MAHALAKSHMI ENGINEERING COLLEGE TIRUCHIRAPALLI

Transcription

1 MAHALAKSHMI EGIEERIG COLLEGE TIRUCHIRAALLI 6 QUESTIO BAK - ASWERS -SEMESTER: V MA 6 - ROBABILITY AD QUEUEIG THEORY UIT IV:QUEUEIG THEORY ART-A Quesio : AUC M / J Wha are he haraerisis of a queueig heory? Aswer : ().Arrival paer of usomers ().Servie paer of servers (). umber of servie haels (4).Sysem apaiy (5).Queue disiplie Quesio : AUC M / J Wha is he probabiliy ha a usomer has o wai more ha 5 miues o ge his servie ompleed i a M/M/ Aswer : X k X 5 Quesio : AUC M / J queueig sysem, if 6 per hour ad per hour? M/M/ GD Sae Lile's formula for ; / / Aswer : L s k Lq Ws Lq () Quesio : 4 AUC M / J Defie seady sae ad rasie sae i Queueig Aswer : queueig model heory Seady sae : If he haraerisis of a queueig sysem are idepede of ime Trasie sae : If he haraerisis of a queueig sysem are depede of ime M.MARIA AROCKIA RAJ A/MATHEMATICS MEC TRICHY

2 Quesio : 5 AUC M / J Arrival rae of elephoe alls a a elephoe booh are aordig o oisso disribuio wih a average ime of 9 miues bewee wo oseuive alls arrival. The legh of phoe alls is assumed o be expoeially disribued wih mea miues. ().Deermie i he probabiliy ha a perso arrivig a he booh will have o wai Aswer : p / m p / m 9 probabiliy ha a perso arrivig a he booh will have o wai. 9 Quesio : 6 AUC M / J Trais arrive a he yard a every 5 miues ad he servie ime is miues.if he lie apaiy of he yard is limied o 4 rais, fid he probabiliy ha he yard is empy Aswer :, K probabiliy ha he yard is empy.7 K Quesio : 7 AUC A / M Suppose ha usomers arrive a a oisso rae of oe per every miues ad ha he servie ime is expoeial rae of oe servie per 8 miues.wha is he average umber of usomers i he sysem? Aswer : 8 L s Quesio : 8 AUC A / M Defie M/M/ queueig model.why he oaio M is used? Aswer : M/M/ is he wo server oisso queue model, whe he arrival rae follows oisso proess ad servie rae follows oisso proess. The oaio M sads for Markov Quesio : 9 AUC / D Give a real life siuaio i whih (a).usomers are osidered for servie i las i firs ou queue disiplie.(b).the sysem wih ifiie umber of servers M.MARIA AROCKIA RAJ A/MATHEMATICS MEC TRICHY

3 Aswer: ().I a mos argo hadlig siuaios where he las iem loaded is removed firs beause i redues hadlig ad raspor os, he las oes beig easier o reah loser I he produio proess iems arrive a a work plae ad are soked oe o op of he oher Iem o he op of he sak is ake firs for proessig whih is he las oe o have arrived firs for servie. (b).if every railway ouer is allowed he persos o wai uil servie is provided M.MARIA AROCKIA RAJ A/MATHEMATICS MEC TRICHY

4 Quesio : AUC M / J ART B There are hree ypiss i a offie. Eah ypis a ype a average of 6 leer perhour. If leers arrive for beig yped a he rae of 5 leer per hour, wha fraio of imes all he ypiss will be busy? Wha is he average umber of leers waiig o be yped? Aswer : Here, 6, 5, 8,!!() !! 6! 6! 6! 6! 5 8 ()! 6(8 5) all he ypiss are busy.7!()! 6(8 5) 5 8 Lq ( )!()()! 6 (8 5) Quesio : AUC M / J Derive (). L, average umber of usomers i he sysem(). L, average umber of usomers S i he queue for he queueig model M / M / ; / FIFO Aswer : This is a poisso queue wih limied queue size, ha is he maximum umber of usomers allowed i he sysem is. Le represes he probabiliy of havig usomers i he sysem. he here are hree ases,, Case : () ()() () () ()() ()() ()() ()()()() ()()()() ()() ()() '()()()() q M.MARIA AROCKIA RAJ A/MATHEMATICS MEC TRICHY 4

5 Case : ()() () () () () ()()() Omiig higher powers of we ge () () ()() ()() ()() ()()()()() ()()()()() ()() ()()() ()() lim lim()()() '()()()()() Case : () () ()() () () () ()()()()() Omiig higher powers of we ge ()()()() ()()()() ()() ()() '()()()...() () ad (),() are alled differeial differee equaios i rasie sae probabiliy () To fid he seady sae equaios : As,() ad '(), he () ad () ad ( ) beomes...(4)...(5)...(6) (4),(5) ad (6) are alled seady sae equaios From (4), u i (5) u i (5) 5 M.MARIA AROCKIA RAJ A/MATHEMATICS MEC TRICHY

6 u i (5) To fid , Quesio : AUC M / J Cusomers arrive a a oe ma barber shop aordig o a oisso proess wih a mea ier arrival ime of miues.cusomers sped a average of 5 miues i he barber hair If a hour is used as a ui of ime. ().Wha i is he probabiliy ha a usomer eed o wai for a hair uy? ().Wha ii is he expeed umber of usomers i he barber shop ad i he queue? ( iii).how muh ime a a usomer expe o sped i he barber shop? ().Fid iv he average ime ha he usomer sped i he queue? ().Wha v is he probabiliy ha he re will be 6 or more usomers waiig for servie? Aswer : 5 p / m, p / m ().robabiliy ha a usomer eed o wai for a hair uy ().Expeed umber of usomers i he barber shop L s M.MARIA AROCKIA RAJ A/MATHEMATICS MEC TRICHY 6

7 7.5.5 Lq. () Ls.5 (). Ws 65mi.5 Lq. (4). Wq 47 mi.5 (5).robabiliy ha here will be 6 or more usomers waiig for servie Quesio : 4 AUC M / J A supermarke has girls ruig up sales a he ouers.if he servie ime for eah usomer is expoeial wih mea 4 miues ad if people arrive i oisso fashio a he rae of per hour fid he followig: (). Wha is he probabiliy of havig o wai for servie?.67(m/j/) (). Wha is he expeed pereage of idle ime for eah girl?.67 (). Wha is he expeed legh of usomer's waiig ime? mi Aswer : Here, p / m, p / m, 4 hour 6 mi 6.5,.5..!!().7.7!!!!!()! (). usomers havig o wai for servie.5.8!()!. ().The fraio of ime whe he girls are busy The fraio of ime whe he girls are idle The expeed % of idle ime for eah girl 67 M.MARIA AROCKIA RAJ A/MATHEMATICS MEC TRICHY 7

8 Quesio : 5 AUC M / J A T.V repairma fids ha he ime sped o his job has a expoeial disribuio wih mea miues.if he repair ses i he order i whih hey ame i ad if he arival of ses is approximaely oisso wih a average rae of per 8 hour day, wha is he repairma's expeed idle ime eah day? How may jobs are ahead of average se jus brough? Aswer : = p / mi, p / mi,.65 8hour 8 6 mi Repairma is idle Average umber of jobs ahead of average se jus brough L Quesio : 6 AUC M / J q job () Trais arrive a he yard a every 5 miues ad he servie ime is miues.if he lie apaiy of he yard is limied o 5 rais, fid he probabiliy ha he yard is empy ad he average umber of rais i he sysem. Aswer : = per mi per mi,.,. 5. robabiliy ha he yard is empy. k 6. ( ) Average umber of rais i he sysem Ls 4. Quesio : 7 AUC M / J Fid he seady sae soluio for he muliserver M/M/C model ad hee fid Lq, Wq, Ws ad Ls by usig lile formula. Aswer : This is a poisso queue wih ulimied queue legh havig o server/haels I is a muli server model where C.The followig hree ases are o be disussed... Case : ()() () () ()() ()() ()() () M.MARIA AROCKIA RAJ A/MATHEMATICS MEC TRICHY 8

9 ()()()() ()()()() ()() ()() '()()()() Case : Le deoe he servie rae per server.whe he umber of usomers () is he servie rae of he sysem is.whe he umber of usomers is aleas () he servie rae of he sysem is () ( )() () ()()() ()() () Omiig higher powers of we ge ) ()() ()()( () ()()( )() ()()()()( )() ()()()()( )() ()() ()( )( )() ()() lim lim()()( )() '()()()( )()() Case : Here he servie rae of he sysem is () () () () () () ()()() Omiig higher powers of we ge ()() ()() () ()()() ()()()()() ()()()()() ()() ( )()( )() ()() lim lim()()() M.MARIA AROCKIA RAJ A/MATHEMATICS MEC TRICHY 9

10 '()()()()...() (),() ad () are alled he rasie sae equaios To fid he seady sae equaios : As,() ad '(), he (...(4) ( )...(5) ) ad () ad () beomes......(6) From (4), his implies u i (5) we ge u i (5) we ge Similarlly 4! ( )! 4 4 u i (5) 6! ( )( ) ( ) ( ) ( )! ( ) ( )!( )! ( ) ( )! ( ) ( )! ( )! ( )!( )! ( )! M.MARIA AROCKIA RAJ A/MATHEMATICS MEC TRICHY

11 ( )! ( )!( )!! u i (6) we ge!!( )!!( )!! ( )!!! Here i, u i (6) we ge!!!!!!!! Here i,! I geeral ;! ;! To fid W. K. T M.MARIA AROCKIA RAJ A/MATHEMATICS MEC TRICHY

12 M.MARIA AROCKIA RAJ A/MATHEMATICS MEC TRICHY......

13 Quesio : 8 AUC M / J Show ha for he (M/M/):(FCFS/ /), he disribuio of waiig ime i he sysem is ()() w () e,. Aswer : Le W be he oiuous RV ha represes he waiig ime of a usomer i he S sysem, he ime bewee he arrival ad ompleio of servie.le is pdf be () f wad le ( f w/) be he desiyfuio of Wsu bje o he odiio ha here are ' ' usomers i he queueig sysem whe he usomer arrives, he ()( f w /) f w w f () w e w ( /) f w! w e ( /) w f w! w e w! () () e,. S M.MARIA AROCKIA RAJ A/MATHEMATICS MEC TRICHY