CC BY
70
Modified Group Lottery Scheduling Algorithm for Ready Queue Mean Time Estimation in Multiprocessor
Environment
Diwakar Shukla and Sarla More
Department of Computer Science and Applications Dr. Harisingh Gour University, Sagar, (MP), India diwakarshukla@rediffmail.com, sarlamore@gmail.com
Abstract
The problem of ready queue mean time estimation in the multiprocessor environment was discussed by Shukla et. al. [5] and several others. In recent years, most of the existing and relating contributions assume that all processes in the ready queue might have been completed before a particular instant of time occur like a sudden failure or interrupt. Due to this, data of time consumed by processes remain available. The idea of improvement in this paper is to assume that at the instant of occurrence of breakdown, some processes are partially completed and remaining is completely processed. Under this situation, the time computation and allocation strategies need to be re-designed. Therefore this has been taken into account in this paper with a proposal of a modified scheme. It contains arbitrary, Type-A, and Type- B allocations of sample units to the processors. Confidence intervals for the sample mean values are calculated and simulated over many samples using cumulative probabilities. It was found that Type-A allocation has the lowest variance.
Keywords: CPU, Scheduling, Lottery Scheduling, Estimation, Sampling, Probability, Allocation, Simulation.
I. Introduction
The challenging task of an operating system is CPU scheduling algorithms where various non-probabilities based traditional schemes are operational. These can simply be handled easily by processors while probabilistic scheduling schemes have to face the difficulty of resource management, system performance, and low system overhead. Lottery scheduling is one such probability-based scheme first introduced by Carl A. Waldspurger [12]. Shukla, Jain, and Choudhary [4] have initiated the problem of estimation of ready queue processing time by suggesting SL scheduling algorithm in a multiprocessor environment. The contribution contains a sample-based estimation of ready queue mean time which likely to be spent while completes exhaust of ready queue occurs. It reveals the approach of systematic sampling which has some limitations in terms of efficiency of the predicted value. Shukla et. al. [6] extended similar problem under the approach of lottery scheduling. Content of contribution stands for randomly selected processes from the ready queue for forecasting the sample-based mean time. The limitation of lottery scheduling appears due to the reason that processes happen to be of any size may appear in any order before multiprocessors. Shukla and Jain [7] extended the ready queue processing time estimation approach to the care of probability proportional to size-dependent lottery scheduling which provides better prediction than earlier. Following the similar approach, Shukla and Jain [8] used factor type estimation method for estimating mean ready queue processing time in setup of
lottery scheduling under a multiprocessor environment. Shukla and Jain [9] extended approach using ratio type estimation method and advocated for better efficiency under constraints. A similar approach adopted in Jain and Shukla [10] and Shukla and Jain [11] with additive features. An exhaustive review of the problem of ready queue mean time estimation is due to Shukla and More [1] and some suggestive contributions are due to Shukla and More [2] [3]. Sampling technique concepts and applications are in Cochran [13].
Shukla D., Jain, and Choudhary [5] discussed GL scheduling which assumes the processes present in all processors in the time session (0-T) have been completely processed at instant T and their compound predictive estimate of average processing time could be obtained. Such an estimate is useful for forecasting the expected time required to vacate the entire ready queue. This helps in backup management while sudden failure (or disaster) occurs. But it doesn't cover the case when a sudden failure occurs during the processing of these jobs (processes). How estimation will be in a situation when the last process is partially processed and kept on hold. This paper takes into account this problem and provides a solution
II. GL Scheduling Scheme (due to Shukla, Jain, and Choudhary [5]):
Step 1: Assume multiple processors Q1, Q2, Q3......Qr, each draws random samples of jobs from
corresponding ready queues. Processes in the ith ready queue are homogeneous concerning certain characteristics whereas in the usual waiting queue they are present in any order of size measure.
Step 2: The CPU restricts a session of time duration T. All N ready queue processes are divided into r groups each of size containing Ni processes ( £ Ni = N). This division is based on size measure.
Step 3: All N processes are allotted token of numbers and each processor draws a random number. If the random number of ith processor matches the allotted random number to the jth process of the ith group then it is selected for processing (i=1, 2, 3.. ..r, j=1, 2, 3.....Ni).
Step 4: Let k1 processes received from the first group, k2 processes from the second group, and so on, the krth received processes from rth group in a random manner using lottery procedure [ £ ki = k] in a session of fixed time T where k is the total sample size.
Step 5: At the end of a session, the CPU provides processed time data for k1, k2, k3....kr jobs as (tn, t12, t13......t21, t22, t23...., .ti1, ti2, ti3...) where tij are the time consumed by jth job.
III. Modified Group Lottery Scheduling (MGLS) Scheme
The proposed contribution is an extension of the previous algorithm suggested by Shukla et. al. [5],
with the idea of improvement to include the processing time of those processes that remained
partially processed due to sudden system breakdown or occurrence of an interrupt. Following are
steps of the proposed scheme:
Step 1: Assume r processors Q1, Q2, Q3, Q4..........Qr, in a system each, receives random samples from
corresponding linked ready queues. Processes in corresponding ready queues are of homogeneous concerning a specific characteristic. If any event wait appears, that process moves to a waiting/blocked/suspended queue.
Step 2: Total N processes assumed present in the system are divided into r groups of ready queues with the assumption that ith group (or ready queue) has Ni processes (£ Ni = N).
Step 3: All N processes in the system are assigned token of numbers. Processors generate random numbers whose matching occurs with token assigned to processes. If ith processor random number matches to the token number of jth process then jth assigns to ith processor.
Step 4: Using (3), suppose total kr processes selected from rth group of the ready queue in a
random manner and assigned to Qrth processor. The total sample size is k =£ ki where i
=1,2,3'.......r, j = 1,2,3'.........Ni
Step 5: Let tij denote time consumed by the jth process assigned to ith processor. Step 6: At instant time T, out of total ki processes present in ith processor, assume ki-i have completely processed but the last one is partially processed with time ti* in all Qi, Q2, Q3
....Qr. The set of time (ti*, t2*, t3*.......tr*) is the time consumed by partially processed jobs.
Step 7: Processes within the processor are divided into two parts. The Part A being sub-group of
completely processed and part B for unprocessed (ti*)
1 ^ . -r 1 ^N. x , v o „ 1
Step 8: Overall mean time, mt = 1 tij, mtj = ^ H^Oij) (for ith ready queue), Si2= -— H^i^tjj -mti )2 (for ith ready queue) and S2 = ¿[=1 1 (tjj - mt )2 under assumption while all N completely processed before occurring T but under step (6) it does not happen. Note: The steps 5, 6, and 7 are the idea of improvement in this paper over the Shukla et. al. [5].
Pi P2 P3 P4
( Proces V UnProc sed \ ( Proce esse J V TTnPro ' i ssed A ( Proc *cesse J V TTnPro r 1 essed A ! Process \cesse y \ Unnroce ' \ ed
00 M W («0
Figure 1: Setup of ready queue and multiprocessor environment
IV. Estimation Procedure under Arbitrary Allocation
The Modified Group Lottery Scheduling algorithm (MGLS) provides the estimation of mean time likely to consume by the N processes in the ready queue while occurrences of time T. For ith ready queue (group), the mean time is spited into:
(a) ti' = ( ^ ) YJjL 11 (tjj) (for processed part A of sample not including unprocessed)
_ * _ 1 *
(b) t = — Ylj=1 (tj ) (for unprocessed part B jobs in all r samples)
MODIFIED GROUP LOTTERY SCHEDULING ALGORITHM FOR READY lS^T^ N° ^olo QUEUE MEAN TIME ESTIMATION IN MULTIPROCESSOR ENVIRONMENT_°ume ' ecem er
_ ' _ * N-
(c) The mean time estimator is u = [ £[=1wjtj + t ] / 2 where wj '
N
(d) The mean square of time q for ith group is Si2 = ^^ f^i^ij - fi)2 = f ^i^ij -
mt;)2 Where tt = ± SN=ii tij
(e) S2 = ^ £=1 £|=i(tjj - t ) where t = 1 £=1 ^(j US
(f) Variance of estimator u is V(u)arbit = V [ £f=1 wjtj' + t *] = £r=1 wj2 V (tj') + V(t *)
= MikV ¿w Sj2 + [ (1- n)s2 ] (4.1)
This estimator u and variance V (u) arbit is based on arbitrary allocation of processes to the processors.
V. Types of Allocations:
Type-A Allocation: Based on prior information of processor speed
The choice of ki depends on the speed of processors. A fast processor can randomly pick a larger
number of jobs from the group of ready queue samples. Let priority known processor speed are Si*,
S2*' S3*.......Sr* for Qi, Q2, Q3 .. ..Qr respectively, and £r=1 Sj = S holds.
* * * k * Let ki a s, ' kj = ms, , £kj = ems, , k = M S*, M = (k/s*), kj = ( k ^ (M is any constant)
(5.1) Substituting (5.1) in (4.1) one can get
V (u) I = f.,
I F )Si -1
(1» N) S ]
— IW S, N I i i
w2 S2
+ [ c--N)s2 ]=K^- NH S,2 ] +
N
V (U) I = f-.i [ ( S^ )] - N Ï-.1W, Si2 + [ (- - N) S2 ] (5.2)
Type-B Allocation: Based on prior information of variation (Si2 ) in ready queue:
The Si2 for ith group is defined in section 4.0 as under Si2 = (tij-fi )= fa^ f^j-^2
Consider ki a Sj * and ki a Sj together where Si refers to variability among processes in ith queue related to a characteristic (e.g. expected time of process) and assumed known. Then, ki a Sj * Sj, ki= M * Sj * Sj where M is constant £ki= M * £ Sj * Sj,
M *=—^r- and ki= [ —k— ] S * S fc ^ 2 S S LSSV j j (5)
The variance under Type-B allocation could be obtained by substituting (5.3) in expression (4.1)
V (U)II = a., [(i^-f-f^ )w,2 s,2] - [N X-=1 WjSj2 ] + [ (1- N)S2 ] (5.4)
VI. Numerical Illustration:
Consider a small data setup with 30 processes in the ready queue whose expected processing time (tij) are given in table 1. This numerical table 1 is to justify the computations, expressions, results.
1
Diwakar Shukla, Sarla More
MODIFIED GROUP LOTTERY SCHEDULING ALGORITHM FOR READY QUEUE MEAN TIME ESTIMATION IN MULTIPROCESSOR ENVIRONMENT
Table 1: Total Processes Data Total Processes Data
Process CPU Process CPU Process CPU Process CPU Process CPU Process CPU
Time Time Time Time Time Time
Proc1 30 Proc6 60 Proc11 138 Proc16 89 Proc21 143 Proc26 79
Proc2 20 Proc7 33 Proc12 43 Proc17 123 Proc22 29 Proc27 46
Proc3 142 Procs 43 Proc13 109 Proc18 67 Proc23 147 Proc28 59
Proc4 40 Proc9 101 Proc14 26 Proc19 58 Proc24 94 Proc29 72
Proc5 59 Proc10 69 Proc11 138 Proc16 89 Proc21 143 Proc26 79
Assume there are three processors Qi, Q2, Q3 (r=3) having known processing speed Si*, S2*, S3* respectively. Ready queues are divided into three groups as under as in Table 2, Table 3 and 4.
_Table 2: First Group Data (below 50 CPU time)_
_Ready Queue Group 1_
Process Proci Proc2 Proc4 Proc7 Proc8 Proc12 Proc14 Proc22 Proc27 Proc30 CPUTime 30 20 40 33 43 43 26 29 46 22
_Table 3: Second Group Data (above 50 but below 100 CPU time)_
_Ready Queue Group 2_
Process Proc5 Proc6 Proc10 Proc15 Proc16 Proc18 Proc19 Proc20 Proc24 Proc26 Proc28 Proc29 CPUTime 59 60 69 74 89 67 58 84 94 79 59 72
_Table 4: Third Group Data (above 100 CPU time)_
_Ready Queue Group 3_
Process Proc3 Proc9 Proc11 Proc13 Proc17 Proc21 Proc23 Proc25
CPUTime 112 101 138 109 123 143 147 131
_Table 5: Available Speed of the Processor_
Processor's Speeds
Processors Q1 Q2 Q3 Total available speed
Speed S1* = 2.5 S2* = 3.0 S3* = 5.5 11.0
Table 6: Parameters of all N Processes in System
Parameters of all N Processes in System
Complete N
Group 1 (Table 6.2)
Group 2 (Table 6.3)
Group 3 (Table 6.4)
Mean time t = tjy =
73.33 '
N1 _ N
0.33
w2 = —2 = 0.4
N
w3 =0.26
N
Mean square Mean time (mt-i ) = ,, ,-s _ _„ „ Mean time
S2 = 146L8484 M =33.20 ' Mean time (mt2 ) = ^^ 0^)^=125.50
Square of mean time Square of mean time (^ )2 = 5184 SqUare of mean time ^ )2 )2 = 1102.24 q ( 2 ) = 15750.25
Total sum of Total sum of square ^ t2j 2 = The total sum of
square 2 =11804 63890 square Sy=1t3j 2 = 128018
Mean square S12 = 86. 8444 Mean square S22 = 152. 9090 and S2 Mean square S32 = 288
and S1 = 9.32 =12.37 and S3 = 16.97
VII. Calculation for Arbitrary Allocation
Table 6 reveals parametric values of all three queues assuming if all N have been processed before occurrences of instant breakdown T. Parameters Si2, S2, t^, ¿2 , ¿3, and t have been calculated at the entire level. Moving on at the sample level, the arbitrary allocation ki, k2, k3 is adopted for sample size k =£kj =12. In table 7, sample values ki = 4, k2 = 4, k3 = 4 considered for total random sample size k=12 drawn from N=30.
Variance of estimator u is V (u) arbit = V [ £f=1wjtj' + t ] = £f=1wj2 V (tj') + V(t )
= zr=1^- ^w2 s,2 + [ (1- n)s2 ]
Table 7: Variances Calculation under Arbitrary Allocations (Si2 and S2 known)
Variance under Arbitrary Allocation
k1 =4, k2= 4, k3= 4
V(u) = 446.442 arfeit
Calculation for Type-A and Type-B allocations:
Consider following available data for variability and processor speed, both are assumed priory known. Table 8 has similar content relating to Si*
Table 8: Prior knowledge of Speed and Variability
Prior knowledge of Speed and Variability
Processors Speed (Si*) Variability (Si) Si*Si
Processor 1 S1* = = 2.5 S1 = 9.3 23.25
Processor 2 S2* = = 3.0 S2 = 12.3 36.9
Processor 3 S3* = 5.5 S3 = = 16.9 92.95
Total (S*) =11.0 ^ Si*Si =153.1
**
Case 1: For Type-A allocation using (5.1), kj =( k /S )Sj , S*= £Si*, k = £ ki, For pre-fixed k = 12, its division in three parts is in table 9 .
Table 9: Allocation under Type -A
Allocation under Type -A
k1 = (k/S*)S1* = 2.72 = 3 (from first ready queue)
k2 = (k/S*)S2* = 3.27 = 3 (from second ready queue)
k3 = (k/S*)S13 = 6.0 = 6 (from third ready queue)
Total k = (k1+k2+k3) k = 12
Case 2: For Type-B allocation using (5.3), ki = [ ] (Sj * Sj ), and k = 12 is divided in three parts as shown in table 10.
Table 10: Allocation under Type- B
Allocation under Type-B
ki " [k/(£Si"Si)]= 2.20
k2 S [k/(£Si"Si)]= 1.98
k3 S [k/(£Si"Si)]= 7.87 Total k = (ki+k2+k3)
= 2 (from first ready queue) = 2 (from second ready queue) = 8 (from third ready queue) k = 12
Calculation of Variance under Type-A allocation:
V (u)i = zr=1 [S * (wi2Si2) / (kSi * - S*)]- i^Wi Si2 + (1- 1) S2
= S*{[wi2Si2/ (kSi* - S*)] + [W22S22/ (kS2* - S*)] + [wa2Sa2/ (kSa* - S*)]} - 1 [wiSi2 +W2S22 +W3S32]
+ ([- N) N-i [2f=iSN=ii(tij- t- )] when r = 3 7.\)
Calculation of Variance under Type-B allocation:
V (u)n = sr=i[(kSi *Si-s Si* Si)/s Si * Si] wi2 Si2 - Nsr=i wiSi2 + [(1 - 9 S2]
= [(kSi*Si - ZSi*Si)/ £Si*Si] wi2Si2 + [(kS2*S2 - £S2*S2)/ £S2*S2] w22S22 + [(kS3*S3 - £S3*S3)/ £&>"&]
w32S32 -1 [wiSi2 +w2S22 +w3S32] +g - 1) ^ EU ji^ty - t )] when r = 3 (7.2)
Table 11: Comparison of Variances under different Allocations
Comparison of Variances under different Allocations
Variance under Type-A Variance under Type-B Variance under Arbitrary
Allocation Allocation Allocation
ki =3, k2= 3, k3= 6 ki =2, k2= 2, k3= 8 ki =4, k2= 4, k3= 4
V(u ) = 442.08 V(u ^ = 6ii.452 V(u) = 446.442 ar&it
Table 8 contains the assumption that three Si2 (i = 1, 2, 3) are priory known (or guessed) and so the variance V (u) is lowest under the type-A allocation (while Si2 and S2 known) in comparison to Type-B and Arbitrary allocation.
Estimate of Variance :
The value Si2 = H^^ij - ^¿)2 suppose not known then they are to be replaced by sample
value estimates. The sample based estimate of S2 and Si2 are defined like (es) 2 and (esi) 2 with expressions are as under:
(esi) 2 = fez^ ZMtj - h) and (es)2 = S^-^ij - H) 2
Est[ V(U)arbit] = (65^)2 + [(1- £ (65)2]
Est[ V(U)j] = Sf=i[S * (wi2 (es)2) /( kSi *-S* )]- i^Wi (es,)2 + (i- 1) (es)2 Est[V(U)jj] = [( Sr=i[k Si*(esi)-SSi *(esi))/ £Si* (esi) ]wi2 (esj)2- i^w^es;)2 +
[(i- ^)2]
Calculations of estimated values are in table 7.6 and 7.7 on the 10 samples.
(7.3.1)
(7.3.2)
(7.3.3)
(7.3.4)
Table 12: Calculations of Sample Mean and Estimate of Variance under Arbitrary Allocation (Section 4.0) in 10 samples (when Si2 and S2 unknown)
(*Partially processed job containing a part of the processing time and unprocessed due time)
Calculations of Sample Mean and Estimate of Variance under Arbitrary Allocation
Random Sample No. Sampled Selected with Processing Time (k=9) Processed E wfr Unprocessed (ti*+t2*+t3*)/3 es2^ Zr=I(ti *-f*)2 Sample Mean V(u)arbit
Groupl Ki=4 Group2 K2=4 Group3 K3=4 (Û)
30,43,33,30* 60,84,6 7,59* i38,ii2,i09,i0i* 56.24
1. Mean=35.33 ti*=25 (esi)2=46.33 Mean=70.33 t2*=39 (es2)2=i52.33 Mean=ii9.6 t3*=6i (es3)2=254.33 70.88 4i.6 (es)2=37.66 ii2.478
33,46,40,20* 69,58,59,60* i09,i0i,ii2,i43*
2. Mean=39.6 ti*=15 (esi)2=50.26 Mean=62 t2*=35 (es2)2= 37 Mean=i07.33 t3*=88S (es3)2= 32.33 65.77 46 (es)2 =i423 55.88 430.07
20,46,30,40* 59,72,79,69* i47,i38,i0i,i23*
3. Mean=32 ti*=25 (esi)2=172 Mean=70 t2*=39 (es2)2=i03 Mean=i28.6 t3*=56 (es3)2=594.33 7i.99 40 (es)2 =24i 55.99 86.66
40,22,26,33* 74,84,60,58* i3i,i09,i23,ii2*
4. Mean=29.33 ti* =23 (esi)2=89.33 Mean=72.66 t2*=29 (es2)2=i46.79 Mean=i2i t3*=67 (es3)2=i24 70.20 39.77 (es)2 =557 54.98 i76.44
43,29,30,20* 79,67,58,60* i23,i43,ii2,i0i*
5. Mean=34 ti*= i5 (esi)2 = 6i Mean=68 t2*=35 (es2)2= iii Mean= i26 t3*=65 (es3)2=247 7i.i8 38.33 (es)2 =634 54.75 i98.63
20,22,29,43* 59,72,84,67* i0i,i09,i23,i3i*
6. Mean=23.66 ti*=28 Mean=7i.66 t2*=47 Mean= iii t3*=8i 65.33 52 (es)2 =72i 58.66 224.36
(esi)2=22.80 (es2)2=i56.33 (es3)2=i24
30,29,20,26* 59,69,72,58* i0i,i47,i09,ii2*
7. Mean=26.33 ti*=i9 (esi)2=30.33 Mean=66.66 t2*=38 (es2)2=46.33 Mean=ii9 t3*=66 (es3)2=604 66.29 4i (es)2=559 53.64 i76.34
30,26,33,29* 72,58,74,60* ii2,i3i,i0i,i23*
8. Mean=29.66 ti* =24 Mean=68 t2*=44 Mean=ii4.66 t3*=68 66.79 45.33 (es)2 =486 56.06 i5i.44
(esi)2= i2.33 (es2)2=76 (es3)2=230.33
40,29,30,46* 60,58,67,79* i09,ii2,i3i,i0i*
9. Mean=33 ti*=26 (esi)2=37 Mean=6i.66 t2*= 49 (es2)2=23.57 Mean= ii7.33 t3*=79 (es3)2= i42.33 66.05 5i.33 (es)2 =707 58.69 2i5.38
20,43,40,22* 79,58,60,59* i23,i0i,ii2,i43*
10. Mean=34.33 ti*=i6 (esi)2=i56.5 Mean=65.66 t2*=34 (es2)2=i34.33 Mean= ii2 t3*=73 (ess)2 =i2i 66.7i 4i (es)2 =849 53.85 265.i9
Table 13: Estimated values of Variances over 10 samples as per table 6.7 (when Si2 and S2 are unknown)
Sample Number i
Sample Mean (u ) 56.24
Est[V(u)arbit ] ii2.478
Est[V(u) ] ii3.65
Est[V(u )ii] 242.29
2 3 4
55.88 55.99 54.98
430.07 86.66 i76.44
43i.86 90.26 i80.95
453.07 333.ii 26i.55
5 6 7
54.75 58.66 53.64
i98.63 224.36 i76.34
20i.02 227.ii i75.22
3i7.58 308.78 405.65
8 9 i0
56.06 58.69 53.85
i5i.44 2i5.38 265.i9
i5i.93 2i6.ii 27i.09
253.46 273.94 349.22
Calculation of Confidence Interval (CI):
A. The 95% Confidence Interval of the sample mean u is defined as:
Probability [(u) ± 1.96 Vv(u) ] = 0.95. The interpretation of C.I. is that it is an interval where the chance of laying the unknown true value of mean time is 95%.
In another way, the 95% chance is that unknown mean processing time of all N processes will lie in the confidence interval.
Table 8, 9, and 10 present the computation of confidence intervals for different types of allocations. When Si2, S2 treated unknown.
B
C.
Table 14: Confidence Interval Calculation under Arbitrary Allocation [using Table 6 and 7]
Sample Number i 2 3 4 5 6 7 8 9 i0
Sample Mean (u) 56.24 55.88 55.99 54.98 54.75 58.66 53.64 56.06 58.69 53.85
Est.[ V(u)arbit] ii2.478 430.07 86.66 i76.44 i98.63 224.36 i76.34 i5i.44 2i5.38 265.i9
Estimate of Confidence Interval for Est[ V(u)arbit ] (35.45, 77.02) (i5.23, 8i.28) (37.74, 74.23) (28.94, 8i.0i) (27.i2, 82.37) (29.30, 88.0i) (27.6i, 79.66) (3i.94, 80.i7) (29.92, 87.45) (2i.93, 85.76)
Table 15: Confidence Interval Calculation for Type-A Allocation [using Table 9 and 10]
Sample Number i 2 3 4 5 6 7 8 9 i0
Sample Mean (u) 56.24 55.88 55.99 54.98 54.75 58.66 53.64 56.06 58.69 53.85
Est.V(u)i ii3.65 43i.86 90.26 i80.95 20i.02 227.ii i75.22 i5i.93 2i6.ii 27i.09
Estimate of Confidence Interval for Est[ V(u)i ] (35.34, 77.i3) (i5.i4, 96.6i) (37.36, 74.6i) (28.6i, 8i.34) (26.96, 82.53) (29.i2, 88.i9) (27.69, 79.58) (3i.90, 80.2i) (29.87, 87.5) (2i.57, 86.i2)
Table 16: Confidence Interval Calculation for Type-B Allocation [using Table 11 and 12]
Sample Number i 2 3 4 5 6 7 8 9 i0
Sample Mean (u) 56.24 55.88 55.99 54.98 54.75 58.66 53.64 56.06 58.69 53.85
Est.[V(u)n] 242.29 453.07 333.ii 26i.55 3i7.58 308.78 405.65 253.46 273.94 349.22
Estimate of Confidence (25.73, (i4.i6, (20.2i, (23.28, (i9.82, (24.2i, (i4.i6, (24.85, (26.24, (i7.22,
Interval for Est[ V(u)ii] 86.74) 97.59) 9i.76) 86.67) 89.67) 93.i) 93.ii) 87.26) 9i.i3) 90.47)
Random Sample Number Random Sample Number Random Sample Number
Fig. 2: Fig. 3: & Fig 4: Graphical Representation of Estimated CI under Arbitrary, Type-A and Type-B
Allocation over 10 samples
The graphical representation in Fig. 2, 3, 4 shows wide gap between the upper and lower limit. The
Fig 2 shows the smallest length interval.
8.1 Simulation of Confidence Interval under Arbitrary Allocation: 8.1.1 Simulation Algorithm:
Step I: Draw a random sample of size k.
Step II: Compute the lower limit and upper limit of confidence interval (CI) under three allocations.
Step III: Repeat step I and II for d times (here d =200 considered)
Step IV: Let fi be the frequency of ith class interval for lower limit (LL) of CI over d=200 samples. Calculate probabilities pi = (fi/d) = (frequency of class interval /Total frequency d). Similar is for upper limit (UL) CI.
Step V: Compute the Less than Type (LTT) and more than Type (MTT) cumulative
probabilities overall d samples for lower limit (LL) and upper limit (UL) of confidence intervals.
Step VI: Plot data of step IV on the graph. The perpendicular from point of intersection on the x-axis is the simulated value of lower limit and upper limit of a confidence interval for unknown parameters required to be estimated.
Table 17: Cumulative Probability-based Simulation for Arbitrary Allocation (over d=200)
The lower limit of Confidence Interval_The upper limit of Confidence Interval
Class Interval (LL) Mid- Cumulative Class Interval (UL) Mid- Cumulative
value of class interval Probability Pi probabilities LTT MTT value of class interval Probability Pi probabilities LTT MTT
10-15 12.5 0.01 0.0i i 70-75 72.5 0.09 0.09 i
15-20 17.5 0.12 0.i3 0.99 75-80 77.5 0.23 0.32 0.9i
20-25 22.5 0.15 0.28 0.87 80-85 82.5 0.42 0.74 0.68
25-30 27.5 0.43 0.7i 0.72 85-90 87.5 0.23 0.97 0.26
30-35 32.5 0.18 0.89 0.29 90-95 92.5 0.03 i.00 0.03
35-40 37.5 0.10 0.99 0.0i Total i.00
40-45 42.5 0.01 i.00 0
Total 1.00
iy
'S
Xb
o —
Ph
OJ >
13 S s
u
Class Interval (using Table 8.4 Lower Limit)
¿y
'S b o r
Ph
e iv lati ul
um u
C
Class Interval (using Table 8.4 Upper Limit)
Fig 5: & Fig 6: Graphical representation for LTT & MTT for Arbitrary Allocation Table 18: Simulated values of C I under Arbitrary Allocation (using Table 12, Fig 5 & Fig. 6)
Simulated values of Lower Limit of C I Simulated values of Upper Limit of C I
24.5 79.5
Fig. 5.and Fig. 6 is revealing point of intersection of two curves. The final value is determined by perpendicular drawn on the X-axis. The table i8 contains the estimated value, based on perpendicular, which is (24.5, 79.5).
Simulation of Confidence Interval under Type-A Allocation:
Table 19 Sample mean and variance calculation for Type-A allocation (over 10 samples)
Sampled Selected with Processing Time TT >
r B Unprocessed
Sample Number
(k=9)
Groupl Ki=(3)
Group2 K2=(3)
Group3 K3=(6)
Processed
E wfr
(tl*+t2*+t3*)/3
es2=--- Iï=1(t,*-t*)2
Sample Mean (û)
V(ü)i
30,43,33* Mean=36.5 ti*=25 (esi)2= 42.25 33,46,40* Mean=39.5 ti*=20 (esi)2= 42.25
20,46,30* Mean=33 ti*=20 (esi)2= 169
40,22,26* Mean=31 ti* =20 (es1)2=81
43,29,30* Mean=39 t1*= 15 (es1)2 = 176
20,22,29* Mean=21
60,84,67* Mean=72 t2*=37 (es2)2= 144
69,58,59* Mean=63.5 t2*=34 (es2)2=
30.25
59,72,79*
Mean=65.5
t2*=49
(es2)2=
42.25
74,84,60* Mean=79 t2*=31 (es2)2= 25
79,67,58* Mean=73 t2*=35 (es2)2= 36
59,72,84* Mean=65.5
138,112,109, 101,143,123* Mean=120.6 t3*=83
(es3)2= 279.44
109,101,112,
143,147,131*
Mean=122.4
t3*=81
(es3)2= 355.04
147,138,101,
123,112,109*
Mean=124.2
t3*=59
(es3)2=279.76
131,109,123,
112,101,143*
Mean=115.2
t3*=100
(es3)2=112.19
123,143,112,
101,109,147*
Mean=117.6
t3*=75
(es3)2=211.04 101,109,123, 131,143,112*
72.19
70.25
68.91
71.78
72.64
64.69
48.33
(es)2=937.8
45
(es)2= 1021
42.66
(es)2= 274.12
50.33
(es)2=1880.83
41.66
(es)2=934.16 52
(es)2=964
60.26
57.62
55.78
61.05
57.15
58.34
293.31
312.19
94.93
570.43
292.54
356.33
10.
ti*=20 (esi)2= 1
30,29,20* Mean=29.5 ti*=25 (esi)2= 0.25
30,26,33* Mean=28 ti* =22 (esi)2= 4
40,29,30* Mean=34.5 ti*=2i (esi)2= 30.25 20,43,40* Mean=3i.5 ti*=30
(esi)2=
i32.25
t2*=54 (es2)2=
42.25
59,69,72* Mean=64 t2*=42 (es2)2= 25
72,58,74* Mean=65 t2*=50 (es2)2= 49
60,58,67* Mean=59 t2*= 47
(es2)2= i
79,58,60*
Mean=68.5
t2*=35
(es2)2 =
ii0.25
Meani2i.4 t3*=82
(es3)2=226.24 i0i,i47,i09, ii2,i38,i23* Mean=i2i.4 t3*=73
(es3)2=3i7.84 ii2,i3i,i0i, i23,i09,i3i* Mean=ii5.2 t3*=90
(es3)2=ii2.i6 i09,ii2,i3i, i23,i43,i0i* Mean=i23.6 t3*=79
(es3)2=i55.84 i23,i0i,ii2, i43,i47,i38* Mean=i25.2 t3*=78
(ess)2 = 3ii.36
66.89
65.i9
67.ii
66.i2
46.66
(es)2=593.26
54
(es)2=ii68
49
(es)2= 844
47.66
(es)2=697.28
56.77
59.59
58.05
56.89
i92.63
353.95
255.55
223.97
Table 20: Confidence Interval for Type-A Allocation (using Table 19)
Confidence Interval for Type-A Allocation
Sample Number i 2 3 4 5 6 7 8 9 i0
Sample Mean (u ) 60.26 57.62 55.78 6i.05 57.i5 58.34 56.77 59.59 58.05 56.89
Est.[V(ü)i ] 293.3i 3i2.i9 94.93 570.43 292.54 356.33 i92.63 353.95 255.55 223.97
confidence interval for Est[ V(u)i ] (26.69, (22.98, (36.68, (i4.23, (23.62, (2i.34, (29.56, (22.7i, (26.7i, (27.55,
93.82) 92.25) 74.87) i06.6i) 90.67) 95.33) 83.97) 96.46) 89.38) 86.22)
u
MH
o
CD
OJ
13 >
120 100 80 60 40 20 0
lower limit
upper limit
123456789 10
Sample Number using Table i9 Fig 7: Graphical Representation of Confidence Interval for Type-A Allocation
_Table 21: Cumulative Probabilities Simulation for Type-A Allocation (over d=200)
The lower limit of Confidence Interval The upper limit of Confidence Interval
Class Mid-value Probabilit Cumulative Class Mid-value Probabi Cumulative
Interval of class y probabilities Interval of class lity Probabilities
(LL) interval Pi LTT MTT (UL) interval Pi LTT MTT
i0-i5 i2.5 0.0i 0.0i i 70-75 72.5 0.02 0.02 i
i5-20 i7.5 0.i8 0.i9 0.99 75-80 77.5 0.i5 0.i7 0.98
20-25 22.5 0.22 0.4i 0.8i 80-85 82.5 0.i7 0.34 0.83
25-30 27.5 0.32 0.73 0.59 85-90 87.5 0.35 0.69 0.66
30-35 32.5 0.i5 0.88 0.27 90-95 92.5 0.3i i.00 0.3i
35-40 37.5 0.i2 i.00 0.i2 Total i.00
Total i.0
e
iv lat
ul
m u
C
1,2 1 0,8 0,6 0,4 0,2 0
LTT MTT
12,5 17,5 22,5 27,5 32,5 37,5
Class Interval (using Table 21 Lower Limit)
Class Interval (using Table 21 Upper Limit)
Fig 8: & Fig 9: Graphical representation for Lower limit & Upper limit for Type-A allocation
Table 22: Simulated values of CI under Type-A Allocation (using Table 9, Fig 8 & Fig. 9)
Simulated values of Lower Simulated values of
Limit of C I Upper Limit of C I
23.5 83.5
Fig. 8 and Fig. 9 are revealing point of intersection of two curves. The final value is determined by perpendicular drawn on the X-axis. Table 22 contains the estimated value, based on the perpendicular, which is (23.5, 83.5).
Simulation of Confidence Interval for Type-B Allocation:
Table 23: Sample Mean and Variance Calculation for Type-B Allocation (over 10 samples)
Random
Sampled Selected with Processing Time (k=9)
Processed
sample Groupl Group2 Group3 E Wjfj-
Ki=(2) K2=(2) K3=(8)
30,20* 59,60* 123,101,112,143,
Mean=30 Mean=59 147,138,109,131*
1. t1*=20 t2*=60 Mean=124.71 65.92
(es1)2=30 (es2)2=59 t3*=131 (es3)2=331.48
40,33* 69,74* 123,101,112,143,
Mean=40 Mean=69 147,138,131,109*
2. t1*=33 t2*=74 Mean=127.85 74.04
(es1)2=40 (es2)2=69 t3*=109 (es3)2= 286.27
43,20* 67,58* 123,101,112,143,
Mean=43 Mean=67 147,109,131,138*
3. t1*=20 t2*=58 Mean=123.71 73.15
(es1)2=43 (es2)2=67 t3*=138 (es3)2= 306.23
40,29* 33,58* 123,101,112,143,
Mean=40 Mean=33 138,109,131,147*
4. t1*=29 t2*=58 Mean=122.42 58.22
(es1)2=40 (es2)2=33 t3*=147 (es3)2= 247.95
46,22* 58,59* 123,101,112,147,
Mean=46 Mean=58 138,109,131,143*
5. t1*=22 t2*=59 Mean= 123 70.36
(es1)2=46 (es2)2=58 t3*=143 (es3)2= 277.66
30,40* 59,72* 101,143,147,138,
Mean=30 Mean=59 109,131,143,112*
6. t1*=40 t2*=72 Mean=130.28 67.37
(es1)2=30 (es2)2=59 t3*=112 (es3)2= 328.90
Unprocessed Sample
(ti*+t2*+t3*)/3 Mean 1 * - *
es2=--- Zi=1(t1 -t*)2 (a)
V(ü)n
51.66
(es)2=1909.36
56.66
(es)2=1234.46
53.33
(es)2=2033.86
53.33
(es)2=2158.86
51.66
58.79 579.42
65.35 377.46
63.24
55.77
56.66
(es)2=759.46
617.62
652.91
61.01 677.44
62.01 234.36
43,26* 59,69* ii2,i43,i47,i38,
Mean=43 Mean=59 i09,i3i,i0i,i23*
7. ti*=26 t2*=69 Mean= i25.85
(esi)2=43 (es2)2=59 t3*=i23
(es3)2= 336.90
26,30* 69,58* i23,i0i,ii2,i43,
Mean=26 Mean=69 i47,i38,i09,i3i*
8. ti*=30 t2*=58 Mean=i24.7i
(esi)2=26 (es2)2=69 t3*=i3i
(es3)2= 33i.48
22,29* 94,59* i23,i0i,ii2,i43,
Mean=22 Mean=94 i47,i38, i3i,i09*
9. ti*=29 t2*=59 Mean= i27.85
(esi)2=22 (es2)2=94 t3*=i09
(es3)2=286.27
20,33* 59,79* i23,i0i,ii2,i43,
Mean=20 Mean=59 i47,i09,i3i,i38*
i0. ti*=33 t2*=79 Mean=i23.7i
(esi)2=20 (es2)2=59 t3*=i38
(es3)2= 307.47
70.5i
68.60
78.i0
62.36
60
(es)2=2575
55
(es)2=i975
5i.66
(es)2=i259.36
64
(es)2=i948
65.25
6i.8
64.88
63.i8
779.92
60i.96
385.75
590.45
Table 24: Confidence Interval for Type-B Allocation (using Table 10.1)
Confidence Interval for Type-B Allocation
Random sample Sample Mean (u) Est.[V(u)n] Estimate of confidence interval for Est.[V(u)n ]
i
58.79 579.42
(ii.6i, i05.96)
2
65.35 377.46
(27.27, i03.42)
3
63.24 6i7.62
(i4.53, iii.94)
4
55.77 652.9i
(5.68,i05.85 )
5
6i.0i 677.44
(9.99, ii2.02)
6 7 8 9 i0
62.0i 65.25 6i.8 64.88 63.i8
234.36 779.92 60i.96 385.75 590.45
(32.00, 92.0i) (i0.5i, ii9.98) (i3.7i, ii0.5) (26.38, i03.37) (i5.55, iii.26)
u
MH
o
n
o
w
Sample Number using Table i0.i Fig 10: Graphical Representation for Type-B Allocation Table 25: Cumulative Probabilities Simulation for Type-B Allocation (over d=200)
The lower limit of the confidence interval The upper limit of the confidence interval
Class Mid-value Probability P Cumulative Class Mid-value Probability P Cumulative
Interval of class probabilities Interval of class probabilities
(LL) interval LTT MTT (UL) interval LTT MTT
i0-i5 i2.5 0.04 0.04 i 70-75 72.5 0.0i 0.0i i
i0-i5 i7.5 0.i5 0.i9 0.96 75-80 77.5 0.i2 0.i3 0.99
i5-20 22.5 0.i7 0.36 0.8i 80-85 82.5 0.2i 0.34 0.87
20-25 27.5 0.20 0.56 0.64 85-90 87.5 0.32 0.66 0.66
25-30 32.5 0.25 0.8i 0.44 90-95 92.5 0.34 i.00 0.34
30-35 37.5 0.i9 i.00 0.i9 Total i.00
Total i
LTT MTT
Class Interval (using Table 10.3 Lower Limit)
o
e t at
w
1,2 1 0,8 0,6 0,4 0,2 0
LTT MTT
72,5 77,5 82,5 87,5 92,5
Class Interval (using Table 10.3 Upper Limit)
Fig 11: & Fig 12: Graphical representation for Lower limit & Upper limit Type-B allocation
Confidence Interval
Table 26: Simulated values of CI under Type-B Allocation
Simulated values of Lower Limit of C I_Simulated values of Upper Limit of C I
25.5 84.5
Fig. 11 and Fig. 12 are revealing point of intersection of two curves. Final value is determined by perpendicular drawn on the X-axis. Table 26 contains the estimated value, based on the perpendicular, which is (25.5, 84.5).
11. Results, Discussion and Conclusion:
The comparative analysis is stated in table 27
Table 27: Comparative Analysis of Variance and Confidence Interval Range
Strategy Variance of Mean 95% Confidence Interval CI
OJ of Mean
Arbitrary allocation 73.33 450.92 [24.5, 79.5]
Type-A allocation 73.33 442.08 [23.5, 83.5]
Type-B allocation_73.33_611.452_[25.5, 84.5]_
Algorithm MGLS considers a possibility that some processes remain unprocessed while time instant T occurs which was not considered in GL scheduling [5]. As a consequence, the processes in a sample drawn are divided into two parts A and B. The part A incorporates those who processed and part B has partially processed at the breakdown instant T.
Specific assumption herein is that the last process remains unfinished while T appears in every processor. Estimation procedure proposed herein is such as from whole population of jobs in system, some processes are randomly selected and using the sample estimates mean time and variance of the mean time of processed jobs, as well as the variance of partially processed jobs. The estimation procedure is categorized for arbitrary allocation of sample units to processors.
Further, content has two special cases Type-A allocation and Type-B allocation. The Type-A allocation is based on available prior information of processor speed and Type-B allocation is based on available prior information of variability along with processor speed. In all types of allocations, attempt has been made to find out which allocation will provide the lowest variance (efficient).
For the sake of convenience and simplicity, 30 processes present in system have been considered where groups of ready queues are formed. In particular, three groups Group 1, Group 2, and
MODIFIED GROUP LOTTERY SCHEDULING ALGORITHM FOR READY N° ^olo
QUEUE MEAN TIME ESTIMATION IN MULTIPROCESSOR ENVIRONMENT_°ume , ecem er
Group 3 are formed having some processes according to pre-determined CPU time. Table 5 shows the pre-defined speed of processors. For the arbitrary allocation of sampled processes, the sample mean and variance are calculated with the setup shown in table 12 and subsequently in table 19 and table 23. For the special cases, the processor speed and variability of processors is considered. The variance of the Type-A and Type-B allocation is calculated and compared. This can be seen in Table 4. Table 5 which reveal the comparison between them relating to variance of allocations.
The simulation procedure is proposed and the confidence intervals Prob.[(u ) ± 1.96 V V(u)] are calculated and represented in graphical form. Over a large number of samples, the confidence interval of Type-A and Type-B allocation are calculated and displayed in graphical representation. For obtaining a single-valued result, it has been introduced the calculation of cumulative probabilities and the LTT and MTT probabilities of lower and upper limits of the confidence interval are measured. Observing all the calculated data and the final table, one can conclude that the Type-A allocation is an efficient scheme to find out the predictive estimate and it is the best one among all who tested.
It was found that estimation of mean times lies within the length of the confidence interval. The improvement suggests over [5] is fruitful and provides better results. The sample-based procedure of estimation of the mean time is more efficient under the Type-A allocation scheme. Such estimates are useful when the system fails suddenly and the system manager needs time estimation for processing the remaining jobs in the queue. This approach helps in the immediate arrangement of resources while disaster management required.
References
[1] More S, and, Shukla D. (2020) 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, and Available at doi.org/10.1007/978-981-15-2071-6_54, 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, Available at SSRN: https://ssrn.com/ abstract = 3356312 or https:// dx.doi.org/ 10.2139/ SSRN 3356312, February 26-28, 2019.
[3] More, Sarla and Shukla, Diwakar "A Review on Ready Queue processing time estimation problem and methodologies used in multiprocessor environment". International Journal of computer science and engineering, Available at https://doi.org/10.26438/ijcse/v6i5.11511155, Vol.6, Issue 5, pp. 1186-1191, 2018
[4] Shukla D., 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, ISSN: 1985-1553, volume 4, Issue 1, 2010.
[5] Shukla D., Jain Anjali and Choudhary Amita, "Estimation of ready queue processing time under usual group lottery scheduling (GLS) in multiprocessor environment", International Journal of Computer Applications, Vol.8, No.14, 2010.
[6] Shukla D., Jain Anjali and Choudhary Amita, "Prediction of Ready Queue Processing Time in Multiprocessor Environment using Lottery Scheduling (ULS)", International Journal of Computer Internet and Management, Vol.18, No.3, pp 58-65, 2010.
[7] Shukla D., and Jain Anjali, "Analysis of Ready Queue Processing Time under PPS-LS and SRS-LS Scheme in Multiprocessing Environment", GESJ: Computer Science and Telecommunications, vol. 33, No.1, 2012.
[8] Shukla D., 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. Vol. 48, No.16, 2012.
[9] Shukla D. and Jain Anjali, "Ready Queue Mean Time Estimation in Lottery Scheduling using Auxiliary Variables in Multiprocessor Environment", International Journal of Computer Applications, Vol. 55, No.13, 2012.
[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, Vol. 2, Issue 8, 2013.
[11] Shukla D., 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 (0975 - 8887), Volume 79, No 16, 2013.
[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, 1994.
[13] Cochran, W.G, "Sampling Technique", Wiley Eastern Publication, New Delhi, 2005.
Received: August 27, 2020 Accepted: November 15, 2020
