Sampled Ready Queue Processing Time Estimation Using Size Measure Information In Multiprocessor Environment Текст научной статьи по специальности «Компьютерные и информационные науки»

Sampled Ready Queue Processing Time Estimation Using Size Measure Information In Multiprocessor Environment

Sarla More and Diwakar Shukla

Dr. Harisingh Gour University, Sagar, MP, India sarlamore@gmail.com, diwakarshukla@rediffmail.com

Abstract

In a multiprocessor computer system, there exist a ready queue of large number of processes waiting for computing resources allocation by the processors. These jobs may have size measure, which are additional information priory known while entry to the ready queue. Suppose the sudden system breakdown occurs and recovery management is required immediately. At this stage, one can find some processes who are completely finished, some partially processed, some blocked by processors and remaining waiting for allocation in the ready queue. Prime act of a system manager is to evaluate the maximum time required to process all the remaining jobs. This paper presents an estimation strategy for such, derived by applying the lottery scheduling, sampling technique and imputation methodology. Expressions for mean squared error of the proposed strategy are derived and optimized for suitable selection of system parameters. Three cases are discussed and compared and consequent results are numerically supported. It is found that at the optimal choice of constants in the estimation methodology, the shortest confidence interval can be predicted estimating the remaining required time. Such findings are useful as a part of disaster management of a cloud based multiprocessor data centre.

Keywords: Ready Queue, Lottery scheduling, Multiprocessors, Simulation, Sampling, Random, Estimation

I. Introduction

Assume a computer system equipped with several processors having a ready queue, where processes are waiting for allocation of resources. Lottery scheduling is a type of priority scheduling where the computer system resources are allocated randomly to the waiting processes. In this, a bunch of token numbers are assigned to processes and multi-processors used to issue random numbers. A process who contains the token of issued number receives first the desired system resource. Using this, every process has chance of allocation of resources, sooner or later, therefore, probability of starvation vanishes. A process in a ready queue may have a predetermined measure of its size in terms of bytes which is additional information, can be used for better prediction of the remaining mean time of ready queue processing. This paper presents a strategy for effective use of known size measure of processes.

Let ti, t2, t3.......tk be the time of k processes and xi, X2, X3.........xk be their size measure. Figure

1 shows r processors (r < k) and waiting queue.

Processi

Time ti

Size xi

1/

Process;

Time h

Size x 2

/ / /

Processi Processt

Time tî Time tt

Size X3 n^i-171 / Size xt

Processors

Qi

Q-

Qs

Figure 1: Ready queue with limiting Processes and Multiprocessor

Further assume processes Ai, Ai, A3............are of small size with time consumption tii, ti2,

ti3............. and size measure xn, X12, xb...... (see figure 2). The large size processes areBi, B2,

B3..........with time consumption hi, tzi, t23..........and size measure X21, X22, X23...........(see figure 3).

All the Ai and Bi are to be processed by r processors, under the size measures. The case of partially processed and completely processed [23] exists when sudden breakdown occurs. One can further think of possibilities as under:

(a) inside multi-processors, some are completely processed,

(b) some are partially processed,

(c) some are blocked, and

(d) size measure of processes are known.

This paper extends approach of [23] in collective presence of (a), (b), (c) & (d) when multiprocessor computer system fails at an instant. The issue of estimation (prediction) of recovery time duration of the remaining is focused.

Processors

Process Bi Time t2i

Size x21

(loo-lio)

Process B2

Time Î22

Size x22 /111 1 n\

Process B3

Time t23

Size x23

1 1 1 1 -7\

Figure 2: Small size processes and Multiprocessors

Processors

Process Bi Time Í2i

Size X2i

I, nr. 1in\

Process B?

Time Í22

Size X22

Process B3

Time Í23

Size x23

Qr

Figure 3: Big size processes and Multiprocessors

II. A Review

Lottery scheduling is a resource sharing technique [12] like a particular case of priority scheduling where the processes in ready queue are allotted bunches of ticket numbers. Process who receives maximum count of tickets has highest priority of being allocated the demanded system resources. Lottery scheduling is efficient and effective [22] in the framework of LINUX kernel also. It could be used as a model tool [4], [6] for estimating the mean time of processing of a ready queue where large number of jobs are in waiting but only some have processed. Completed jobs could be used as a sample just like a preliminary source of information for prediction. Concept of grouping of homogeneous jobs together [5] came into existence which has improved the prediction. Units in sample may have additional correlated variables which could be utilized for efficient computing ([7]/ [8], [9], [10], [11]) along with precise prediction. An exhaustive review [3] on the similar problem contributes few recent aspects of solutions extendable into [1] and [2]. Some authors have extended the lottery scheduling variants [20], [21] in the form of hybrid multi-level structure using Markov chain model along with analysis and chance based prediction.

Sampling techniques are useful tools for parameter estimation and value prediction. Random sampling schemes exist in statistical literature ([13], [15], [16]) who are widely used for parameter evaluation of a finite collection. Some popular schemes are like stratified sampling, cluster sampling, two stage sampling, systematic sampling, successive sampling etc.( [14], [17], [18]) useful in varying situations of the aggregate . Moreover, such needs appropriate selection of methods also [19] to provide accurate confidence interval for unknown parameter.

Imputation is a methodology used when one or more values in a sample are found missing (or non-responded). For example, if a processor blocks a process then mean time parameter remains unpredicted using sample from the ready-queue. However, some processes may be blocked after the partial processing. For completely blocked processes, Random Imputation methods ([25], [26], [27]) could be used to recover information. In this, the missing values are selected randomly from the available part of sample values and replaced. Other popular imputation methods are mean imputation, deductive imputation, mean imputation within classes, deductive imputation within class, hot deck imputation, cold deck imputation etc. ([28], [29], [30], [31]). This paper considers the approach of [6] and [23] and extends using [10], [11], assuming situation when one process is blocked, one is partially processed and remaining others in a

processor are completed before the occurrence of sudden breakdown.

I. Remaining Time Estimation Problem

Assume a large number of processes (say N) present in a ready queue of a multiprocessor computer system and only few of them (say n, n<N) have been processed before a fixed time instant. The remaining in the ready queue are (N-n) for whom the expected time computation is required. If sample mean time of those who already processed is A then remaining time estimate is S = [(N-n) A] which is an unknown quantity. For any two real numbers 'a' and 'b', if A is predicted as AG (a,b) where (a,b) is an interval containing A with very high probability, then Si =[(N-n) a]is lowest, Si =[(N-n) b] is the value of highest expected 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 undertaken in this paper for strategy formation in the multiprocessor setup with the consideration of multiple real life possibilities.

II. Confidence Interval (CI)

It is a statistical tool for evaluating the precision of mean time estimate. If catches the true unknown value then it is termed as a confidence interval. Let P[A] denotes the probability of happening of event A. In statistical theory, for any two real numbers a', b', the 95% confidence interval is defined as P[a' < true unknown value < b'] = 0.95. Define length of CI= 1 = (b'-a1). Let one confidence interval has length liobtained through a methodand other has length ^obtained by another method. If h< lithen second one is said to be better than the first in terms of efficient prediction.

III. Motivation

Earlier contributions (specially [6], [23]) were under assumption that processes present in a multiprocessors system are completely processed before sudden failure. But this is not a practical reality. While sudden failure, 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 [23] [see figure (5)]. This paper extends the approach of [23] by applying the tools of random imputation method against the blocked processes.

Long Term Scheduling

1119

T, T

•'1 •'2 •'3 JN

,ong Term Scheduling

Medium Term Scheduling

1113

HIN

Medium Term Scheduling

P2

P3

Pr

Short Term Scheduling

Medium term scheduling

Li

L2

n.

Blocked/Suspended

Figure 4: Ready Queue Processing under Lottery Scheduling (due to [6])

Pl P: P3 I t

-O

Figure 5: Setup of ready queue and multiprocessor environment (due to[23])

III. Proposed Computational Setup

Assume the existence a virtual sampled ready queue in a system of 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 ofthe virtual sampled ready queue. Figure 6 shows basic setup of this approach without size measure while figure 5 shows the earlier approaches [4], [5], [6], [23]. Moreover, figure 7 reveals the special case when all sample units processed before the occurrence of breakdown.

Sampled Ready Multiprocessor Queue k Qr

System Queue of size N

P2

P3

ki

k2

Jö—►

kr

k,-?

k2

k3-2

k,-2 1 1

kr-2 1 1

Figure 6: Sampled Ready Queue Processing Time Estimation setup ivithout size measure

Sampled Ready Multiprocessors Queue k Qr

ki

System Queue

of size N

Pi

P2

P3

№

k2

k3

kr

k2

k3

kr

Figure 7: Sampled Ready Queue Processing Time Estimation when all processed before breakdown

I. Assumption and Model

In view of figure 6, let the selection of processes is as per priority scheduling, in particular, as per 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 in system and a ready queue of N processes denoted as [Pi, Pi, P3.........Pr-j]who are 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 then time consumed by them are [ti, fa, t3,.......tr-j] and each

process has size measure [xi, x?, x3,.......xn] who are priory known.

4). Define the whole ready queue mean timet= ~Hi=i t[, size measure meanx= xi and respective mean squares S<2 = j-^T1'j'=1(ti - t)2, Sx2 = - x)2

The process Pi of known size Xi consumes time ti( i = 1,2,3,......N).

5). Hereby denote r multiprocessors as Qi, Q2, Q3.....Qr, (r < N) and time consumed by the ith

process in the jth processor is fa with corresponding size measures xij (j = 1,2,3,......r)

6). 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 Q generates a random number. A process who receives the random number gets the desired resource from Qj.

7). 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 =Yj=i kj )' who is allocated to the jth processor Q in sequential manner.

8). In sample let sxji denotes the file size measure and stji denotes time consumed by ith process in Qi(l = 1,2,3,...kj) when all processed completely who are included in the sample of size k.

(i). Sample mean of time st= Y!\Li stji

(ii). Sample mean square of time, (es)r = ~~ st )2

(iii). Sample mean of size, ( sx)= -¡^l/jLi X'z^sxji)

(iv). Sample mean square of size, (es)x2 = ^I^4i(5Xjl _ ^ )2 The termst, sx, (es)t2, (es)x2hold when system runs without failure.

9). Assume system breakdown occurs at the time instant T and there are (kj - 2) processes who are finished in Qj, but one remain partially processed and one remain unprocessed (blocked). This is an assumed model sh own in fig. 6 and fig.8.

10). Let (st')ji is time consumed by the lth process in the processor Qj [l =1, 2, 3... (kj - 2)],who is among those processed completely before the occurrence of T.

11). Some sample mean related measures are:

(i). Sample mean of (kj - 2) prore^ (5-t')j=((1_2) 9 ¡!;_2)(st,jl)

(ii). Sample mean square, (es').2 = 9(=_2(5t'jl - (—')j)2

(iii). Similar is for size measure also as (sx'jj) represents size of lth process who is in Qj before T.

(iv). Sample mean,(SX') j= 9($_2(SX'jl)

i , '

(v). (sx)j= —- 9i$1(5X jl)is sample mean of all kj known values related to x in jth segment of ready queue.

(vi). Sample mean square, (ex')^. 2 = (c^—) 9($_2(5X jl - ip88)])2

(vii). Sample Covariance, (es'x) 9(=_2(5t jl - (—')j) (5X jl - 088')j)

12). Assumei0 is partially processed time of a process in Qj (j = m =1,2,3....r) whose sample mean under T is

(t /T)= & Em=1 ^m

11

Variance (t * / T) = V(t * / T) = ( - - ) St2, where St2 is the conditional ready queue mean

square of the remaining unsampled part [N-K+r] expressed as: St2 = ^^i^ )2where

1

tx =

N -k + r

i=i

Herein to mention that St2 and ^contain time tionly from non-sampled processes (N-k) of the main ready queue with the addition of those r who partially processed. For such, the size

converts from N into (N-k+r) and only those processes are the part of ^T and ST2who are in (N-K+r).

13). The r blocked processes are imputed by Random Imputation Method using random selection of a process among (kj-2) relating to Qj. Let for Qj this imputed time is denoted as fm .

(i). Sample mean of imputed time,t ** = &S0=1 £0*

(ii). Variance of imputation under T,V(t**/T) =(! - 1)(es)2, r < k.

14). Sample based estimate of (es)2can 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

denoted as (es*)2 and mathematically expressed as

1 ( * _ * 2 _ *

(es*)2 = — X'=1 S;=1(st jl - st )2where (st*ji) and st include completely processed time

st*ij , partially processed tm* and imputedtm**.

15). The sample estimate of St2 is (es')2 = 90=1(t0-f * )2 ]

16). Bias of estimation strategy is assumed negligible wherever appears and applicable in mathematical expressions.

IV. Computational Set-up

The objective is to compute the remaining ready queue processing time while occurrence of

sudden failure of system at time instant T. This is subject to condition that r processes are partially processed, r are unprocessed (blocked) and remaining (K-2r) are fully completed. Blocked and partially processed are one each from every Qj and the available size measures are the part of computation. Some frequently used symbols for process time t and process size measure X are as under:

"t= "p=1 ti = " I I t, .^i)

t = & Xm=1 ^m .(4.2)

t ** = & D& 1 if .(4.3)

& m=1

(-)j= ((■_) S(=_12(st'jl) ...<4.4)

(«;j= ((-_) S(=_12(sx'jl) .(4.5)

O88) j= (C1) 9(=1^ jl) .(4.6)

(es')j2 = 1/(kj-3) Z(L_2(st'jl - (s-0j)2 .(4.7)

(ex')j2 = 1/(kj-3) SH_12(sx'jl - (<88'jj)2 .(4.8)

(es'x) j^ 9(=_2(5t j - (-')j) )l - (^88'))j .(4.9)

(es*)2 = D&=1 (st * jl - 888 *)2 .(4.10)

.(4.11)

Rn. j

try

(5Q;

(-);J _

[(st) ((g)') aj], aj being constant, (0<aj«~) .(4.12)

I. Estimation Strategy

The sample based proposed estimation strategy for mean time is: (tmean/T) = €l [ S&=1 Wj (try/T) ] + €2 (t */ T) + (1- €l - €2) (t **/T)

with condition that 9"=i €p= 1 and €pdenotes constants to be determine suitability and wj= (kj/k) is known weight (Iwj =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[ €l [ £% = $ Wj (try /T)] + €2 (t */T) + (1- €l - €2) (t **/T)]

=€1 [ Z!=1 WjE (tr;- /T)] + €2 E (t */T) + (1- €1 - €2)E (t **/T)] ^twhich shows estimator (t mean /T) is biased.

II. 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[ E&=1 Wj2 MSE(try/T)]+ €22V (t */T) + (1- €1 - €2)V (t **/T)]

= €i2 - () wj2 { (e5'>2+«j2 j> -2aj RNj(esV)y}]+€22 [g - "_1t&)St2]

+(1- €1 - €2)2D & (l - Wj(es')j2(as per Cochran[16] page24, page29 and page164) The expressions P, Q, R are in the sample based estimate form of population parameters

Let P

Z-i ferhrk) Wj2{ (e5'>2+af R2Ni(ex')i2-2«j RNj(es'x')y}

St"

Q=(i--J_)

^ \r N-k+rJ

R=Z-=1(l"^)wi2(es')j2

The above expression is re-written as:

V[tmean/T] = [€i2 P + €.i2 Q+ (1- €1 - €2)2R ]ignoring the covariance terms due to independency. For optimum variance, differentiate V[tmean/T] with respect to €1 and €23 nd equate to zero, one gets (€1) oPt= (QR) / [PQ+PR+QR] = QM

(€2) opt= PQ/ [PQ+PR+QR] = PM where M = R/ [PQ+PR+QR] One can differentiate the variance expression byajalso to get optimum value which is (aj)opt=[(es'x')y/(RNj*(ex )j2)] Substituting optimum choices in expression, the optimum variance is: V[tmean/T]oPt = (€1) 2oPt P + (€2) 2oPt Q + (1- (€i)opt- (€2) opt) 2fi] with («j)oPt as above.

Virtual Sampled Ready Multiprocessors Queue k Qr

System Main Ready Queue of size N

Pi

P2

P3

Pn

k2

k3

kr

E>

o

k2

kj-2, (staXsxy)

k3

k3-

t,"t

kr

l&-2,(st'ti)(sx'ti)

tk =

tk"=trr

Figure 8: Proposed Model of Virtual sampled Ready Queue Processing Time Estimation with size and Imputation

V. Numerical Illustrations

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 Ji J? J3 J4 Js J* Jy J« J» J10 Jn J12 Jl3 Jl4 Jis

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 Ji« Jl7 Ji" J» J20 J21 J22 J23 J24 J25 J* 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) 122.51

3 Total sum of square =Iti2 2697717

4 Mean square St2_3080.62_

Assume that there are three processors Q1, Q2, Q3in the system (r = 3) and a random sample of k = 30 is drawn from N = 150 by lottery scheduling. The sample k = 30 is divided into k1 = 12, k2= 10, k3 = 8 in sequential manner for virtual sampled ready queue. The kj process are assigned to Qj(j = 1, 2, 3). Calculation is performed on 10 random samples each of size 30. Computation for only one sample is presented below:

I. CASE I: aj = 0 (ai = 0, a2 = 0, a3 = 0)

Calculation for Sample No. 1 where sample size k=30

ki:(Jo1, 30,41), (J31,61,172), J1, 118,313), (J91,120,247), (Jm,235,287), J,102,153), (J32,52,243), J81,194), (J92,112,79), (Jm,142,278), (J3,42,103), J,72,253)

Partial Processed = (J03, 42), (Processed= 22, unprocessed=20), Blocked = (J33,72), Blocked replaced a' =(J8,48)

[stf = 104.9, from eq.(4.4),(es )12= 3317.65, from eq.(4.7)],

[s-a=2363/12 = 196.91, from eq. (4.5),sx1'=2007/10= 200.7, from eq. (4.6)], (ex ')12=8158.45,

from eq.(4.8)] [R^=

OA

(sx')!

=0.52,from eq.(4.10)],[(es'x')i = 2982.52,from eq.(4.9)]

k2:(J49,100,254), (J 34,75,262), (J 64,105,255), (J 94,135,276), (J 124,165,197), (Ju5,241,230), (J 35,89,83), (J 65,119,225) (J 95,149,285), (J 125,179,249)

Partial Processed=(J 95,149),(Processed=100, unprocessed=49), Blocked=(J125,179), Blocked replaced p' =(J38, 78)

[<Tt2' = 128.62, from eq. (4.4), (es')22= 2843.98 from eq.(4.7)]

[sx2=2316/10 = 231.6, from eq. (4.5),sx2' = — = 222.75, from eq. (4.6), (ex >2=3806.21, from eq.(4.8)]

[Rl JifLk

[rn2=.P8-.

ka:(J29, 111,271), (J59, 141,309), J 171,300), (J 96,125,257), (J 119,201,306), (J149, 231,315), (J 67, 88,206), (J97, 115, 56) Partial Processed = (J 67, 88), (Processed=40, unprocessed=48), Blocked = (J97, 115), Blocked replaced y'=(J10, 60)

[st3 = 163.33, from eq.(4.4), (es>2= 2152.66 from eq.(4.7)]

[sx3=2020/8=252.5,from eq. (4.5),sx3' = ^ = 293, from eq. (4.6), (ex >2= 547.6, from eq.(4.8)]

= 0.57, from eq.(4.10)], [(es'x')2 = 281.75, from eq.(4.9)]

[RN3:

(st%

0.55, from eq.(4.10)],[(es'x')3 = 841.2, from eq.(4.9)]

t* = (22+100+40)/3 = 54

t" = (a' + p' + y')/3 = (48+78+60) / 3 = 62

Estimated St2 = 1658 (using point 15)

Let P = D& ((^-() wj2 {(es')j2+aj2 R2Nj(ex ')j2 -2ajRNj(es'x')y} Q = ^ (;- "—b)[ estimated st2 ]

R = (1—-W-Oj2 R = X(1- (j—-H«2

Calculation of P, Q, R at ai=a2 =a3 = 0

P = (!?-3?) (0.4)2{3317.65}+(!-3?) (0.33)2 {2843.98 }+(!-3?) (0.26)2 {2152.66}

= 0.06(3 *0.16*3317.65+ 0.092*0.1089*2843.98 +0.133*0.0676*2152.66= 82.88 Q = (!--1-) (1658)= (0.3252 *1658) =539.18

^ v3 150—30+3 v ' v '

R = ( 1 - 1 ) (0.4)2 *3317.65+ (1-!) (0.33)2 *2843.98 +(l-!) (0.26)2 *2152.66 = 0.9 *0.16*3317.65+ 0.875*0.1089*2843.98 +0.833*0.0676*2152.66 = 869.95 Calculation of Mean and Variance V[tmean/T]

(€1)opt= (QR) / [PQ+PR+QR] = QM = 539.18*869.95/[82.88*539.18+82.88*869.95+539.18*869.95]

= 469059.641/585848.3354= 0.8006 (€2)opt= PQ/ [PQ+PR+QR] = PM =82.88*539.18/[82.88*539.18+82.88*869.95+539.18*869.95] = 44687.2384/ 585848.3354= 0.0762

tmean/T= (€1)opt [ £[=1 Wjtj-j] + (€2)opt (t *) + (1- (€1)opt - (€2)opt) (t **)

tmean/T= 0.8006 [0.4*104.7(196.91/200.7)+0.33*128.62(231.6/222.75) +0.26*163.33(252.5/293)] +0.0762*54 +0.1232*62 = 0.8006 [41.08 +44.13 +36.59] +4.11+1.88= 97.51+4.11+7.63= 109.25

V[tmean

/T] = (€1) 2opt P + (€2) 2opt Q + (1- (€1)opt- (€2)opt) 2R]

V[tmean/T] = [(0.8006)2 *82.88+ (0.0762)2*539.18+ 0.0152*869.95] = 53.12+3.13+13.22 = 69.47 The 95% confidence intervals for t, P [(tmean/T) ± 1.96^[ V (tmean/T)] = 0.95 = 109.25± 1.96V6947 = 109.25± 16.33 = (92.92, 125.58)

Table 3: Estimated Sample Mean, Variance and Confidence Interval(CI) of Ten Random Samples _CASE I: At (€i)opt, (€2)0?, , a = 0 fa = 0, a2 = 0, a3 = 0)_

S.No. True Mean Estimated Sample Mean V[tmean/T] 95% Confidence Interval (CI) CI Length

1 122.51 109.25 69.47 (92.92, 125.58) 32.66

2 122.51 123.30 61.64 (107.92, 138.68) 30.76

3 122.51 107.67 75.92 (90.59, 124.74) 34.15

4 122.51 114 289.87 (80.63, 147.37) 66.74

5 122.51 128.09 285.83 (94.95, 161.22) 66.27

6 122.51 113.82 30.09 (103.07, 124.57) 21.50

7 122.51 119.23 39.79 (106.87, 131.59) 24.72

8 122.51 113.51 185.98 (86.78, 140.23) 53.45

9 122.51 133.73 175.83 (107.74, 159.30) 51.56

10 122.51 111.47 56.65 (96.72, 126.22) 29.5

Average Length (411.31/10)_41.13

CASE II: aj = 1 (ai = 1, a2 = 1, a3 = 1)

Calculation for Sample No. 1 , k=30, on above sample and P, Q, R at (ai = 1, a2 = 1, a3 = 1)

P = (^ - (0.4)2 {3317.65+1*0.52*0.52*8158.45-2*1*0.52*2982.52}

+(- - (0.33)2 {2843.98+1*0.57*0.57*3806.21-2*1*0.57*281.75}

+(- - (0.26)2 {2152.66+1*0.55*0.55*547.6-2*1*0.55*841.2}

= 0.066 *0.16*2421.87+ 0.092*0.1089*3759.42 +0.133*0.0676*1392.98 = 75.76

Q = (-----) 1658= 0.3252 *1658= 539.1816

v3 150-30+3'

R = ( 1 - 1 ) (0.4)2 *3317.65+ (1-(0.33)2 *2843.98+(l-(0.26)2 *2152.66 = 0.9 *0.16*3317.65+ 0.875*0.1089*2843.98+0.833*0.0676*2152.66 = 869.95 Calculation of Mean and Variance V[tmean/T] at (ai = 1, a2 = 1, a3 = 1)

(€i)opt= (QR) / [PQ+PR+QR] = QM = 539.1816*869.95/[75.76*539.1816+75.76*869.95+539.1816*869.95] = 469061.03292/575816.842936= 0.8146

(€2)opt= PQ/ [PQ+PR+QR] = PM =75.76*539.1816/[75.76*539.1816+75.76*869.95+539.1816*869.95] = 40848.398016/ 575816.842936= 0.0709

(tmean/T) = (€i)opt[ £&=1 Wj try] + (€2)opt (t *) + (1- (€i)opt- (€2)opt) (t **)

(tmean/T)= 0.8146[0.4*104.7(196.91/200.7)+0.33*128.62(231.6/222.75) +0.26*163.33(252.5/293)] +0.0709*54 +0.1145*62 = 0.8146[41.08+44.13 +36.59] +3.82+7.09=99.21+4.44+7.09= 110.74 V[tmean/T] = (€1) 2opt P + (€2) 2opt Q + (1- (€i)opt- (€2)opt) 2fl] with aj = 1for all j = 1,2,3 V[tmean/T] = [(0.8146)2 *75.76+ (0.0709)2*539.1816+ 0.0131*869.95] = 50.27+2.71+11.39=64.37 The 95% confidence intervals for t ,P [(tmean/T)± 1.96^[ V (tmean/T)] = 0.95 =110.74±1.96V6437 =110.74±15.73 = (95.01, 126.46)

Table 4: CASE II: At (€i)p, (€i)oPt , a= 1 (ai = 1, a2= 1, a?, = 1) Sample Mean, Variance and Confidence Interval(CI) of Ten Random Samples

S.No. True Mean Estimated Sample Mean V[tmean/T] 95% Confidence Interval CI Length

1 122.51 110.74 64.37 (95.01, 126.46) 31.45

2 122.51 125.72 49.78 (111.89, 139.55) 27.66

3 122.51 112.61 54.66 (98.11, 127.10) 28.99

4 122.51 113.87 305.14 (79.63, 148.10) 68.47

5 122.51 127.45 235.98 (97.35, 157.55) 60.2

6 122.51 113.63 62.92 (98.08, 129.18) 31.1

7 122.51 119.64 37.80 (107.58, 131.69) 24.11

8 122.51 144.02 144.34 (120.48, 167.57) 47.09

9 122.51 133.45 171.53 (107.77, 159.12) 51.35

10 122.51 122.85 40.12 (110.44, 135.26) 24.82

Average Length (395.2/10) 39.52

III. CASE III: aj = aopt where (aopt)j= (es'x')j/ (Rnj* (ex )j2 )

Calculation for Sample No. 1, sample size k=30, ai = (aopt)i= 0.70, a2 = (aopt)2 = 0.13, a3 = (aopt)3 = 2.79and P, Q, R at aj = (a0pt)j[ai = (aopt)i , a2 = (aopt)2 , a3 = (aopt)3] P = (^ - w) (0.4)2 {3317.65+0.70*0.70*0.52*0.52*8158.45-2*0.70*0.52*2982.52} +(- - w) (0.33)2 {2843.98+0.13*0.13*0.57*0.57*3806.21-2*0.13*0.57*281.75} +(- - w) (0.26)2 {2152.66+2.79*2.79*0.55*0.55*547.6-2*2.79*0.55*841.2} = 0.066 *0.16*2227.34+ 0.092*0.1089*2823.12 +0.13*0.0676*860.45 = 59.36 Q = (---1-) 1658= 0.3252 *1658 = 539.1816

v3 150-30+3'

R = ( 1 - 1 ) (0.4)2 *3317.65+ (l-(0.33)2 *2843.98 +(l-(0.26)2 *2152.66 = 0.9 *0.16*3317.65+ 0.875*0.1089*2843.98 +0.833*0.0676*2152.66 = 869.95 Calculation of Mean and Variance V[tmean/T]ata = (a0pt)j

(€i)opt= (QR) / [PQ+PR+QR] = QM = 539.1816*869.95/[59.36*539.1816+59.36*869.95+539.1816*869.95]

= 469061.03292/552707.084696= 0.8486 (€2)opt= PQ/ [PQ+PR+QR] = PM =59.36*539.1816/[59.36*539.1816+59.36*869.95+539.1816*869.95] = 32005.819776/ 552707.084696= 0.0579

(tmean/T)= (€i)opt[ £&=1 Wj fry] + (€2)opt (t *) + (1- (€i)opt- (€2)opt) (t **)

(tmean/T)= 0.8486[0.4*104.7(196.91/200.7)+0.33*128.62(231.6/222.75) +0.26*163.33(252.5/293)]

+0.0579*54 +0.0935*62 = 0.8486[41.08+44.13 +36.59] +3.13+5.797= 103.35+3.13+5.79= 112.27

V[tmean /T] = (€i) 2opt P + (€2) 2opt Q + (1- (€i)opt- (€2)opt) 2fl]

V[tmean/T] = [(0.8486)2 *59.36+ (0.0579)2*539.1816+ 0.0087*869.95] = 42.74+1.80+7.56=52.10 The 95% confidence intervals for t P [(tmean/T) ± 1.96^[ V (tmean/T)] = 0.95 = 112.27± 1.96V52.10 = 112.27± 14.14 = (98.13, 126.41)

Table 5: CASE III: At (€i)opt, (€i)oV> , a= (aopt)j (j = 1, 2, 3) Estimated Sample Mean, Variance and Confidence Interval (CI) of Ten Random Samples

S.No. True Estimated V[tmean/T] 95% Confidence CI

Mean Sample Mean Interval (CI) Length

1 122.51 112.27 52.10 (98.13, 126.41) 28.28

2 122.51 126.91 44.04 (113.90, 139.91) 26.01

3 122.51 110.92 50.71 (96.96, 124.88) 27.92

4 122.51 114.44 32.79 (103.22, 125.66) 22.44

5 122.51 127.37 230.85 (97.59, 157.14) 59.55

6 122.51 114.59 26.79 (104.44, 124.74) 20.30

7 122.51 121.13 30.35 (110.33, 131.93) 21.60

8 122.51 139.34 105.75 (119.18, 159.49) 40.31

9 122.51 129.54 84.13 (111.56, 147.52) 35.96

10 122.51 123.66 31.31 (112.69, 134.62) 21.93

Average Length ( 304.3/10) 30.43

IV. CASE IV: At €1 = 1, €2 =0 , with aj =( aopt )

It is the case when no imputation used and partially processed situation not considered. But it is away from practical situation.

(aop0j= (e5'*');/ (Rj >2)

Calculation for Sample No. 1, sample size k=30, when €1 = 1, €2 =0 with aj = aopt and P, Q, R at (aopt)j

P = 59.36, Q = 539.1816, R = 869.95

Calculation of Mean and Variance V[tmean/T] at (a0pt)jwith (€1= 1, €2 =0) (tmean/T)= €1 [ £&=1 Wj fry] + €2 (t *) + (1- €1 - €2) (t **)

(tmean/T)= 1* [0.4*104.7(196.91/200.7)+0.33*128.62(231.6/222.75) +0.26*163.33(252.5/293)]

= [41.08+44.13 +36.59] +0+0= 121.8

V[tmean/T] = [(1)2 *59.36+ (0)2*539.1816+ 0*869.95] = 59.36

The 95% confidence intervals for tare P [tmean/T ± 1.96^[ V (tmean/T)] = 0.95

= 121.8± 1.96V5936 = 121.8± 15.10 = (106.70, 136.90)

Table 6: CASE IV: when[€i = 1, €2 =0 aj = (aopt)j ] Estimated Sample Mean, Variance and Confidence Interval (CI)of Ten Random Samples

S.No. True Estimated V[tmean/T] 95% Confidence CI

Mean Sample Mean Interval (CI) Length

1 122.51 121.8 59.36 (106.70, 136.90) 30.20

2 122.51 136.51 50.11 (121.63, 150.38) 28.75

3 122.51 117.57 56.95 (60.62, 132.36) 71.74

4 122.51 119.77 47.64 (106.24, 133.29) 27.05

5 122.51 121.23 44.23 (108.19, 134.27) 26.08

6 122.51 125.01 27.56 (114.72, 135.30) 20.58

7 122.51 127.21 34.62 (115.67, 138.74) 23.07

8 122.51 116.64 58.63 (101.63, 131.64) 30.01

9 122.51 114 41.37 (101.40, 126.60) 25.20

10 122.51 127.23 30.69 (116.37, 138.09) 21.72

Average Length 30.44

MAX.

(a) Case I

(b) Case II

Min.

Max.

(c) Case III

(d) Case IV

Figure 9: (a), (b), (c), (d) are graphical representation of Confidence Interval range of Ten Random Samples for four different cases of Table 3,4,5 and 6 (X-axis has sample number as shown in table 3,4,5,6) Table 7: Comparison Between Cases I,II, and III

S.

No.

CASE 1

a.) = 0 (m =02 =«3 =0)

95%Confi dence Length Interval

CASE II Oj = 1 (m = 02 = «3 =1)

95%Confiden ce Interval

Length

CASE III (a)j=(aopt)f 95%Confi dence Length Interval

CASE IV [€i=1,€2=0 ] with a.) = aopt

95%Confiden ce Interval

Length

1. (92.92, 125.58) 32.66 (95.01,126.46) 31.45 (98.13, 126.41) 28.28 (106.70, 136.90) 30.20

2. (107.92, 138.68) 30.76 (111.89, 139.55) 27.66 (113.90, 139.91) 26.01 (121.63, 150.38) 28.75

3. (90.59, 124.74) 34.15 (98.11,127.10) 28.99 (96.96, 124.88) 27.92 (60.62,132.36) 71.74

4. (80.63, 147.37) 66.74 (79.63,148.10) 68.47 (103.22, 125.66) 22.44 (106.24, 133.29) 27.05

5. (94.95, 161.22) 66.27 (97.35,157.55) 60.2 (97.59, 157.14) 59.55 (108.19, 134.27) 26.08

6. (103.07, 124.57) 21.50 (98.08,129.18) 31.1 (104.44, 124.74) 20.30 (114.72, 135.30) 20.58

7. (106.87, 131.59) 24.72 (107.58, 131.69) 24.11 (110.33, 131.93) 21.60 (115.67, 138.74) 23.07

8 (86.78, 140.23) 53.45 (120.48, 167.57) 47.09 (119.18, 159.49) 40.31 (101.63, 131.64) 30.01

9. (107.74, 159.30) 51.56 (107.77, 159.12) 51.35 (111.56, 147.52) 35.96 (101.40, 126.60) 25.20

10. (96.72, 126.22) 29.5 (110.44, 135.26) 24.82 (112.69, 134.62) 21.93 (116.37, 138.09) 21.72

Average Length 41.13

39.52

30.43

30.44

VI. Comparison and Discussion

The proposed setup has three parameters €1, €2and a (0< a < «>) whose suitable choices provide the best estimate. The case I has aj= 0 (for all j) which means there is no consideration of size measure in the strategy. Case II considers aj= 1 (for all j) indicating for the presence of size measure x in the estimation strategy but at a particular choice. Case III considers aj= (aopt)j (for all j) where size measure is at the best (optimal) fractional level incorporated in strategy of prediction. All the three cases (see table 7) are showing the average length of confidence intervals, but smallest average interval length is 30.43 obtained by the case III where choices (€1)opt , (€2)opt and(aopt)j are used. The ten sample average confidence intervals are in table 8. Fig.9 shows smooth, increasing, condensed and controlled variations of lower and upper limits of CI, best found in the case III which deserved for recommendation.

Table8: Ten Sample average Confidence Interval & estimated total processing Remaining time for Recovery Management

Case I Case II Case III True

(Without size measure) (With size measure) (With size measure) Value

Average Interval (Over 10 samples) (96.8 - 137.9) (102.6 - 142.1) (106.8 - 137.2) 122.51

CI Length 41.1 39.5 30.4

Lowest Predicted Remaining time (N-k)*96.8 = 11,616 units (N-k)*102.6 = 12312 units (N-k)*106.8 = 12816 units

Highest Predicted Remaining time (N-k)*137.9 =16,548 units (N-k)*142.1 = 17052 units (N-k)*137.2 = 16464 units

To note that average intervals (table 8) are producing the same length as shown in table 7.Define relative efficiency measure in terms of percentage as:

Percentage Relative Efficiency (PRE) = [[ LengthofCIefcasel ]-[ LengthofCIofethercases ] ] X 100

° LengthofCIefcasel '

Table 9: Percentage Relative efficiency (PRE)

Case II with respect to Case I Case III with respect to Case I

PRE = 3.91 % PRE = 26.01 %

The case III is more efficient (26.02%) than the case II with respect to case I as base where no size measure considered for estimation. In fact, all the sample computed confidence intervals are catching the true value (122.51) which is the strength of the proposed method. The minimal highest predicted time required to process the remaining jobs in ready queue (after breakdown)is 16464 units which is in case III ( see table 8).

VII. Conclusion

On recapitulation, the paper considers the practical problem of remaining time estimation of processes in ready queue, after the occurrence of system failure in a multiprocessor computer system. While sudden breakdown how much backup time and computer related infrastructure required? This time duration and maximum time estimation are done using the tools of sampling theory and assumption of lottery scheduling. This scheduling opens avenues for application of random sampling tools and techniques. A concept of virtual ready queue is added as a new feature, who found useful in allocating the processes to multi-processors. The virtual ready queue along with lottery scheduling have created environment for the estimation of remaining time of

main ready queue. The proposed estimation strategy is capable enough to predict for mean time. For efficient estimation (€i)opt , (€2 )opt and (aopt) are used who provide the lowest length confidence interval. The Case III found best estimated and predicted than case I and case II. The case III also provides prediction indicating the minimal highest remaining time to arrange backup accordingly while failure. Imputation has improved the level of estimation and use of additional information (size measure) contributed a lot for higher precision. Such estimates are useful for backup and recovery management while the occurrence of system breakdown. Such findings are useful for risk evaluation and disaster management in setup of cloud computing and data centre.

References

[1] More Sarla, and Shukla Diwakar, Some new methods for ready queue processing time estimation problem in a multiprocessor environment, Social Networking and Computational Intelligence, Lecture notes in Networks and Systems, Springer, Singapore, March 2020; 100: 661-670.

[2] More, Sarla and Shukla Diwakar, Analysis, and extension of methods in ready queue processing time Estimation in Multiprocessor Environment, Proceedings of International Conference on Sustainable Computing in Science, Technology and Management (SUSCOM), Amity university Rajasthan, Jaipur-India, February 2019; 1558-1563.

[3] More, Sarla and Shukla Diwakar, "A review on ready queue processing time estimation problem and methodologies used in multiprocessor environment", International Journal of Commuter Science and Engineering, 2018; 6(5): 1186-1191.

[4] Shukla Diwakar, Jain Anjali, and Choudhary Amita, Estimation of ready queue processing time under SL scheduling scheme in multiprocessors environment, International Journal of Computer Science and Security, 2010; 4(1): 74-81.

[5] Shukla Diwakar, Jain Anjali and Choudhary Amita, "Estimation of ready queue Processing time under usual group lottery scheduling (GLS) in multiprocessor environment", International Journal of Commuter Applications, 2010; 8(14): 39-45.

[6] Shukla Diwakar, Jain Anjali and Choudhary Amita, Prediction of ready queue processing time in multiprocessor environment using lottery scheduling (ULS), International Journal of Commuter Internet and Management, 2010; 18 (3):58-65.

[7] Shukla Diwakar, and Jain Anjali, Analysis of ready queue processing time under PPS-LS and SRS-LS scheme in multiprocessing environment, GESJ: Computer Science and Telecommunications, 2012; 33(1): 54-61.

[8] Shukla Diwakar and Jain Anjali, Estimation of ready queue processing time using efficient factor type estimator (E-F-T) in multiprocessor environment, International Journal of Computer Applications, 2012; 48(16): 20-27.

[9] Shukla Diwakar and Jain Anjali, Ready queue mean time estimation in lottery scheduling using auxiliary variables in multiprocessor environment, International Journal of Commuter Applications, 2012; 55(13):13-19.

[10] Jain, Anjali and Shukla, Diwakar, Estimation of ready queue processing time using factor type (F-T) estimator in multiprocessor environment, COMPUSOFT, An International Journal of Advanced Computer Technology, 2013; 2(8): 256-260.

[11] Shukla Diwakar , Jain Anjali and Verma Kapil, Estimation of ready queue processing time using transformed factor- type (T-F-T) estimator in multiprocessor environment, International Journal of Computer Applications, 2013;79(16): 40-48.

[12] Carl A. Waldspurger and E William Weihl, Lottery Scheduling: Flexible proportional share resource management, The 1994 Operating systems design and implementation conference (OSDI '94), Monterey, California.

[13] Johnnie Daniel, Sampling Essentials: Practical Guidelines for Making Sampling Choices, Sage Publication, 2011.

[14] Paul S. Levy and Stanley Lemeshow, Sampling of Populations: Methods and Applications, Volume 543 of Wiley Series in Survey Methodology, Wiley, 2008.

[15] Sampath, S.,Sampling Theory and Methods, Alpha Science International Publication, 2005.

[16] Cochran, W.G, Sampling Technique, Wiley Eastern publication, New Delhi,2005.

[17] Poduri S. R. S. Rao, Sampling Methodologies with Applications, Texts in Statistical Science, Chapman and Hall/CRC Press, 2000.

[18] Ranjan K. Som, Practical Sampling Techniques, Second Edition Statistics: A Series of Textbooks and Monographs, CRC Press, 1995.

[19] Steven K. Thompson, Sampling, Volume 272 of Wiley Series in Probability and Statistics, 1992.

[20] Shweta Ojha, Saurabh Jain and Diwakar Shukla, Hybrid lottery multi-level queue scheduling with a Markovian model, GESJ: Computer Science and Telecommunications, 2011; 3(32): 86-97.

[21] Pradeep kumarJatav, Rahul Singhai, and Saurabh Jain, Analysis of hybrid lottery scheduling algorithm using Markov chain model, International Journal for Research in Engineering Application and Management (IJREAM), 2018; .04 (09): 180-192.

[22] MaríaMejía, Adriana Morales-Betancourt and Tapasya Patki, Lottery scheduler for the Linux kernel, The author; licensee universidad Nacional de Colombia. Dyna 82 (189), Medellín. Printed, Online DOI: http://dx.doi.org/10.15446/dyna.v82n189.43068, 2015; 216-225.

[23] Diwakar Shukla and Sarla More, Modified group lottery scheduling algorithm for ready queue mean time estimation in multiprocessor environment, Reliability: Theory & Applications (RT&A), 2020; 15, 4(59): 69-85.

[24] Carl A. Waldspurger, Lottery and stride scheduling: Flexible proportional-share resource management, Massachusetts Institute of Technology, September 1995.

[25] Singh H.P., Gupta A. and Tailor R., Estimation of population mean using a difference-type exponential imputation method, https://doi.org/10.1007/s42519-020-00151-2, Journal of Statistical Theory and Practice 2021; 15(19).

[26] Kone Dramane, Goore Bi Tra, and Dr. Kimou Kouadio Prosper, New hybrid method for efficient imputation of discrete missing attributes, International Journal of Innovative Science and Research Technology, 2020; 5(11): 983-991.

[27] GaribNath Singh, Mohd. Khalid and Jong-Min Kim, Some imputation methods to deal with the problems of missing data in two-occasion successive sampling, Communications in Statistics-Simulation and Computation, 2019; 50(2):557-580.

[28] Diwakar Shukla, Narendra Singh Thakur and Sharad Pathak, Some new aspects on imputation in sampling, African Journal of Mathematics and Computer Science Research, 2013; 6(1): 5-15.

[29] Diwakar Shukla, D. S. Thakur and N. S. Thakur, Utilization of mixture of x , xi, and X2 in imputation for missing data in post-stratification, African Journal of Mathematics and Computer Science Research, 2012; 5(4):78-89.

[30] Diwakar Shukla, Narendra Singh Thakur, Dharmendra Singh Thakur and Sharad Pathak, Linear combination based imputation method for missing data in sample, International Journal of Modern Engineering Research (IJMER) 2011;1(2): 580-596.

[31] Diwakar Shukla and Narendra Singh Thakur, Imputation Methods in Sampling Hardcover, Aman Prakashan, January 2014, available at Amazon:https://www.amazon.in/Imputation-Methods-Sampling-DiwakarShukla/dp/B07VY2B59D.