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Ismail Tukur, Kabiru H. Ibrahim, Muhammad Salihu Isa, Ibrahim Yusuf AVAILABILITY AND PERFORMANCE ANALYSIS OF COMPUTER
NETWORK WITH DUAL-SERVER USING GUMBEL-HOUGAARD RT&A, No 3 (63) FAMILY COPULA DISTRIBUTION_Volume 16, September 2021
Availability And Performance Analysis Of Computer Network With Dual-Server Using Gumbel-Hougaard Family Copula Distribution
Ismail Tukur
Department of Art and Humanities, Kano State Polytechnic, Kano, Nigeria. ismailatukur35@gmail.com
Kabiru H. Ibrahim
Department of Mathematics, Aminu Kano College of Islamic and Legal Studies Kano, Nigeria
kbrhamis032@gmial.com
Muhammad Salihu Isa
Department of Mathematics, Federal University Dutse, Jigawa, Nigeria salihu.muhd.isa@gmail.com
Ibrahim Yusuf
Department of Mathematical Sciences, Bayero University, Kano, Nigeria iyusuf.mth@buk.edu.ng
Abstract
The determination of this paper is to study reliability measures and routine analysis of computer network, which is a combination of four subsystems A, B, C and D and all the subsystem connected in series parallel pattern, the subsystem A is client, the subsystem B is load balancer and subsystem C is servers which is divided in to two subsystem (i.e. subsystem C1 and subsystem C2) and C1andC2 served as computer servers together with two unit each and working 1-out-of-2: G policy, and subsystem D is centralized server. The system has two types of failure, degraded and complete failure. The system can completely fail due to failure of one of the following subsystems A, B, C and D. The system is at partial failed state if at least one unit is working in either subsystem C1 or subsystem C2.The system is examined using supplementary variables techniques and Laplace transform. General distribution and copula family are employed to restore degraded and complete failed state respectively. Calculated results have been highlighted by the means of tables and graphs.
Keywords: Reliability, Availability, Sensitivity, (MTTF) mean time to system failure, Gumbel - Hougaard, family, cost Analysis and supplementary variable techniques.
Ismail Tukur, Kabiru H. Ibrahim, Muhammad Salihu Isa, Ibrahim Yusuf AVAILABILITY AND PERFORMANCE ANALYSIS OF COMPUTER NETWORK WITH DUAL-SERVER USING GUMBEL-HOUGAARD FAMILY COPULA DISTRIBUTION
RT&A, No 3 (63) Volume 16, September 2021
I. Introduction
The study of reliability modeling was started During the World War II in 1939, later then, substantial efforts have been made in this direction to create the comprehensive theoretical background for reliability modeling. The discipline is mainly anxious with requirement and valuations of the probability of device performing its purpose sufficiently for the period intended under the encountered functioning condition. Reliability has introduced new dimensions in recent years because of the difficulty of larger systems and suggestions of their failure. Undependability in the modern age of technology causes inefficiency of a system, overgenerous maintenance and can also risk human life. In today's technological world, nearly everyone be subject to upon the continued function of a wide array of machinery and equipment for our safety, security, mobility and financial welfare. We receive our electronic agreements from illuminations, hospital monitoring control, next generation aircraft, nuclear power plants data exchange system and aerospace application, to function whenever we need them when they fail, the results can be a catastrophic, injurious or eventual loss of life. The theory of reliability is the scientific discipline that studies the general regularity that must be maintained under design, research, manufacture, acceptance, and use of units/ components to acquire the highest effectiveness of their use.
The most common vital mode of redundancy is fc-out-of-n redundancy. This type of redundancy is more branded into two classifications, these are fc-out-of-n: G and fc-out-of-n: F. A fc-out-of-n: G redundancy indicates that for successful operation of the system at least fc units out of n units are essential to be good (i.e. essential to work properly). If less than fc units are good then the system fails. A fc-out-of-n: F redundancy indicates that if fc units out of n units have failed then the system has failed. The system reliability has been widely studied and used by numerous authors like, Yusuf et al [1] studied performance analysis of multi computer system consisting of active parallel homogeneous, Abubakar and Singh [2] have studied assessment and performance of industrial system using Gumbel Hougaard copula approach, Singh et al. [3] have studied the reliability characteristic for Internet data center with a redundant server including a main mail sever, M. Ram et.al. [4] Have discussed the reliability of a system with different failure and common cause failure under the preemptive resume policy using Gumbel-Hougaard family copula distribution. Minjae Park [5] analyzed the multi-component system with imperfect repair during warranty using renewal process. Zhang [6] analyzed on computer network reliability analysis based on intelligent cloud computing method. Kudeep et al [7] studied tree topology network environment analysis under reliability approach, nonlinear. Muhammad et al [8] studied cost benefit analysis of tree different series parallel dynamo configurations. Dillon et al. [9] have discussed the common cause's failure analysis of k-out-of-n: G systems which consisting repairable units. Ibrahim Yusuf [10] evaluates the performance of a repairable system with the concept of minor deterioration under imperfect repair. Tseng- Chang Yen et al. [11] studied Reliability and sensitivity analysis of the controllable repair system with warm standbys and working breakdown. v. v. Singh et al [12] evaluates the performance and cost assessment of repairable complex system with two subsystems connected in series configuration. v. v. Singh and Jyoti Gulati. [13] Studied performance assessment of computer Centre at Yobe state university Nigerian under different policies using copula. Kabiru H. Ibrahim et al. [14] have studied the availability and cost analysis of complex tree topology of computer network with multi-server using Gumbel-Hougaard family approach. Geon Yoon, Dae Hyun et al [15] focused on ring topology-based redundancy Ethernet for industrial network. Pratap et al [16] have examined on the assessment of complex system with two subsystems and multi types failure and repair. Rawal et al. [17] have discussed the reliability of Internet Data Centre having one mail server and one redundant server especially for the use of Interne.
Ismail Tukur, Kabiru H. Ibrahim, Muhammad Salihu Isa, Ibrahim Yusuf AVAILABILITY AND PERFORMANCE ANALYSIS OF COMPUTER
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Ibrahim Yusuf and Hussaini [18] examine a three-unit redundant system with three types of failure and general repair. Negi and Singh [19] studied reliability characteristics of a non-repairable complex system connected in series. C. K. Goel.et al. [20] performance assessment of repairable system in series configuration under different types of failure and repair policies using copula linguistics. However, researchers have considered different models and scrutinized the performances and availability of a complex system to forecast better performance. Most of the authors studied the complex repairable systems that are treated as single repair between two contiguous transition states. In the present paper, several reliability measures of a complex repairable system consist of four subsystems together with k-out-of-n; G configuration using two types of repair have been studied. The designed structure of the system consist of four subsystems A, B, C, and D. where C consist of C1 and C2 served as computer servers together with two unit each and working 1-out-of-2: G policy respectively. The system both are working in a series and parallel arrangement. Gumbel Hougaard family copula distribution employed for calculation and illustration.
Lastly, {S1, S3, S5, S6} represents the states of operation in degraded/partial failure while {S2, S4, S7, S8 and S9} are entirely failed states and So is at flawless operational state. The degraded points have repaired by general repair and completely failed states have repaired under Gumbel Hougaard family copula. the supplementary variables and Laplace transformation use analyze the system. Measures in reliability among availability, reliability, and MTTF and cost analysis are all treated by the means of tables and graphs.
II. State Description, Assumption and Notations
Tablel: State description
State State Description
So The state So indicates a perfect state in which both the subsystems are in outstanding working condition.
Si The state Si reveals a degraded state with partial failure in subsystem C, due to the failure of first server in subsystem Ci.
S2 In this state S2 indicate a complete failure state, due to the failure of first and second servers in subsystem Ci. The system is under repair using copula.
S3 The state S3 account a degraded state with partial failure in subsystem C, due to the failure of third server in subsystem C2.
S4 In this state S4 indicate a complete failure state, due to the failure of third and fourth servers in subsystem C2. The system is under repair using copula.
S5 The state S5 account a degraded state with partial failure in subsystem C, due to the failure of third server in subsystem C2.
S6 The state S6 represents a degraded state with partial failure in subsystem C, due to the failure of first server in subsystem Ci.
S7 The state S7 represents a complete failed state, due to the failure of subsystem D.The system is under repair using copula.
Ss The state S8 represents a complete failed state, due to the failure of subsystem A. The system is under repair using copula.
S9 The state S9 represents a complete failed state, due to the failure of subsystem B.The system is under repair using copula.
Ismail Tukur, Kabiru H. Ibrahim, Muhammad Salihu Isa, Ibrahim Yusuf AVAILABILITY AND PERFORMANCE ANALYSIS OF COMPUTER
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The state description indicates, that So is a perfect state where both subsystems are in good working condition. The states Si, S3, S4, and S6 are the operational states. The states S2, S4, S8, and S9 of this modeling are a complete failed state in which the system is inoperative mode. I. Assumption
The conventions of the model is discussion below:
• At initial state, all subsystems are in good working condition.
• One unit of each subsystem C1 and subsystem C2 is necessary for operational mode.
• The system will be at complete fail state if both subsystems and C1 and C2 fail.
• Failure of all if subsystems A, B, C and D will damage the entire system.
• Failed unit of the system can be repaired when it is in operative or failed state.
• All failure rates are constant and assumed to follow exponential distribution
• The repairs follow a general distribution.
• It is assumed that a repaired system works like a new system and no damage appears during repair.
• As soon as the failed unit gets repaired, it is ready to perform the task.
Table2: Notations
t: Time variable on time scale.
S: A variable for Laplace transform for all expressions.
1 / 1s : Si s2 Failure rates of servers of subsystems C1
l / l : V s4 Failure rates of servers of subsystems C2
1 / 1 / ¿Cs : Failure rates of subsystems B, A, and D
f ( X ) : z ) Repair rates for all subsystems i.e. A, B, C and D
«0 (x), y), M{a): Repair rates for complete failed states.
Pi (X*): The probability that the system is in Si state at instant's' for I =0 to 9.
Pi ( s ) Laplace transformation of state transition probability P (t).
EP (t) Expected profit during the time interval [0, t).
K1, K2: Revenue and service cost per unit time in the interval [0, t) respectively.
sf ( x ) Standard repair distribution function Sf (x) = f (x) f *)s(
The expression of joint probability (failed state Si to good state S0)
u0 (x)= Ce(u{^x)) according to Gumbel Hougaard family copula is given as
Ce (u (x) ,u2 (x)) = exp[xq +{log^(
where, u1 = f (x), and u2 = ex, where 6 as a parameter, 1< 6<
Ismail Tukur, Kabiru H. Ibrahim, Muhammad Salihu Isa, Ibrahim Yusuf AVAILABILITY AND PERFORMANCE ANALYSIS OF COMPUTER
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Figure 1: Transition Diagram
III. Formulation and Solution of Mathematical Model
By the related literature review, the following differential equation were generated for the mathematical classical.
{Yt+ASl + *s3 )p0(t)
f& f& f&
= I <Po(x)Pl(x,t)dx + I &o(x)P3(x,t)dx + I noP4(x,t)dx
Jo Jo Jo
f & f&
+ I ^oP2(x,t)dx + I ¡uoP7(z,t)dz + I ¡uoP*(y,t)dy Jo 'o 'o
p &
+ I ^oPB(a,t)da (1)
o
(h + IL + Âs2+Â!3 +àb + ¿a +& to) Pi(x, t) = 0 (2)
(é + h + V (*)y2(x,t) = 0 (3)
(h + h + ^^Sl +ÀB +Àc + Àcs +& to) P#(X, t) = 0 (4)
(é+h + V ( * ) ^8P'(x,t) = ° (5)
(h + h +ÀB +Àc + Àcs +& to) P5(X, t) = 0 (6)
(h + h + +ÀB +Àc + Acs +# (x))p6(x,t) = 0 (7)
h + ± + ^ ( 2) )P7(z,t) = 0 (8)
h + J5 + V (a))p+(a,t) = ° (9)
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(é+56 + ( y ) )^Cy,t) = ° (10)
Boundary conditions
P1(0,t)=AslPo(t) (11)
P2(0,t)=Â21ÂslPo(t) (12)
P3(0,t)=As3Po(t) (13)
P4(0,t) = Âs4Âs3Po(t) (14)
P5(0,t)=Âs3ÂsiPo(t) (15)
P6(0,t)=^S3Po(t) (16)
P7(0, t) = (1 + + As# + 2ÂS1ÂS3)ÂCSPo(t) (17)
P8(0, t) = (Asi + As# + 2ÂsiÂs#)ÂcPo(t) (18)
P9(0, t) = (Asi + As# + 2AsiAs#)Ab Po(t) (19)
I. Solution of the Model
Equation 20 to 38 below is obtained by taking Laplace transformation of equations 1 to 19 with the help of initial conditionP0(0) = 1, (s + A^ + AB + 2AC + 2AD )P%(S) =1
+ + +
ç OO r> OO r> oo
I 0o(x)P"(x, s)dx + I i>o(x)P3(x,s)dx + I no(x)P2(x,s)dx
ooo /■ & /■ & /■ &
■ I ;Uo(x)P4(x,s)dx + I ^o(z)P7(z, s)dz + I ^o(a)P8(a,s)da
-'o -'o -'o
&
■ I ^o(y)^*(y,5)dy (20) o
(* + + ^S2+ + ¿c + ¿cs + ¿b + «»oto) ^ifo s) = 0 (21)
(s + + ^o(x))i;2(x,5) = 0 (22)
(s + + AS4 ■+ Asi + Ac + AcS + + 0o M)^ (x, 5) = 0 (23)
(s+ + ^M)iU*,s) = 0 (24)
(s + 5. + + Asi + Ac + A„ + + 0o(^)) P5(x,s) = 0 (25)
(s + 5. + + AS4 + Ac + A„ + AB + foW) ¿U*,*) = 0 (26)
(s+ 5. + ^o(^))i;7(2,5) = 0 * (27)
(s+ 5. + ^o(«))i58(«,^) = 0 (28)
(* + 5. + ^(y))iUy,s) = 0 (29) The Laplace transformations of the boundary conditions are:
Pi(0,s)=AsiP;(s) (30)
/;2(0,s)=AS2A1i/;0(s) (31)
P3(0,s)=AS3P0(s) (32)
/04(0,s) =As4As3/0o(S) (33)
/05(0,s)=AS3Asi/0o(s) (34)
P6(0,s)=AsiAS3Po(s) _ (35)
P)(0,s) = ( Asi + As# + 2AsiAs#) Acs № (36)
P8 (0, s) = ( Asi + As# + 2AsiAs#) Ac (s) (37)
P9(0,s) = ( Asi + As# + 2AsiAs#) Ab Po(s) (38)
Now solving equations (20) to (38) with the help of equations (11) to (19), yields,
^ = (39)
p (s) = @i~>#(>+ <!$ + <S3 + <B + <c+ <cs)J (40)
i(5) = 9(s) [ >+ <s2+ <s3+ <B+ <C+ <CS J ( )
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<s1<s2 fl->@oW)
L(s)
P2(s) =
p (5) = <s" {1-s#(s+ <s*+ <s1+ <b+ <c+ <cs)}
#( J L(s) { S+ <s*+ <s1+ <B + <C+ <CS }
P'(s)= ^
P5(s)
= <s1<s3 f1-S#(S+ <si+ <B+ <C+ <CS^ L(s) { S+ <s1+ <B+ <C+ <CS } <s1<s3 f1-S#(S+ <s*+ <B+ <C+ <CS)~)
L(s) { S+ <s*+ <B+ <C+ <CS }
P7(s) = (1+ XS1 + As# + 2Asi_As#) {i-^}
P8(s) = l<+) ( *si + ¿s# + 2ÄsiÄs3) {1-S@°(x^
L(s) <B
S
1-S@o(.)1
P*(S) = ITT) ^ ^ + +
Pup(s) = P0(s) +_P"(S) + P2(s) + P3(s) + P5(s) + P6(s)
Pdown 1 Pup
Where,
(41)
(42)
(43)
(44)
(45)
(46)
(47)
(48)
(49)
(50)
L( s) =
(f
(s + Xsl + XS3 + Xcs+)-
A (s+As+As+A+1c+As)+A3(s+A4+As+A+A+As) +AnAnßiJs (s)+A}AsSß (s)+(1+As+A}+sAiAs) A,Si(s)+
(1 + Ass + AS3 + 2 Ass AS3) A S^ (s)+(1 + AS1 + AS3 + 2ASSAS3) ABSi(s )
,ö ö +
0 0
(51)
The Pup (s) and Pdawn (s) are the system Laplace transform of the state probabilities in operative and failed state. Then,
__12___ _
Pp (s) = Zp (s) and Pdawn (s) =1 - Pup (s)
Pp (') =
. . i1!(s+^ +13 +1 +1 +1) f1-s* (s+A*+1 +1 +1 +1) 1 + 1C1 -;--- + 1 -
(S + 12 + 13 + 1ß + + 1cS ) ~Sf (s +11 +1 +1 +1
(s +11 +1 +1 +1)
1-Sf (s +11 +1 +1 +1)ö+ , , f1-Sf (s + A4 +1 +1 +1)
( s + ^ 4 + A, j + ģ + 1 + 1 )
(S + 14 + ^^ + 1C + ^^^ ( s + A, 4 + 1 + 1 + ^ )
(52)
III. Numerical Study of the Model
I. Availability Analysis By Setting Sßo(s) =
_ exp[X+ {logjx)}q]1/q ^
j
^xp[xq+{iogjW}qr^ ^ + exp[xq+{logj(x)}q]1/q j ' s + jS
The expression of availability is obtained by taking the inverse Laplace transform of equation 52 together with the values of failure rates, Asi=0.0001, AS2=0.0002, , AS3=0.0003, , As4=0.0004,Ab=0.0005, Ac=0.0006, Acs = 0.0007 at f (= 0 = x= 1 and Mo(x) = fi0(^^ = 2.781
A( x) =
f-5.99867865610-VL00™ +1.000000381e-°000699997787364i + 3.16650339610-7 e-2-718300861i | I -6.76892596410-Y1 00269936" -1.49939921910-8e-L001900000i
The values of Pup(t) through variation of time t= 0, 10, 20, 30, 40, 50, 60, 70, 80, 90. as shown in Table 1 and figure 2.
i=0
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Table 3:Variation of Availability with respect to time (t)
Time(t) Availability
0 1
10 0.993
20 0.986
30 0.979
40 0.972
50 0.965
60 0.958
70 0.952
80 0.945
90 0.938
100 0.932
-Ava ila bil rty
- Linear
(Availability]
50 IOO 1 50 Availability
Figure 2 ¡Variation of Availability with respect to time (t)
II. Reliability Analysis
Compelling all repair rates to zero with the same value of failure and repair rates in equation 53, i.e ^(.T)and//Q (jc)and Asi=0.0001, As2=0.0002, As3=0.0003, Asi=0.0004 Ab=0.0005, Ac=0.0006, AHX0007,
and then taking inverse Laplace transform, we obtained the expression of reliability.
_ i-O.OOOOSTSOOOOOOe-0 00190000000' -0.3333333333e-°°™00' -0.00002727272727e-00™°00' {+1.333398106e-°00110—
Aimed at different values of time t= 0, 10, 20, 30, 40, 50, 60, 70, 80, 90 units of time, we get different values of Reliability as shown in Table 2. and Figure. 3.
(54)
Table 4: Variation of reliability with respect to time (t)
Figure 3: Variation of reliability with respect to time (t)
III. Mean Time To Failure (MTTF) Analysis
The expression for MTTF is found by pleasing all repairs zero in equation (53), and set the limit of s tends to zero:
M.T.T.F. = MPts)=-.-!-r 1+-i-+-i-+-ii-+-ii-
"" {(4+4+4)} [ 4+4+4+4+4 44+4+4+l+4 4+4+4+4 4+4+l.+4j Location Asi=0.0001, AS2=0.0002, AS3=0.0003, As4= 0.0004, Ab=0.0005, Ac=0.0006, A&=0.0007 and changing
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Aa,Ab, Ac and Ad respectively as, 0.1,0.2,0.3, 0.4,0.5,0.6,0.7,0.8,0.9, in (53), the variation of M.T.T.F. is found with respect to failure rates as shown in Table.3 and corresponding Figure.4.
Table 5: Variation ofMTTF with respect to failure rate
1----- Failure MTTF MTTF MTTF MTT MTTF MTTF MTTF 1200 _ a ~1 rn _ _ -♦-MTTFAsl
Rate F =4= ■ — — — H
Xsl Xs2 Iß Xs4 Xb k ks 1000 E Ep WW -■-MnFXs2
0.01 488.6 1035.2 496 971.6 939.9 940.2 99.47
- 800 ; z -*-MTTFAs3
0.02 462.5 1031.8 466.5 961.1 925.8 925.9 49.93 X
u 600 _ H—
0.03 453.5 11Q 0 1030.5 1 mo Q 456.2 Kfi Û 957.2 920.5 01 7 C 920.6 017 0 33.31 11 00 ^e-MTTFAs4
3 400 — jp H H H H H H HeMTTFAb -•-MTTF Ac
U.U4 0.05 446.1 lUiy.y 1029.4 447.8 yjj.i 953.9 yl 1.0 916.1 yl /.0 916.1 ll.yy 19.91 200 — = - — — = E -
0.06 444.3 1029.2 445.6 953 915 915 16.66
0 — —tr — — — — —
0.07 442.9 1028.1 444.1 952 914.1 914.2 14.28 0 C05 0J —t—MTTF Xcs
0.08 441.9 1028.8 443 952 913.5 913.5 12.41 MTTF
0.09 441.2 1028.7 442.1 951.6 913.1 913.1 11.11
Figure 4: Variation ofMTTF with respect to failure rate
IV. Sensitivity Analysis of MTTF
The calculation of sensitivity MTTF is studied through the partial differentiation of MTTF with respect to the failure rates of the system, by introducing the set of parametric variation of the failure rates Asi = 0.0001 As2=0.0002, As3=0.03, Asi=0.0004„ Ab=0.0005, Ac=0.0006, AcS=0.0001, from the resulting expression, the MTTF sensitivity is calculated as shown in Table 4 and the corresponding value in Figure.5 However the failure rate with respect to Ab=0.0005 and As3=0.03 that is the load balancer and a server from subsystem two must be given a special consideration.
Table 6: Sensitivity ofMTTF as a function of failure rates
----- Failure Rate MTTF SITTF itTTF MTTF MTTF *ÎTTF m
[/ Xsl u Xsl As3 CXs4 I Ab u Ac Acs
0.01 -5058 -621 -5726 -1926 -2611 -2657 -9850
0.02 -1342 -186 -1528 -569 -765 -772 -2490
0.03 -607 -88 -694 -268 -360 -362 -1109
0.04 -345 -51 -395 -155 -208 -209 -624
0.05 -222 -33 -254 -101 -136 -136 -400
0.06 -155 -24 -177 -71.2 -95.2 -95.5 -278
0.07 -114 -17 -131 -52.8 -70.4 -70.7 -204
0.08 -87.3 -13 -100 -40.7 -54.4 -54.5 -156
0.09 -69.1 -11 -79.4 -32.3 -43.2 -43.3 -123
partial Deravrtive of MTTF
Figure 5: Sensitivity ofMTTF as a function of failure rates
V. Cost Analysis
The predictable profit over the time interval [0, t), can be estimate by the following relation
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t
EpPt) = KjPp)dt — K2t. If the service facility of the system is always available, where ki is
0
revenue generated and k2 service cost per unit time. For the same set of the parameter of failure and repair rates in (53), the expression of cost benefit analysis is attained.
i 5.9855105331CT9 e_1'0022000000f —1429.023674e_0'0006997787364f +6.75070337910-7 e-1-002699361 f|
-1.16488334410"7 e-2718300861, +1.49655576310~
—1.001900000?
+ 1429.023673
By fixing the revenue Ki= 1 and taking the values K2= 0.6,0.5,0.4,0.3,0.2 and 0.1 respectively together with the variation of t =0, 1, 2, 3, 4, 5, 6, 7, 8, 9, Units of time, we obtained the results for expected profit as shown in Table 5 and Figure.6
Table 7: Expected Profit in [0,t) t=0,l ,2, 3, 4, 5,6.7,8,9 & 10
Time (t) Ep(t): K2=0.6 Ep(t): K:=0.5 Ep(t): K2=0.4 Ep(t): K:=0.3 Ep(t): K:=0.2 Ep<t): K;=0.1
0 0 0 0 0 0 0
1 0.4 0.5 0.6 0.7 0.8 0.9
2 0.8 1 1.2 1.4 1.6 1.8
3 1.2 1.5 1.8 2.1 2.4 2.7
4 1.59 1.99 2 39 2.79 3.19 3.59
5 1.99 2.49 2.99 3.49 3.99 4.49
6 2.39 2.99 3.59 4.19 4.79 5.39
7 2.78 3.48 4.18 4.88 5.58 6.28
8 3.18 3.98 4.78 5.58 6.38 7.18
9 3.57 4.47 5.37 6.27 7.17 8.07
10 3.97 4.97 5.97 6.97 7.97 8.97
5 10 15 Expected profit
-EP[t): K 2=0.6
-EPlt): K 2=0.5
-EP(t): K2=0.4
-EP(t|: K2=0.3
-EP(t): K2=0.2
-EP(t): K2=0.1
Figure 6: Expected Profit in [0,t) t=0,l ,2, 3, 4, 5, 6, 8,9 & 10
IV. Discussion
The presentation of the system under the valuation of reliability measures for different values of failure and repair rates. Table.1 and figure 2 shows the evidence of availability of the complex with respect to time when the failure rates are fixed at different values mainly, Asi=0.0001, As2=0.0002, , As3=0.0003, , As4=0.0004,Ab=0.0005, Ac=0.0006, Ac* = 0.0007. The availability of the system decreases slowly, as the probability of failure increases, the system availability will tend to zero at t very large. However, one can simply predict the future behavior of the complex system at any stage for any given set of parametric values.
Table.2 and figure.3 figure out the reliability of the system if there is no repair. The figure shown clearly that the reliability of the system is decreasing faster compare to availability, which evidently proved that when the repairs provided the performance of the system is quite better.
Table.3 and figure.4 assess the information of mean time to failure of the system (MTTF) with respect to variation of failure rates. The value change of MTTF is directly propotional to the system reliability. The computations MTTF for different values of failure rates, Asi, As2, , As3, As4,A, Ac, and Acs from the figure the variation in MTTF corresponding to failure rates Asi is high compared to other failure which indicates that the system will not be affected with higher variations in values.
Table.4 and figure.5 demonstrations of sensitivity MTTF with respect to the values of parameters. which obtained from partial derivative of MTTF with respect to the corresponding failure rate, Moreover the variation of sensitivity MTTF corresponding to failure rates Ab=0.0005 and As3=0.03 that is the load balancer and a server from subsystem two, must be given a special consideration.
Ismail Tukur, Kabiru H. Ibrahim, Muhammad Salihu Isa, Ibrahim Yusuf AVAILABILITY AND PERFORMANCE ANALYSIS OF COMPUTER
NETWORK WITH DUAL-SERVER USING GUMBEL-HOUGAARD RT&A, No 3 (63) FAMILY COPULA DISTRIBUTION_Volume 16, September 2021
Table.5 and figure.6 afford the information on how the profit has been generated, by fixing revenue
cost per unit time Ki= 1, and varies the service costs K2 = 0.6, 05. 0.4, 0.3, 0.2 and 0.1, if we examine
critically from Figure.6 we can reveals that the expected profit increases for low service cost. Which
finally shows the Networking system of tree topology system is reliable.
Reference
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Ismail Tukur, Kabiru H. Ibrahim, Muhammad Salihu Isa, Ibrahim Yusuf AVAILABILITY AND PERFORMANCE ANALYSIS OF COMPUTER
NETWORK WITH DUAL-SERVER USING GUMBEL-HOUGAARD RT&A, No 3 (63) FAMILY COPULA DISTRIBUTION_Volume 16, September 2021
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