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A GENERALIZED APPROACH IN MULTIPROCESSOR ENVIRONMENT USING REGRESSION TYPE ESTIMATOR AND COST ANALYSIS
Sarla More, Diwakar Shukla
Department of Computer Science and Applications Dr. Harisingh Gour University Sagar (MP) India sarlamore@gmail.com, diwakarshukla@rediffmail.com
Abstract
Consider a multi-processors computer system consisting of a ready queue of different jobs to be executed/processed. Lottery scheduling is fair enough to schedule the resources for each and every job. The research idea assumes condition where one can observe some processes to be fully executed; some partially executed few blocked/suspended/ terminated, after sudden system breakdown. An estimation strategy has been designed for the estimation of the total time required to process all these types of processes (processed, partially processed and blocked processes). How much time is required to process the remaining in any hazardous situation? A regression type estimator of sampling theory is used to perform this task. This remaining time estimation technique deals with the backup cost and recovery management as well. Sampling techniques are used in proposed approach for the testing purpose and a simulation has been performed. Another tool adopted is the confidence intervals which are calculated and gives proper précised values in comparison to the true mean for the total remaining time. The linear, square root and square cost function model are adopted for the calculation of backup cost and recovery management. In addition some auxiliary information is also incorporated in the form of size measure of the processes which is an effective approach to calculate the complete remaining time of the processes in multiprocessor environment. The purpose of the proposed research has been served effectively as one can observe the results of disaster and recovery management of the computer system.
Keywords: Ready Queue, Lottery scheduling, Multiprocessors, Simulation, Random Sampling, Estimation, Confidence Interval, Jobs(Processes), Size measure, Estimator
I. Introduction
In the scenario of cloud computing, ready queue is a setup among many servers and processors. For optimal resource allocation there exists several priority scheduling methodologies in the literature of scheduling schemes. In same way lottery scheduling scheme works on randomness of selection of process and distribution of resources providing fair chances. A random number is generated by processors in multiprocessor environment and some token numbers are assigned to each of the process. The execution of process depends upon the condition when the token number of a process is matches with the token number of the processor. The process which has the highest number of tokens has the chance to be allocated the resource for execution of the task. The jobs waiting in the queue always have the chance to be allocated the resource. lottery scheduling maintains the fairness between processes and gives equal chance to each and every process to be allocated the resource. Due to this reason Lottery scheduling is also known as starvation free scheme. In multiprocessor cloud based environment working of Lottery scheduling scheme is
Sarla Mor, Diwakar Shukla
GENERALIZED APPROACH IN MULTIPROCESSORS USING RT&A, No 2 (68)
REGRESSION ESTIMATOR AND COST ANALYSIS_Volume 17, June 2022
similar to draw a random sample through the sampling technique. The remaining time parameter estimation of the ready queue can be executed using the sampling techniques. A job in the ready queue has its process ID, the CPU time(in terms of bytes) as well as the process size (in terms of bytes). With the use of information of process size, it is expected to estimate better the unknown parameter. This paper exploit the approach of use of size measure information for efficient prediction.
Let (ti, xi), (t2, X2), (t3, x3)............( ti, xi)........(tk, xk) be the time consumed by ith process in the
waiting queue having size measure xi. Further let Qi, Q2, Q3,........Qr be the r processors ( r < k) in a
computer system who generate random numbers to select processes for resource allocation. Figure 1 describes the general setup of multiprocessors and ready queue. The Figure 2 and 3 are showing the same but in the classified and categorized manner.
Figure 1: Ready queue with waiting Processes and Multiprocessor, Figure 2: Small size processes and Multiprocessors
This paper takes into account the approach of [4] but adds additional feature of partially processed, blocked processes and size measure of processes for time estimation. All these features are under assumption that the multiprocessor computer system fails at an instant due to unavoidable reasons and backup/recovery management is required. How much the backup cost is needed while sudden breakdown is a question of interest and can be predicted by using the suggested methodology of this paper.
Figure 3: Big size processes and Multiprocessors
II. A Review
The priority scheduling is used when any of the jobs is to prefer over others in the waiting queue. Lottery scheduling is one such similar [8] where the job having highest number of tickets has the high chance of being allocated the desired resource. In Linux kernel setup, the lottery scheduling is useful [18] and it could be utilized as a framework [5, 7] for applying the sampling techniques. The similar job group formation scheme for mean time estimation of a ready queue [6] came into picture using lottery scheduling. A review on ready queue mean estimation [3] has opened up avenues for developing new methods in this area. The lottery scheduling types and model based utilization [16, 17] exists in literature as hybrid multilevel structure using Markov chain model along with analysis and chance based prediction. A sample can be used as a suitable input source for mean value prediction [9, 11, 16]. Many various sampling methodologies exist [10, 13, 14] who are comparatively better over another. The best method of selection among them [15] is always possible for precise prediction of unknown parameter. For missing data, the imputation techniques are popular who to replace the non-responding units [19, 20, 21] by known values. Some of most popular imputation methods are mean imputation, deductive imputation, mean imputation within classes, deductive imputation within classes, hot deck imputation, cold deck imputation etc. ([22, 23, 24, 25]). The content of this paper follows idea of [5] and [4] and uses them as input sources in order to resolve the issue of remaining time estimation in presence of sudden breakdown of the system. The contribution in [26] has opened up avenues to think for the use of size measure of processes.
I. Remaining Time Estimation Problem
Let there are finite number of N processes in a ready queue and n (n < N) have been processed completely before the system breakdown, obviously (N-n) are still in waiting to get signal for resource allocation. One can assume that n processes are just like a random sample selected from ready queue of size N using lottery scheduling. If 9 is mean time obtained through sample then remaining total time estimate is A = [(N-n) 9] which is an unknown quantity. For numbers 'c' and 'd', if A is predicted as Ae (c, d) who is an interval containing A with very high probability, then A 1 = [(N-n) c] is lowest, A 2 = [(N-n) d] is upper expected remaining time. If highest expected time is precisely estimated then it could be used for backup management during system failure. The efficient estimation of this expected range is a problem which is chosen in this paper for strategy formation in the multiprocessor setup with the consideration of multiple real life possibilities.
II. Confidence Interval (CI)
A confidence interval is a kind of predictive range for catching of unknown parameter. The feature of a confidence interval is that it contains the true value with 95% precision. Let P[A] denotes the probability of happening of an event A. In statistical theory, contains for any two real numbers a', b', the 95% confidence interval is defined as P [a' < true unknown value < b'] = 0.95. It could be interpreted as chance of being true value within a', b' is 95 percent. The length of confidence interval is a tool for measure of betterment. It is a difference of lower limit and upper limit. Let there are m different confidence intervals of length (l1, l2, l3, l4 ... lm) who all catch the true value than an efficiency measure is: Best Confidence Interval = min [l1, l2, l3, l4 ... lm]
III. Simulated Cost Aspect
Let C0 be the fixed cost and C1 be the cost per unit predicted time. If £1 is the minimum and 82 is the maximum remaining time after the occurrence of breakdown than
(a) Linear cost function is total cost (Tc)1A = C0 + C1 * 81 and (Tc)2A = C0 + C1 * 82
(b) Square root cost function (Tc)1B = C0 + C1 1 and (Tc)2B = C0 + C1 2
Sarla Mor, Diwakar Shukla
GENERALIZED APPROACH IN MULTIPROCESSORS USING RT&A, No 2 (68)
REGRESSION ESTIMATOR AND COST ANALYSIS_Volume 17, June 2022
(c) Squared cost function is (Tc)ic = Co + Ci * Si2 and (Tc)2c = Co + Ci * fo2
Overall average cost = [Linear cost + Square root cost + Squared cost] / 3 The average cost is likely to incur in the recovery management of resources after the system breakdown. Averaging over linear, squared function and square-root function is taken to control the sampling fluctuations due to lottery scheduling sample.
IV. Sample based Estimation Method
Let (Yi, Xi), (Y2, X2), (Y3, X3).........(Yn, Xn) be the data of totality of size N where Y is variable of
main interest and X is the support correlated information. For example, the Y may be expenditure of army officers in a country while x is income data which is known from the salary register of organization/head quarter. The mean of population is Y = (1/N) £ Yi and X = (1/N) £Xi
Figure 4: Sample selection from Aggregate (n<N)
A sample of size n (n<N) is drawn randomly from N by simple random sampling without replacement method. Sample values are (yi, xi), (y2, x2), (y3, x3) ... (yn, xn). Sample mean are y = (1/n) £ yi and x = (1/n) £xi
The objective is to estimate unknown parameter Y using known X along with sample means y and x. Some well known estimators are:
• Sample mean estimator: y
• Ratio-estimator: = y (X/x)
• Difference estimator: y^ = y + d (X — x)
III. Motivation
Earlier contributions (specially [4], [5]) were under assumption that processes who exist in a multiprocessors system are completed before sudden failure. But this is not a practical reality. Since some jobs may complete, some may partially processed and some may blocked by the processors [see figure 4]. The processed and unprocessed case was considered in [4] [see figure (6)]. This paper extends the approach of [4] and [26] by applying the tools of random imputation method against the blocked processes.
Figure 5: Ready Queue Processing under Lottery Scheduling (due to [6])
Figure 6: Setup of ready queue and multiprocessor environment (due to[23])
IV. Proposed Generalized Computational Setup
Assume the existence a virtual sampled ready queue in a computer system having multiprocessors environment. Some jobs are randomly selected using lottery scheduling from the ready queue and placed in the sampled ready queue from top to bottom in the sequential manner of their selection. Processors are assigned processes in the ordered manner from top to bottom of the virtual sampled ready queue. Figure 5 shows basic setup of this approach but without the size measure while figure 5 shows the earlier approaches [4], [5], [6], [7]. Moreover, figure 6 reveals the special case when all sample units processed before the occurrence of breakdown.
Figure 7: Sampled Ready Queue Processing Time Estimation setup without size measure
V. Generalized Assumption and Model
As per figure 7, let the selection of processes is according to lottery scheduling. The process who selects first is placed at the top of the virtual queue who is segment or group of processes likely to allocate to the multi-processors.
1. Assume r processors and a ready queue of N processes in a system like denoted as [Pi, P2, P3.........Pn] waiting for allocation of resources.
2. The selection of process for resource allocation is on priority basis using lottery scheduling.
3. If all N are processed completely, time consumed are [ti, t2, t3 ...Tn] who has known size measure [xi, x2, X3 ....xn].
4. Overall ready queue mean time t= -£¿=1 tj, mean size measure x = 1 Xj mean squares
N '
St2 = ¿SLi^-f)2, Sx2= -i-^i^-x-)2.
5. The Pi of known size Xi consumes time ti( i = 1,2,3,......N) when all assumed processed.
6. Consider r multiprocessors Qi, Q2, Q3.....Qr, (r < N) and time consumed by the ith process in
the jth processor is tij with corresponding size measures xij (j = 1,2,3,......r)
7. The unknown total completion time of ready queue is NT, which is an unknown quantity. This paper is focused to estimate such using sampling methodology. Lottery scheduling is a tool for such estimation where process Pi has a bunch of token numbers and Qj generates a random number. A process who receives the random number gets the desired resource from Qj. This scheduling produces a random sample.
8. A virtual ready queue of size k (k < N, k>3r) exists to store sequentially the records of randomly selected k processes from N. The jth segment of virtual sampled queue is kj( k =£;=1 kj ), who is allocated to the jth processor Qj in sequential manner.
9. In sample, let sxjl denotes the file size measure and stjl denotes time consumed by ith process in Qj (l = 1,2,3,...kj) when all processed completely who are included in the sample of size k.
■ Sample mean of time st= £fcii stjl
1 w v^fc ,■
Sample mean square of time, (es)t2 = — £J=1 £ ¿¿^j — st )
fc_i' 1 v^fc; v^fc;
Sample mean square of size, (es)x2 = — £J=1 £ ¿^(sxji — sx )2
Sample mean of size, ( sx)= ¿ii(sxji)
1 vr vfcj — £j = 1£ ¿=i(
i. The term iFt, sx, (es)t2 , (es)x2 hold when system runs without failure.
10. Assume system breakdown occurs at the time instant T and there are (kj - n'j - n"j) processes completed in Qj, but n j remain partially processed and n j remain unprocessed (blocked). This is an assumed generalized model shown in figure 7. Define g = £J=1 n 'j and u = £J=in"j
11. Let (st')jl is time consumed by the lth process in the processor Qj [l =1, 2, 3... (kj - n'j -n' 'j)],who is among those processed completely before the occurrence of T.
12. Some sample mean related measures are:
■ Sample mean of (kj - n'j - n' 'j) process, (T-1—— Z(kj n j n j)(st'ji)
j (kj - n j - n j) i_1 jI
■ Sample mean square, (es'^. 2 = £[Lin j - n j)(st'ji — (st0j)2
■ Similar is for size measure also as (sx'jj) represents size of lth process who is in Qj before T.
■ Sample mean,(sx') j= (kj _n^n,tj) £ ¡Li- n j - n j)(sx 'jl)
■ (sx)j= --1—— Z(_i n j n j)(sx'ji)is sample mean of all kj known values related to x
(kj - n'j - n' j) i_1 J
in jth segment of ready queue.
■ Sample mean square, (ex,)^.2 = ^ ZiLi- n j - n — («0j)2
■ Sample Covariance, (es x) j = (k - ^ Z (=j1- n j - n j)(^t ;7 — (st0j)(5x ;7 — (sx ' )j)
Sarla Mor, Diwakar Shukla
GENERALIZED APPROACH IN MULTIPROCESSORS USING RT&A, No 2 (68) REGRESSION ESTIMATOR AND COST ANALYSIS_Volume 17, June 2022
13. Assume is partially processed time of a process in Qj (j = m =1,2,3....r) whose sample mean under T is
14. (t */T) = i£m=iC, Variance (t * / T) = V(t * / T) = ( 1 — ) St2, where St2 is the conditional ready queue mean square of the remaining un-sampled part [N- k + g] expressed as:
St2 = —1--S^-fc+s(ti-tT )2 where tT = (t;) where g = E^nj
15. Herein to mention that St2 and ^contain time ti only from non-sampled processes (N-k) of the main ready queue with the addition of those g who partially processed. For such, the
size converts from N into (N - k + g) and only those processes are the part of tT and St2 who are in (N - k + g).
16. The u blocked processes are imputed by Random Imputation Method using random selection of a process among (kj - n'j - n''j) relating to Qj. Let from Qj all random imputed time are denoted as tm**.
1
■ Sample mean of all random imputed time, t ** = -Em=1 *
■ Variance of imputation under T,V(t **/T) =(1 -1) (es)2, u < k.
17. Sample based estimate of (es)2 can be obtained by using all k values of time consumption in sample including the partially processed time tm* and imputed time value tm**. It is
1 fc * _ * 9
denoted as (es*)2 and mathematically expressed as (es*)2 = — E^I^i^st ji — st )2
_*
where (st*jl) and st include completely processed time st*ij , partially processed tm* and imputed tm**.
18. The sample estimate of St2 is (es')2 = ^[ -t* )2 ]
19. Bias of estimation strategy is assumed negligible wherever appears and applicable in mathematical expressions
I. Computational Set-up
Aim is to compute the remaining ready queue processing time after occurrence of sudden failure of system at time instant T. This is subject to condition that r processes are partially processed, r is unprocessed (blocked) and remaining fully completed. Blocked and partially processed are nj' and nj'' from every Qj and known size measures are the part of computation. Some frequently used symbols for process time t and process size measure X are as under:
t= 121=1^« = £ I I (1)
£ = (2)
t **=¿SUC * (3)
(*Oj= nj - n''p(stji) (4)
Mj= (jj^Zj n'j - n'j)(sxji) (5)
1 T-fc;
j= (¿¡jl^'ji) (6)
(es')j2 = 1/(kj - n'j - n''j — 1) E(=1- n'j - n"i)(st'ji — (s"t )j)2 (7)
(ex')j2 = 1/(kj - n'j - n'j — 1) E(=1- nj - n"i)(sx'ji — (sx')j)2 (8)
(es'x)j (k- n._ nV1) ^(=j- n'j - "'V )l — (^')j) (« )l — W)j (9)
J J J . *
(es*)2 = 2fc=1 (st *ji — "st *)2 (10)
t ■ = [(.st )j + dj{(sx)j — (s^')j}], dj being constant, (0< dj < (11)
Note: The tr. is a Difference type estimator as stated in subsection IV of section II.
II. Estimation Strategy
The sample based proposed estimation strategy for mean time is:
(tmean/T) = €l [ ^Wj (t^/T) ] + €2 (t */ T) + (1- €1 - €2) (t **/T) (12)
with condition that Ep=i €p= 1 and €pdenotes constants to be determine suitability and wj= (kj/k) is known weight (£wj =1). With the help of Cochran [16; see page 166, page 27, 29] for tmean, the expected value E[.] is expressed as:
E [tmean/T] =E[ €1 [ Wj (try /T)] + €2 (t */T) + (1- €1 - €2) (t **/T)]
=€1 [ Ey=i Wyf (try /T)] + €2 E (t */T) + (1- €1 - €2)E (t "/T)j (13)
*t which shows estimator (tmean/T) is biased.
III. Mean Squared Error
Let MSE (.), V (.) and B (.) denote mean squared error, variance and bias respectively. One can express
MSE (tmean/T) = Variance (tmean/T) + [Bias (tmean/T)]2 which holds in general. Assume the bias is small, therefore negligible (as in assumption no. 16)
MSE (tmean/T) = Variance (tmean/T) = €i2[ Wj2 V(try/T)]+ €22V (t */T) + (1- €1 - €2)V (t **/T)]
= €12 Z=1 j^S Wj^O^ftex> - js'OyH+^g - ^>t2]+(1- €1
- €2)2 Z^J1^-^^")) Wj (es')j2] (as per Cochran[12] page 24, page 29
and page 164) (14)
The expressions P, Q, R are in the sample based estimate form of population parameters
Let P = fejjrS j («> -2dj V)A
Q = p--1 )st2
R = Vr (l--1-) w;2(es');2
Zj^V (kj - nj - n' j)/ j ^ J j
The above expression is re-written as:
V[tmean/T] = [€12 P + €22 Q+ (1- €1 - €2)^ ] ignoring the covariance terms due to independency. For optimum variance, differentiate V[tmean/T] with respect to €1 and €2 and equate to zero, one gets (€1) opt = (QR) / [PQ+PR+QR] = QM (15)
(€2) opt = PQ/ [PQ+PR+QR] = PM where M = R/ [PQ+PR+QR] (16)
One can differentiate the variance expression by dj also to get optimum value which is (dj)opt=[(es'x')y/(ex )j2] Substituting optimum choices in expression, the optimum variance is: V[tmean/T]opt = (€1) 2opt P + (€2) 2opt Q + (1- (€1)opt- (€2)opt) 2fl] with (dj)opt (17)
VI. Numerical Illustration Consider the 150 processes with processed CPU time whose details are in table 1 with assumption that all 150 processes have been completed.
Table 1: System Ready Queue Processes with time (N = 150)
Process J1 J2 J3 J4 J5 J6 J7 J8 J9 J10 J11 J12 J13 J14 J15
CPU Time 30 20 42 45 59 35 25 48 50 60 32 55 62 47 69
Process Size 41 71 103 142 316 82 199 163 220 127 76 192 251 52 133
Process J16 J17 J18 J19 J20 J21 J22 J23 J24 J25 J26 J27 J28 J29 J30
CPU Time 34 24 44 70 57 65 38 84 101 66 80 90 92 111 85
Process Size 318 202 106 181 242 148 46 252 136 222 261 97 109 271 116
Process J31 J32 J33 J34 J35 J36 J37 J38 J39 J40 J41 J42 J43 J44 J45
CPU Time 61 52 72 75 89 67 51 78 80 91 63 86 93 77 99
Process Size 172 243 253 262 83 203 183 166 219 193 223 272 281 301 289
Process J46 J47 J48 J49 J50 J51 J52 J53 J54 J55 J56 J57 J58 J59 J60
CPU Time 64 54 74 100 87 95 68 114 131 96 110 123 122 141 49
Process Size 205 244 223 254 146 263 53 218 273 139 282 302 173 309 290
Process J61 J62 J63 J64 J65 J66 J67 J68 J69 J70 J71 J72 J73 J74 J75
CPU Time 118 81 102 105 119 97 88 108 110 121 240 113 122 107 129
Process Size 313 194 153 255 225 169 206 264 58 274 283 303 184 291 216
Process J76 J77 J78 J79 J80 J81 J82 J83 J84 J85 J86 J87 J88 J89 J90
CPU Time 94 73 104 130 117 234 98 237 161 126 143 236 152 171 233
Process Size 207 246 228 360 256 275 217 265 226 195 284 292 304 300 280
Process J91 J92 J93 J94 J95 J96 J97 J98 J99 J100 J101 J102 J103 J104 J105
CPU Time 120 112 132 135 149 125 115 138 140 150 122 232 152 137 159
Process Size 247 79 208 276 285 257 56 293 266 187 305 178 310 299 215
Process J106 J107 J108 J109 J110 J111 J112 J113 J114 J115 J116 J117 J118 J119 J120
CPU Time 124 114 134 160 147 155 128 174 191 156 170 180 182 201 175
Process Size 277 286 211 248 227 294 157 258 229 267 196 298 188 306 270
Process J121 J122 J123 J124 J125 J126 J127 J128 J129 J130 J131 J132 J133 J134 J135
CPU Time 235 142 162 165 179 151 145 168 171 238 152 175 189 167 241
Process Size 287 278 295 197 249 307 268 311 213 350 112 314 259 297 230
Process J136 J137 J138 J139 J140 J141 J142 J143 J144 J145 J146 J147 J148 J149 J150
CPU Time 154 144 164 190 177 185 158 204 221 186 200 210 212 231 209
Process Size 214 250 260 279 288 296 308 269 312 245 317 198 319 315 239
Table 2: Descriptive Statistics of Table 1
S. No. Parameters Name Calculated value
1 Number of Processes N 150
2 Mean time (t) 122.56
I. Case-I: where each sample size k=40, and dj = 0 (di = 0, d2 = 0, d3 :
Table 3: Calculation for Sample No. 1
ki:16
k2:13
0)
ks:11
{(j0i),(30),(41)},{(j3i),(61),(172)},
{(j6i),(118),(313)},{(j9i),(120),(247)},
{(ji2i,(235),287)},{(j63),(102),(153)},
{(J32),(52),(243)},{(J62),(81),(194)},
{(j4(112),(79)},{(ji22),(142),(278)},
{(ja),(42),103)},{(j33),(72),(253)},
{(ji4i),(185),(296)},{(j2i),(65),(148)},
{(j86),(143),(284)},{(ji00),(150),(187)}
ni' = 2, ni'' = 3
Partial Processed
={(j33),(72)(253)}{(ji4i),(185),(296)}
(Processed=50 unprocessed=22)
(Processed=90 unprocessed=95)
Blocked = {(j2i),(65),(148)},
{(j86),(143),(284)},{(ji00),(150),(187)}
Blocked replaced
ai' = {(j9i),(120),(247)}
a2' ={(j32),(52),(243)}
a3 = {(j0i),(30),(41)}
[«ft! = 99.54, from eq.(4.4),
(es ')i2 = 3330.87, from eq.(4.7)],
[sxi=3583/16 = 223.94, from
eq.(4.5)
Sxi'=2110/11=191.81 from eq. (4.6)],
[(ex ')i2 =8210.36, from eq.(4.8)] [(es'x')i = 3230.60, from eq.(4.9)]
{(J49),(100),(254)},{(J 34),(75),(262)}, {(J29),(111),(271)},{(JS9),(141),(309)}
{(J 64),(105),(255)},{(J94),(135),(276)}, {(J28),(92),(109)},{(J96),(125),(257)}
{(Ji24),(165),(197)},{(Ji35),(241),(230)} {(Jii9)(201)(306)},{(Ji49)(231)(315)},
{(J 35),(89),(83)},{(J65),(119),(225)}, {(Ji42),(158),(308)},{(Js7),(115),(56)},
{(J95),(149),(285)},{(Ji50),(209),(239)}, {(Ji08),(134),(211)},{(Jii2)(128)(157)},
{(J99),(140),(266)},{(J143),(204),(269)}, {(J120), (175), (270)} {(J116),(170),(196)}
ni' = 2, ni'' = 2
Partial Processed={(ji50)(209)(239)} {(j99),(i40),(266)}
(Processed=i20, unprocessed=89)
(Processed=90, unprocessed=50),
Blocked={(ji43),(204),(269)},
{(jii6),(i70),(i96)}
Blocked replaced
Pi' ={(j64),(105),(255)}
(32 ={(ji35),(241),(230)}
[«to' = 130.88, from eq. (4.4), (es>2 = 2534.61 from eq.(4.7)] [sx2 =3149/13 =242.23, from eq.(4.5),
_ , 2067 SX2 = - =
9
229.66, from eq. (4.6), (ex )22 =3761,from eq.(4.8)] [(es'x')2 = 387.45, from eq.(4.9)]
ni' = 2, ni'' = 3
Partial Processed =
{(j142)(158)(308)} {(j97)(115)(56)}
(Processed=110unprocessed=48),
(Processed=65 unprocessed=55),
Blocked={(j108),(134),(211)},
{(j112)(128)(157)},{(j120)(175)(270)}
Blocked replaced
Y1' = {(ju9)(201)(306)}
72' = {(j59),(141),(309)}
y£ ={(j29),(111),(271)}
[<to = 150.16, from eq.(4.4),
(es')32 = 2950.56 from eq.(4.7)]
[SX3 =2641/11=240.09,from eq.(4.5)
SX3= "56" = 261.16, from eq. (4.6)
[(ex )32 = 6092.96, from eq.(4.8)]
[(es'x')3 = 2952.56, from eq.(4.9)]
t* = (50+90+120+90+110+65)/6 = 87.5
f* = (a' + P' + y')/8 = (120+52+30+105+241+201+141+111) / 8 = 125.13 Estimated [st2 = 2,204.16] (using point 15) St2 is (es')2 = ^ [ -t * )2 ]
[(50-87.5)2 +(90-87.5)2 +(190-87.5)2 +(110-87.5)2 +(140-87.5)2+(95-87.5)2]/5 =[4,333.58+1,167.58+4,117.78+250.58+200.78+950.48]= 2,204.16
" =p-"} wj2 {(es')j2+dj2 (e* ')j2 -2rij (esx'- ~
—-}w,2(es'>2
- n'j - n"jV j V J.
Let P = Y' (-1--1)
Z—i_/=i Vkj - n'j - n''j) k)
r ■ z;=, (=
W-fc+3
)sT2
P = I
J7' = 1
1
j - n'j -
( 16-b-40) (0-4)2 *{3330-87}+ ( 1
1
y, Q=(1-:
40) (O.33)2 {2534-61}+ ( C0-28)" {295°.56)
16-2-3 40/ v ' V 13-2-2 40/ v ' V 11-2-3 40
0.0659 *0.16*3330.87+ 0.0861*0.1089*2534.61+0.1416*0.0784*2950.56= 91.64
Q = ( 1--1-) 2,204.16 = 0.3245 *2,204.16 = 715.25
^ v 3 150-40+3
R = ( 1 _ I^i ) (0.4)2*3330.87+ ( 1-75^) (0.33)2 *2534.61 +( 1-77^) (0.28)2 *2950.56
= 0.9091 *0.16*3330.87+ 0.8889*0.1089*2534.61 +0.8334*0.0784*2950.56 = 922.63 Calculation of mean and Variance V[tmean/T] at dj = 0 (for all j = 1,2,3) (€1)opt= (QR) / [PQ+PR+QR] = QM = 715.25*922.63/[91.64*715.25+91.64*922.63+715.25*922.63]
= 659911.1075/810006.4307 = 0.8147 (€2)opt= PQ/ [PQ+PR+QR] = PM = 91.64*715.25/[91.64*715.25+91.64*870.50+715.25*870.50] = 65545.51/ 810006.4307= 0.0809
(tmean/T) = (€:)opt [ trj] + (€2)opt (t *) + (1- (€:)opt - (€2)opt) (t **)
trj= [(iTt')j + dj{(sx)j - (sx)j)],
trj= [0.4*99.54+0*(223.94-191.81)]+[0.33*130.88+0*(242.45-229.66)] +[0.28*150.16+0*(240.09-261.16)]
= 39.82+43.19+42.04 = 125.05 (tmean/T) = 0.8147*125.05+0.0809*87.5+0.1044*125.13 = 122.02
V[tmean /T] = (€1) 2opt P + (€2) 2opt Q + (1- (€1)opt- (€2)opt) 2R]
V[tmean/T] = [(0.8147)2 *91.64+ (0.0809)2*715.25 + 0.0108*922.63] = 60.82+4.68+9.96 = 75.46 The 95% confidence intervals for t, P [(tmean/T) ± 1.96^[ V (tmean/T)] = 0.95 = 122.02± 1.96V75.46 = 122.02 ± 17.02 = (104.99, 139.04)
Table 4: Estimated Sample Mean, Variance and Confidence Interval (CI) of Ten Random Samples
Case-I: At (€1)opt, (€2)opt, dj = 0 (d1 = 0, d2 = 0, d3 = 0) where True mean = 122.51
S.No. Estimated Sample Mean V[tmean/T] 95% Confidence Interval (CI) CI Length
1 122.02 75.46 (104.99, 139.04) 34.05
2 134.58 64.83 (118.79, 150.36) 31.57
3 117.56 74.36 (100.66, 134.46) 33.80
4 113.89 48.45 (100.25, 127.53) 27.28
5 127.00 85.37 (108.89, 145.11) 36.22
6 119.27 46.42 (105.92, 132.62) 26.70
7 123.39 45.41 (110.18, 136.60) 26.42
8 113.12 97.36 (93.78, 132.46) 38.68
9 115.01 53.05 (100.73, 129.28) 28.55
10 120.21 60.91 (104.91, 135.51) 30.60
Average Length (3138/10) = 31.38
Figure 8: Graphical representation of Confidence Interval range of Ten Random Samples for Case-I of Table 4 ( X-axis has sample number as shown in table 4)
II. Case-II: where each sample size k=40, and (dopt)j = (es'x')j / (ex )
Table 5: Calculation for Sample No. 1
ki:16
k2:13
ks:11
{(J49),(100),(254)},{(J 34),(75),(262)},
{(J 64),(105),(255)},{(J94),(135),(276)},
{(Ji24),(165),(197)},{(Ji35),(241),(230)}
{(J 35),(89),(83)},{(J65),(119),(225)},
{(J95),(149),(285)},{(JI50),(209),(239)},
{(J99),(140),(266)},{(JI43),(204),(269)},
{(Jii6),(170),(196)}
ni
: 2
{(Joi),(30),(41)},{(J3I),(61),(172)}, {(J61),(118),(313)},{(J91),(120),(247)}, {(J121,(235),287)},{(J63),(102),(153)}, {(J32),(52),(243)},{(J62),(81),(194)}, {(J92),(112),(79)},{(J122),(142),(278)}, {(J3),(42),103)},{(J33),(72),(253)}, {(J141),(185),(296)},{(J21),(65),(148)}, {(J86),(143),(284)},{(J1OO),(150),(187)} ni'= 2, ni''= 3
Partial Processed = {(j33),(72),(253)}, {(J141),(185),(296)} (Processed=50, unprocessed=22), (Processed=90, unprocessed=95),
Blocked = {(J21),(65),(148)}, {(J86),(143),(284)},{(J1OO),(150),(187)} Blocked replaced ai={(j91),(120),(247)},a2'={(j32),(52),(243 ={(J64),(105),(255)}
2, ni' =
Partial
Processed={(j15o),(209),(239)}, {(j99),(140),(266)}
(Processed=120, unprocessed=89) (Processed=90, unprocessed=50), Blocked={(j143),(204),(269)}, {(JU6),(170),(196)} Blocked replaced
)}
«3 = {(Jo1),(30),(41)} [«ft! = 99.54, from eq.(4.4), (es')12 = 3330.87, from eq.(4.7)],[sx1=3583/16 = 223.94, from eq.(4.5), Sx1=2110/11=191.81from eq. (4.6)], [(ex>2 =8210.36, from eq.(4.8)] ,[(es'x')1 = 3230.60, from eq.(4.9)],(dopt)1 = (es'x')1 / (ex')12 = 3230.60/8210.36 =0.3935
(32' ={(j135),(241),(230)} [«to' = 130.88,
from eq. (4.4), (es>2 = 2534.61 from eq.(4.7)],[Sx2 =3149/13 =242.23, from eq.(4.5),sx2 = ^ = 229.66, from eq. (4.6), (ex>2 =3761,from eq.(4.8)],[(es'x')2 = 387.45, from eq.(4.9)],(dopt)2= (es'x')2 / (ex ')22 = 387.45/3761 = 0.1030
{(J29),(111),(271)},{(J59),(141),(309)}
{(J28),(92),(109)},{(J96),(125),(257)}
{(Jii9)(201)(306)},{(Ji49)(231)(315)},
{(JI42),(158),(308)},{(J97),(115),(56)},
{(Ji08),(134),(211)},{(Jii2)(128)(157)},
{(J120), (175), (270)}
ni'= 2, ni''= 3 Partial Processed = {(j142),(158),(308)},{(j97)(115)(56)} (Processed=110, unprocessed=48), (Processed=65, unprocessed=55),
Blocked={(j108),(134),(211)}, {(j112)(128)(157)},{(j12o)(175)(270)} Blocked replaced
y1'={(j119)(201)(306)},y2'={(j59),(141),(309 )}
Y3 ={(j29),(111),(271)}
[<to' = 150.16, from eq.(4.4),(es')32 =
2950.56 from eq.(4.7)] ,[sx3
=2641/11=240.09,from eq.(4.5),sx3=
1567 = 261.16, from eq. (4.6), [(ex>2 = 6
6092.96, from eq.(4.8)],[(es'x')3 = 2952.56, from eq.(4.9)],(dopt)3= (es'x')3 / (ex )32 =2952.56/6092.96 = 0.48
t* = (50+90+120+90+110+65)/6 = 87.5
t** = (a' + (' + y')/8 = (120+52+30+105+241+201+141+111) / 8 = 125.13 Estimated [st2 = 2,204.16] (using point 15) St2 is (es ')2 = ^ [ (C-t * )2 ]
[(50-87.5)2 +(90-87.5)2 +(190-87.5)2 +(110-87.5)2 +(140-87.5)2+(95-87.5)2]/5 =[4,333.58+1,167.58+4,117.78+250.58+200.78+950.48]= 2,204.16
Let P
= 1 Vkj - n'j - n''j) R = Vr (l--1-) wj2(es')j2
/_,,=-, V (kj - n'j - n'' j)J j ^ Jj
i) wj2 {(es')j2+dj2 (ex')j2 -2dj (es'x')^ Q = (1- ^->t2
P =
J7' = l
1
( —1--—) (0.4)2 *{3330.87+0.39*0.39*8210.36 - 2*0.39*3230.60}+ ( —1--—) (0.33)2
V 16-2-3 40/ V 13-2-2 40/ v '
{2534.61+0.10*0.10*3761- 2*0.10*387.45}+ ( (0.28)2 {2950.56 +0.48*0.48*6092.96
2*0.48*2952.56} 11-2-3 40
= 0.0659 *0.16*2059.79+ 0.0861*0.1089*2494.73+0.1416*0.0784*1519.92 = 61.98 Q = ( 1--1-) 2,204.16 = 0.3245 *2,204.16 = 715.25
^ v 3 150-40+3
2
1
) (0.4)2*3330.87+ ( (0.33)2 *2534.61 +( 1-77^} (0.28)2 *2950.56
R = ( 1
= 0.9091 *0.16*3330.87+ 0.8889*0.1089*2534.61 +0.8334*0.0784*2950.56 = 922.63 Calculation of mean and Variance V[tmean/T] at dj = (dopt)j
(€1)opt= (QR) / [PQ+PR+QR] = QM = 715.25*922.63/[61.98*715.25+61.98*922.63+715.25*922.63]
= 659911.1075/761426.9099= 0.8666 (€2)opt= PQ/ [PQ+PR+QR] = PM = 61.98*715.25/[61.98*715.25+61.98*870.50+715.25*870.50] = 44331.195/ 761426.9099 = 0.0582
(tmean/T) = (€1)opt [ E^Wj trj] + (€2)opt (t *) + (1- (€1)opt - (€2)opt) (t **)
tjr [(iTt )j + dj{(sx)j - (sx')j}],
trj= [0.4*99.54+0.39*(223.94-191.81)]+[0.33*130.88+0.10*(242.45-229.66)] +[0.28*150.16+0.48*(240.09-261.16)] = [0.4*99.54+12.64]+[0.33*130.88+2.11] +[0.28*163.33-45.50] = 52.45+44.47+31.93 = 128.85 (tmean/T) = 0.8666*128.85+0.0582*87.5+0.0752*125.13 = 126.16
V[tmean /T] = (€1) 2opt P + (€2) 2opt Q + (1- (€1)opt- (€2)opt) 2R]
V[tmean/T] = [(0.8666)2 *61.98+ (0.0582)2*715.25 + 0.0056*922.63] = 46.54+2.42+5.17 = 54.13 The 95% confidence intervals for t, P [(tmean/T) ± 1.96^[ V (tmean/T)] = 0.95 = 126.16 ± 1.96V54I3 = 126.16 ± 14.42 = (111.74, 140.58)
Table 6: Estimated Sample Mean, Variance and Confidence Interval (CI) of Ten Random Samples
Case-II: At (€1)opt, (€2)opt, (dopt)j = S.No. Estimated Sample Mean V[t m
(es'x')j / (ex )j2where True mean = 122.51 an/T] 95% Confidence Interval (CI) CI Length
1 126.16 54.13 (111.74, 140.58) 28.84
2 130.78 39.24 (118.50, 143.06) 24.56
3 125.24 48.98 (111.52, 138.96) 27.44
4 124.84 45 (111.70, 137.99) 26.29
5 128.89 53.58 (114.54, 143.24) 28.7
6 140.30 100.86 (120.62, 159.98) 39.36
7 125.99 29.81 (115.29, 136.69) 21.4
8 110.79 77.25 (93.56, 128.02) 34.46
9 128.36 50.01 (114.50, 142.22) 27.72
10 128.07 38.42 (115.92, 140.22) 24.3
Average Length (28307/10) = 28.30
Figure 9: Graphical representation of Confidence Interval range of Ten Random Samples for Case-II of Table 6 (X-axis has sample number as shown in table 6)
Table 7: Comparison between Case-I and Case-II
S. NO CASE-I CASE-II
dj = 0 (di = =0, d2 =0, d3 =0) (d)j =(dopt)j
95% Confidence Interval Length 95% Confidence Interval Length
1. (104.99, 139.04) 34.05 (111.74, 140.58) 28.84
2. (118.79, 150.36) 31.57 (118.50, 143.06) 24.56
3. (100.66, 134.46) 33.8 (111.52, 138.96) 27.44
4. (100.25, 127.53) 27.28 (111.70, 137.99) 26.29
5. (108.89, 145.11) 36.22 (114.54, 143.24) 28.7
6. (105.92, 132.62) 26.7 (120.62, 159.98) 39.36
7. (110.18, 136.60) 26.42 (115.29, 136.69) 21.4
8 (93.78, 132.46) 38.68 (93.56, 128.02) 34.46
9. (100.73, 129.28) 28.55 (114.50, 142.22) 27.72
10. (104.91, 135.51) 30.6 (115.92, 140.22) 24.3
Average Length (3138/10) 31.38 Average Length (2830/10) 28.30
Table 8: Case-I: Cost aspect when Co = 100 units, Ci = 10 units
C.I C I & 52 Total cost Total cost
S. NO Lower Limit Upper Limit 52 (Tc)1A (Tc)1B (Tc)1C (Tc)2A (Tc)2B (Tc)2C
1 104.9 139.0 11,548. 15,294. 115589 1174. 1333771 1530 1336.7 2339186
9 4 90 40 65 012 44 053 814
2 118.7 9 150.3 13,066. 6 90 16,539. 60 130769 1243. 10 1707438 856 1654 96 1386.0 63762 2735583 782
3 100.6 134.4 11,072. 14,790. 110826 1152. 1226024 1480 1316.1 2187618
6 6 60 60 26 808 06 66107 584
4 100.2 127.5 11,027. 14,028. 110375 1150. 1216057 1403 1284.4 1967932
5 3 50 30 11 663 83 11246 109
5 108.8 145.1 11,977. 15,962. 119879 1194. 1434700 1597 1363.4 2547886
9 1 90 10 43 984 21 12047 464
6 105.9 132.6 11,651. 14,588. 116612 1179. 1357504 1459 1307.8 2128155
2 2 20 20 40 714 82 16211 892
7 110.1 8 12,119. 136.6 80 15,026. 00 121298 1200. 89 1468895 620 1503 60 1325.8 05857 2257806 860
8 93.78 132.4 10,315. 6 80 14,570. 60 103258 1115. 66 1064157 396 1458 06 1307.0 87404 2123023 944
9 100.7 3 129.2 11,080. 8 30 14,220. 80 110903 1152. 63 1227730 581 1423 08 1292.5 09958 2022311 626
10 104.9 135.5 11,540. 14,906. 115501 1174. 1331739 1491 1320.9 2221918
1 1 10 10 24 180 61 05402 272
Average 115501 1173. 74 1336802 081 1500 26.7 1324.0 88329 2253142 435
NOTE 8.1: Overall average cost by lower limit = (115501 +1173.743546+ 1336802081)/3
= 445639585.25 units
NOTE 8.2: Overall average cost by upper limit = (150026.7 + 1324.088329+ 2253142435)/3
= 751097928.59 units
Table 9: Case- II: Cost aspect when C0 = 100 units, C1 = 10 units
C.I
S. Lower NO Limit
C I Upper Limit
51 52 Total cost Total cost
51 52 (Tc)1A (Tc)1B (Tc)1C (Tc)2A (Tc)2B (Tc)2C
12,291 15,463 123014 1208.66 15107 1547 1343.5 23912
.40 .80 5865 85240 38 35283 91204
13,035 15,736 130450 1241.70 16991 1574 1354.4 24764
.00 .60 9245 12350 66 56057 05896
12,267 15,285 122772 1207.57 15048 1529 1336.3 23364
.20 .60 3925 42058 56 49465 95774
12,287 15,178 122970 1208.46 15097 1518 1332.0 23039
.00 .90 741 03790 89 26785 90152
12,599 15,756 126094 1222.47 15874 1576 1355.2 24826
.40 .40 049 48904 64 44996 41510
13,268 17,597 132782 1251.87 17604 1760 1426.5 30968
.20 .80 673 51412 78 66998 25748
12,681 15,035 126919 1226.13 16083 1504 1326.2 22607
.90 .90 9423 05976 59 09607 82988
10,291 14,082 103016 1114.47 10591 1409 1286.6 19830
.60 .20 5234 70406 22 84457 83668
12,595 15,644 126050 1222.27 15863 1565 1350.7 24474
.00 .20 4476 40350 42 67764 10036
12,751 15,424 127612 1229.21 16259 1543 1341.9 23790
.20 .20 2115 31114 42 42028 59556
Average value 124167. 9 1213.28 6491 15452 09160 15530 5.6 1345.3 78344 24157 98653
1 111.74
2 118.5
3 111.52
4 111.7
5 114.54
6 120.62
7 115.29
8 93.56
9 114.5
10 115.92
140.58 143.06 138.96 137.99 143.24 159.98 136.69 128.02 142.22 140.22
NOTE 9.1: Overall average cost by lower limit = (124167.9+1213.286491+ 1545209160)/3
= 515111513.72 units
NOTE 9.2: Overall average cost by upper limit = (155305.6+ 1345.378344+ 2415798653)/3
= 805318434.65 units
900000000
aoooooooo
700000000 600000000 Vi 500000000
o
• J 400000000 300000000 200000000 100000000 0
Caieil Casel
erase L
■
1 1
Figure 10: Pair of graph lines for Case-I and Case-II
VII. Discussion
In Section VI the data description is in Table 1 and 2 where 150 processes are presented assuming all finished before T. Their total processing time and size process measures are noted. The proposed estimate tmean has unknown constants €i, €2 and d whose suitable values need to be obtained for obtaining a best estimate. Two cases are considered herein as Case I: €1 = (€i)opt, €2 = (€2)opt, and di = 0, d2 = 0, d3 = 0.
This case indicates for no use of size measure in the estimation strategy at the optimum choice of €1 and €2. The average confidence interval length, under Case-I is 31.38 as evident form table 3. The lowest predicted total remaining time is 11540.1 units while highest is 14991.9 units (table 10). Average cost consumption for lowest estimated time is 445639585.25 units and at highest time level it is 751097928.59 units (table 8).
Case II: €1 = (€1)opt, €2 = (€2)opt, and d1 = (dopt)1, d2 = (dopt)2, d3 = (dopt)3
This case contains choice of all constants at the optimum level and size measure information x has also been used. The impact of using the support information seems positive since the average length reduced to 30.53 in this case with respect to Case-I while simulated over 10 samples. Figure 9 also reveals for more condensed pair of graph lines for Case-II. Lowest predicted remaining time is 12406.9 units and highest is 15521 units (Table 10). Average cost likely to consume is 515111513.72 units as minimum whereas 805318434.65 units as highest (Table 9).
The percentage relative efficiency of Case-II with respect to Case-I is 9.82 % which supports the use of size measure in estimation (Table 2). The highest cost by Case-I and lowest by Case-II are the recommended cost required for infrastructure creation for backup management (Figure 10).
Table 10: Ten Sample average Confidence Interval and estimated total Remaining time of processing _for Recovery Management_
Case-I Case-II True
(Without size measure) (With size measure) Value
Average Interval (Over 10 samples) (104.91 - 136.29) (112.79 - 141.10) _ „ 122.51
CILength 31.38 28.30
Lowest Predicted (N-k)* 104.91 = 11540.1 (N-k)* 112.79 = 12406.9
Remaining time units units ------
Highest Predicted (N-k)* 136.29= 14991.9 (N-k)* 141.10 = 15521
Remaining time units units
Percentage Relative Efficiency (PRE) = [ [ length of ci of case-]-[ Length of ci of other cases] ] x 1QQ
0 J Length of CI of case-I
Table 11: Percentage Relative Efficiency (PRE)
Case-II with respect to Case-I
PRE = 9.82 %
VIII. Conclusion
In case when the sudden breakdown occurs in a multiprocessor computer system this paper represents an idea of calculating the ready queue remaining processing time. The paper assumes that (kj - nj' - nj' ') processes are completely finished before breakdown, nj' are partially processed and nj'' are blocked by jth processor. Under this an estimation strategy is proposed for estimating the total remaining time of jobs to be processed in waiting ready queue. The proposed generalized strategy contains constants whose optimum values are derived and used. Two cases are compared where the first case is having no consideration of size measure of jobs in waiting queue whereas
Sarla Mor, Diwakar Shukla
GENERALIZED APPROACH IN MULTIPROCESSORS USING RT&A, No 2 (68) REGRESSION ESTIMATOR AND COST ANALYSIS_Volume 17, June 2022
the second case considers the additional features of size measure of processes. The confidence
interval is used as a tool for predicting about the unknown with 95% accuracy. Three cost
functions are suggested for predicting about the backup infrastructure cost needed for recovery
management after system breakdown. The proposed methodology under Case-II performs better
than Case-I by comparing the length of confidence intervals. The highest predicted remaining time
under ten considered samples is 15521 units, under Case-II. Moreover, the Case-II is 9.82 % more
efficient than Case-I. The average cost required for recovery after occurrence of failure is also lower
in Case-II. Overall it is found that the suggested estimation strategy is effective for predicting the
remaining total time with high efficiency. The suggested is a new methodological approach for
predicting the unknown using sampling methodology in the multiprocessor environment.
Proposed advocates for the use of size measure of processes, if available for predicting unknown
parameters like remaining time of a ready queue.
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