EP2200 Queueg theory ad teletraffc systems Queueg etworks Vktora Fodor
Ope ad closed queug etworks Queug etwork: etwork of queug systems E.g. data packets traversg the etwork from router to router Ope ad closed etworks Ope queug etwork: customers arrve ad leave the etwork (typcal applcato: data commucato) Closed queueg etworks: ad out flows are mssg costat umber of customers crculate the etwork (applcato: computer systems) γ p q q 2 q 3 p 2 p 23 γ 2 q 4 p 34 2
Ope queug etworks- A tadem system q q 2 Posso() Exp(µ ) Exp(µ 2 ) The most smple ope queug etwork Assume a Posso arrval process ad depedet expoetally dstrbuted servce tmes What s the departure process from queue? Iterdeparture tme: Customer leaves queue behd: tme of servce of ext customer Customer leaves empty system behd: tme to ext arrval + tme of servce L( f τ µ µ ( t)) ρ + ( ρ) s + µ s + s + µ 2 2 ρµ ( s + ) + µ ρµ s + + µ ( s + )( s + µ ) ( s + )( s + µ ) s + Departure process: Posso ()! Same for //m but ot for systems wth losses ad ot for /G/m systems 3
A tadem system q q 2 Posso() Exp(µ ) Exp(µ 2 ) Tadem system Queue s a // queue Departure process from Queue s Posso Thus Queue 2 s also a // queue State of the tadem queue: N( 2 ) p()p( 2 ) Jackso theorem: the etwork behaves as f set of depedet queues that s: p( 2 ) p( )p( 2 ) Proof: see Vrtamo otes 4
odelg commucato etworks - ote o the depedece assumpto terarrval tme packet sze correlato! The product form p( 2 ) p p 2 apples oly f the arrval ad servce processes are depedet For two trasmsso lks seres queue 2 s ot a //-queue Correlato betwee servce tmes of a customer the two queues determed by the packet legth ad the lk trasmsso rate Correlato betwee arrval ad servce tmes For two cosecutve packets the terarrval tme at the secod queue ca ot be smaller tha the servce (that s trasmsso) tme of the frst packet at the frst queue E.g. there wll ot be ay queug queue 2 f the trasmsso rate at queue 2 s larger Product form soluto does ot apply 5
odelg commucato etworks - ote o the depdece assumpto Klerock s assumpto o depedece Traffc to a queue comes from several upstream queues Superposto of Posso processes gve a Posso process Traffc from a queue s spread radomly to several dowstream queues Partal processes are Posso wth testy p (Σ p ) It s assumed to create depedet arrval ad servce processes Product form soluto apples E.g. etwork of large routers 6
Ope Jackso s queug etworks where the product form works Ope queug etwork arrvals to the etwork from all arrval pot a departure pot s reachable queues wth fte storage ad m expoetal servers Eve fte storage f last queue the etworks Customers from outsde of the etwork arrve to ode as a Posso process wth testy γ 0 The servce tmes are depedet of the arrval process (ad servce tmes other queues) A customer comes from ode to ode after servce wth the probablty p or leaves the etwork wth the probablty p 0 - p. Note t allows feedback (e.g p ). The arrval process ot Posso aymore but the queue behaves as f the arrval would be Possoa. Network s stable f all the queues are stable. γ p q q 2 q p p 3 2 23 p 34 γ 2 q 4 7
Ope Jackso s queug etworks Flow coservato: arrval testy to ode s γ + p Jacksos theorem: The dstrbuto of umber of customers the etwork has product form queues behave as depedet //m queues! (we do ot prove same as for tadem queues) p ( ) p ( ) p ( ) 2 γ p q q 2 q p p 3 2 23 p 34 γ 4 q 4 8
Ope Jackso s queug etworks Flow coservato: arrval testy to ode : γ + Example : sgle feedback queue p p γ µ Performace measures as f t would be // Though the arrval process s ot Posso Stablty: / µ < γ + p γ p ρ µ p( k) k ( ρ) ρ 9
Ope Jackso s queug etworks Arrval testy ad state probablty + γ p ( ) ( ) ( ) P 2 For the // case: P P ( ) ( ρ ) ρ ad ρ / < P µ Example 2 calculate arrval testes gve the stablty rego the possble arrval rates whe the etwork s stable calculate the probablty that the etwork s empty calculate the probablty that there s oe customer the etwork 0
Lttle s theorem apples to the etre etwork! Good because T s hard to calculate f there are feedback loops. The mea umber of customers the etwork ad the average tme spet the etwork are (e.g. // case) The mea umber of odes a customer vsts before leavg: {Sum arrval testy to the queues} / {arrval testy to the etwork} N T N N / γ ρ ρ + p V / γ γ Ope Jackso s queug etworks ea performace measures
Closed Jackso s queug etworks Not exam materal ths year Closed queug etwork queues wth fte storage ad m expoetal servers K customers crculatg the etwork o arrvals ad departures The servce tmes are depedet of the arrval process (ad servce tmes other queues) A customer comes from ode to ode after servce wth the probablty p Queues ca ot be depedet sce there s a fxed umber of customers p q q 2 q p p 3 2 23 p 34 p 42 p 4 q 4 2
3 Flow coservato: arrval testy to ode the problem s that oe of the -s are kow: Lmted set of states sce the sum of the customers s costat K: C based soluto: state: vector of umber of customers per queue - complex Algorthmc soluto e.g. // (*) gves a set of depedet equatos wth soluto of e.g.: we have to select the oe that gves sum of etwork state probabltes equal to oe Gordo-Newell: state probabltes wthout calculatg arrval testes (wthout proof) ) ( p Closed Jackso s queug etworks ( ) } 0 { 2 K S }... { }... { 4 3 2 4 3 2 e e e e α S K K e G e G P ) ( µ µ
Summary Queug etworks: set of queug systems customers move from queue to queue Appled to etworkg problems: depedece of queues have to be esured Ope queug etworks Burke: Output process of a //m queue s Possoa Jackso theorem: etwork state probablty has product form f //m queues Closed queug etworks ot exam materal Number of customers costat State of queues s depedet Gordo-Newell ormalzato 4