Advanced Queueing Theory. M/G/1 Queueing Systems

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1 Advand Quung Thory Ths slds ar rad by Dr. Yh Huang of Gorg Mason Unvrsy. Sudns rgsrd n Dr. Huang's ourss a GMU an ma a sngl mahn-radabl opy and prn a sngl opy of ah sld for hr own rfrn, so long as ah sld onans h opyrgh samn, and GMU fals ar no usd o produ papr ops. rmsson for any ohr us, hr n mahn-radabl or prnd form, mus b oband from h auhor n wrng. CS 756 M/G/ Quung Sysms Srv ms hav a gnral dsrbuon. Ohr assumpons of M/M/ ar rand. Implaons: W an no longr rly on h h mmorylss propry of srv ms. If w wr o us h sa ranson dagram approah, hn ah sa mus onan N, A, whr N s h numbr of usomrs a m and A rprsns how long h usomr n h srvr has bn srvd up o m. Can you xplan why w don' nd M/M/*? A CS 756 n

2 Noaons W --- wang m n quu of h h usomr R --- rsdual srv m of h urrnly srvd usomr upon h arrval of h h usomr N --- srv m of h h usomr --- numbr of usomrs found wang n quu by h h usomr upon hs arrval CS By dfnon, W W W W R R N µ R λw, µ R ρ N Analyss { } { } { E W E R E } E{ R} E{ } E{ N } Q, N by ang h lm by Ll's Thorm whr R s h avrag rsdual srv m. CS 756 4

3 Rsdual Tms r --- h rsdual srv m a m m --- h numbr of srv omplons up o m. Rs dual Sr v T m r m T m CS L us ompu h m avrag of h nrval, : lm R λ m m r s ds lm lm m, whr E[ r n m m m r s ds m Rallng ha W R / ρ, w now hav h ollaz-khnhn -K formula: λ W ρ CS ] 3

4 Avrag m n sysm / µ s h avrag srv m: T W λ ρ Avrag numbr of usomrs n quu: N Q λw λ ρ Avrag numbr of usomrs n sysm: N λt λ ρ ρ CS G/M/ Quung Sysms Inrarrval ms form a gnral dsrbuon wh pdf g. All ohr M/M/ assumpons ar rand. As n h as of M/G/ quus, w anno summarz h sa of h nr sysm n a sngl numbr, h numbr of usomrs n sysm. Insad, w wll fous on h bhavor of h sysm a som spal momns whn h sa of h sysm an b summarzd n on numbr. CS

5 Sa robably Rvsd For M/M/ quus, w solvd, h probably for h sysm o b n sa. If you hn arfully, h sa probably hangs ovr m. onsdr a barbrshop whos λ 4 usomrs pr hour and whos µ 5 usomrs pr hour aordng o our formula, howvr, wha s h han of sng 3 usomrs n h shop n h frs sond? mus b vry vry small! ranly lss han % Wha dos rally man? CS L b h probably of havng usomrs durng h nrval [,], ha s h poron n [, ] whn h sysm has usomrs h lngh of h nrval, s dfnd as lm Tha s, s h avrag probably of sa ovr an ndfnly long prod of m, ang all m pons no aoun. I urns ou ha G/M/ quus. s dfful o oban wh CS 756 5

6 Rahr han fndng h probably ovr all m pons, w shall onn ourslvs wh h sysm's bhavor only a h momns of usomr arrvals. rsly, l b h probably ha an arrvng usomr ss usomrs n h sysm. Can you s ha h nowldg of s rahr lmd? wh, w now h probably of sa a all ms, as long as h sysm has bn runnng long nough wh, w now h probably of sa only a h momns of usomr arrvals On h posv sd, h sysm CAN b summarzd n a sngl numbr a suh momns. Why? CS 756 Sa Transon Dagram 33, ,-,-3,-4,-, Al l h way s of l a v ng s a, A ranson rprsns a usomr arrval., rprsns h probably of movng from o upon a nw arrval. CS 756 6

7 Radng h Dagram How do w nr sa from? h sysm had usomrs whn h prvous usomr arrvd has usomrs whn h nx usomr arrvs hs mans ha xaly on usomr has bn srvd and lf h sysm bwn h wo arrvals In gnral, a ranson from o - mans usomrs hav bn srvd bwn wo onsuv arrvals. CS I an b shown ha s Appndx: whr u W an oban h valu of hrough numr mhods. CS

8 No ha, whn an arrval ss usomrs n sysm, hn h spnds srv prods n h sysm, mplyng T µ µ µ Fnally, h avrag numbr of usomrs n h sysm s λ N λt µ Amazngly, h abov T and N formulas ar unondonal, ha s, hy ar vald a all ms, no us h momns of arrvals. CS By dfnon, Appndx By h naur of, w hav and u u g d,,,,...,!,,, CS

9 9 CS Th abov an b rdud o and L us ry a soluon of h form Tha s,,! u u.!! u u µ µ CS Howvr, W an oban h valu of hrough numr mhods. Sn w hav I follows ha h ondonal probably!! u u µ µ µ,.

Consider a system of 2 simultaneous first order linear equations

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