate of occurrence of minor faults. Further from the graph it may also be noticed that for λ₂ =0.8, the profit is < or = or > 0 according as C₁ is < or = or > 193.36. Hence in this case the system is profitable to the company whenever C₁ ≥ 193.36. Similarly, for λ₂ =1.4 and λ₂ =2.0, the profit is < or = or > 0 according as C₁ is < or = or > 370.35 and 441.18, respectively. Hence in these cases the system is profitable to the company whenever C₁ ≥ 370.35 and 441.18 respectively.

 

COMPARATIVE ANALYSIS

            The proposed model say model I is compared to the model given in Kumar and Kapoor (2013) say model II for the above mentioned particular cases with respect to various measures of system performance and profits.

            The various measures of system performance obtained in Kumar and Kapoor (2013) are as under:

Mean Time to System Failure (T₂) = N₂/D₁

Expected Uptime of the System (UT₂) = N₁₁/ D₁₂

Expected Degradation Time of the System (DT₂) = N₂₂/ D₁₂

Expected Congestion Time of the System (CT₂) = N₃₁/ D₁₂

Busy Period of Repairman (inspection time only) (BI₂) =

N₄₁/ D₁₂

Busy Period of Repairman (repair time only) (BR₂) =

N₅₁/ D₁₂

 

where  

N₂ = µ₀ + p₀₂ µ₂+ p₀₃ µ₃+ p₀₂ p₂₈ µ₈ + (p₀₂ p₂₈ p₈₉ + p₀₂ p₂₉) µ_{9 +} (p₀₂ p₂₈ p₈₉ + p₀₂ p₂₉ + p₀₂ p₂₁₀) µ₁₀ ,

N₂₂ = p₀₂µ₂ + p₀₂ p₂₈µ₈ + p₀₂ (p₂₈ p₈₉ + p₂₉₎ µ₉ + p₀₂ (p₂₈ p_{89 +} p_{29 +} p₂₁₀) µ₁₀ ,

D₁₂ = µ₀ + p₀₁µ₁ + p₀₂µ₂ + p₀₃µ₃ + p₀₁p₁₄µ₄ + p₀₁ (p₁₄ p₄₅ + p₁₅) µ₅ + p₀₁ (p₁₄ p₄₅ + p₁₅ + p₁₆) µ₆ + p₀₁ p₁₇µ_{7 +} p₀₂p₂₈ µ₈ + p₀₂ (p₂₈p₈₉ + p₂₉) µ₉+ p₀₂ (p₂₈ p₈₉ + p₂₉ + p₂₁₀) µ₁₀

D₁, N₁₁, N₃₁, N₄₁, N₅₁ are already specified in model I.

Also the expected profit incurred from the system using the model II is given by

P₁ = C₀ UT₂ + C₁DT₂ + C₂ CT₂ - C₃ BI₂ - C₄

 BR₂ – C

where

C₀, C₁, C₂, C_3, C₄, C are the costs as specified in model I.

 

Figure 11 shows the behavior of difference between the mean times to system failure of the model I and model II, i.e. T₁-T₂ with respect to the rate of occurrence of software based minor hardware faults (λ₄) for different values rate of occurrence of hardware based minor software faults (λ₆). The graph reveals that the difference of mean times to system failure of the two models decreases with increase in the values of rate of occurrence of software based minor hardware faults and has smaller values for higher values of rate of occurrence of hardware based minor faults. It may also be observed from the graph that the model II has more reliability than model I for a fixed value of hardware based minor software fault.

Figure 11: Difference of mean times to system failure v/s rate of software based minor hardware faults for different values of rate of hardware based minor software faults

 

The curve in the Figure 12 shows the behavior of the difference of expected uptimes (UT₁-UT₂) of the models with respect to the rate of occurrence of software based major hardware faults (λ₃) for different values of rate of occurrence of major faults (λ₁). It is evident from the graph that difference of expected uptimes decreases with the increase in the values of rate of occurrence of software based major hardware faults and has lower values for upper values of the rate of major faults. From the Figure 12 it may also be observed that for λ₁= 0.003, the difference of expected uptimes is > or = or < 0 according as is λ₃ < or = or > 0.001. Hence the model I is better or equally good or worse than model II whenever λ₃ < or = or > 0.001. Similarly, for λ₁ = 0.004 and λ₁ =0.005, the difference of profits is > or = or < 0 according as is λ₃ < or = or > and 0.00067 and 0.00051 respectively. Thus, in these cases, model I is better or equally good or worse than model II whenever λ₃ < or = or > 0.00067 and λ₃ < or = or >0.00051.

Figure 12: Difference of expected uptimes v/s rate of software based major hardware faults for different values of rate of major faults

 

The curves in the Figure 13 shows the behavior of the difference of expected uptimes(UT₁-UT₂) of the two models with respect to the rate of occurrence of software based minor hardware faults(λ₄) for different values of rate of occurrence of hardware based minor software faults(λ₆). It is concluded from the graph that the difference of expected uptimes decreases with the increase in the values of rate of occurrence of software based minor hardware faults and has smaller values for higher values of the rate of hardware based minor software faults. From the Figure 13 it may also be observed that for λ₆= 0.0005, the difference of expected uptimes is > or = or < 0 according as is λ₄ < or = or > 0.00124. Hence the model I is better or equally good or worse than model II whenever λ₄ < or = or > 0.00124. Similarly, for λ₆ = 0.0009 and λ₆ =0.0013, the difference of expected uptimes is > or = or < 0 according as is λ₄ < or = or > and 0.00096 and 0.00082 respectively. Thus, in these cases, model I is better or equally good or worse than model II whenever λ₄ < or = or > 0.00096 and λ₄ < or = or >0.00082 respectively.

 

Figure 13: Difference of expected uptimes v/s rate of software based minor hardware faults for different values of rate of hardware based minor software faults

 

The curves in the Figure 14 shows the behavior of the difference of profits (P₁-P₂) of the models with respect to the rate of occurrence of software based minor hardware faults (λ₄) for different values of rate of occurrence of minor faults (λ₂). It is evident from the graph that difference of profits decreases with the increase in the values of rate of occurrence of software based minor hardware faults and has lower values for higher values of the rate of minor faults. From the Figure 14 it may also be observed that for λ₂= 0.005, the difference of profits is > or = or < 0 according as is λ₄ < or = or > 0.00310. Hence the model I is better or equally good or worse than model II whenever λ₄ < or = or > 0.00310. Similarly, for λ₂ = 0.0065 and λ₂ =0.008, the difference of profits is > or = or < 0 according as is λ₄ < or = or > and 0.00244 and 0.00214 respectively. Thus, in these cases, model I is better or equally good or worse than model II whenever λ₄ < or = or > 0.00244 and λ₄ < or = or 0.00214 respectively.

 

Figure 14: Difference of profits v/s rate of software based major hardware faults for different values of rate of minor faults

 

            The curves in the Figure 15 shows the behavior of the difference of profits (P₁-P₂) of the models with respect to the rate of occurrence of software based major hardware faults(λ₃) for different values of revenue per unit uptime of the system(C₀). It is observed from the graph that difference of profits decreases with the increase in the values of rate of occurrence of software based major hardware faults and has higher values for higher values of the revenue per unit uptime of the system. From the Figure 15 it may also be observed that for C₀=200, the difference of profits is > or = or < 0 according as is λ₃ < or = or > 0.00081. Hence the model I is better or equally good or worse than model II whenever λ₃ < or = or > 0.00081. Similarly, for C₀ = 450 and C₀ =700, the difference of profits is > or = or < 0 according as is λ₃ < or = or > and 0.00117 and 0.00142 respectively. Thus, in these cases, model I is better or equally good or worse than model II whenever λ₃ < or = or > 0.00117 and λ₃ < or = or > 0.00142 respectively.

Figure 15: Difference of profits v/s rate of software based major hardware faults for different values of revenue per unit uptime of the system

 

            The curves in the Figure 16 shows the behavior of the difference of profits (P₁-P₂) with respect to the cost per unit replacement time of the system (C₅) for different values of cost per unit repair time of the system (C₄). It is evident from the graph that difference of profits decreases with the increase in the values of cost per unit replacement time of the system (C₅) and has larger values for larger values of the cost per unit repair of the system. From the Figure 16 it may also be observed that for C₄=100, the difference of profits is > or = or < 0 according as is C₅ < or = or > 1831.84. Hence the model I is better or equally good or worse than model II whenever C₅ < or = or > 1831.84. Similarly, for C₄ = 250 and C₄ =400, the difference of profits is > or = or < 0 according as is C₅ < or = or > 1954.49 and 2077.13 respectively. Thus, in these cases, model I is better or equally good or worse than model II whenever C₅ < or = or >1954.49 and C₅ < or = or 2077.13 respectively.

 

Figure 16: Difference of profits v/s cost per unit replacement time of the system for different values of cost per unit repair time of the system

 

CONCLUSION

From the graphical analysis, it may be concluded for the proposed model that the mean time to system failure (MTSF) decreases with increase in the values of probability of minor hardware faults. It is also observed that MTSF and expected uptime of the BTS decreases with the increase in the values of the rates of occurrence of major as well minor faults. Further it is observed that the MTSF and expected uptime decreases with the increase in the values of rates of occurrence of hardware based software/software based hardware faults whereas expected degradation time of the system increases with the increase in values of rates of hardware based software/software based hardware faults.  The expected congestion time of the system increases with the increase in values of the traffic congestion rate and decreases with the increase in the values of rate of automatic restoration. The profit of the system increases with the increase in the values of revenues per unit up time, degradation time and congestion time of the system but decreases with increase in the values of rates of occurrence of major and minor hardware/software faults and also decreases with rate of occurrence of hardware based software/software based hardware faults. Various cutoff points for revenue per unit uptime/degradation time, rate of hardware based software/software based hardware faults can be obtained.  From the comparative analysis between the proposed model and the existing model it can be concluded that the difference of MTSF of the models decreases with increase in the rate of occurrence of minor faults as well as with rate of occurrence of software based minor hardware faults. It is also concluded that the model II is better than model I in terms of mean times to system failures for the considered values of the parameters. It is also observed that the difference of the expected uptimes decreases with increase in the values of occurrence of rate of major/minor faults as well as with rate of software based hardware and hardware based software faults. Further for fixed values of rate of major and minor faults, cutoff points for rate of occurrence of software based major/minor faults and revenue per unit uptime of the system can be obtained. For fixed values of cost per unit repair time of the system, cutoff points for cost per unit replacement time of the system can also be obtained.

 

REFERENCES

1. Ahsan, Q., Hasan, K.N., Hussain, M. and Saifullah, K., "Reliability analysis of mobile communication system of Bangladesh”, Proceedings of International Conference on Electrical and Computer Engineering, pp.45-48, 2010.
2. Bothwell, R., Donthamsetty, R., Kania, Z., Wesoloski, R., “Reliability evaluation: a field experience from Motorola's cellular base transceiver systems”, Proceedings of Reliability and Maintainability Symposium, International Symposium on Product Quality and Integrity, pp.348-359, 1996.
3. Ever, E., Kirsal, Y., Gemikonakli, O., "Performability modeling of handoff in wireless cellular networks and the exact solution of system models with service rates dependent on numbers of originating and handoff calls”, Proceedings of International Conference on Computational Intelligence, Modeling and Simulation, pp.282-287, 2009.
4. Kumar, R. and Bhatia, P., “Reliability and cost analysis of one unit centrifuge system single repairman and inspection”, Pure and Applied Mathematika Sciences, Vol.74, No. 1-2, pp.113-121, 2011.
5. Kumar, R. and Kumar, M., “Performance and cost-benefit analysis of a hardware-software system considering hardware based software interaction failures and different types of recovery”, International Journal of computer applications, Vol.53, No. 17, pp. 25-32, 2012.
6. Kumar, R. and Kapoor, S., “Economic and performance evaluation of stochastic model on a base transceiver system considering various operational modes and catastrophic failures”, International Journal of Mathematics and Statistics, Vol.9, pp. 198-207,2013
7. Kumar, R. and Batra, S., “Cost benefit analysis of a reliability model on printed circuit boards manufacturing system”, International Journal of Advances in Science and Technology, Vol. 6, No. 4, pp. 56-64,2013
8. Purohit, N. and Tokekar, S., “Performability and survivability analysis of GSM”, Proceedings of Fourth International Conference on Wireless Communication and Sensor Networks, pp. 196-200, 2008.
9. Rizwan, S.M. and Taneja, G., “Profit analysis of system with perfect repair at partial failure or complete failure”, Pure Applied Mathematika Sciences, Vol. LII, No.1-2, pp. 7-14, 2000.
10. Taneja, G., Tayagi, V.K and Bhardwaj, “Profit analysis of single unit programmable logic controller”, Pure and Applied Mathematica Sciences, Vol. LX, No.1-2, pp. 55-71, 2004.
11. Teng, X., Pham, H. and Jeske, D.R., “Reliability modeling of hardware and software interactions, and its applications”, IEEE Transactions on Reliability, Vol. 55, No. 4, pp. 571-577,2006.
12. Tumer, I.Y. and Smidts, C.S., “Integrated design-stage failure analysis of software driven hardware systems”, IEEE Transactions on Computers, Vol. 60, No.8, 1072-1084, 2011.
13. Tuteja, R.K., Arora, R.T. and Taneja, G., “Analysis of two-unit system with partial failures and three types of repair”, Reliability Engineering and System Safety, Vol. 33, pp. 199-214, 1991.
14. Welke, S.R., Johnson, B.W. and Aylor, J.H., “Reliability modeling of hardware/software systems”, IEEE Transactions of Reliability, Vol. 44, No.3, pp. 413-418, 1995.

 

Copyrights � statperson consultancy www.statperson.com  2013. All Rights Reserved.

Developer Details