Performance of a Queuing System with Exceptional Service

Transcription

1 Iteratoal Joural o Eeer ad Matheatcal Sceces Ja.- Jue 0, Volue, Issue, pp ISSN Prt , Ole All rhts reserved IJEMS Abstract Perorace o a Queu Syste wth Exceptoal Servce Dr. Habeer Sh, Dr. Sajay Ja, Dr. A.K. Shrvastav 3 Assocate Proessor, Deptt. o Math, RIET, Fardabad, Haryaa drhabeer@al.co Assstat Proessor, Deptt o Statstcs, St. Joh s Collee, Ara Assstat Proessor, Deptt. o Math, RIET, Fardabad, Haryaa 3 I ths paper a attept has bee ade to calculate the characterstcs o M/G/ odel wth exceptoal arrval/servce.e. statoary wat te, sojour te, queue leth ad soe other characterstcs. A M/G/ odel, whch servce te dstrbuto each busy perod ay deped o the uber o custoers who have bee served the sae busy perod, s called exceptoal servce odel. We use the reeeratve approach; wth ths approach we et characterstcs tractable ors. Itroducto I ths paper we studed a No-Markova cha wth exceptoal servces. I whch arrval ollows Posso asho ad servce dstrbuto ay deped o the uber o custoers who have bee served the curret busy perod. Our odel s also looks lke queu systes wth vacato. By a vacato we ea that the server wll ot wat or custoers ater the copleto o busy perod. By exceptoal servce we eas that servce te dstrbuto each busy perods ay deped the uber o custoers who have bee served the sae busy perod. Kella ad Yechall 988 studed prortes M/G/ queues wth server vacatos. Lee 989 studed M/G//N queue wth vacato te ad lted servce dscple. Heker 990 studed a ote o sojour tes queu etworks wth ult-server odes. Taka 99 studed aalyss o a M/G//N queue wth ultple server vacato ad ts applcatos to a poll odel. Dosh 996 surveyed the queu systes wth vacato. Iak et. al. 998 studed o a eeralzed M/G/ queue wth servce deradato/eorceet Ahahru ad Ftzatrck 999 studed wat te dstrbuto o a Fo/ Lo M/D/ queue. Keaku ad Myazawa 000 also studed a reeeratve cycle approach to a M/G/ queue wth exceptoal servce. Tooyuk ad Myazawa 00 studed a M/G/ queue wth Markov-depedet exceptoal servce tes. Basal 003 studed aalyss o the M/G/ processor-shar queue wth bulk arrvals. Yechal 004 studed O the MX/G/ Queue wth a Wat Server ad Vacato. Perry & Stadje 006 studed a cotrolled M/G/ workload process wth a applcato to pershable vetory systes. Boxa, O.J., Bru, J. 66

2 Dr. Habeer Sh, Dr. Sajay Ja, Dr. A.K. Shrvastav Fralx. B.H. 009 studed wat tes poll systes wth varous servce dscples. es Deso & Seva Sheer 00 also studed Global ad local asyptotcs or the busy perod o a M/G/ queue. I ths preset work we used reeeratve cycle approach ad studed wat te, statoary wat te, sojour te ad queue leths wth soe uercal exaples. Such type o study s useul coputer systes where a server ay eed ore or less te ater coplet all jobs.e. dl to be servce to the ext batch o jobs. Notato Used Mea arrval rate. W The wat te o the -arrv custoer. W + W + S T 0, N > T Iter-arrval betwee th ad +th arrv custoer. N Total uber o custoers that arrves rst busy perod. Laplace trasor o G. Laplace trasor o the statoary wat te dstrbuto. S The servce te o the th arrv custoer the rst busy perod,. G Dstrbuto o S. Nuber o custoers. Nuber o custoers who et exceptoal servce. S Subjects to the dstrbuto G. U Rado varable subject to the statoary sojour te dstrbuto. U Sojour te to the th arrv custoer. h Laplace trasor o the dstrbuto 0,. q + Queue leth just ater the th custoer copletes ths servce. q Rado varable subject to the Statoary queue leth dstrbuto at a arbtrary te stat. Q M / G / k Queue leth ot clud a custoer be served the correspod M/G/ queue. A arbtrary postve teer. Trac testy The Queue Model ad Wat Te We have studed the M/G/, queu odel wth exceptoal servces as the exceptoal servce s related wth busy perod. We have started a busy perod wth a epty state.e. there s o custoer the syste. Ths epty state s cosdered as a reeerato epoch. S s assued to be a sequece o depedet rado varable. That are depedet o arrval process ad ES s te or all,, s a teer, dstrbuto o servce te s eeral G. Suppose that rst custoers who 67

3 Perorace o a Queu Syste wth Exceptoal Servce receved exceptoal servce FIFO.e. rst -rst out asho. G be the servce te dstrbuto o a custoer other tha these -custoer, s depedet o ; ay be te. As we have cosdered that the syste bes wth the epty state zero dee as. E e W ; N, 0, E N... W S T E e, N,... W S w S E e e N...3 W ad S are depedet & W = 0 so, we have By equato 3,...4 Where = e S I = 0, the eq. 4 ples. = 0 + 0,...5 Wth the help o eq. 3 we et,...6 To d Laplace trasor te. o the statoary wat te dstrbuto I case queue s stable, EN s Us cycle orula N - E N W E E N e - W N, Where W s a dscrete te reeeratve process 68

4 Dr. Habeer Sh, Dr. Sajay Ja, Dr. A.K. Shrvastav E N...7 Statoary Wat Te As we are assue that oly rst custoers et exceptoal servce, de...8 For each = 0, ad E S...9 Fro eq. 7 we have, E N...0 Frst we have to calculate. wth the help o eq 5, eq 4 ca be wrtte as, or we have Ad or Su o over ad wth the help o eq, we have, =+, + 0 Substtut to ths orula, we have Also or

5 Perorace o a Queu Syste wth Exceptoal Servce Su eq 3 & 4 or =,,.., Wth act F = leads to Above eq. ples...5 Above eq. 5 wll provde us 0 Hece we calculate statoary wat te by eq. 5 & 6 W Both odels have the sae arrval rate ad the sae dstrbuto or the o- exceptoal servce te s.e. W M / G /...8 W WM G / 0 /...9 Further we studed the statoary probablty wth the help o eq. 7, eq. 4 ples P W 0 E N 70

6 Dr. Habeer Sh, Dr. Sajay Ja, Dr. A.K. Shrvastav Us eq. 9, E S + ad E S k + or =, are te, K s a arbtrary teer. k k k k j E W E W M / G / j E WM / G / j 0 k j 0 k j k j k 0 j 0 E S E S... Above eq. Ca be wrtte as ' 0 0 E S E S E S E W... 0 Where 0 s o eatve, ad calculated as ' 0 l 0 0,,, Sojour Te ad Queue Leth Sojour Te: I ths secto we calculate sojour te.e. the te spet by a custoer the syste ro arrval to al departure ad queue leth that eat the total uber o custoers clud a custoer be served. We assue the teess o the uber o the exceptoal servces ad testablty codto < as the prevous secto. For N, we et u ; N...4 Because S s depedet o W 7

7 Perorace o a Queu Syste wth Exceptoal Servce Slar to recurrece relato U 0 5 Wth the help o eq. eq., we have, 6 By the relato 5, 6 becoes U 0 Put the value o & 0 by eq. 5 & 6 above eq. We et U Ths s the sojour te 0 7 We et the ollow or by the help o eq. 7 W M/G/ the correspod M/G/ queue. Uder the assupto o eq. 8, we have, / u WM / G / 8 0 Queue Leth The uber o custoers who arrved dur the th custoer be the syste.e. q +. E z q ; N W W S S k = z ; N! 0 7

8 Dr. Habeer Sh, Dr. Sajay Ja, Dr. A.K. Shrvastav = z z 9 Ths s the codto eerat ucto I eq. 8, put = -z E z q N, throuh E z Q M z / G / 30 z z Where Q M/G/ be the queue leth ot clud a custoer be served the correspod M/G/ queue. Fro eq. 8 ad the dstrbuto Lttle s law.e. E z q z U QM / G / z WM / G / z E 3 By the eq. 8 wth = -z, we et E q QM G / [ z ] E z z z z z z z / 3 0 by eq. 3, we have ' We deretate eq. 3 at z = ad us the act 33 we et, 0 0 E S E S E S E[ q]

9 Perorace o a Queu Syste wth Exceptoal Servce Calculato o s As we copute 0 s or s. For deterato as well as to copute the eas o the wat te ad queue leth 0 are coputed by putt = 0 eq. 6 We et 0, 35 For o-eatve teer. Dee the operator D ad D ~ as D h h 36 ~ D h h 37! Where, D ad D ~ dstrbuto o [0, ]. be the dervatves o Laplace trasors at ; h be a Laplace trasor o a Multply both sde o eq. 6 or + stead o by yelds. j j j j j j 38 Sce, or ad k D k kk!, k 0, k 39 Apply D to both sdes o eq

10 Dr. Habeer Sh, Dr. Sajay Ja, Dr. A.K. Shrvastav! D j j j j j j! 40 j Where =j s dropped the suato. Dvd eq. 40 both sdes by! The we et ~ ~ j j j j D D a 4 j Cocluso I ths paper, we have dscussed statoary wat te eq. 7 ad sojour te, queue leth ad calculatos s equato 7, 3, 4 respectvely. The expected values o the codtoal wat te are preseted or 5 custoers, whe all the servce te are expoetally dstrbuted raph ad raph. I raph 4 ad 6 all S s are deterstc, whle raph 3 ad 5 all S s are expoetally dstrbuted. But raph 3 to 6 the expected values o the statoary wat te are preseted or the case that S s expoetally dstrbuted. I all raphs Case assues that E S, ES 0 5 =,,3,4 Whle case assues that E S, ES 0 5 =,,3,4 75

11 Perorace o a Queu Syste wth Exceptoal Servce Sce ES = so =. I raph & whe the uber o custoers creases, codtoal wat te also creases. I raph 3 & 4 whe the ubers o custoers, who et exceptoal servces are creased, statoary wat te decreases. I raph 5 & 6 as the value o teds towards oe, statoary wat te creases sharply. Graphs Graph Graph - 76

12 Dr. Habeer Sh, Dr. Sajay Ja, Dr. A.K. Shrvastav Graph - 3 Graph-4 77

13 Perorace o a Queu Syste wth Exceptoal Servce Graph - 5 Graph - 6 Graph 78

14 Dr. Habeer Sh, Dr. Sajay Ja, Dr. A.K. Shrvastav Reereces. Kella, O. ad Yechall, U. : Prortes M/G/ queues wth server vacatos, Navel Research Lostcs, Lee, T.T.: M/G//N queue wth vacato te ad lted servce dscple, Perorace Eval-I, Heker, J.: A ote o sojour tes queue etworks wth ult-server odes, J. Appl. Prob., 7, Taka H.: Aalyss o a M/G//N queue wth ultple server vacato ad ts applcatos to a poll odel, J. o Oper. Res. Soc. o Japa, 35, 3, Dosh B.T.: Queue syste wth vacato a survey, Queue systes., Iak, N. Suta, U. ad Kowada, M.: O a eeralzed M/G/ queue wth servce deradato/eorceet, Joural o Operatos Research Socety o Japa, 4, Ahahru, S.A. ad Ftzatrck, G.J.: Wat te dstrbuto o a Fo/ Lo M/D/ queue, INFOR, 37, Keaku H. ad Myazawa M.: A reeeratve cycle approach to a M/G/ queue wth exceptoal servce, Joural o the Operatos Research Socety o Japa., 43, 4, Tooyuk, K. ad Myazawa, M.: A M/G/ queue wth Markov-depedet exceptoal servce tes, O. R. Letters, 30, Basal, N.: Aalyss o the M/G/ processor-shar queue wth bulk arrvals, O.R. Letters, 3, Yechal U.: O the M X /G/ Queue wth a Wat Server ad Vacato, Sakhya: the Ida joural o stat., 66, part-, Perry D., Stadje, W.: A cotrolled M/G/ workload process wth a applcato to pershable vetory systes. Math.Methods Oper.Res. 643, Boxa, O. J., Bru, J. Fralx.B. H.: Wat tes poll systes wth varous servce dscples. Peror.Eval Des Deso & Seva Sheer: Global ad local asyptotcs or the busy perod o a M/G/ queue. Queu Syst