International Journal of Management, IT & Engineering (ISSN: ) CONTENTS TITLE & NAME OF THE AUTHOR (S)

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1 Internatonal Journal of anageent, IT & Engneerng (ISS: ) COTETS Sr. o. TITLE & AE OF THE AUTHOR (S) Page o. 2 3 Hoogeneous arkov Processes On Dscrete Te Queues To Contnuous Te. Dr. ayank Pawar and Dr. Bhuvnender Chaudhary An Enhancng Productvty And Perforance Of The Eployees By Explorng Eployee Satsfacton: An Effcacous Tool For It Sector In Inda. Archana Sngh, Lokendra Vkra Sngh and Dr. Peeyush Khare Applcaton Of Analytcal Tools In Student Retenton Syste. Vbha Gupta, Yasn Shakh and Dr. Geeta eea V.I.E.T A Case of Leadershp Gone Wrong. Rash Shara The Steady-State Soluton Of ultple Parallel Channels In Seres And on-seral Servers Wth Balkng & Renegng Due To Long Queue And Soe Urgent essage. eenu Gupta, an Sngh and Deepak Gupta A Study On Personal Fnancal Plannng For It Sector Investor In Pune. Pravn arayan ahaun, Santosh Kalabhau Apte and Dr. Anand Ganpatrao Jule Industral Energy: what role for Polces? Dr.. Sugunatha Reddy and Dr. B. Raa Bhupal Reddy cro fnance Role of Bankng nteredares n Inclusve Econoc Growth. Suresha B and Dr. Gaendra adu A Coparatve Study Of Publc (Sb & Ut) &Prvate (Hdfc & Icc) Asset anageent Copanes Funds (Balanced, Glt (Long Ter &Short Ter) On The Bass Of av, Fund Average Return, Rsk. Shelly Snghal, Sav Chanana and Gaurav Kabo Perforance Issues of Indvdual and Tea Gae Plannng. ata Jangra easurng Quanttatve antanablty Of Conceptual odel For Re-Engneerng Process. Shabana Kausar, r. Ahed ateen and r. Ahsan Raza Sattar

2 20 IJIE Volue, Issue 5 ISS: Chef Patron Dr. JOSE G. VARGAS-HERADEZ eber of the atonal Syste of Researchers, exco Research professor at Unversty Center of Econoc and anageral Scences, Unversty of Guadalaara Drector of ass eda at Ayuntaento de Cd. Guzan Ex. drector of Centro de Capactacon y Adestraento Patron Dr. ohaad Reza oruz PhD: Publc Adnstraton, Publc Sector Polcy akng anageent, Tarbat odarres Unversty, Tehran, Iran Faculty of Econocs and anageent, Tarbat odarres Unversty, Tehran, Iran Young Researchers' Club eber, Islac Azad Unversty, Bonab, Iran Chef Advsors Dr. AGEDRA. S. Senor Asst. Professor, Departent of BA, angalore Insttute of Technology and Engneerng, oodabdr Dr. SUIL KUAR ISHRA Assocate Professor, Dronacharya College of Engneerng, Gurgaon, IDIA r. GARRY TA WEI HA Lecturer and Charperson (Centre for Busness and anageent), Departent of arketng, Unversty Tunku Abdul Rahan, ALAYSIA Edtoral Board Dr. CRAIG E. REESE Professor, School of Busness, St. Thoas Unversty, a Gardens A onthly Double-Blnd Peer Revewed Refereed Open Access Internatonal e-journal - Included n the Internatonal Seral Drectores Internatonal Journal of anageent, IT and Engneerng 55

3 20 IJIE Volue, Issue 5 ISS: Dr. S.. TAKALIKAR Prncpal, St. Johns Insttute of Engneerng, PALGHAR (.S.) Dr. RAPRATAP SIGH Professor, Bangalore Insttute of Internatonal anageent, KARATAKA Dr. P. ALYADRI Prncpal, Governent Degree College, Osana Unversty, TADUR Dr. Y. LOKESWARA CHOUDARY Asst. Professor Cu, SR B-School, SR Unversty, CHEAI Prof. Dr. TEKI SURAYYA Professor, Adkav annaya Unversty, ADHRA PRADESH, IDIA Dr. T. DULABABU Prncpal, The Oxford College of Busness anageent, BAGALORE Dr. A. ARUL LAWRECE SELVAKUAR Professor, Adhparasakth Engneerng College, ELARAVATHUR, T Dr. S. D. SURYAWASHI Lecturer, College of Engneerng Pune, SHIVAJIAGAR Dr. S. KALIYAOORTHY Professor & Drector, Alagappa Insttute of anageent, KARAIKUDI Prof S. R. BADRIARAYA Snhgad Insttute for anageent & Coputer Applcatons, PUE r. GURSEL ILIPIAR ESADE Busness School, Departent of arketng, SPAI r. ZEESHA AHED Software Research Eng, Departent of Bonforatcs, GERAY r. SAJAY ASATI Dept of E,. Patel Insttute of Engg. & Tech., GODIA(.S.) r. G. Y. KUDALE..D. College of anageent and Research, GODIA(.S.) A onthly Double-Blnd Peer Revewed Refereed Open Access Internatonal e-journal - Included n the Internatonal Seral Drectores Internatonal Journal of anageent, IT and Engneerng 56

4 20 IJIE Volue, Issue 5 ISS: Edtoral Advsory Board Dr. AJIT DAS Assstant Professor, Deptt. of Econocs,.C.College, ASSA Dr. ROLI PRADHA aulana Azad atonal Insttute of Technology, BHOPAL Dr.. KAVITHA Assstant Professor, Departent of anageent, ekelle Unversty, ETHIOPIA Prof C.. ARA Assstant Professor (Senor), VIT Busness School, TAIL ADU Dr. RAJIV KHOSLA Assocate Professor and Head, Chandgarh Busness School, OHALI Dr. S. K. SIGH Asst. Professor, R. D. Foundaton Group of Insttutons, ODIAGAR Dr. (rs.) AISHA. PALIWAL Assocate Professor, Snhgad Insttute of anageent, PUE Dr. (rs.) ARCHAA ARJU GHATULE Drector, SPSP, SK Snhgad Busness School, AHARASHTRA Dr. EELA RAI DHADA Assocate Professor, Departent of Coerce, kuk, HARYAA Dr. FARAH AAZ GAURI Assocate Professor, Departent of Coerce, Dr. Babasaheb Abedkar arathwada Unversty, AURAGABAD Prof. Dr. BADAR ALA IQBAL Assocate Professor, Departent of Coerce, Algarh usl Unversty, UP A onthly Double-Blnd Peer Revewed Refereed Open Access Internatonal e-journal - Included n the Internatonal Seral Drectores Internatonal Journal of anageent, IT and Engneerng 57

5 20 IJIE Volue, Issue 5 ISS: Assocate Edtors Dr. SAJAY J. BHAYAI Assocate Professor,Departent of Busness anageent, RAJKOT (IDIA) OID UDDI AHAD Assstant Professor, Japura Insttute of anageent, OIDA Dr. SUEEL ARORA Assstant Professor, G D Goenka World Insttute, Lancaster Unversty, EW DELHI r. P. PRABHU Assstant Professor, Alagappa Unversty, KARAIKUDI r. AISH KUAR Assstant Professor, DBIT, Deptt. Of BA, DEHRADU rs. BABITA VERA Assstant Professor, Bhla Insttute Of Technology, DURG s. OIKA BHATAGAR Assstant Professor, Technocrat Insttute of Technology, BHOPAL s. SUPRIYA RAHEJA Assstant Professor, CSE Departent of IT Unversty, GURGAO A onthly Double-Blnd Peer Revewed Refereed Open Access Internatonal e-journal - Included n the Internatonal Seral Drectores Internatonal Journal of anageent, IT and Engneerng 58

6 20 IJIE Volue, Issue 5 ISS: Ttle The steady-state soluton of ultple parallel channels n seres and non-seral servers wth balkng & renegng due to long ueue and soe urgent essage Author(s) eenu Gupta Asstt. Prof., Dept. of Appled Scences, H.I.E.T.,Kathal (Haryana) Inda an Sngh Prof. (Retd.), CCS,Haryana Agrcultural Unversty, Hsar Inda Deepak Gupta Prof.& Head, Dept. of atheatcs,..engg.college, ullana(abala)- Inda A onthly Double-Blnd Peer Revewed Refereed Open Access Internatonal e-journal - Included n the Internatonal Seral Drectores Internatonal Journal of anageent, IT and Engneerng 59

7 IJIE Volue, Issue 5 ISS: Abstract: Ths paper consders the ost approprate & ore general ueung odel n respect of custoers whch are allowed to leave the syste at any stage wth or wthout gettng servce. The paper consders the steady-state behavor of the ueung processes when servce channels n seres, are lnked wth non-seral channels havng balkng & renegng phenoenon, wheren: Each of servce channels has dentcal ultple parallel channels. Posson arrvals & exponental servce tes are followed. The servce dscplne follows SIRO rule (servce n rando order) nstead of FIFO rule (frst n frst out). The custoer becoes patent n the ueue after soete and ay leave the syste wthout gettng servce and can renege due to urgent essage. Watng space s nfnte. Keywords: Posson strea, Renegng, Balkng, Traffc ntensty, Steady-state, Parallel channels, Urgent essage. Introducton: The proble of seral ueues was studed by O Bren (954), Jackson (954), Barrer (955), Hunt(955) and aggu (970) n steady-state wth Posson assuptons wth the restrcton that the custoer ust go through each servce channel before leavng the syste. Sngh (984) studed the proble of seral ueues ntroducng the concept of renegng. The steady-state solutons of ultple parallel channels n seres wth patent custoers are obtaned by Sngh & Ahua (995). The solutons of seral and non-seral ueung processes wth renegng and balkng phenoenon have been studed by Vkra& Sngh (998). The steadystate soluton of seral and non-seral ueung processes wth renegng and balkng due to long ueue and soe urgent essage and feedback phenoenon s obtaned by Sngh, Puna & Ashok(2009). In our present socety, the patent custoers generate the ost approprate and A onthly Double-Blnd Peer Revewed Refereed Open Access Internatonal e-journal - Included n the Internatonal Seral Drectores Internatonal Journal of anageent, IT and Engneerng 60

8 20 IJIE Volue, Issue 5 ISS: odern odels n the ueung theory. Incorporatng ths concept, we study the steady-state analyss of general ueung syste n the sense that: servce channels n seres are lnked wth non seral channels havng renegng and balkng phenoenon where each of servce channels has dentcal ultple parallel channels. The nput process s Posson and the servce te dstrbuton s exponental. The servce dscplne follows SIRO-rule(servce n rando order) nstead of FIFO- rule (frst n frst out) The custoer becoes patent n ueue after soete and ay leave the syste wthout gettng servce and can renege due to urgent essage also. The nput process depends upon the ueue sze n non-seral channels. Watng space s nfnte. Forulaton of odel: The syste conssts of Q ( =,2,.) servce phases where each servce phase Q has c ( =,2,.) dentcal parallel servce facltes and Q channels (=,2,.) wth respectve servers S (=,2, ) and S (=,2, ).Custoers deandng dfferent types of servce arrve fro outsde the syste n Posson dstrbuton wth paraeters λ ( =,2,.) at Q servce phase and λ (=,2,..) at Q servce phase respectvely. But the sght of long ueue at Q, ay dscourage the fresh custoers fro onng t and ay decde not to enter the servce channel Q (=,2,.) then the Posson nput rate λ would be where s the ueue sze of Q. Further, the patent custoers onng any seral or non-seral servce channel Q ay leave the ueue wthout gettng servce after a wat of certan te. The servce te dstrbuton for the server S (=,2,) and S (=,2..,) are utually ndependent negatve exponental dstrbuton wth (,2,..., ) and (,2,..., ) respectvely. After the copleton of servce at Q (=,2,..), the custoers ether leave the syste wth probablty p or on the next phase wth probablty such that p + = (=,2..-). After A onthly Double-Blnd Peer Revewed Refereed Open Access Internatonal e-journal - Included n the Internatonal Seral Drectores Internatonal Journal of anageent, IT and Engneerng 6

9 20 IJIE Volue, Issue 5 ISS: copleton of servce at Q, the custoers ether leave the syste wth probablty p or on any of the Q (=,2,.,) wth probablty (=,2,.,) such that p =. If the custoers are ore than c n the Q servce phase,all the c servers wll rean busy and each s puttng out the servce at ean rate and thus the ean servce rate at Q s c,on the other hand f the nuber of custoers s less than c n the Q servce phase,only n out of the c servers wll be busy and thus the ean servce rate at Q s n (=,2,.).It s assued that the servce coences nstantaneously when the custoer arrves at an epty servce channel. Forulaton of euatons: Defne P(n,n 2,. n ;, 2, 3, ;t) as the probablty that at te t, there are n custoers ( whch ay renege or after beng servced by the Q phase ether leave the syste or on the next servce phase ) watng n the Q servce phase (=,2,..,), custoers ( whch ay renege or after beng servced leave the syste ) watng before the servers S (=,2,.). We defne the operators T. and T. to act upon the vector n ( n, n2,..., n ) and T. and T. and T., +. to act upon the vector (, 2,..., ) as follows: T. ( n ) = (n, n 2, n - n ) T. ( n ) = (n, n 2, n + n ) T. ( ) = (, 2, - ) T. ( ) = (, 2, + ) T., +. ( ) = (, 2,, +, ) The custoers ay leave any servce channel wthout gettng servce f they receve urgent call whle watng. A onthly Double-Blnd Peer Revewed Refereed Open Access Internatonal e-journal - Included n the Internatonal Seral Drectores Internatonal Journal of anageent, IT and Engneerng 62

10 20 IJIE Volue, Issue 5 ISS: dp( n, ; t) dt n n n c rn ( ){( ) R } ( ){ ( )} P( n, ; t).. P( T ( n ), ; t)) P( n ; T ( ); t) ( r ) P( T ( n ), ; t) P( T ( n ), ; t) n c n. Followng the procedure gven by Kelly(979), we wrte dfference dfferental euatons : for n 0, 0, ( =,2,. ), ( =,2.). n.. p P( T ( n ), ; t) P( n, n,... n, T ( ); t) n. n 2. ( R ) P( n ; T ( ); t)...(). Where ( x) ( n ) ( n c) when x 0 0 when x 0 0 when n 0 when n 0 0 when n c when n c n n when n c c when n c A onthly Double-Blnd Peer Revewed Refereed Open Access Internatonal e-journal - Included n the Internatonal Seral Drectores Internatonal Journal of anageent, IT and Engneerng 63

11 20 IJIE Volue, Issue 5 ISS: r n R e T n e 0 T0 T0 n e ( e ) ;,2,... e T0 Where r and R are the average rates at whch the custoers renege after a wat of n certan te T0 and T0whenever there are n and custoers n the Q and Q servce phases respectvely. and are the ean renegng rates at seral and non-seral servce phases due to soe urgent essage and P(, n ; t) 0f any of the arguents s negatve Steady-State Euatons: We wrte the steady-state euatons of the ueung odel by euatng the te dervatve to zero n the euaton () n n c n.. ( n ){ ( r )} ( ){( ) R } P( n, ) P( T ( n ), )) P( n ; T ( )) ( r ) P( T ( n ), ) n c n. P( T ( n ), ) n.. n. n 2. p P( T ( n ), ) P( n, n,... n, T ( )) ( R ) P( n ; T ( ))...(2). for n 0, 0 ; ( =,2,., ) ; ( =,2,.,). A onthly Double-Blnd Peer Revewed Refereed Open Access Internatonal e-journal - Included n the Internatonal Seral Drectores Internatonal Journal of anageent, IT and Engneerng 64

12 20 IJIE Volue, Issue 5 ISS: Case I: When n < c For n < c, the resultng euatons (2) reduce to as under:.... ( n )( n ) ( ){( ) R } P( n, ) P( T ( n ), )) P( n ; T ( )) ( n ) P( T ( n ), ) p ( n ) P( T ( n ), ) ( n ) P( n, n2,... n, T. ( )). ( R ) P( n ; T ( ))...(3). The solutons of the steady state euatons (3) can be verfed to be: P( n, ) P(0,0) n n ' 2 2 n n n ' 2 n3... ' n n ( R ) 2 ( 2 2 ) ( ) R2 R ( ) ( )...(4) for n 0, 0 ; ( =,2,., ) ; ( =,2,.,). where A onthly Double-Blnd Peer Revewed Refereed Open Access Internatonal e-journal - Included n the Internatonal Seral Drectores Internatonal Journal of anageent, IT and Engneerng 65

13 20 IJIE Volue, Issue 5 ISS: ' ' ' ' k k k k k 2,3,..., Case II: When n c For n c, the resultng euatons (2) wll reduce to as under: n.. n. ( n )( c r ) ( ){( ) R } P( n, ) P( T ( n ), )) P( n ; T ( )) ( p c r ) P( T ( n ), ) c P( T.. ( n ), ) c P( n, n2,... n, T. ( )). ( R ) P( n ; T ( ))...(5) The solutons of the steady-state euatons can be verfed to be: P( n, ) P(0,0) ( ) n n n2 { ( c r ) c } 2 n ( c r ) ( c r )( c r ) n n2 n2 n3 2 { ( c r ) c } 3 n ( c r ) ( c r ) n 2 n3 n3... n c rn c ' ( c ) n n { ( ) } ( c r ) ( c r ) ni ( R ) ' ( c ) ' 2 ( 2 c 2)... (6) 2 ( R ) ( R ) A onthly Double-Blnd Peer Revewed Refereed Open Access Internatonal e-journal - Included n the Internatonal Seral Drectores Internatonal Journal of anageent, IT and Engneerng 66

14 20 IJIE Volue, Issue 5 ISS: Where ( c r ) c n ' ( c r ) ( c r ) n n k ( c u r ) u c ; k 2,3,...,. k k n k k k k We obtan P(0,0) fro (4) and (6) by the noralzng condton 0 n 0 P( n, ) and wth the restrctons that traffc ntensty of each servce channel of the syste s less than unty Thus P( n, ) s copletely deterned. References: Barrer D.Y.(955) A Watng lne proble characterzed by patent custoers and ndfferent clerk. Journal of operatons research. socety of Aerca.vol.3,360 Hunt, G.C. (955): Seuental arrays of watng lnes Opps. Res. Vol. 4, no. 6. Jackson, R.R. P (954): Queung syste wth phase type servce Operat. Res. Quart. 5, Kelly, F.P. (979): Reversblty and stochastc networks Wley, ew York aggu, P.L.(970): On certan types of ueues n seres. Statstca eerlandca. 24, O Bren,.G. (954): The soluton of soe ueung probles J.Soc. Ind. Appl. ath. Vol. 2, Sngh,. (984); Steady-state behavour of seral ueung processes wth patent custoers. ath. Operatons forsch, U. statst. Ser, statst. Vol. 5. o Sngh, an and Ahua, Asha (995): The steady state soluton of ultple parallel channels A onthly Double-Blnd Peer Revewed Refereed Open Access Internatonal e-journal - Included n the Internatonal Seral Drectores Internatonal Journal of anageent, IT and Engneerng 67

15 20 IJIE Volue, Issue 5 ISS: n seres wth patent custoers. Intnl. J. gt. syst. Vol., o. 2. Sngh, an, Puna and Ashok Kuar (2009): Steady-state solutons of seral and nonseral ueung processes wth renegng and balkng due to long ueue and soe urgent essage and feedback.internatonal Journal of Essental Scence Vol. 2, o.2. Vkra and Sngh, an (998): Steady state soluton of seral and non seral ueung processes wth renegng and balkng phenoenon. recent advances n nforaton theory, statstcs and coputer applcatons,ccs Haryana Agrcultural Unversty Publcaton, Hsar A onthly Double-Blnd Peer Revewed Refereed Open Access Internatonal e-journal - Included n the Internatonal Seral Drectores Internatonal Journal of anageent, IT and Engneerng 68