A Markov Chain Model for the Analysis of Round-Robin Scheduling Scheme

Transcription

1 Vol No page: -7 (29) A Markov Chan Model for the Analy of Round-Robn Schedulng Schee D Shukla Dept of Matheatc and Stattc, Dr HS Gour Unverty, Sagar (MP), 473, INDIA Eal:dwakarhukla@redffalco Saurabh Jan Dept of Cop and Applcaton, Dr HS Gour Unverty, Sagar (MP), 473, INDIA Eal:aaurabh_4@yahoocon Rahul Sngha Dept of Cop and Applcaton, Dr HS Gour Unverty, Sagar (MP), 473, INDIA Eal:ngha_rahul@hotalco Dr RK Agarwal Indra Gandh Engneerng College, Sagar INTRODUCTION I ABSTRACT In the lterature of Round-Robn chedulng chee, each job proceed, one after the another after gvng a fx quantu In cae of Frt-coe frt-erved, each proce executed, f the prevouly arrved proceed copleted Both thee chedulng chee are ued n th paper a t pecal cae A Markov chan odel ued to copare everal chedulng chee of the cla An ndex eaure defned to copare the odel baed effcency of dfferent chedulng chee One chedulng chee whch the xture of FIFO and round robn found effcent n ter of odel baed tudy The yte ulaton procedure ued to derve the concluon of the content Keyword: Proce chedulng, Markov chan odel, State of yte, Deadlock State, Proce queue, Frt-n Frt-out (FIFO), Round-Robn chedulng, Tranton probablty atrx Paper ubtted: 2529 Accepted: n an operatng yte, a large nuber of procee arrve to the cheduler whoe role to anage the proceng of thee job There are any chedulng chee avalable n lterature [ee Slberchatz and Galvn [3], Stallng [7], Tanenbau and Woodhull [8]] lke FIFO, Round robn, Prorty baed, Mult-level queue and o on All thee have oe advantage and dadvantage over each other A unfed tudy for chedulng chee requred under a coon envronent Th otvate to degn a general cla of chedulng chee o that t eber ay poe coon properte of the cla a well a could be utually copared Wth th thought of otvaton, a general cla of chedulng chee degned n th paper contanng oe well-known chee lke FIFO and Round robn a eber chee Shukla and Jan [4] have tuded the ult-level queuechedulng chee n the envronent of Markov chan odel Shukla etal [5] tuded the etup of pace dvon wtche n a Markov chan odel cenaro Shukla and Jan [6] ued Markov chan odel for deadlock-baed tudy of ult-level queue chedulng Soe other related contrbuton are due to Medh [] and Nald [2] In the preent tudy, the degned general cla of chedulng chee exaned through a Markov chan odel n order to perfor the coparatve analy of the perforance of eber chedulng chee The overall recoendaton that, under the Markov chan odel baed tudy, the general cla contan chee-iii a the ot recoendable 2 GENERAL CLASS OF ROUND-ROBIN QUEUE SCHEDULING SCHEME Conder a round-robn chedulng chee hown n fg 2 A general cla lad down below: () The S denote cheduler and there are procee P, P 2, P 3, P n queue; (2) The S provde one quantu of te to each proce and next quantu decded by a rando tral; (3) The S tart fro any proce P n queue and then ove to P j ( j =,2,3 ) ; (4) The new proce enter fro the end e P + placed after P and o on; (5) Suppoe S at any proce P (, 2, 3 ) at the end of a quantu, then n the next quantu

2 Vol No page: -7 (29) 2 (a) S wll be on P + wth prorty p, (b) S wll be on P wth prorty, (c) S wll be on P - wth prorty q, (6) The S becoe dle when there no proce n the queue However t aued that the cheduler S ay be n deadlock n any quantu; (7) Fro th deadlock level, the S could be back alo to the queue n any other quantu for proceng purpoe; (8) There a long watng queue of procee P, P 2 outde the proceng unt and f one proce over nde, then a new proce, watng outde, enter nde o a to antan procee there 2 PROPOSED MARKOV CHAIN MODEL ( ) Let { X, n } n denote a Markov chan wth the tate pace P, P 2, P 3 P, D where D a deadlock tate ued to denote dle, blockng or any dturbance caued n the ( n ) yte, durng job proceng The X the tate of th cheduler of the yte at the end of n quantu (n=,2,3 ) Aue that procee are n yte at a te Further let the tranton of cheduler S rando over th + tate n n quantu The tranton dagra for any three procee P -, P, P + and D gven n fg 2 Defne unt-tep tranton P[X (n+) =P + /X (n) =P ] = p P[X (n+) =P /X (n) =P ] = P[X (n+) =P - /X (n) =P ] = q P[X (n+) =D /X (n) =P ] = r P[X (n+) =P /X (n) =D] = P - p Reark 2 The generalzed expreon for n quantu P[X (n) =P ] = P[X (n-) =P - ]p+ P[X (n-) =P ]+ P[X (n-) =P + ]q P[X (n) n ( n ) ( n ) =D] = P[ X = P ] r + P[ X = D] r q 3 SOME SPECIAL SCHEDULING SCHEMES P Fg 3 (Syte Dagra) D By pong retrcton and condton over the way and procedure, one can generate varou chedulng chee fro the generalzed cla n ecton 2 3 Schee- I [A]: When q =, p =, r=, = r q p r P + Then th general cla ha chedulng chee FIFO for all quantu n Reark 3 The ntal probablte at n= for chee- I[A] P[X () =P ]=pb and ubject to the condton pb = Reark 32 The tate probablte after the frt quantu P[X () =P ]=pb Reark 33 The generalzed expreon of chee-i [A] for n quantu P[X (n) =P ]=pb 3 2 Schee- I [B]: When q =, p =, r+ = Then th general cla ha chedulng chee FIFO for all quantu n Reark 32 The ntal probablte at n= for chee-i [B] P[X () =P ]=pb P[X () =D ]= and ubject to the condton pb = Reark 322 The tate probablte after the frt quantu P[X () =P ]=pb P[X () =D ]= r pb = r Reark323 The tate probablte after the econd quantu P[X (2) =P ]=P[X () =P] P[X (2) ( ) ( ) =D]= P[ X = P ] r + P[ X D] = Reark 324 The generalzed expreon of chee-i [B] for n quantu P[X (n) =P ]=P[X (n-) =P] P[X (n) ( n ) n ( ) =D ]= P[ X = P ] r + P[ X = D] 3 3 Schee-II [A]: when q =, =, r=, p= Th general cla ha chee called Round-Robn Schedulng chee for all quantu n Reark 33 The ntal probablte at n= for chee-ii [A] P[X () =P ]=pb and ubject to the condton pb = Reark 332 The tate probablte after the frt quantu P[X () =P ]=pb - Reark333 The tate probablte after the econd quantu P[X (2) =P ]=pb -2 Reark 334 The generalzed expreon of chee-ii [A] for n quantu P[X (n) =P ]=pb -n 3 4 Schee- II [B]: When q =, =, p + r = Then th general cla ha chedulng chee called Round-Robn Schedulng chee for all quantu n

3 Vol No page: -7 (29) 3 Reark 34 The ntal probablte at n= for chee-i are P[X () =P ]=pb (,2,3 ) P[X () =D ]= and ubject to the condton pb = Reark 342 The tate probablte after the frt quantu P[X () =P ]=pb - p P[X () =D ]= r pb = r Reark 343 The tate probablte after the econd quantu P[X (2) =P ]=P[X () =P - ]p P[X (2) ( ) ( ) =D ]= P[ X = P ] p r + P[ X D] = Reark 344 The generalzed expreon of chee-ii [B] for n quantu P[X (n) =P ]=P[X (n-) =P - ]p P[X (n) ( n ) n ( ) =D ]= P[ X = P ] p r + P[ X = D] 3 5 Schee- III [A]: When q =, r=, p + = Reark 35 The ntal probablte at n= for chee-iii [A] P[X () =P ]=pb and ubject to the condton pb = Reark 352 The tate probablte after the frt quantu P[X () =P ]=pb - p + pb Reark 353 The tate probablte after the econd quantu P[X (2) =P ]=P[X () =P - ]p + P[X () =P ] Reark 354 The generalzed expreon of chee-iii [A] for n quantu P[X (n) =P ]= P[X (n-) =P - ]p + P[X (n-) =P ] 3 6 Schee- III [B]: When q =, p + r + = Reark 36 The ntal probablte at n= for chee-iii [B] are P[X () =P ]=pb (,2,3 ) P[X () =R ]= and ubject to the condton pb = Reark 362 The tate probablte after the frt quantu P[X () =P ]= P[X () =P - ]p + P[X () =P ] P[X () =R ]= r pb = r Reark 363 The tate probablte after the econd quantu P[X (2) =P ]=P[X () =P - ]p + P[X () =P ] P[X (2) =D ]= P X ( ) [ = P ] Reark 364 The generalzed expreon of chee-iii [B] for n quantu P[X (n) =P ]=P[X (n-) =P - ]p + P[X (n-) =P ] P[X (n) =D ]= P X ( n ) [ = P ] 3 7 Schee- IV When q =, =, r=, p= The general cla ha chee called Round-Robn Schedulng chee wth condton that cheduler tart proceng wth frt proce for all quantu n Reark 37 The ntal probablte at n= for chee-ii [A] P[X () =P ]= (when ) and ubject to the condton n pb = Reark 372 The tate probablte after the frt quantu P[X () =P ]= (when 2) Reark 373 The tate probablte after the econd quantu P[X (2) =P ]= (when 3) Reark 374 The generalzed expreon of chee-iv for n quantu P[X (n) =P ]= (when ) 6 SIMULATION STUDY In order to copare all the four chedulng chee under a coon etup of Markov chan odel, the ulaton tudy perfored whoe graphcal output below Under Schee-I[A]: Conder ntal probablte pb =7, pb 2 =5, pb 3 =7, pb 4 =8, pb 5 =3 Here p=q=r=, =and all p are nuber of job p Fg 4[A] Fg 4[B]

4 Vol No page: -7 (29) 4 In lght of fg 4[A] and fg 4[B], th to oberve that the quantu varaton doe not affect the tate probablte P The chee-i[a] purely frt-coe frt-erved (FIFO) wth no chance of deadlock Under Schee-I[B]: Intal probablte are pb =7, pb 2 =5, pb 3 =7, pb 4 =8, pb 5 =3, pb r = wth p = q =, r + = and r = p p p Fg 42[A] Fg 43[A] p4 p Fg 42[B] In reference of fg 43[A] and 43[B], the chedulng followed round robn chee wth the condton that cheduler can tart proceng fro any proce wth no deadlock condton It oberved that at oe pecfed quantu for an pecfed proce, the probablty attan a axu But over a large quantu, the downfall of probablty occur At regular nterval, after fve quantu, the tate probablty bear a chance of cheduler beng tranted over the ae Under Schee-II[B]: Fg 43[B] Fg 42[A] and Fg 42[B], relate to the ae chee but wth the conderaton of deadlock tate There chance that durng proceng of job P (,2,3,4,5), the yte ay trant to tate D and aborbed there It found the probablty that yte urvve on the ae proce over a large nuber of quantu reduce wth a fat rate Th ndcate for hangng chance of proce cheduler f the proce P conue ore te The chance of oveent toward deadlock tate are hgh for chee I[B] Under Schee-II[A]: Conder ntal probablte pb =7, pb 2 =5, pb 3 =7, pb 4 =8, pb 5 =3 wth =q=r=, p= Intal probablte are pb =7, pb 2 =5, pb 3 =7, pb 4 =8, pb 5 =3, pb r = wth = q =, p + r = and r = p p Fg 44[A]

5 Vol No page: -7 (29) p4 p5 R ean every proce ha alot ae chance of beng proceed Under Schee-III[B]: Intal probablte are pb =7, pb 2 =5, pb 3 =7, pb 4 =8, pb 5 =3, pb r = wth q =, p + r + = and r = Fg 44[B] When we refer to fg 44[A] and 44[B], where the chee round robn chedulng wth the effect of deadlock tate, the ncreang nuber of quantu ha ndcaton for the yte to be over the deadlock tate Under Schee-III[A]: Conder ntal probablte pb =7, pb 2 =5, pb 3 =7, pb 4 =8, pb 5 =3 wth =5, p=5, q=r= and p + = p p Fg 46[A] p4 p5 R p Fg 56[B] Fg 45[A] Fg 45[B] The III[A] hown n fg 55[A] and 55[B] nether FIFO nor a round robn chee But t a xture of thee two In th, the quantu dtrbuton take over the ae tate or to the next tate dependng upon the outcoe of the rando experent If the nuber of quantu ncreae, th chee how alot a table pattern of the tate probablte Th Under Schee-IV: Conder ntal probablte pb =, pb 2 =, pb 3 =, pb 4 =, pb 5 = wth p=, q = r = = p Fg 47[A]

6 Vol No page: -7 (29) The chee-iv purely a round robn chee, whch tart fro the frt proce, the tate probablte are n fluctuatng trend a evdent fro fg 47[A] and 47[B] After a contant nterval of quantu each proce bear a hgh probablty of beng proceed The overall vew ndcate that chee-iii bear ore probablty for proceng job n coparon to other chee Becaue of t beng a xture of FIFO and round robn, the proce chedulng apect tronger n th cae 6 CONCLUDING REMARK Fg 47[B] The preent tudy ncorporate a general cla of chedulng chee wth FIFO and round robn a t eber Soe other chee are alo eber of th cla and all thee are condered wth and wthout deadlock tate All the chee are exaned through a coon Markov chan odel The chee-i not a effectve n coparon to other In chee-ii[a], at the regular nterval after fve quantu, the tate probablty bear a hgh chance of cheduler beng tranted over the ae If the nuber of quantu ncreae then chee-iii[a] how alot a table pattern of tate probablte The chee-iii ee a good choce becaue of tablty pattern over job proceng under coon Markov Chan Model etup on Network Securty and Manageent, NCNSM-7, 27, [7] W Stallng, Operatng yte, (Ed5, Pearon Educaton, Sngapore, Indan Edton, New Delh, 24) [8] A Tanenbau, and AS Woodhull, Operatng Syte, (Ed 8, Prentce Hall of Inda, New Delh 2) Author Bography Dr Dwakar Shukla preently workng a a faculty eber n the Departent of Matheatc and Stattc, H S Gour Sagar Unverty, Sagar, MP and havng over 9 year experence of teachng to UG and PG clae He obtaned MSc(tat), PhD(tat) degree fro Banara Hndu Unverty, Varana and erved the Dev Ahlya Unverty, Indore, MP a a peranent Lecturer fro 989 for nne year and obtaned the degree of MTech(Coputer Scence) fro there He joned Sagar Unverty, Sagar a a Reader n tattc n the year 998 Durng PhD fro BHU, he wa junor and enor reearch fellow of CSIR, New Delh through Fellowhp Exanaton (NET) of 983 Tll now, he ha publhed ore than 55 reearch paper n natonal and nternatonal journal and partcpated n ore than 35 enar/conference at the natonal level He alo worked a a Profeor n the Lucknow Unverty, Lucknow, UP, for one (fro june, 27 to 28) year and vted abroad to Sydney (Autrala) and Shangha (Chna) for conference partcpaton and paper preentaton He ha uperved even PhD thee n Stattc and Coputer Scence and eght tudent are preently enrolled for ther doctoral degree under h upervon He a eber of learned bode of Stattc and Coputer Scence at the natonal level The area of reearch he work are Saplng Theory, Graph Theory, Stochatc Modelng, Data nng, Operaton Reearch, Coputer Network and Operatng Syte REFERENCES [] J Medh, Stochatc Procee, (ed 4, Wley lted fourth reprnt, New Delh, 99) [2] M Nald, Internet acce traffc harng n a ult-uer envronent, Coputer Network, Vol 38, 22, pp [3] A Slberchatz, and P Galvn, Operatng Syte Concept, Ed5, John Wley and Son (Aa), Inc, (999) [4] D Shukla, and S Jan, A Markov chan odel for ultlevel queue cheduler n operatng yte, Proceedng of the Internatonal Conference on Matheatc and Coputer Scence, ICMCS-7, 27, pp [5] D Shukla, S Gadewar, RK Pathak, A Stochatc Model For Space-Dvon Swtche In Coputer Network, Appled Matheatc And Coputaton Elever Journal, 84( 2), 27, [6] D Shukla, and S Jan, Deadlock tate tudy n ecurty baed ult-level queue chedulng chee n operatng yte, Proceedng of Natonal Conference Mr Saurabh Jan ha copleted MCA degree fro Dr HS Gour Unverty, Sagar n 25 He preently workng a Lecturer n the departent of Cop Scence & Applcaton n the ae Unverty nce 27 He dd h reearch n the feld of Operatng yte In th feld, he ha authored and coauthored 6 reearch paper n Natonal/Internatonal proceedng and 3 reearch paper n Natonal/Internatonal journal H current reearch nteret to analyze the cheduler perforance under varou algorth Mr Rahul Sngha ha obtaned MCA degree fro HS Gour Unverty, Sagar, MP, n 2 and obtaned MPhl degree n Coputer Scence fro Madura Kaaraj Unverty, Madura, Talnadu n 28 Preently he purung PhD n Coputer Scence fro HS Gour Central Unverty, Sagar H reearch nteret nclude Coputer Network, Data nng

7 Vol No page: -7 (29) 7 & Software Tetng He ha authored and co-authored 8 reearch paper n proceedng & journal Currently, he workng on to develop new putaton baed ethod for fndng ng value n data preproceng of data nng He apponted a a Lecturer n the Deptt of Cop Scence & Applcaton, HS Gour Central Unverty, agar nce 25 Mr Shweta Ojha receved her MCA degree fro Dr HS Gour Unverty, Sagar n 25 She preently workng a Lecturer n the departent of Cop Scence & Applcaton n the ae Unverty nce 27 Her reearch nteret to analyze the cheduler perforance under varou algorth & Traffc analy n varou coputer networkng wtche She ha three reearch paper publhed n Natonal/Internatonal conference Currently he purung PhD n Coputer Sc Fro Dr HSG Central Unverty, Sagar