The influence of error occurrence on branched computer network reliability

Microelectron. Reliab., Vol. 32, No. 4, pp. 487--492,1992. Printed in Great Britain.

0026-2714/9255.00 + .00 © 1992PergamonPressplc

THE INFLUENCE OF ERROR OCCURRENCE ON BRANCHED COMPUTER NETWORK RELIABILITY I~NEUSZ J. J6~WIAK Institute of Engineering Cybernetics, Technical University of Wrochw, Janiszewskiego Str. 11-17, 50-372 Wroclaw, Poland (Received for publication 7 December 1990)

Abstract--A hierarchical computer network with branched structure containing three data processing levels and realizing, in fixed time intervals, a determined set of tasks is considered as a class of such networks. The influence of error occurrence on the number of realized tasks and on the assumed reliability measure is discussed.

1. INTRODUCTION

Failures occur in the devices at three levels of processing and in transmission lines connecting these levels, where 2j is the intensity of the failure of t h e j t h level device, j = 1, 2 . . . . . 5 and gj is the intensity of the renewal of thejth level device,j = 1, 2 . . . . . 5. The process of failures, renewal and task shifts between phases is the Markov process [11-13]. The shifts from one reliability and functional state to another are therefore the results of failures, renewal and realization of the task phases characterized by respective intensities [5, 14]. The reliability and functional models of the branched computer network considered are presented in Refs [5, 14].

This paper discusses a hierarchical computer network (CN) with branched structure composed of three levels at which processing takes place. Typical examples of the systems analysed in the paper are presented in Refs [1-6]. The system under consideration is of a multi-task character which realizes tasks from a definite task set [5]. As a result of the total task processing into the computer network, the task is derived for the user. Computer and transmission lines of the network undergo failures [7]. If a system, which is necessary for processing or transmission of the task, breaks down, the task is stopped. It resumes after the repair 2. TASK REALIZATION AT THE TRANSMISSION of the failed systems. LEVEL Errors often occur when information is transmitted in lines [8-10]. Some of them are not detected and Let the mean length of the packet in the ith phase have no influence on the network operation. The of the task transmission be a U, then immediately the others cause faulty packets to be repeated many times mean number of packets, that is the number of until the correct transmission is obtained, provided dispatches of non-faulty packets, equals n~. The mean that if a correct transmission is not achieved in the length of the ith task phase of transmission at the ith determined number of M repetitions, the errors corre- level before transition to t h e j t h task phase at t h e j t h spond to the line failure. It is assumed that errors level is occur only in transmission and not during processing. dij = niaq. (1) Introduction of the task to the system is realized at three processing levels connected by transmission We assume that the realization intensity z 0 of the lines. A phase---a task component--is subordinated ith task phase of transmission before transition to the to each level of the system structure [4, 5]. Each task j t h task phase is defined as phase possesses a time exponent distribution with the parameter z~j, which is the realization intensity of the z,j = _l = ~ (2) ith phase of the task at the ith level before transfert,j d,j ring from the ith to the j t h level. where t U is the mean realization time of the ith task This paper deals with the effect of errors in task phase before transition to the j t h task phase, and vU transmission between processing levels on the task realization intensity z~j at the ith level, where trans- is the realization velocity of the ith task phase before transition to the j t h task phase. mission takes place. It is assumed that processing Let the transmission time of the information takes place at the levels i = 1, 3, 5, while transmission packet at the ith level before transition to t h e j t h level of the task phase is conducted at the levels i -- 2, 4. be The errors which occur in the transmission lines bring about an increase in the mean transmission aq x~j = - - . (3) time. vii 487

488

IRENEUSZJ. Jr~WlAK

When we use (1) and (3) in (2) we obtain:

zij =

1

i = 2, 4 before transition to the j t h task phase realization equals:

.

ni'cij

(4)

Zij(P)= Zij'j~=lPJ-l/j~=ljpl-1.

3. INFLUENCE OF ERRORS ON THE TASK REALIZATION INTENSITY

4. SYSTEM RELIABILITY MEASURES

As a result of errors occurring with probability p during transmission, there appears the necessity to repeat the faulty packets. The number of packet dispatches increases to the value ni(p), whereas the number of packets remains the same. On the basis of equation (4), the realization intensity of the ith task phase at the ith transmission level including the presence of errors occurring in the network with probability p equals: 1

zij (p ) = ni (p )z--~j'

(5)

Multiplying the numerator and denominator of equation (5) by n; and taking into consideration equation (4), we obtain: ni

zij(p) = zij ni(p)"

(12)

(6)

The basic characteristics of a system is its throughput (or system capability), defined as the quotient of the expected number of tasks realized in an unreliable system in the fixed time t (EN) and the expected number of tasks realized in an ideal system in the fixed time t (ENo) [15]. Let P
[14]. Let P(i) be the probability that the ideal system is in the functional state S(i) of the ith task phase realization at the ith level [14]. For a computer network with a branched structure, the throughput V is given by

Denoting

5

Ei(p) =

ni

(7)

V = i= 1.3.5

nj(p) where E~(p)e (0.1); i = 2, 4 is the transmission error coefficient. Taking equation (7) into consideration, equation (6), the dependence equation, assumes the form

zij(p) = ZijEi(p).

(8)

The coefficient E~(p) depends on the assumed service strategy of the transmission failure. If we assume that each packet with errors is repeated until the correct transmission is obtained, then the mean number of transmission repetitions is m~, i = 2, 4. If a correct transmission is not obtained in M~, i = 2, 4, repeated transmissions, the error corresponds to the failure of the line. Thus,

ni(p) = mini.

(9)

From the assumption that the probability of error occurrence during the transmission of a single packet equals p, we obtain the expected number rn~ of packet repetitions at the ith level: Mi

~ jp,- i -I mi _ -- g"-M-i

(10)

j=0

(13)

°toi•iP(i) i= 1,3,5

However, the expected number of tasks realized in an unreliable system EN is given by

kt EN =Tpp"

i=21,3ollP'M)fli" ,5 j=O~5 ?iJP
(14)

where

fO=~s(i,j) is the state of unfitness, ?iJ= ~l=~S(i,j) is the state of fitness, ~i is the coefficient of the relative speed of processing of the real device at the ith level of the system, • 0i is the coefficient of the relative speed of processing in the ideal device at the ith level of the system, /~i is the coefficient of the processing usability in the device at the ith level of the system, k is the number of users, Tp is the mean time of the effective task processing. The parameters in equations (13) and (14) depend on the values of p, M, 2, # and k.

~ pj-i

j=l

5. INFLUENCE OF ERRORS

When we place (9) and (10) into (7) we obtain:

pj- 1

El(p ) = j=1

1.

(11)

j

After combining equations (11) and (8), the intensity of task realization at the transmission levels for

The quantitative influence of the number of users for various probabilities p of errors during transmission on throughput V is presented in Ref. [16]. Errors occurring during transmission of the task cause an increase in the realizing task time. These

Influence of error occurrence on computer reliability

489

Table 1. Influence of the parameter k on EN for different values of p

EN k

p = 0

p = 0.4

p --- 0.8

1 5 10 15 20 25 30 35 40 45 50

271.52 874.55 1026.29 1061.64 1069.00 1067.08 1061.67 1054.96 1047.89 1040.88 1034.13

265.71 855.86 1004.36 1039.07 1046.31 1044.15 1038.75 1032.10 1025.15 1018.34 1011.74

260.97 841.87 987.03 1021.12 1028.97 1027.15 1021.97 1015.53 1008.73 1002.00 995.52

EN p=0.0

100(

e0(

200 I 10

I 20

I 30

I 40

I 50

P'k

Fig. 1. Function EN(k) for different values ofp. results arise from the fact that more time is needed for retransmission of faulty packets of tasks. The greater the probability p of error occurrence during transmission the smaller expected number EN of the task performed in CN. This is shown in Table 1 and Fig. 1.

Parameters

2j = O.Ol(l/h), j = 1, 2. . . . . 5

/~ = l(1/h), j = l , 2 . . . . . 5

We obtain all practical results for the following data: Tp ffi l(h), t = 100(h), ~q = 1, Gt3 ffi 2, ~q = 10, fll ffi t 3 = t 5 -~" 0 . 8 , D 3 ffi 0.05 [15], zl2 = 2, z21 = z23 ffi z43 = z45 = 100(I/h), z54 = 5(l/h), z32 = zu = 10(l/h). The influence of error occurrence during transmission is presented in Table 2 and Figs 2 and 3. The errors occurring during transmission decrease the number of tasks performed in CN. The functions presented in Figs 2 and 3 show that the dependence of the C N throughput and the expected number of tasks performed in an unreliable network is almost linear. The greater the number of incorrect packets of retransmission the greater the influence of errors. Due to errors retransmissions lengthen the time of performing the tasks in an unreliable CN. As a result of these retransmissions the average number of tasks realized in the network is smaller, and therefore the

Table 2. Influence of the error occurrence probability p on V and EN for different values of M2 = M4 = M

V p 0.0 0.2 0.4 0.6 0.8

EN

M=I

M=2

M=3

M=4

M=I

M=2

M=3

M=4

0.9664 0.9664 0.9664 0.9664 0.9664

0.9664 0.9589 0.9535 0.9496 0.9465

0.9664 0.9562 0.9574 0.9365 0.9289

0.9664 0.9552 0.9539 0.9310 0.9225

271.52 271.52 271.52 271.52 271.52

271.52 269.40 267.90 266.78 265.92

271.52 268.65 265.71 263.12 260.97

271.52 268.36 264.00 261.55 259.18

Parameters: 2j = 0.01(I/h),

gj= 1(l/h), j = 1, 2. . . . . 5, k = l

Mffil

270

0.96

265

0.94

O.

M,.3 2 0

I

I

I

I

0.2

0.4

0.0

0.8

Fig. 2. Function V(p) for different values of M. M R 32/4--D

M,,4

~ p

0

I

I

I

I

0.2

0.4

0.8

0.8

,,.

~ p

Fig. 3. Function EN(p) for different values of M.

490

IRENEUSZ J. J6;~WIAK Table 3. Influence o f the number of packet retransmissions M on V and EN for different values of p

V

EN

M

p = 0.2

p = 0.4

p = 0.8

p = 0.2

p = 0.4

p = 0.8

Parameters

1 2 3 4 5

0.9692 0.9627 0.9605 0.9598 0.9569

0.9586 0.9532 0.9500 0.9498 0.9497

0.9485 0.9429 0.9349 0.9339 0.9338

517.19 513.73 512.52 512.15 512.06

511.52 508.63 506.93 506.82 506.81

506.14 502.14 498.76 498.38 498.36

).j = 0.01(l/h), j = 1, 2 . . . . . 5, #j= l(I/h), j = 1, 2 . . . . . 5, k = 2

Vi 0.97

0.96

EN 520

p=0 2

p-0.2

0.9E

p,,O.4 0.9~

~

p=0.8

0.9', 0

I

I

I

I

I

1

2

3

4

6

1 1

.. M

I 2

I 3

I 4

I 5

=,

M

Fig. 4. Function V(M) for different values of p.

Fig. 5. Function EN(M) for different values of p.

t h r o u g h p u t o f t h e n e t w o r k is lower• T h e s e findings are p r e s e n t e d in T a b l e 3 a n d Figs 4 a n d 5. F o r a n admissible n u m b e r o f r e t r a n s m i s s i o n s M g r e a t e r t h a n 3, the c h a n g e s in t h e values o f V a n d E N

are insignificant. T h e insignificant c h a n g e s in t h e values o f V a n d E N for M > 3 are due to the small c h a n g e s in t h e t r a n s m i s s i o n e r r o r coefficient ~i, i = 2 , 4 .

Table 4. Influence o f the failure intensity on V and EN ).i(1/h)

V

EN

0.01 0.20 0.40 0.60 0.80

0.9692 0.8219 0.6811 0.5625 0.4717

517.17 438.59 363.46 300.17 251.72

i= 3 ).l = ).5 = 0.01

0.20 0.40 0.60 0.80

0.8114 0.7035 0.6271 0.5701

432.98 375.37 333.64 304.19

i= 5 ).l = ).3 --- 0.01

0.20 0.40 0.60 0.80

0.8481 0.7561 0.6999 0.6597

452.57 403.48 373.46 352.02

i= 1 ).3 = )-5 = 0.01

Parameters k=2

M2= M4= M =3 p = 0.01

#j = l(l/h), j=l,2

..... 5

EN

0.8

40c

(e)

(a)

•

0 0

ki[1/h] Fig. 6. Character of the system throughput for varying intensity of failures. (a) i = 5 , ).1= ).3 = 0.01; (b) i = 3 , 21=).5=0.01;(c) i = I , ).3=).5=0.01. 0.2

0.4

0.6

0.8

,

,

,

0.2

04

0.6

0.8

~'-

)~i [1/hi

Fig. 7. Character of the expected number of the realized task for varying intensity o f failures in the following cases: (a) i = 5 , ).t = ).3 ----O,Ol; (b) i = 3 , ).t -- ).5 -- O.Ol; (c) i=1, ).3=).5=0.01.

Influence of error occurrence on computer reliability

491

Table 5. Influence of the renewal intensity of V and E N Pt(1/h)

V

EN

Parameters

i=1 /~3= ~5 = 1.0

0.0

0.4 0.8 1.0

0.0000 0.5117 0.8285 0.9692

0.00 273.04 442.09 517.17

k=2 M2 = M~ -- M = 3 p = 0.01 2j = 0.01(I/h),

i=3 #1 =/Zs = 1.0

0.0 0.4 0.8

0.0000 0.4986 0.7581

0.00 266.08 404.50

i= 5 ~1 = P3 = 1.0

0.0 0.4 0.8

0.0000 0.4885 0.7349

0.00 260.65 392.71

j=l,2 ..... 5

EN~ 517.17

V 0.9692~ 0.8

400

0.8 0.4

2oo

0.2

lOO

0

0.2

0.4

0.6

P

0.8

~Z B/h]

Fig. 8. Character of the system throughput for varying renewal intensity in the following cases: (a) i = 5, #t = #3 = 1.0;

(b)

i = 3,

/A = P5 = 1.0;

(c)

i = I,

/~3=/~5 = 1.0. Due to failure of a device necessary for performing the given task set, the performance is stopped. The task realization time is lengthened by the time needed to renew the failed device. Increasing the failure intensity of the devices at the processing levels causes a decrease of the number of tasks performed in the CN as well as the CN throughput. This is shown in Table 4 and Figs 6 and 7. It can be seen from Figs 6 and 7 that the task performed in CN, at all processing levels, remains for the longest time at level no. 1, and for the shortest time at level no. 5. Therefore, the lower the processing level, the greater the influence of the failure intensity at this level on V and EN. The smaller influence of the devices at higher processing levels on V and E N results from the fact that these devices are used to perform the task only when the processing possibilities of the lower levels are not sufficient. The better the device operating in CN is organized, the shorter the renewal time caused by failures of the CN. The increase in renewal intensity # of the failed devices of the CN shortens the time of realization particular tasks and increases the average number of tasks performed in a real CN and increases the throughput of the network. This is shown in Table 5 and Figs 8 and 9. From the characters in Figs 8 and 9, we can see that devices of processing levels which are at the lowest

o

./~'la) ............... o'2

o',

&

o'.8

'

it i [1/hi

Fig. 9. Character of the expected number of the realized task for varying renewal intensity in the following cases: (a) i = 5 , ~1=/~3=1.0; (b) i = 3 , p l = p s = 1.0; (c) i = 1, /~3=/~5 = 1.0. level have the greatest influence on network reliability. It is impossible to load a task without efficiency at level no. 1. 6. FINAL REMARKS On the basis of the reliability and functional model [14], the reliability of a branched computer network was analysed. Typical limitations applied with reliability analysis were taken into account. The throughput and the expected number of tasks realized in the real branched computer network were the measures of reliability. These measures were computed for different reliability and functional parameters of this network. The reliability analysis of branched computer networks provides a basis for further investigations of CN. The occurrence of many parameters in the network having a significant influence on both the network throughput and the expected number of tasks realized in the network, points to the necessity of working out some methods of reliability and functional optimization of CN. These methods would make a choice of network parameters possible in order to obtain the best possible reliability and functional coefficients. For example, one of the optimization methods could be used to maximize the system throughput

492

IgENEUSZ J. J6~WIAK

provided that the expected number of tasks is at the level determined by the network users. Other methods could choose parameters of the C N which would bring about the highest possible profits.

8. 9.

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