Open and Closed Networks of M/M/m Type Queues (Jackson s Theorem for Open and Closed Networks) Copyright 2015, Sanjay K. Bose 1

Transcription

1 Ope ad Closed Networks of //m Type Qees Jackso s Theorem for Ope ad Closed Networks Copyrght 05, Saay. Bose

2 p osso Rate λp osso rocess Average Rate λ p osso Rate λp N p p N osso Rate λp N Splttg a osso process probablstcally as radom, probablstc rotg leads to processes whch are also osso atre. Copyrght 05, Saay. Bose

3 λp Q λp λ Q λ p -p Q3 Uder eqlbrm codtos, average flow leavg the qee wll eqal the average flow eterg the qee. λ-p λ-p Rotg robabltes are p ad -p For //m/ qees at eqlbrm, Brke s Theorem assres s that the departre process of obs from the etwork wll also be osso. From flow balace, the average flow rate leavg the qee wll also be the same as the average flow rate eterg the qee. Copyrght 05, Saay. Bose 3

4 osso rocess Average Rate λ osso rocess Average Rate λ +λ osso rocess Average Rate λ Combg depedet osso processes leads to a process whch wll also be osso atre. Copyrght 05, Saay. Bose 4

5 Example: λ Q A Acyclc Feedforward Network of //m Qees λ Q Q Exteral arrvals wth rates λ ad λ from osso processes Q3 0.4 robablstc rotg wth the rotg probabltes as show Copyrght 05, Saay. Bose 5

6 Applyg flow balace to each qee, we get λ Q Average ob arrval rate for Q λ λ Q Average ob arrval rate for Q 0.4λ+λ λ Q3 Average ob arrval rate for Q3 0.4λ λ Q4 Average ob arrval rate for Q4 0.84λ+λ Brke s Theorem ad the earler qoted reslts o splttg ad combg of osso processes mply that, der eqlbrm codtos, the arrval process to each qee wll be osso. ve the mea servce tmes at each qee ad sg the stadard reslts for //m qees, we ca the fd the dvdal state probablty dstrbto for each of the qees Ths process may be doe for ay system of //m qees as log as there are o feedback coectos betwee the qees Copyrght 05, Saay. Bose 6

7 It shold be oted that ths aalyss ca oly gve s the state dstrbtos for each of the dvdal qees bt caot really say what wll be the ot state dstrbto of the mber of obs all the qees of the etwork. Jackso s Theorem, preseted sbseqetly, s eeded to get the ot state dstrbto. Ths gves the smple, ad elegat reslt that -,, 3, 4 p p p 3 3 p 4 4 Q Q Q Q rodct Form Solto for Jot State Dstrbto of the Qeeg Network Copyrght 05, Saay. Bose 7

8 Jackso s Theorem for Ope Networks Jackso s Theorem s applcable to a Jackso Network. Ths s a arbtrary ope etwork of //m qees where obs arrve from a osso process to oe or more odes ad are probablstcally roted from oe qee to aother tl they evetally depart from the system. The departres may also happe from oe or more qees The //m odes are sometmes referred to as Jackso Servers Jackso s Theorem states that provded the arrval rate at each qee s sch that eqlbrm exsts, the probablty of the overall system state. for qees wll be gve by the prodct-form expresso,..., p Q Copyrght 05, Saay. Bose 8

9 Jackso Network: Network of //m qees, arbtrarly coected Exteral Arrval to Q : osso process wth average rate Λ At least oe qee Q mst be sch that Λ 0. Note that Λ 0 f there are o exteral arrvals to Q. Ths s becase we are cosderg a Ope Network. Closed Networks are cosdered later. Rotg robabltes: p {a ob served at Q s roted to Q } p {a ob served at Q exts from the etwork} Copyrght 05, Saay. Bose 9

10 Arrval rocess of Jobs to Q [Exteral Arrvals, f ay, to Q ] + Jobs whch fsh servce at Q ad are the roted to Q for the ext stage of servce Let λ Average Arrval Rate of Jobs to Q {exteral ad reroted} ve the exteral arrval rates to each of the qees the system ad the rotg probabltes from each qee to aother, the effectve ob arrval rate to each qee at eqlbrm may be obtaed by solvg the flow balace eqatos for the etwork. Copyrght 05, Saay. Bose 0

11 Flow Balace Codtos at Eqlbrm mply that - λ Λ + λ p for,.., 5. For a Ope Network, at least oe of the Λ s wll be ozero postve The set of eqatos 5. ca therefore be solved to fd the effectve ob arrval rate to each of the qees, der eqlbrm codtos. The etwork wll be at eqlbrm f each of the qees are at eqlbrm. Ths ca happe oly f the effectve traffc offered to each qee s less tha the mber of servers the qee..e. ρ λ /µ < m,.., where m s the mber of servers Q. Copyrght 05, Saay. Bose

12 For a etwork of ths type wth //m/ qees.e. Jackso Servers at each ode, Jackso's Theorem states that provded the arrval rate at each qee s sch that eqlbrm exsts, the probablty of the overall system state,..., wll be - ~,..., p 5.4 wth p { cstomers Q } Ths dvdal qee state probablty may be fod by cosderg the //m/ qee at ode solato, wth ts total average arrval rate λ, ts mea servce tme /µ ad the correspodg reslts for the steady state //m/ qee Copyrght 05, Saay. Bose

13 Stablty reqremet for the exstece of the solto of 5.4 s that - For each qee Q,.., the etwork, the traffc offered shold be sch that ρ λ µ < m where m s the mber of servers the //m/ qee at Q Copyrght 05, Saay. Bose 3

14 Implcatos of Jackso s Theorem - extesos ad geeralzatos cosdered sbseqetly Oce flow balace has bee solved, the dvdal qees may be cosdered solato. The qees behave as f they are depedet of each other eve thogh they really are ot depedet of each other ad the ot state dstrbto may be obtaed as the coted prodct of the dvdal state dstrbtos prodct-form solto The flows eterg the dvdal qees behave as f they are osso, eve thogh they may ot really be osso atre.e. f there s feedback the etwork. Note that Jackso s Theorem does reqre the exteral arrval processes to be osso processes ad the servce tmes at each qee to be expoetally dstrbted atre wth ther respectve, dvdal meas. Copyrght 05, Saay. Bose 4

15 erformace easres Total Throghpt λ Λ 5.5 Average traffc load at ode.e. Q λ ρ 5.6 µ Vst Cot to ode λ V 5.7 λ The vst cots may also be obtaed by drectly solvg the followg lear eqatos - V Λ λ + V p,, 5.8 Scaled Flow Balace Eqatos Copyrght 05, Saay. Bose 5

16 Iterpretato of the Vst Rato V : Average mber of tmes a ob wll vst Q every tme t actally eters the ope qeeg etwork. Usefl to calclate trast soor tmes from dfferet etry pots the etwork Average mber of obs at ode N kp k 5.9 k 0 Average mber of obs system N N 5.0 ea Soor Tme W : The mea total tme spet the system by a ob before t leaves the etwork. N N W ad also λ λ N W λ W VW λ λ Copyrght 05, Saay. Bose 6

17 Whe does the rodct- Form Solto hold? The prodct-form expresso for the ot state probabltes hold for ay ope or closed qeeg etwork where local balace codtos are satsfed. Some other reslts dcate that ths type of solto also hold for somewhat more geeral codtos. Copyrght 05, Saay. Bose 7

18 Specfcally, ope or closed etworks wth the followg types of qees wll have a prodct-form solto -. FCFS qee wth expoetal servce tmes. LCFS qees wth Coxa servce tmes 3. rocessor Sharg S qees wth Coxa servce tmes 4. Ifte Server IS qees wth Coxa servce tmes A Coxa servce tme has a dstrbto of the followg type - L µ LB s γ + ββ... β γ + s µ + wth β γ for L ad γ L+ Copyrght 05, Saay. Bose 8

19 Example: Ope Jackso Network λ λ λ λλ p λ Q // Q // λ Servce Rate of Qµ Servce Rate of Qµ p p p +p λ λ p, ρ ρ ρ ρ λ λ p λ ρ µ p N ρ ρ p λ λ p λ p ρ µ p ea Nmber the Qees N ρ ρ ea Soor Tme W N λ ρ ρ + λ ρ λ ρ Copyrght 05, Saay. Bose 9

20 Example λ + λ p λ r + λ p + λ p r r λ µ Q // r -p λ p µ Q // p p -p r3 5p λ p p r3 5p ρ 4 µ p p Jot robablty Vst Ratos r3 p λ p p, ρ ρ V 3 5p 6 p p r3 p ρ µ p p V ρ ρ 3 p 6 p p ea Nmber Q, Q ad the system ea Soor Tme W N 3r N ρ ρ N ρ ρ N + ρ ρ ρ ρ Copyrght 05, Saay. Bose 0

21 Example λ N Q // Q // Q3 // Q4 // ,, 3, 4 ρ ρ ρ 4 for ρ ρ µ µ µ µ ,, 3, 4 N N 3 ρ ρ N 4 ρ ρ 4 ρ 4 0.λ + 0.λ3 + λ λ λ λ3 0.4λ λ4 0.λ + 0.6λ + 0.6λ ~ λ λ, λ, λ, λ λ,0.476λ,0.476λ,0.8095λ ~ ρ ρ, ρ, ρ, ρ ρ 3.905ρ,0.954ρ,0.954ρ,.690ρ + 3 ρ Note that as λ creases, system evetally becomes stable becase ρ 4 Ths, therefore, mples that ρ 0 <0.676 or λ<0.676μ for the qeeg etwork to be stable. 0 0 Copyrght 05, Saay. Bose

22 Example a Codto for etwork to be stable b Trast Delay throgh the system for ay ob arrval for Λ0.5, μ c Trast Delay throgh the system for ob eterg at A for Λ0.5, μ Copyrght 05, Saay. Bose

23 Solvg Flow Balace, we get - λ.75λ λ.5λ λ.375λ λ.375λ 3 4 ad Λ Λ Λ Λ ρ.375 ρ.5 ρ3.375 ρ4.875 µ µ µ µ Sce all the qees are sgle-server qees, System Stable f Λ.375 < or Λ< 0.77µ µ Copyrght 05, Saay. Bose 3

24 For Λ0.5, μ ρ ρ 0.65 ρ ρ N. N.667 N. N N 7.59 ad N 7.59 W Λ W s the trast tme throgh the system averaged over all obs that eter the system,.e. throgh A or throgh B Copyrght 05, Saay. Bose 4

25 For Λ0.5, μ N N N N W.6 W.667 W 3. W λ λ λ3 λ4 are the delays throgh the dvdal qees Q-Q4 To fd delays for obs eterg from oe partclar etry pot the etwork, we eed to fd how may tmes o the average sch a ob wll vst each qee the etwork. Ths wold reqre fdg the vst ratos to each qee wth oly that flow preset set other flows to zero. Copyrght 05, Saay. Bose 5

26 Cosderg the system wth arrvals comg oly from A, we get - λ.5λ λ 0 λ.5λ λ.5λ 3 4 Vst Ratos.5Λ.5Λ.5Λ V.5 V 0 V V Λ Λ Λ Therefore, Trast Delay for ob eterg at A.5x x x Copyrght 05, Saay. Bose 6

27 Extesos to Jackso s Theorem for Ope Networks [A] Jackso's Theorem wth State depedet Servce Rates at the Qeg Nodes..e. qees wth mltple servers For ths, assme that the servce tmes at Q are expoetally dstrbted wth mea /µ m whe there are m cstomers Q st before the departre of a cstomer. [B] Qeg Networks wth ltple Cstomer Classes For ths, we eed to assme that the servce tme dstrbto at a ode wll be the same for all classes eve thogh they may dffer from oe ode to aother. The servce tmes may be state depedet. The exteral arrval rates ad rotg probabltes wll vary from oe class of cstomers to aother Copyrght 05, Saay. Bose 7

28 Closed Qeeg Networks qees - Q,..., Q the qeeg etwork obs of the same class crclatg the etwork p s the rotg probablty from Q to Q probablstc rotg Sce etwork s a closed etwork p,..., No arrvals from otsde ad o departres from the etwork Flow balace codtos for ths etwork may stll be wrtte as λ λ p,..., 5. Copyrght 05, Saay. Bose 8

29 The eqatos of 5. are ot depedet. Hece, they caot be solved to qely fd the λ s for the qees,,..., Usg ay - eqatos of the eqatos 5., we ca however fd the λ s p to a mltplcatve costat For ths, assme that α s a kow scalar qatty ad let {λ * },..., be a partclar solto of 5. sch that the tre average arrval rates {λ },..., are gve by λ αλ *,..., 5.3 α ad {λ },.., are both fctos of the poplato sze of obs crclatg the closed etwork However, {λ * },..., wll be depedet of Copyrght 05, Saay. Bose 9

30 A alterate, bt eqvalet approach wold be to do the followg - Choose ay qee the etwork say Q as the referece qee ad assme that λ * α Ay vale of α may be chose! A coveet choce s αµ so that ρ λ * /µ Solve the flow balace eqatos of 5.3 to obta the relatve throghpts λ *,λ 3*,...,λ * terms of α. Copyrght 05, Saay. Bose 30

31 Assmg the servce tmes to be expoetally dstrbted recall that we are assmg //m type qees, we ca allow the actal servce rates at each qee to be state depedet µ m servce rate at Q whe Q s state m expoetal servce tmes assmed Usg the relatve throghpts {λ * },..., fod earler, we defe the relatve tlzatos { },..., as - m λ *,., m, 5.4 µ m Copyrght 05, Saay. Bose 3

32 Copyrght 05, Saay. Bose 3 Let Nmber qee Q,., State robablty Vector,..., ~ sch that Total mber of obs Jackso s Theorem for Closed Networks of //- Type Qees ˆ,...,, ~... 0 ˆ where ad ˆ... ˆ ˆ Normalzato Costat 5.7

33 Copyrght 05, Saay. Bose 33 Example Q // Q // q p -q -p µ µ Closed Network wth obs Choose λ * µ The p q p q * * µ λ λ µ µ p q,, + 0 Normalzato Costat

34 Example cot. -p p Q // µ µ Q // q -q Closed Network wth obs {Q s bsy} -0, {Q s bsy} -, 0 Copyrght 05, Saay. Bose 34

35 Vst Ratos The vst rato V of the th qee Q the qeeg etwork s defed as the mea mber of tmes Q s vsted by a ob for every vst t makes to a gve referece qee, say Q. Note that the defto s bascally the same as for a ope etwork. Wth Q as the referece qee, V * λ * λ,., The same reslt wll be obtaed by settg V ad solvg ~ ~ ~ ~ ~ the eqatos V V wth V V,..., V ad [ ] p Copyrght 05, Saay. Bose 35

36 Jackso s Theorem for Closed Networks of lt-server Qees expoetal servce qees the closed etwork wth probablstc rotg gve by { p,,..., µ m µ µ Q has s servers m m, s,, µ servce rate of a sgle server at Q µ m overall state depedet servce rate at Q whe t has a total of m obs watg ad -servce Defe λ * where λ * s the relatve throghpt for Q µ Copyrght 05, Saay. Bose 36

37 Copyrght 05, Saay. Bose 37 Jackso s Theorem for a Closed Network of lt-server Qees Usg these,..., ~ β sch that s s s s s >!! β ad β

38 These expressos may be wrtte a smpler form for a Closed Network of Sgle Server Qees wth Expoetal Servce Closed Network of Sgle Server Qees wth Expoetal Servce Tmes ~,..., sch that Copyrght 05, Saay. Bose 38

39 The maor comptatoal dffclty wth fdg the state probablty dstrbto of a Closed Network s that of fdg the vale of the ormalzato costat Ths complexty creases rapdly wth larger etworks creasg vales of ad larger poplato of crclatg obs creasg vales of may be calclated drectly oly for very small etworks wth a very small mber of crclatg obs. For larger etworks, the Covolto Algorthm shold be sed to calclate. If mea performace parameters are desred rather tha the actal state probablty, the the ea Vale Algorthm may be drectly sed to fd these wthot fdg at all. Copyrght 05, Saay. Bose 39

40 Copyrght 05, Saay. Bose 40 For a closed etwork of sgle server qees wth obs crclatg *,...,...,..., ~ µ λ

41 Copyrght 05, Saay. Bose 4 The } { factorg ot Normalzato costat of the qeeg etwork wth fewer cstomers,.e. -

42 Copyrght 05, Saay. Bose 4 Therefore } { ad + } { } { } { argal dstrbto of the th qee Note that } { } { ad therefore E } { ea mber Q

43 Copyrght 05, Saay. Bose 43 Note that the departre rate from Q wll always be µ wheever Q has oe or more obs. Therefore, the actal throghpt λ of Q wll be gve by } { µ µ λ The actal tlzato ρ of Q wll the be } { or ρ ρ µ λ ρ

44 Copyrght 05, Saay. Bose 44 Example Q // Q // q p -q -p µ µ obs the system For Q, choose * µ λ p q p q p q + * * * * * µ λ λ λ λ λ Therefore, µ µ p q State robablty Dstrbto at eqlbrm,, Normalzato Costat + 0 {Q s bsy} -0, {Q s bsy} -,0 p q V V 0 N N N + + ad,

45 Example obs the system μ Q // p -p Q3 // μ 3 μ Q // λ * µ For Q, choose the * * λ pµ λ3 p µ Therefore, ad, pµ p µ 3 µ µ 3 V V p V p 3 State robablty Dstrbto at eqlbrm + It wold be easer to se VA or the Covolto Algorthm to calclate or the qee parameters rather tha tryg to compte them drectly. These algorthms are dscssed ext wth,, 3 0,..., ,, 0,..., Normalzato Costat Copyrght 05, Saay. Bose 45