Journal of Natural Sciences Research ISSN (Paper) ISSN (Online) Vol.5, No.3, 2015

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1 Journal of atural Scences Research ISS (Paper ISS 5-09 (Onlne Vol.5, o.3, 05 Analyss of Queung Syste Consstng of ultple Parallel Channels n Seres Connected to on-seral Servers wth Fnte Watng Space Balkng, Renegng eenu Gupta an Sngh Deepak Gupta 3.Asstt. Prof., Dept. of atheatcs, Dr. B. R. A. Govt. College,Kathal-3607 (Haryana Inda. Prof. (Retd., CCS, Haryana Agrcultural Unversty, Hsar Inda 3. Prof. & Head, Dept. of atheatcs,..engg.college, ullana (Abala- Inda Abstract : Ths chapter provdes the steady-state analyss of ore general ueung syste n the sense that servce channels n seres are lnked wth non-seral channels havng renegng balkng phenoenon where each of servce channels has dentcal ultple parallel channels. The nput process s Posson the servce te dstrbuton s exponental. The servce dscplne follows SIRO-rule (Servce n ro order nstead of FIFO-rule (frst n frst out. The custoer becoes patent n ueue after soete ay leave the syste wthout gettng servce. The nput process depends upon the ueue sze n non-seral channels. Watng space s fnte. The practcal stuatons where such a odel fnds applcaton are of coon occurrence. For exaple, consder the adnstraton of a partcular state at the level of dstrct head uarter consstng of Patwars, Kanoongoes, Tehsldars, Sub-dvsonal agstrates, dstrct cosoner etc. These offcers correspond to the servers of seral channels. Educaton departent, Health departent, Irrtaton departent etc. connected wth the last server of seral ueue correspond to non-seral channels. The people eet the offcers of the dstrct n connecton wth ther probles. It s also a coon practce that the offcers call the people hearng roly.further Dstrct cosoner ay send the custoers to dfferent departents such as educaton, health, rrgaton etc. f ther probles are related to such departents.. Introducton The proble of seral ueues studed by [], [], [3], [5], [6] n steady-state wth Posson assuptons wth the restrcton that the custoer ust go through each servce channel bee leavng the syste. [7] 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 [8]. The solutons of seral non-seral ueung processes wth renegng balkng phenoenon have been studed by [0]. The steady-state soluton of seral non-seral ueung processes wth renegng balkng due to long ueue soe urgent essage feedback phenoenon s obtaned by [9].Ths chapter provdes the steady-state analyss of ore general ueung syste n the sense that servce channels n seres are lnked wth non-seral channels havng renegng balkng phenoenon where each of servce channels has dentcal ultple parallel channels. 6

2 Journal of atural Scences Research ISS (Paper ISS 5-09 (Onlne Vol.5, o.3, 05 The nput process s Posson the servce te dstrbuton s exponental. The servce dscplne follows SIRO-rule (Servce n ro order nstead of FIFO-rule (frst n frst out. The custoer becoes patent n ueue after soete ay leave the syste wthout gettng servce. The nput process depends upon the ueue sze n non-seral channels. Watng space s fnte.. Forulaton of odel The syste conssts of Q ( =,,. servce phases where each servce phase Q has c ( =,,. dentcal parallel servce facltes Q channels (=,,. wth respectve servers S (=,, S (=,,.Custoers deng dfferent types of servce arrve fro outsde the syste n Posson dstrbuton wth paraeters ( =,,. at Q servce phase (=,,.. at Q servce phase respectvely. But the sght of long ueue at Q, ay dscourage the fresh custoers fro onng t ay decde not to enter the servce channel Q (=,,. then the Posson nput rate would be where s the ueue sze of Q. Further, the patent custoers onng any servce channel ay leave the ueue wthout gettng servce after a wat of certan te. The servce te dstrbuton the servers S (=,, S (=,.., are utually ndependent negatve exponental dstrbuton wth servce rates ( =,,..., (,,..., = respectvely. After the copleton of servce at Q (=,,.., the custoer ether leaves the syste wth probablty p or ons the next phase wth probablty such that p = (=,..-. After copleton of servce at Q, the custoer ether leaves the syste wth probablty p or ons any of the Q (=,,., wth probablty such that p = (=,,., =. If the custoers are ore than c n the Q servce phase,all the c servers wll rean busy each s thus the ean servce rate at Q s c,on the other h f the puttng out the servce at ean rate nuber of custoers s less than c n the Q servce phase,only n out of the c servers wll be busy thus the ean servce rate at Q s n (=,,..It s assued that the servce coences nstantaneously when the custoer arrves at an epty servce channel. = = Here we assue that f at any nstant, there are K custoers n the syste, then the custoers arrvng at that nstant wll not be allowed to on the syste t s consdered as a ced balkng lost the syste. n = K 3. Forulaton of Euatons Defne P(n,n,. n ;,, 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 (=,,..,, custoers ( whch ay balk or renege or after beng servced leave the syste watng bee the servers S (=,,.. We defne the operators T, T T, to act upon the vector T T, to act upon the vector T (n% = (n, n, n - n T (n% = (n, n, n n T, (n% = (n, n,.,n, n -,.,n T (% = (,, - % = (,,..., n% = ( n, n,..., n as follows: T 7

3 Journal of atural Scences Research ISS (Paper ISS 5-09 (Onlne Vol.5, o.3, 05 T. (% = (,, T, (% = (,,,,. -. The followng dfference-dfferental euatons hold: d P ( n %, t %, = δ ( n ( n δ ( ( (, ; n c r n δ R P n % t % dt = = = = --( PT ( n, t ; P( nt, ; t (% % % ( % = = ( ( δ r PT ( n%, t %; PT ( n%, t %; n c n n, = = p PT ( n, t ; Pn (, n,, n, T ; t (% % K ( % n n = = = n n 0 0, = = ( % ( % R PnT (, ; t < K; ( =,,3,,; =,,3,,, d Pnt (, ; n r R Pnt (, ; dt % % = δ δ δ % % = = PT ( ( n%, t %; P( nt %, ( % ; t = = ( ( n ( ( n c n n PT (, (, ; n% t % n ( K % = = n = K n 0 0 ; = = P( n, n,, n, T ; t ; (=,, 3,, ; =,, 3,, ( 8

4 Journal of atural Scences Research ISS (Paper ISS 5-09 (Onlne Vol.5, o.3, 05 δ ( x w hen x 0 = 0 w hen x = 0 δ ( n c n 0 w hen n < c = w hen n c n w hen n < c = c w hen n c 0 n e r = ; =,,... n T 0 n ( e T 0 e R = ; =,,... T T 0 ( e rn R are the average rates at whch the custoers renege after a wat of certan te T0 T0 whenever there are n custoers n the Q Q servce phases respectvely P( %, nt %; = 0 f any of the arguents s negatve. 4. Steady-State Euatons: We wrte the followng Steady-State euatons of the ueung odel by euatng the te dervatves to zero n the euatons ( ( δ ( n ( n δ ( ( (, n cr n δ R P n % % = = = = = PT ( n, (, (% % PnT % ( % = = ( ( δ r PT ( n%, % PT ( n%, % n c n n, = = p PT ( ( n%, % n P( n, n,, n, ( K T % = = n = ( R PnT % ( % (, n < K; n 0 0 ; = = (3 9

5 Journal of atural Scences Research ISS (Paper ISS 5-09 (Onlne Vol.5, o.3, 05 δ n δ r δ R Pn (%, % = = = PT ( ( n%, % P( nt %, ( % ( ( n ( ( n c n = = n PT (, (, n% % n ( P n n K n T % = = n = K; n 0 0 ; = = (,,,, (4 n Two cases arse dependng upon the nuber of custoers nuber of channels c at Q phase (=,, 3,, 5. CASE( Q When the nuber of custoers n bee phase s less than the nuber of dentcal servce channels c (.e. n c < ; =,, 3,,, then there s no renegng n Q the servce s edately avalable to the δn custoers on arrval. Then under such stuaton c = 0 n = n Steady State Euatons: The euatons (3 (4 reduce to n δ R P( n%, % = = = = = PT ( ( n%, % P( nt %, ( % = = n PT, ( n% % ( (, = ( ( p ( n PT ( ( n%, % n Pn n K n T ( = = ( R PnT % ( % (, ( (,,,, % = n < K; n 0 0 ; = = n δ R Pn (%, % = = = PT ( ( n%, % P( nt %, ( % ( ( = = (5 ( n PT (, ( n%, % n Pn n K n T ( = n 0 0 ; ( (,,,, % = = Steady State Solutons: The Steady State solutons of the above euatons (5 (6 can be verfed to be = n = K; (6 30

6 Journal of atural Scences Research ISS (Paper ISS 5-09 (Onlne Vol.5, o.3, 05 n n α P( n %, % = P( 0,0 % % n n where ρ α 3 3 α n n α L n3 3 n ( ρ ( ρ ( R ( R = = ( ρ L ( R = α, α, α, Kα 3 = = ρ are the sae as entoned below. α (7 α = α ; k =, 3,..., k k k k 6. CASE (II Q When the nuber of custoers bee phase s ore than or eual to the nuber of dentcal servce c channels (.e. n c, then there s renegng n Q δ (=,,,. So under such stuaton n c = n = c δ ( n =. Steady State Euatons: The euatons (3 (4 reduce to c rn (, δ R P n% % = = = = = PT ( ( n%, % PnT (%, ( % = = ( ( ( ( ( r PT ( n%, % c PT ( n%, % n, = = p cpt ( ( n%, % cp( n, n, K, n, T ( % = = = ( R PnT % ( % (, n 0 = = < K; (8 3

7 Journal of atural Scences Research ISS (Paper ISS 5-09 (Onlne Vol.5, o.3, 05 c r R Pn (%, % = = = PT ( ( n%, % PnT (%, ( % ( n δ ( ( = = ( ( r PT ( n%, % c PT ( n%, % n, = = c Pn (, n, K, n, T ( % = 0 n = K; = = (9 Steady State Soluton: The Steady state soluton of the above euatons can be verfed to be: (%, % = ( 0,0 % % { ( c rn } c α n n ( n n P n P n ( c r ( c r ( c r n = = n3 3 ( c rn c α ( c rn c α = K = n3 n n 3 n ( c3 3 r3 ( c rn ( c r ( c rn = = = = { c ρ } c ρ ( R R = = c ρ... R = α, α, α, K α 3 ρ { } ( { } ( (0 ρ are the sae as entoned below: ( c r c α n = = α = ( c r ( c r n n = k k k n k k k k = α = ( c r α c ; k =,3,...,. n 3

8 Journal of atural Scences Research ISS (Paper ISS 5-09 (Onlne Vol.5, o.3, 05 Here, t s entoned that the custoers leave the syste at constant rate as long as there s a lne provded that R = R the custoers are served n the order n whch they arrve. Puttng (=,,3,, n euatons (5, rn = r (6,(8 (9 n euatons (8 (9, the steady-state solutons (7 (0 reduce to (%, % ( 0,0 % % n n α P n = P n n 3 3 α n n α L n3 3 n ρ ρ... R R ρ... ( R n < c, 0 ;(=,,3,, ; =,,3,...,,, K ρ are the sae as entoned bee. α α α α 3 n ( (%, % ( 0,0 % % ( n c r c α P n = P c r c r c r ( ( n3 3 ( c r c α ( c r c α = = K 3 ( c r ( c r = = c ρ c ρ ( R ( R c ρ... ( R ( n c, 0 ;(=,,3,, ; =,,3,... α, α, α, K α 3 ( 0,0 We obtan ρ are the sae as entoned bee. K K ( n P % % P n %, % = n = K fro the noralzng condtonn % = 0 % % = 0 % = l = wth the restrctons that the traffc ntensty of each servce channel of the syste s less than unty. References. Barrer D.Y.(955 A Watng lne proble characterzed by patent custoers 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 Jackson, R.R. P (954: Queung syste wth phase type servce Operat. Res. Quart. 5, Kelly, F.P. (979: Reversblty stochastc networks Wley, ew York 5. aggu, P.L.(970: On certan types of ueues n seres. Statstca eerlca. 4, O Bren,.G. (954: The soluton of soe ueung probles J.Soc. Ind. Appl. ath. Vol.,

9 Journal of atural Scences Research ISS (Paper ISS 5-09 (Onlne Vol.5, o.3, Sngh,. (984; Steady-state behavour of seral ueung processes wth patent custoers. ath. Operatons sch, U. statst. Ser, statst. Vol. 5. o Sngh, an Ahua, Asha (995: The steady state soluton of ultple parallel channels n seres wth patent custoers. Intnl. J. gt. syst. Vol., o.. 9. Sngh, an, Puna Ashok Kuar (009: Steady-state solutons of seral non-seral ueung processes wth renegng balkng due to long ueue soe urgent essage feedback.internatonal Journal of Essental Scence Vol., o.. 0. Vkra Sngh, an (998: Steady state soluton of seral non seral ueung processes wth renegng balkng phenoenon. recent advances n naton theory, statstcs coputer applcatons,ccs Haryana Agrcultural Unversty Publcaton, Hsar

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