An analysis of multi-server queueing system with finite capacity using iterative method

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1 Intrnational Journal of Ali Rarh 26 2(): ISS Print: ISS Onlin: Imat Fator: 5.2 IJAR 26 2(): Riv: At: Dhanh Garg Maharihi Markanhwar Univrit, Mullana, Ambala- 3323, Harana, Inia An anali of multi-rvr uuing tm with finit aait uing itrativ mtho Dhanh Garg Abtrat Thi ar rib an analz of ingl uu, multi-rvr with limit tm aait unr firt om firt rv iilin uing itrativ mtho. Th arrival of utomr an rvi tim of utomr ar aum Poion an onntial itribution rtivl. Thi uuing mol i an tnion of ingl uu, ingl rvr with limit tm aait. Prforman maur of th mol, uh a th t numbr of utomr in th uu an in th tm, th t waiting tim of utomr in th uu an in th tm rtivl wr riv. Th rforman maur o riv wr omar with that of ingl uu, ingl rvr with limit aait {M/M/: (/FCFS)} mol. Kwor: Itrativ mtho, ingl uu, multi-rvr uu mol, limit aait Corronn Dhanh Garg Maharihi Markanhwar Univrit, Mullana, Ambala- 3323, Harana, Inia. Introution A ommon ituation that our in vra lif i that of uuing or waiting in a lin. Quu ar uuall n at bu to, tikt booth, rtaurant, urmarkt, otor lini, bank ountr, an traffi light an o on. In gnral, a uu i form at a uuing tm whn ithr th utomr ruiring rvi wait u to th fat that th numbr of utomr th numbr of rvi failiti, or failiti o not work ffiintl an tak mor tim than rrib to rv a utomr. Th rinial ator in a uuing tm ar th utomr (tho maning for rvi) an th rvr (tho roviing th rvi at th rvi failit). Th two mot imortant haratriti of a uuing tm ar th uu lngth an th waiting tim of utomr-what houl b on to ru uu lngth an waiting tim of utomr. Thr ar a whr on an b in a uu without hiall bing viibl to othr iniviual mmbr of th uu. An aml i th a of allr tring to ommuniat through a tlhon withboar. In othr a, non-human ommoiti an alo b in a uu, for aml, a omutr tm manag uu of omutr rogram. Job an uu waiting to b attn to t. In thi ar, w tu th itrativ mtho of analing a uuing tm whr thr i a ingl uu with man rvi oint an th tm ha a limit aait. Thi uuing mol i an tnion of ingl uu, ingl rvr with limit aait. Singl uu, multi-rv with limit aait an b n in hair ring alon, Barbr ho, vhil arking ara, an roution failit, (Sharma, 2) [2]. In thi uuing mol, th traffi intnit n not b l than on, (Guta, 28) [8]. Conuntl, thi ar u itrativ mtho to obtain th robabilit of utomr, in th tm, th robabilit of no utomr, in th tm, an th t numbr of utomr in th tm an uu rtivl in a ingl uu, multi-rvr with limit tm aait. In th tu, th arrival ro i Poion an th rvi tim onntial. Th uuing mol i rrnt a {M/M/C : (/FCFS)}, whr M i Poion arrival, M i onntial rvi tim, i th tm aait, an FCFS i firt om firt rv iilin. Charan an Mahu (2) [] tui ingl rvr uuing mol with -oli an rmovabl rvr. Th invtigat M/E K/ uuing tm unr -oli, b uing th mtho of ntro maimization. Th intrarrival tim an rvi tim wr aum to follow ngativ onntial itribution an Erlang k itribution rtivl. Sultan t al. (25) [3] anal a multil rvr bulk arrival with two mo of rvr brakown. ~ 297 ~

2 Intrnational Journal of Ali Rarh Ouali an Al (24) [] anal a multil riorit, multi-rvr uu with imatin. Th onir Markovian multi-rvr uu with two t of imatint utomr: High an low riorit on, whr th firt t of utomr ha a non-r-mtiv riorit ovr th othr t. Th alo onir two a whr th iilin of rvi within ah utomr t i FCFS or CFS. Mor t al. (23) [] anal multi-rvr uuing tm with multil riorit la, whr th rnt th firt nar-at anali of an M/PH/K uu with m 2 r-mtiv-rum riorit la. Thir anali introu a nw thniu, whih th rfrr to a Ruriv Dimnionalit Rution (RDR). Th k ia in RDR i that m-imnionall infinit Markov hain, rrnting th m la tat a, i rurivl ru to a - imnionall infinit Markov hain that i ail an uikl olv. Marl t al. (25) [9] onir multi- rvr tanm uu with finit buffr an bloking. Th rvi tim wr gnrall itribut. Th vlo an ffiint aroimation mtho for trmining rforman haratriti uh a th throughut an man ojourn tim. Th mtho wa ba on omoition into two-tation ubtm, th aramtr of whih ar trmin b itration. Ezliora t al (24) anal ingl lin, multi- rvr tm uing Shorit Plaza, Enugu Stat, igria a a a tu. Th thniu u for th anali wa an infinit ingl-lin multil hannl thniu. Eknong an Uoh (2) [2] tn an imrov on th rforman maur of th ingl rvr, ingl uu tm with multil ha. Th tnion rult in a nw uuing tm of multi-rvr with multil ha unr th onition of firt om firt rv, infinit oulation our, Poion arrival an Erlang rvi tim. 2. Quuing Mol: M/M/: (/FCFS) Quu an b n a a form of birth an ath ro whr arrival an b likn to birth an artur likn to ath. Conir t to b th robabilit that thr ar utomr in th tm at tim t. W rall that, unr tationar or uilibrium uuing mol, th following aumtion ar ma. i) t ii) t W alo not hr that, in gnral birth an ath ro whr birth an ath aramtr ar an rtivl, th iffrntial uation i t t t t Uing th aumtion, w hav For, uation () giv () For, 2 For 2, ~ 298 ~

3 Intrnational Journal of Ali Rarh In gnral, (2) For uu of {M/M/ :( /FCFS)} Hr, w obtain th robabilit of utomr, For For an, w hav, in th tm , an w hav,..... i i Thrfor, Hr, th intrt i to obtain th valu of (robabilit of no utomr in th tm). (3) Rall for irt itribution. Thn, (4) ~ 299 ~

4 ~ 3 ~ Intrnational Journal of Ali Rarh W hav, (5) (6) (7) Putting uation (7) into (6), w hav, (8) Thn, utting uation (8) into (3), w hav, 3. Prforman of Maur To trmin th t numbr of utomr in th tm, th t waiting tim in th tm W an uu W rtivl, w fin lot a lot an a, whr i th fftiv arrival rat. Th fftiv traffi intnit, 4. Et umbr of Cutomr in th Quu, To o thi, w rall that For irt itribution (9) P X E

5 ~ 3 ~ Intrnational Journal of Ali Rarh Thrfor, (), () 2... um of -+ trm of GP (2) Putting uation (2) into (), w hav, (3) Diffrntiating with rt to, w hav, 2 (4) But Thrfor, 2

6 ~ 32 ~ Intrnational Journal of Ali Rarh 5. Et umbr of Cutomr in th Stm (5) 6. Et Waiting Tim in th Stm W W (6) 7. Et Waiting Tim in th Quu W W W (7) 8. Th Fration of Srvr Il Tim I Whr Thrfor, I (8) 9. Aknowlgmnt Author i thankful to rfr for thir ontrutiv uggtion an imrovmnt.. Rfrn. Charan JS, Mahu J. Singl Srvr Quuing Mol with -Poli an Rmovabl Srvr. Canaian Ali Mathmati Quartrl. 2 9(): Eknong EJ, Uoh S. Anali of Multi-Srvr Singl Quu with Multil Pha. Pak. J tat. or. r. 2 7(2):35-34.

7 Intrnational Journal of Ali Rarh 3. Ezliora CD, Ogunoh AV, Umh M, Mblogu. Anali of Quuing Stm Uing Singl-in Multil Srvr Stm. A Ca Stu of Shorit Plaza Enugu Stat, igria). Intrnational Journal of Sintifi an Thnolog Rarh. 24 3(3): Guta PK. Oration Rarh. S. Chan & Co. t, w Dlhi, Garg D. Two Srvr Quuing Stm Rah it Caait in Tim t. Intrnational Journal of Rarh in Avnt Thnolog. 24 2(5): Garg D. Sta tat olution of M/M/2/ uuing tm with atatroh an rtoration, Intrnational Journal of Enginring Rarh an Managmnt. 24 (6): Garg D. Aroimat anali of an M/M/ Markovian uu uing unit t funtion, On A ibrar Journal. 24 (973): Garg D. Th numbr of tim a uuing tm rah it aait in tim t, Autralian Soit for Oration Rarh Bulltin ():-. 9. Marl VV, Ivo JBF, Aam, Simon AE. Ring-San. Prforman Anali of Multi-rvr Tanm Quu with Finit Buffr an Bloking. OR Strum : Mor HB, Takauki O, Alan S, Aam W. Multi-rvr Quuing Stm with Multil Priorit Cla. Suort b SF arr Grant CCR-3377, SF Thor CCR-3383, SF ITR CCR-3348, an IBM Cororation via Pittbugh Digital Grnhou Grant, 23.. Ouali J, Al R. On Multil Priorit Multi-Srvr Quu with Imatin. Journal of Oration Rarh Soit : Sharma JK. Oration Rarh: Thor an Aliation, 4 th ition. Mamillan Inia, w Dlhi, Sultan AM, Haan A, Elham M. Comutational Anali of a Multi-Srvr Bulk Arrival with Two Mo Srvr Brakown. Mathmatial an Comutational Aliation. 25 (2): ~ 33 ~