TELCOM 2130 Time Varying Queues. David Tipper Associate Professor Graduate Telecommunications and Networking Program University of Pittsburgh Slides 7
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1 TELOM 3 Tme Vryng Queues Dvd Tpper Assote Professor Grdute Teleommuntons nd Networkng Progrm Unversty of Pttsburgh ldes 7 Tme Vryng Behvor Teletrff typlly hs lrge tme of dy vrtons Men number of lls per mnute t entrl offe swth mesured n 5 mnute ntervls verged over work dys Assoted Men ll holdng tmes oure: ITU Teletrff Hndbook
2 Tme Vryng Behvor However queueng results thus fr re for stedy stte Fous on stedy stte probbltes π = lm t P{n( = } tedy stte men behvor L, W, et. Wht bout behvor s funton of tme? Trnsent: ystem gong from one sttonry stte to nother Nonsttonry: ystem wth ontnuous vrton n rrvl nd/or serve rtes When does tme vryng/trnsent behvor mtter? If lod s dynm n omprson to queue settlng tmes If tme vryng serve rte from resoures beng swthed on nd off, dynm bndwdth lloton, et. erve rte must hnge s rpd s queue After flure ondtons Approxmton Approhes mple ttonry Approxmton (A gnore vrtons n lod/serve rtes use verge vlues n stedy stte queueng model smple nd pplble to wde rnge of queueng systems Good for smll systems wth low vrton Pek Approxmton (PA use pek/mxmum vlue nsted of the verge lod wdely used pproh n teleom Qus-tt Approxmton (QA Montor tme vryng prmeters over set of tme ntervls Assume stt ondtons durng eh tme ntervl nd pply stedy stte results for eh perod usng men of prmeters n eh perod Pek Vlue x Pontwse ttonry Approxmton (PA Use smpled vlues of tme vryng prmeters to evlute stedy stte t eh smpled tmepont
3 Exmple onsder M/M/ wth =, = +.5 sn(π Fous on men number n system L = /(- Men number n ystem L Method Tme (.5 Tme(.5,.5 Tme(.5,.75 Tme(.75, A =, = PA =, =.5 QA =, =.5, 5.5,.75,.75 PA =, =.35,.35,.646, T = T = T = T =.875 tedy tte Behvor Remember bs pproh s to solve system of equtons derved from Mrkov Proess model of queue together wth normlzton ondton Exmple: Erlng B queueng model M/M// queue dentl servers proess ustomers n prllel. ustomers rrve ordng to Posson proess wth men rte tht s ndependent of tme ustomer serve tmes exponentlly dstrbuted wth men rte tht s ndependent of tme The system hs fnte pty of sze, ustomers rrvng when ll servers busy re dropped Bloked lls lered model (B e ( P b e P b 3
4 M/M// tedy tte Anlyze usng Mrkov Proess of n( number of ustomers n the system t tme t Let be the stedy stte probblty of ustomers n the system, then the stte trnston dgrm nd flow blne equtons re gven below 3 ( flow out stte j = flow n stte j j ( j j j ( j j j ( j Normlzton ondton j j M/M// olvng the equtons for, note tht the bs equtons re the sme s for the M/M/ wth j<. Followng the nlyss n prevous slde set! Pluggng nto the normlzton ondton j j One gets n n n!!,,... n! n! n 4
5 Erlng B Formul Bs Qo metr s probblty of ustomer beng bloked B(, B(,! n n n! Vld for M/G// queue B(, Erlng s B formul Erlng s blokng formul Erlngs frst formul In the telephone system, B(, represents bloked ll lered (B model. Tme Vryng Behvor Very lmted set of ext results for tme vryng nlyss Bs pproh s to study system of dfferentl equtons derved from Mrkov Proess model of queue Exmple: Erlng B queueng model M/M// queue dentl servers proess ustomers n prllel. ustomers rrve ordng to Posson proess wth men rte tht s funton of tme ustomer serve tmes exponentlly dstrbuted wth men rte tht s funton of tme The system hs fnte pty of sze, ustomers rrvng when ll servers busy re dropped Bloked lls lered model (B e ( P b e P b 5
6 dp M/M/// Tme Vryng Model Let p ( denote the stte probblty of ustomers n the system, from the stte trnston dgrm for n( 3 ( Rte of hnge of probblty of beng n stte j = - flow out stte j + flow n stte j ( / dt ( p ( ( p ( j t dpj ( / dt ( pj ( ( ( j( pj ( ( j ( pj ( j dp ( / dt ( p ( ( p ( j The hpmn-kolmogorov dfferentl equton model Note, f set left hnd sde to zero get stedy stte flow blne equtons nd n solve for stedy stte results Tme Vryng Model losed form nlytl soluton of -K model not possble due to tme vryng oeffents n be solved numerlly to determne stte probbltes vs. tme usng stndrd numerl ntegrton tehnque lke Runge-Kutt Numerl soluton tehnque n be wrtten n lgorthm form over [t, t f ] Intlzton: set urrent tme t, to t = t estblsh the ntl stte probbltes p(t = [p (t, =,, ] nd spefy tme step Δt Approxmte the rrvl rte λ( by onstnt λ over [t, t+δt] wth λ = λ(t+δt/ nd ( by onstnt over [t, t+δt] wth = (t+δt/ 3 Numerlly solve the system of dfferentl equtons over the smll tme ntervl Δt, nd get the new system stte probbltes p( t tme t+δt; p(t+δ. 4 Inrement tme, t = t+δt. Ift<t t f, go to, else stop. Note, number of equtons grows wth systems pty ( > n n optl network lnk Wll be dffult to study networks of lnks Need n urte pproxmton 6
7 Flud Flow modelng onsder sngle trnsmsson lnk f n ( = flow n to the queueng systems x( = men number of ustomers t queue t tme t f out (= flow out of queueng system x f out ( t f ( t n Expresson for flow n nd flow out wll depend on system under study (e.g., M/M/, M/G/, et. n pproxmte flow n/flow out by mthng equlbrum pont of flud model wth equvlent queueng model stedy stte result ee W. Wng, et.l., IEEE Infoom 95 For M/M// queue Flud Flow model x ( t f ( t f ( t f n out ( t ( t ( p ( t n f out ( t p ( t p ( t... p ( t x ( t x ( t x ( t ( t ( p ( t ( How to fnd p ( (? Mth tedy stte results Pontwse ttonry Flud Flow Approxmton (PFFA 3 ( 7
8 Flud Flow Model At stedy stte dx/dt = nd probblty of ustomer beng bloked p ( = B(, Erlng B Model x ( t ( t ( p ( t ( t x (t( t ( p ( t B (, p! n n n! (3 Fxed pont problem only one vlue of nd p ( wll work - solve tertvely untl onverges or untl hnge n n two terton < ϵ n numerlly solve flud model ( together wth fxed pont equtons ( nd (3 to study queue behvor Flud Model oluton Numerl soluton tehnque n lgorthm form over [t, t f ] Intlzton: set urrent tme t, to t = t estblsh the ntl system oupny x( = x(t, nd spefy tme step Δt Approxmte the rrvl rte λ( by onstnt λ over [t, t+δt] wth λ = λ(t+δt/ nd ( by onstnt over [t, t+δt] wth = (t+δt/ 3 Approxmte p ( over [t, t+δt] by onstnt p by solvng ( nd (3 tertvely untl the hnge n n terton (x( does not exeed prespefed ϵ vlue. 4 Utlzng x(, λ nd (from step, p (from step 3, numerlly solve the dfferentl equton gven by ( over the smll tme ntervl Δt, nd get the new system oupny t tme t+δt; x(t+δ. 5 Inrement tme, t = t+δt. If t < t f, go to, else stop. 8
9 Flow hrt of oluton method trt Intlzton et urrent tme t t Intl vlue x( t x( t nd spefy tme step t Inrement tme t tt Approxmte rrvl rte nd serve rte olve fxed pont eqs Over smll ntervl ( tt/ Determne p ( t t / olve dff. equtons Usng x(, p olve for x( t No End of smulton? END Yes Numerl Results hek the ury of flud flow model vs. ext hpmn-kolmogrov model Numerlly ntegrte ext model ompre results wth flud flow model Results shown for = 4 (e.g., T lnk ( = sn(.(t+ 9
10 Flud Flow Model Model In generl for nfnte pty queues f n ( = flow n to the queueng systems x( = men number of ustomers t queue t tme t f out (= flow out of queueng system x f out ( f ( For nfnte buffer queues : f n ( = (, f out (=( then t stedy stte hve dx/dt = nd x = G ( Assumng G ( s numerlly nvertble = G - (x get n x ( ( x G ( x( ( t Flud Flow Model Model onsder M/G/ queue t stedy stte ( x x x x ( whh yelds x x x x x G ( x( ( t
11 M/G/ Model x x x x Tme Vryng Queueng Models Mny other queueng models n the lterture for tme vryng behvor fous on numerl soluton not losed form results Multple trff lsses Generl erve tmes Generl rrvl proess Network results for smple Jkson type networks 3
M/G/1/GD/ / System. ! Pollaczek-Khinchin (PK) Equation. ! Steady-state probabilities. ! Finding L, W q, W. ! π 0 = 1 ρ
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