Queueing Networks. γ 3

Transcription

1 Queueg Networks Systes odeled by queueg etworks ca roughly be grouped to four categores. Ope etworks Custoers arrve fro outsde the syste are served ad the depart. Exaple: acket swtched data etwork. γ µ γ µ µ γ. Networks wth populato costrats Custoers arrve fro outsde the syste f there s roo the queues. They eter served ad the depart. Exaple: queues sharg a coo buffer pool. µ γ µ µ. Closed etworks Fxed uber of custoers ( are trapped the syste ad crculate aog the queues. Exaple: CU job schedulg proble µ µ µ k 4. xed etwork Ay cobato of the types above. Exaple: sple odel of vrtual crcut that s wdow flow cotrolled.

2 µ µ µ k Several features ca occur queueg etworks that do ot occur sgle queue.. Jockg Custoers ovg aog parallel queues.. Blockg Custoer watg depart a server ad jo ext queue s uable to due to lted watg space ad therefore stays server (blockg t.. Forkg Custoer leavg a queue cloes to ultple custoers gog alog dfferet routes. 4. Jog ultple streas of custoers are cobed to a sgle strea. Forkg ad jog are used odels of parallel processg systes.. Ope etworks The splest type of etwork s a ope etwork wth the followg assuptos (called Jackso Network Let Assue arbtrary etwork of queues Servce te of queue s expoetally dstrbuted wth rate Arrvals fro outsde the etwork to queue are a osso process wth ea rate γ. r j routg probablty that a custoer copletg servce at queue goes to queue j. r routg probablty that a custoer copletg servce at queue leaves the etwork. Note r j j Let be the total ea custoer arrval rate to queue. µ.

3 r γ γ ( r r r µ r r r ad µ We ca see that at each queue γ r j j j Note that the flow coservato equato holds ay ope etwork regardless of arrval ad servce dstrbutos. Ths equato ca easly be solved for. Let [ ] γ [ γ γ γ ] R j R does ot clude r [ r ] j The flow coservato equato ca be wrtte atrx vector for as slovg γ ( I - R R γ - γ ( I - R Now cosder queue the Jackso etwork fro the aalyss of the sgle // queue we kow. ergg of depedet osso processes s osso wth rate equal to the su of the dvdual rates. That s osso process.

4 . The departure process of a // queue s osso wth rate equal to put rate of queue.. robablstc splttg of a osso process results a osso process. p p p p p p Cobg these results we ca see that the put ad output processes of each queue the etwork s osso process. Let ~ ( t be the uber of custoers the syste at queue at the te t. The state of the etwork s defed by the vector ( ~ ( t ~ ( t ~ ( t assuptos above the process {( ~ ( t ~ ( t ~ ( t t 0} uder the s a desoal arkov process. Steady state ca be detered by lettg ( deote steady state probablty. Let l { ~ ( t ~ ( t ~ ( t } ( tf ad - l { ~ ( t ~ ( t ~ ( t - ~ ( t } ( tf decrease by the th queue Slarly { ~ ( t ~ ( t ~ ( t ~ ( t } ( l tf crease by the th queue Wrtg the steady state flow balace equato

5 rate to state rate out of state ( ( ( ( r r j j j j ß ø Œ Œ º Ø - - µ µ µ γ Frst ter o the left had sde s trasto fro - to caused by exteral arrval. Secod ter o left had sde s trasto fro to by departure of custoer fro etwork. Thrd ter o left had sde s trasto fro to by departure fro the th queue wth to the th queue wth -. Rght had sde all ways ca leave state wth arrval or departure. The flow balace equato above ca easly be verfed by costructg the state trasto dagra for the two queues case (.e.. The soluto to the steady state flow balace equato s the well kow roduct For. C C ( L where µ ad C s a costat. rove by substtutg C ( to the steady state flow balace equato ad show that t s a soluto. C s detered by C L L results - C ( Hece - ( ( for stablty " < ; essetally the product for of depedet // queues steady state probabltes π ( - (// steady state

6 ( π Fro the state probabltes oe ca easly detere perforace easures. ow that each queue s a // queue wth L - W (all // easures apply µ - W q µ - π ( - etc For the etwork as a whole let LN Average uber of custoers etwork. LN L - γn total average load o etwork. γ N γ WN Average delay through etwork. WN LN W γ N γ N - N γ Note that applyg ths soluto to packet swtched etworks µc where µ average packet legth C capacty of lk ca also add a deterstc delay d j correspodg to the te t takes a custoer to ove fro the th queue to the jth queue (propagato delay ad stll get Jackso etwork as above oly WN chages. Ø WN ŒW N γ Œ º ø rj dj j ß Exaple: Three ode etwork show below assug osso exteral arrvals ad expoetal servce at each queue.

7 γ µ γ µ µ γ Gve γ 0.5 γ 0.5 γ 0.5 γ N γ µ µ µ Fro the dagra r 0.4 r 0.6 r 0.5 r 4.0 r Solvg the flow coservato equato for γ [ ] R Ø0 Œ Œ 0 Œº ø 0 0 ß - γ ( I - R usg atlab to solve results [ ] µ ; < ;" stable syste WN.646 γ N - ay slght geeralzatos to ope Jackso etworks exst. The ost wdely kow are BC etworks. BC etworks also have a product for : ( C However for of C depeds o the syste odel. Soe of the addtoal features that ca be odeled clude: ultple classes of jobs state depedet expoetal servers

8 coxa servce dstrbuto wth uber of servers etc. A good dscusso ca be foud Ja s textbook.. Networks wth populato costrats Cosder queue syste results oly for sple cases. Custoers arrve fro outsde the etwork accordg to a osso process wth rate to queue ad expoetal servce dstrbuto wth rate µ at queue. Sple exaple: output queues at a output buffer of a packet swtch. µ µ µ As ope etwork case study ( ~ ( t ~ ( t ~ ( t. Ths process s a fte state space desoal arkov process wth state space S ( : 0 B " ; The steady state probablty B { ~ ( t ~ ( t ~ ( t } ( l tf aga has a product for ( where G s the oralzato costat foud by G ( G S S I geeral G ust be detered uercally. Ths becoes dffcult whe S s large. Fro ( oe ca detere varous ea perforace easures. L Average uber of custoers queue.

9 L B j ( j 0 Ł j; S ł LN L Exaple: B State dagra ( : S { ( ; 0 0 < } µ µ Q µ µ µ µ µ µ µ µ µ µ Q For ths exaple ( G We ca wrte out G G G Let 0.5 µ µ 0.5 G 6.5 ( (0.5 ( 6.5 ( L ( j 0 Ł j ; S ł L Slarly L.65 LN.747 ( (0 ( ( ( (0 ( (0

10 For a good dscusso of addtoal geeralzatos see S. La ad J. Wog Queueg Network odels of acket Swtchg Networks art : Network wth opulato Sze Costrats erforace Evaluato vol. 98 pp Note that the coputato of G s dffcult for large S. The other a applcato of etworks wth populato costrats s crcut swtched etworks where a swtch ca be odeled as a ultrate loss syste. Cosder a sgle lk a oder crcut swtched etwork lke ISDN. Varous servces are offered ad each servce has dfferet characterstcs (call arrval rate holdg te badwdth. Assue types of coectos each type arrves accordg to a osso process rate each type coecto holdg te expoetally dstrbuted wth rate µ (results hold for geeral holdg te. Each type coecto requres basc uts of badwdth. The total badwdth avalable s C uts. Let ~ ( t uber of type coecto syste at te t. The tuple ( ~ ( t ~ ( t ~ ( t defes a desoal arkov process wth fte state space S where 0 º C / ß ad C The exaple of the state trasto dagra for the case of C 0 s show below

11 The steady state probabltes { } t t t t f ( ~ ( ~ ( ~ l ( have the product for as before whe µ k G! ( ( whe get Erlag B odel /G/C/C ad S k G! ( aly terested coecto blockg rates B ( S B where type blocked su over states where < - j j j C For state trasto exaple (00 (90 (8 (7 (6 (5 (4 ( (4 (4 (05 B

12 B (05 (4 (4 (6 (8 (00 Nuercal exaple: C 48 k voce 64 bps k H ISDN vdeo 84 bps µ µ 0.5 offered load 0 µ B B Closed Queueg Networks Splest case custoers crculatg aog queues. Each queue has expoetally dstrbuted servce te µ. The routg probablty for a custoer copletg servce at queue to go to queue j s r j ad r j j As before state of etwork defed by ( ~ ( t ~ ( t ~ ( arkov process. The state space S s detered by S ( : 0 " ; For exaple etwork below t whch s desoal µ µ ( state dagra µ µ µ µ µ µ

13 Aga cosder steady state probabltes { ~ ( t ~ ( t ~ ( t } ( l tf Flow balace equato steady state rate rate out rj µ j ( j - µ j Ł ł ( The left had sde s the sae ope etwork except reovg ters for exteral arrvals ad departures. Notce that does ot appear the equato or the state dagra of the exaple above. The soluto of the flow balace equato s oce aga a product for wth ( where G( µ ad G ( s a oralzato costat so that s gve by S G( S I order to detere G ( ad ( eed ;" Flow coservato equato s r j j sae as ope etwork case wthout exteral arrvals or departures. j rocedure s to arbtrarly set or µ ad detere ; > relatve to value. Chagg value wll result chage of G (. ( wll rea costat. For exaple cosder the tade queue odel wth. Custoer wth µ µ µ µ Fro the dagra r r

14 State space S { (0 ( ( (0 } G ( G ( S choosg 0.5 G (.875 ad results ( G( (0 / G( ( / G( 0. ( / G( (0 / G( 0. 5 To llustrate the arbtrary value for let G ( 0.5 as before whe Fro ( oe ca copute the stadard ea perforace easures L j ( ; ote L j 0 Ł j; S ł Fro the exaple above L ( ( (0.667 L ( ( (0 0.7 Note that to fd W oe eeds to fd the effectve arrval rate e µ Ł - ( 0; S ł The effectve server utlzato For the two queues exaple above ( - ( e e - ( µ Ł 0; S e µ e 0. 9 ł ( - ( e µ e

15 Note that oe ca also fd e by frst deterg e the usg flow coservato equato. W W L / e L / e As the case of etworks wth populato costrats the coputato of G ( s dffcult whe the state space becoe large. Ca show that for a closed etwork of queues wth custoers the uber of states s gve by Nuber of states - Ł - ł For eve sall etworks ths s large. For exaple states Several algorths have bee proposed for coputatoal effcecy deterg G (. Oe popular techque s Buze s algorth (also called covoluto algorth. Buze developed sple algorth by otg G( G( - G( - roof s show Ja s text book. Ths ca be coputed recursvely by otg G ( 0 k G( k k Ths ca be coputed a sple tabular for 0 Ø Œ Œ Œ Œ Œ Œº ( ø ß The j eleet the table s coputed by takg the (j- eleet addg ( - j eleet For the two queue exaple prevously dscussed.

16 Oe of the advatages of ths techque s that the perforace easures ca be wrtte ters of G ( k L G( - k G( k e G( - G( k G( - k ( k G( Exaple : Cosder the sple odel of a coputer syste show below queue the CU queue dsk drve ad queue I/O. µ µ µ Fro the dagra r 0. r 0.6 r 0. r r µ µ µ Choosg 0 6 ad. Coputg G(4.

17 G( e G( Slarly e.665 ad k L G(4 G(4 G(4 k G L Slary L ad L. 65 W L / e W e - k [ G( G( G ( (0] Let s look at two rather typcal approxato approaches. The frst s wdely used for packet etworks ad the secod for crcut swtched etworks.. Whtt s ethod for ope s etwork of G/G/ queue (QNA The basc dea s to use the LB G/G/ two oet approxato at each queue the etwork. The odel of queue s slar to the arbtrary queue studed Jackso etworks. Assue arbtrary etwork of queues defe Total ea custoer arrval rate to queue. γ ea arrval rate fro outsde of etwork to queue. r j Routg probablty custoer leavg queue goes to queue j.

18 r robablty custoer leavg queue exts the etwork. µ ea servce rate at queue. Co Squared coeffcet of varato of outsde arrvals to. Cs Squared coeffcet of varato of servce process at. C A Squared coeffcet of varato of arrval process at. r γ Co γ ( r r r C A µ Cs r r r As the Ope Jackso etwork case fd ea arrval rate at each queue by the flow coservato equato γ r j j j - γ ( I - R atrx vector soluto To apply LB equato eed approxatos (parallel to Jackso etwork approach for. Departure process approxato C A at each queue. Ths requres the applcato of three Ca µ Cs ea departure rate C d squared coeffcet of varato of departure process. Cd Cs ( - CA» based o reewal process approxato

19 . Splttg: If a process wth ea ad C s probablstcally splt to strea wth probabltes p ( p. We ca approxate C as below p p C C C p k k Ck Where p C» p C ( - p. ergg: The C of a erger of streas s approxated by C C C µ C s k C k ø [ rj ( jcsj ( - j CAj ( - rj ] Ø γ - Œ rj CA W W Co Œ º j yelds a syste of lear equatos to solve for where C A ß W - Ø ( ø Œ rj 4( - - Œ º Ł j łß

20 Ths approxato teds to do pretty well o etwork-wde easures LN WN etc but ot so well for dvdual queues.