Workload Process, Waiting Times, And Sojourn Times In A Discrete Time MMAP[K]/SM[K]/1/FCFS Queue
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- Randolf Whitehead
- 5 years ago
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1 Worod roc W d Soour cr MM[K]/SM[K]//FCFS Quu QM HE r of dur Er hou Ury Hfx No Sco Cd B3 X4 QM.H@d.c d Schoo of Ecooc d M hu Ury B Ch brc: h r w udy h o worod roc d w uu y wh u y of cuor d frcofrrd rc dc. M/G/ y Mro ch whch coy rd o h o worod h uu y corucd. hod dod for cou h dy drbuo of h Mro ch. U h dy drbuo h drbuo of o worod bch w d w of ddu y of cuor r obd. Cord o h G/M/ d QB roch for w d oour dcr uu h do of h rx boc od h M/G/ roch c b fcy r. Ky word: Quu y M/G/ y Mro ch w oour worod drf hod rx yc hod rodcy d Mro ch.. roduco cr uu y r of c r o h rforc y of couco y d ufcur y. For c dcr uu r ub for y M couco f [] Coro. [4] d rfrc hr. hrfor h udy of dcr uu h rcd r o fro rrchr d rcor. h r w roduc d udy dcr uu od wh u y of cuor. Such uu od r ury fro odr couco wor h r rurd o hd dffr y of d fro ho rc do cofrc hh ou f rfr rc hh dfo V drbuo rc c. BSN uch x. For uch y dffr rc y h dffr rur r of dy d o r. o ufy uy of rc QoS for ddu rc hr d o dffr h d o crry ou rforc y h ddu rc.
2 For uu od wh cofrrd LCFS rc dc h y of w uu r d buy rod c b crrd ou hrouh h u of Mro ch wh r rucur.. HE d f [3]. Howr h y of h uu r uu od wh frcofrrd FCFS rc dc dffcu c w u rc of h y of ch cuor uu. Noh fc ror h b d for couou uu od wh u y of cuor.. HE [9 ] HE d f [3] [3 4] d d Hw [5]. h udy of dcr uu wh u y of cuor d xc for V Houd d Bod [6 7 8]. h r w y dcr uu od wh rr u y of cuor d FCFS rc dc. h foc o o h w of ddu y of cuor. h uo h couou c by coruc M/G/ y Mro ch ocd wh h o worod h y w r b o fd h dy drbuo of h o worod h y rbrry. h b u o fd drbuo of h w d oour for bch d for ddu y of cuor. h udy of w uu x.. f [] Coh [3] Gr d [8] Nu [8] Su [] [4 5] d rfrc hr. For dcr uu ubr of hod h b ud o udy w. Gr d [8] ud h WrHof fcoro hod o do orh for cou h drbuo of w. Rcy h rx yc hod h b ud o udy dcr uu f [] V Houd d Bod [6 7 8] Y d Chudhry [3] c.. Such hod d of h rucur of h Mro ch udr codro d do ffc orh for cou y rforc ur. h rucur of r cud h M/G/ y d G/M/ y Nu [7 9]. V Houd d Bod [6 7 8] G/M/ y Mro ch d QB roc wr corucd for udy h w d oour of dcr uu wh Mro rr roc wh rd ro d Hdrbud rc. HE [] G/M/ Mro ch corucd for w d oour of dcr uu wh Mro rr roc d Hdrbud rc. HE [] G/M/ Mro roc corucd for w d oour of couou uu wh Mro rr roc d Hdrbud rc. h r h bc oo r rx yc hod. W foru M/G/ y Mro ch o ob h dy drbuo of w d oour. h M/G/ y Mro ch h b ud h udy of rou uu od. For c h cc M/G/ uu d r h M/G/ uu d r c b yd ffcy by u M/G/ y Mro ch. h bc hory cocr M/G/ y Mro ch c b foud G. [6] Louch d Rw [5] Nu [7 9] d Rw []. [] couou ro of h M/G/ y Mro roc w roducd d d. co h roc w ud h udy of h o worod d w of couou M/G/ uu. h r codr xo of h d ud [] fro h couou c o h dcr c. h or dffrc r w codr dcr uu od; w codr
3 uu y wh u y of cuor; d 3 h M/G/ y Mro ch h r h o boudry h ro. h roch dod h r M/G/ roch for uu wh r rc whr h hod G/M/ or QB dod HE [] d V Houd d Bod [6 7 8] do o y. E for h MM[K]/H[K]/ uu for whch hr hod y h M/G/ roch h d. w od by V Houd d Bod [8] h do of h rc od wh h G/M/ d QB roch c b u r whch cr couo d c rur fcy. h of ro boc h M/G/ roch c b fcy r. h r of h r ord foow. Sco h dcr MM[K]/SM[K]/ uu roducd. Sco 3 M/G/ y Mro ch ocd wh h o worod dfd d rodcy codo foud. h dy drbuo of h Mro ch h o worod w oour of bch of cuor r Sco 4. Sco 5 wo c c h MM[K]/G[K]/ uu d h MM[K]/H[K]/ uu r codrd d or dd ru r obd. Fy Sco 6 urc x h rod h o h rforc of h uu od of r r rd.. h cr MM[K]/SM[K]//FCFS Quu h uu y of r h K y of cuor whr K o r. cuor rrd of hr y o uu d r rd by rr o frcofrrd FCFS b. h rdr of h co w dcrb h cuor rr roc h uu roc d h rc roc d. h cuor rr roc h cuor rr roc dcr Mro rr roc wh rd ro MM[K] u d Koo [] HE [9 ] d HE d Nu [4]. Cuor r duhd o K y d rr bch. o chrcr h bch of cuor w roduc of r of r dod by À : L K ³.. For h uu y h r L rr bch wh cuor. h cuor r of h y d rcy. Wh h bch h cuor r ordrd d. W c r rro of h bch. h y u obrd r och. L b h h of h udry Mro ch of h rr roc h rod [ whch w b cd rod d X b h r rro of h bch ocd wh h ro rod f hr rr; ohrw X. h w df 3
4 X d ³ Î ÈÀ. whr h ubr of h of h udry Mro ch. h co d h robby h bch rr rod d h h of h udry Mro ch rod h h h w rod. L b rx wh h d. L b ubochc rx for o rr rod.. X. h of rc rod foro bou h Mro rr roc wh rd ro. L b h u of rc h. h rx h ro robby rx of h udry Mro ch. W u h rrducb d ¹. L b h uu r robby cor of h ochc rx.. d whr cou cor wh b o. W o h h rrducby of h h rrofrobu u of.. h u wh h r oduu h o c ¹. W rfr rdr o Gchr [7] for or bou o rc. dy h o rr r of bch of cuor by. L for.. h r rr r of bch of h y. h rr r of y cuor by N K.3 whr N h ubr of rc of h r h r. Foru for h rr r c b drd by u h cc r fuco roch S HE [] d Nu [7 9]. h uu roc fr bch of cuor h rrd cuor o uu ccord o h ordr h bch. bch r rd by rr o FCFS b. Wh ch bch cuor r rd ccord o hr ordr h bch. L b r of r for h uu rod whch obd fr ob rc coo d rr rod. Suo h h rc of ddu cuor bch c b duhd. f L h hr r cuor h y h cuor rc of y h fr cuor w uu of y d h cuor w uu of y. h cuor h b rd h ordr. f bch h h Lh rr x h uu bco L h h Lh. f h rc of cuor cod x h uu bco 3 L d h rr r r cuor. f rc r dfd for bch oy h h uu roc c b dcrbd r of bch of cuor r r. ccord o HE [] h rc ordr of cuor wh bch c b cfd y by rr h r h corrod r. For c f y cuor h rc rory or ohr cuor wh bch h h r cd hd of ohr h r fc HE [] h how 4
5 h h r rro rod r fxby od wh rc o h rc ordr wh bch of cuor. h rc roc W u h h rc roc d h rr roc r dd. h rc roc of bch ord by Mro ch wh. L h b h h of h Mro roc fr h h ro rc. L X h r rro of h h bch rc. h w u h X c ³ Î.4 X h À whr h rc of bch d o r. W u h h rc of y bch o. uo.4 h curr rc h h d h h bco fr h rc coo whch h h of h x rc. No h f h uu y y h h of h udry Mro ch r h. h co c h robby h h rc d h h bco fr h coo of h rc h h curr h d h bch rc of y. L C b rx wh h c. h of rc C ³ rod foro bou h rc roc. For y w do C C C C ³..5 h rx C h ro robby rx of h udry Mro ch h ³ whch dd of.. h y of bch rc h o c o h ro. W u h C rrducb. L b h uu r robby cor of h ochc rx C.. C d. dy h rc of bch c b ccud : C. h rc r of bch of h y E E of... h r ubr of y bch h c b rd r rod h rffc y of h uu y dfd : r /..6 ccord o Loy [6] h uu y c rch dy f d oy f r <. hrfor hrouhou h r w u r <. h bo dfo of h uu od h rc roc r. Lr h r w h codr o c c uch h c wh dd rc MM[K]/G[K]/ d h c wh Hdrbud rc MM[K]/H[K]/. 5
6 3. h Grd o Worod roc h bc d o y h w or fro h foow fud roh for w uu. L w b h cu w of h h bch. h w h w x w X ³ 3. whr h h of h bw h h bch d h bch X h r rro of h h bch d h rc of h h bch. h roc w X ³ h r rucur whch dffcu o d wh Zho. [9]. o udy h w w coruc o roc coy rd o w ³ whch c b d wh udr cr codo o h rr or h rc roc. So x of uch roc r h roc d h o worod h ru w roc. h o worod roc ud h r d h roc ud HE [] for h udy of w d oour. Bd o uo 3. w roduc roc rd o h o worod h uu y X w. 3. whr rr h ord ubr of h bch rrd or bfor rod d h rr of h h bch. rdy h w X d ud hr u f bch rr rod. h roh bw w how Fur 3.. X X d rr of h h bch X w Fur 3. Vrb w d X wh >. h roc ³ o foow. bch rr och y h h bch h o worod h uu y ccud w whch h u of h rod. ur h rr of h x bch h u of dcr by o r rod. f ³ rr h o worod h uu y rod. f < h uu y y rod. h u dc how o h curr d rod X 6
7 h b or h of h d rod rod. Wh h x bch rr h u of udd h o worod h rod d h x cyc b. No h f d hr rr h u uwrd o h rc of h rrd bch. Cord o h roc w ³ ³ rod foro bou h worod y och. fc w ³ c b codrd bddd roc of ³ bch rr och. ordr o coruc Mro ch rd o ³ w roduc h foow uxry rb. W df roc ³ fro h Mro ch h ³ foow: h f h h bch h o rrd or bfor rod. hu co bw wo cocu bch rr. Rc h X h bch h rrd rod. f hr o rr rod h X. h uo 3. d o ïì f X ; í 3.3 ïî x X f X ¹. No h uo 3.3. Sc h rr roc ord by Mro ch h roc ³ h h Mro rory dur o rr rod. h h ch h rc roc drd by h Mro ch h ³. h rc of h bch dd o h Mro ch h ³ w. hu h roc ³ h h Mro rory bch rr rod. hrfor h roc ³ Mro ch. Sc h u of c dcr o by o r rod h h fr o h f rory. hrfor ³ Mro ch of M/G/ y wh o boudry h ro. h rb d r u d hr r r: < < d. yc h of how Fur 3.. Src of bch h u rrd Sy d Fur 3. h of h ro robby rx of ³ 7
8 O O O L L L L O L M M M M O O M M L L 3.4 whr ; ³ C 3.5 whr h oo for Krocr roduc of rx Gchr [7]. No h h u of dr h of h Mro ch. o o h h rrofrobu u of u o h rrofrobu u of whch h o. h ro robb c b rfd foow: î í ì ¹ '. f '; f ' ' ' 3.6. ' ' ; ' ' ' ' ' ' ' ' ³ ³ ³ C C W h u h dy drbuo of h Mro ch ³ o fd h drbuo of w. For h uro w how h uch dy drbuo x f h uu y b.. r <. Fr w ro h foow ror ocd wh h ro boc. f ; ³ 3.7 f h uo r wdfd. By dfo 3.5 w h
9 C C. 3.8 Rr 3. Bfor rocd w o dcu h rrducby of h rx d h Mro ch ³ c h dy y of h uu y dd ry o h uo h d ³ r rrducb. Uforuy hr or ³ urd o b rrducb udr h rrducby uo o h rc d C d Sco. For c : C ¹ for x x o. 3.9 ob h x. f x h h wh ³ r o rchb fro wh. Furhror ob for d ³ o h r cod c of. For x f..9 C 3. for whch h rx rrducb d rodc d C rrducb h rx h wo cod ub d. o y h Mro ch w u dfy h cod c of d ro r. h w cocr o h rrducb ub. hrfor h r w h u h r h b rod d h Mro ch ³ rrducb d rodc. o by c h rrofrobu u of h o rx. L u d b h f d rh cor corrod o c rcy. h wo cor u d r ord by u d u. f rrducb h rrducb d h of h cor u d r o for >. ccord o Nu [7] h fuco c d h cor u d c b cho dffrb fuco. L 3. u h rrducb. h cor h r robby cor of. w h c d S c r. Couy c < f d oy f r <. No h c h fr dr of h fuco c. roof. Fro h xro of uo 3.8 y o rfy h h cor h r robby cor of h ochc rx. o y o c. Furhror w h u d. By u w h u. By dr o boh d of u cu w ob u u c u cu. L d uy o boh d of h uo w ob u c.. c whch c b ud 9
10 . / r À Î E C C C 3. hrfor c r. h c < f d oy f r <. h co h roof of L 3.. o by G rx h h o ouo o G C G G. 3. W rfr o Louch d Rw [5] d Nu [7 9] for or bou h rx G. L b h uu r robby cor of G f G ochc. hor 3. f h Mro ch ³ rrducb o rcurr f d oy f r <. roof. y o fro h rucur of h ro rx uo 3.4 h u b rrducb f h Mro ch rrducb. hrfor by L 3. h cor h r robby cor of. W u h drf hod o ro h hor. Fr w ro h cy of h codo r <. No h h of of h Mro ch drd by h u of. W codr h wh o. h ro o h r h cc M/G/ y Mro ch. f h Mro ch ³ o rcurr h h fr fro y o o h ro u b f. By Nu [7] w u h c <. hrfor w u h r <. For co w h foow roof bd o h drf hod. L b h cou cor fr fro h o h for ³. h cor dd of h > bcu of h M/G/ rucur. Sc h Mro ch o rcurr ry of o d f. By codo o h fr ro h foow uo c b bhd for : G G. 3.3
11 By uo 3.3 h rrofrobu u of h rx G u b h o. Sc h Mro ch o rcurr G ochc rx. c b how h G rb. By rou ccuo w ob G G G. 3.4 ruy o boh d of uo 3.4 w ob G. 3.5 ouy h rh cor corrod o h rrofrobu u of G o boh d of uo 3.5 w fd h S < whch c <. hrfor by L 3. w u h r <. Now w u r <. By L 3. w h c <. hu w h c < for o > d co o o. h w h c <. Sc h rro Frobu u of h o bcu h rrofrobu u of h o h rx rb. Ery of h cor f d o. Wh d w df cor Lyuo fuco fuco foow: for h ïî ï í ì < ³. ; f 3.6 No: Hr w bu h oo b; w u o rr h. Sc rrducb ry of h cor o. hrfor f. L c > d. W choo r ouh o h for >. 3.7 h drf o ro ccud foow. For ³. f f E 3.8 For < w h
12 . f f E 3.9 W o h for. < E f 3. hu h drf wy fro ro wh rc o h Lyuo fuco uo 3.6 r h xc for f ubr of. hrfor h Mro roc o rcurr by For' crro hor..3 Fyo. [5]. h co h roof of hor Sdy S rbuo h co w codr h ory drbuo of ³ udr r < d h rrducby codo d hor 3.. o by h dy drbuo of ³.. whr row cor of h d 4. W furhr u h h Mro ch ³ rodc o h h dy drbuo r robby ur uu. Fr w fd r of. By w h for ³ whch d o for ³. 4. hu o fd w oy d o cocr o. fc w c coruc ohr Mro ch whch rd o h cu worod roc o fd. h coruco of h Mro ch foow. By w h... ³. 4.3
13 o by ] o. h h robby cor ] drbuo of h foow ro robby rx. No h h rx f d h dy 3 L 3 L O. 4.4 O O O For h Mro ch ³ f w oy obr h roc wh ³ w ob roc ³ whr h cu o worod h y. y o h ³ M/G/ y Mro ch d ro robby rx by uo 4.4. No h o ob h dy w drbuo w c drcy codr h Mro ch ³. By codr h Mro ch ³ howr w or foro bou h uu roc cy bou h d rod. o by ] foow uo for :. Fro uo 4.4 y o ob h ]. 4.5 hor 4. f h Mro ch ³ rrducb d rodc d r < h d ] r r whr h r robby cor of G. roof. L o o o boh d of uo 4.5 w ob. Sc robby cor u b h r robby cor of. By L 3.. d boh d of uo 4.5 by d uy boh d by yd ]. 4.6 L o o o boh d of uo 4.6 d u Hô Ru by L 3. w ob 3
14 ] c r. 4.7 Codr h bddd Mro ch h och h Mro ch ³ r h ro. h w u h ] ]. By codo o h fr ro fro h ro w ob G G. 4.8 hrfor w u h ] c whr c co. Sc ] r h c r. h co h roof of hor 4.. Oc ] foud ohr cor c b coud by b rcuro dod Rw []: for ³ ] 4.9 whr G for ³. Rr 4. h ru obd hor 4. d h bo orh y b d for h or r c. For c f x dfd uo 3.9 h Mro ch ³ rducb. Howr f c b obd h c b coud by u uo 4.9. For h c for ³. Euo 4.9 for ³. Now w rur o h o worod d h w of rbrry bch. L b h rc rdo rb for h o worod h y dy w b h rc rdo rb for h w of rbrry bch.. h w of h fr cuor h bch w b h rc rdo rb for h w of rbrry bch d b h rc rdo rb for h oour of rbrry bch h o h who bch h y d d b h rc rdo rb for h oour of rbrry bch. W o do by h rc rdo rb of h udry Mro ch of h rr roc dy d by h rc rdo rb of h rc roc dy. 4
15 5 hor 4. f h uu y b d h Mro ch ³ rrducb d rodc h drbuo of h o worod w d oour r by: < <. 4. ï ï î ï í ì ³. ; w r 4. ï ï î ï ï í ì ³. ; w r 4. ³ C C d r. 4.3 ³ C C d r. 4.4 roof. Euo 4. obou. Rc h f < h uu y y h rod d h y h b d for rod of. Euo 3. how h h w of bch h rd o worod rh bfor rr. hrfor h drbuo of w w of rbrry bch of y c b obd by codo o h rr of bch rbrry whch c b obd foow for ³ w x rr Bch x rr Bch x rr Bch x rr Bch 4.5 x rr Bch x rr Bch
16 whch d o h cod xro uo 4.. f w h w Bch bch rr x whch d o h fr xro uo 4.. h drbuo of w c b obd ry. h drbuo of h oour r obd h coouo of h drbuo of w d rc by h h of h rc roc o codro. h co h roof of hor 4.. No h h couo of h o of h w c b do by u x foru for h M/G/ y Mro ch Nu [7 9]. r od. h couo of h w drbuo of rbrry y cuor c b b od c h rc of cuor bch y o b dd. ohr word f h rc dfd h bch h w drbuo of y cuor co b foud xcy. Noh f or foro o h rc roc b h b ob o fd h drbuo of w of ddu y of cuor. x Sco 5. o d h co w o ou h hr co ro bw h roc of h bch rc d h o worod roc. W df h of bch rod h o h bch h b h uu y h h bch h y rod. h rd roc ³ of h bch rc or o b rd x f h y y dfd w X 4.6 whr h ord ubr of h bch rd or bfor rod d h drur rod of h h bch. h u of w d r udd f drur occur rod whr w c b coud by u uo 3.. ccord o Corory 4. HE [] d h h dy drbuo. hu uo 4. c b ud for cou h dy drbuo of h of h bch rc rbrry w. 5. h cr MM[K]/G[K]/ Quu For h dcr MM[K]/G[K]/ uu h rc of ddu cuor r dd wh r drbuo... Couy h rc of bch r dd w. o by h rc of y cuor. h h rc r of y cuor by /E. h rc of bch L wh E E. By dfo.6 w h 6
17 7. / / À Î À Î À Î K K K E N E N E r 5. hu for h MM[K]/G[K]/ uu dfo.6 for h rffc y co wh h cc dfo. h drbuo of h o worod d h w of bch c b ccud by u foru 4. d 4.. For h c c w r b o cou h w of ddu y of cuor. L w b h rc rdo rb for h w of rbrry y cuor. Bd o uo 4. w h for ³ d K À Î À Î À Î w w d d d r 5. whr d. h dcor fuco. L d b h rc rdo rb for h oour of rbrry y cuor. No h for h dfo of d w u h cuor h y dy fr rc. W h for ³ d K.... À Î w d d 5.3 or c c h dcr MM[K]/H[K]/ uu whr h rc of ddu cuor r dd d h Hdrbuo wh rx rro K whr h ubr of h of h Hdrbuo h robby cor d ubochc rx. W u h h rc of y cuor o.. K. S Nu [7] for or bou Hdrbuo. o by. Sc Hdrbuo r cod udr coouo h rc of bch o h dcr Hdrbuo wh rx rro whr for
18 8....; ; 3 M M O O 5.4 ccord o Nu [7] h drbuo of rc r by. ³ C 5.5 h rx boc of h ro rx of ³ uo 3.5 c b coud y by ³. 5.6 h dy drbuo of w c b coud ccordy by u foru rodd Sco 3 d 4. No h for h c. So rrducb f d oy f rrducb. W o ou h for h dcr MM[K]/H[K]/ uu c boh h rr roc d h rc roc r ord by Mro ch QB roc c b roducd o dcrb h o worod roc d h roc of h bch rc. h roch w ud HE [] d V Houd d Bod [8]. h d of h QB roch h h dy drbuo of h QB roc h rx orc drbuo whch y o cou. O h ohr hd w od ou by V Houd d Bod [8] h do of h rx boc od c b u r for h QB roch whch À Î. 5.7 h ubr c b fcy rr h h do of h boc for h M/G/ y Mro ch dod h r. E f w codr h couo of Hdrbuo ud coruc rc ³ uo 5.6 h r do of rc od wh h M/G/ y Mro ch x whch c b fcy r h h ubr uo 5.7. hrfor h M/G/ roch c y or ro h y of h w d oour for h MM[K]/H[K]/ c. Mor dcuo o h u Sco 6.
19 6. Nurc Ex h co w ou ch for cou drbuo of w d oour d brfy dcu hr urc x. Fr w ur h for cou rforc ur h foow x. Cou cor d h rr r rr r d r by uo.6. Cou ro boc by uo Cou h rx G by uo 3. d h cor. 4 Cou h cor ] d by hor Cou cor by uo Cou cor by h rcuro uo r drbuo of h o worod w d oour by foru Corory 4. or uo 5. d 5.3. Mo of h bo c b do rhforwrd r xc h couo of h rx G d cor. Noh for h couo of G d ffc orh r b h rur Nu [9] d Rw []. Ex 6. Codr MM[]/H[]/ uu wh foow y rr: K.6.4.4; ; h uu y h h rffc wh r.539. For h uu y y cuor oy y o b foowd by y cuor. Sc y cuor h owr rc r xcd h y cuor fc or w h y cuor whch how b 6.. h drbuo of w d oour r b 6.. 9
20 b 6. rbuo of w d oour w w w d d d r o h.486 r whch uch r h w.85. fc for y urc x» w. h ro h for dcr uu rr d drur c occur uouy. hrfor f h o worod o h w of h x rr cuy ro. Nx w x uu y hy rffc. Ex 6. Codr MM[5]/H[5]/ uu wh foow y rr: K ; ; h rffc y of h uu y r.934. h drbuo of w r b 6..
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