APPENDIX A: ELEMENTS OF QUEUEING THEORY

Transcription

1 APPENDIX A: ELEMENTS OF QUEUEING THEORY I a pacet rado etwor, pacets/messages are forwarded from ode to ode through the etwor by eterg a buffer (queue) of a certa legth each ode ad watg for ther tur to be trasmtted to the ext ode. Also, call access to a etwor wth a gve capacty s modeled as a process wth certa dstrbuto of call arrval tmes ad certa dstrbuto of call servce tmes. A umber of dfferet classes of queueg models are used ths feld to aalyze queueg delays or server (or etwor access) blocg probabltes. I order to geeralze the presetato the elemets the queue (pacets, messages, calls, etc.) wll be referred to as customers. The problems owadays become eve more challegg wreless cellular etwors wth hadoffs, multmeda traffc ad dfferetated QoS. The ey elemet ths aalyss s a proper model for these dstrbutos. The very frst results ths feld are based o the most commo stochastc queueg models whch assume that message (call) terarrval ad servce tmes obey the expoetal dstrbuto or, equvaletly, that the arrval rate ad servce rate follow a Posso dstrbuto. I other words, f the arrval rate s λ, the probablty of havg arrvals a terval t s gve as p ( t ) = [(λt) /!]exp( λt). Process wth such expoetal dstrbuto also has Marova or memoryless propertes. I other words, the probablty that a customer curretly servce has t uts of remag servce s depedet of how log t has already bee servce. The state of the queue s characterzed by the state probablty p () t = Pr( pacets the queue at tmet ). I more geeral terms ths s also referred to as populato of sze at tme t. Owg to Marova assumpto, stataeous chages the system state ca oly amout to a crease (brth) or decrease (death) of oe hece the ame brth-death process. Therefore, the probablty of a brth occurrg a small terval of legth t, gve the system s state, s assumed to be λ + t o( t) (>0), ad smlarly a probablty of a death the same state s µ t + o( t), where µ / λ s the Advaced Wreless Networs: 4G Techologes Savo G. Glsc 006 Joh Wley & Sos, Ltd

2 servce/arrval rate of the process at state. As a startg pot let us see what would be the probablty that the system s state at tme t + t. Ths probablty ca be wrtte as p ( t + t) = p ( t )[ λ t][ µ t] + p ( t )[ λ t][ µ t] + p + ( t )[ λ + t][µ + t] + p ( t )[ λ t][ µ t] + o( t); (C.) For =0 we have p 0 (t + t) = p 0 ( t )[ λ 0 t] + p ( t )[ λ t][ µ t] + o( t) (C.) By movg p ( t ) from the rght- to the left-had sde, dvdg through by t ad tag the lmt as t 0 we have dp () t = (λ + µ ) p () t + µ + p + () t + λ p (); t dt dp 0 () t = λ 0 p 0 () t + µ p ( t) dt The statoary soluto s obtaed for dp ( t ) / dt = 0 resultg 0 = (λ + µ ) p + µ + p + + λ p ; 0 = λ p + µ p 0 0 Equato (C.4) ca be rewrtte as (C.3) (C.4) p + = λ + µ p λ p ; µ + µ + (C.5) λ 0 p = p 0 µ I a specal case whe λ = λ ad µ = µ we have 0 = (λ + µ) p + µ p + + λ p ; 0 = λ p + µ p 0 0 ad p + = λ + µ p λ p ; µ µ λ p = p 0 µ (C.6) (C.7)

3 I order to dscuss the elemets of traffc modelg for advaced wreless commucato systems, we wll frst revew the basc results of covetoal queueg theory begg wth the classcal Posso put (Marova-M), expoetal-servce (M), sgle-server () M/M/ queue. C The M/M/ model The desty fuctos for the terarrval tmes ad servce tmes for the M/M/ queue are gve, respectvely, as at () = λe λt (C.8) bt () = µe µt where / λ s the mea terarrval tme ad / µ the mea servce tme. Iterarrval tmes, as well as servce tmes, are assumed to be statstcally depedet. Sce terarrval ad servce tmes are expoetal, ad arrval ad codtoal servce rates Posso, we have Pr{a arrval occurs a ftesmal terval of legth t } = λ t +ο( t) ; Pr{more tha oe arrval occurs t } = ο( t) Pr{a servce completo t, gve system s ot empty} = µ t +ο( t);ad Pr{more tha oe servce completo t, gve more tha oe system} = ο ( t). resultg a brth-death process. There are a umber of ways to solve the brth-death equatos le the teratve method, by usg geeratg fucto or usg delay operator. We wll ot deal wth these methods explctly but rather expla them through dfferet examples below. By usg Equato (C.7) teratvely we get p = p 0 λ = p 0 (λ / µ ) ( ) (C.9) j= µ where p 0 s obtaed from the fact that probabltes must sum to, resultg = x λ p 0 (C.0) =0 µ 3

4 At ths stage we defe traffc testy or utlzato rate ρ as the rato λ / µ, for sgle-server queues ad get from Equato (C.0) p 0 = =0 ρ ρ s the geometrc seres + ρ + ρ + ρ that coverges to =0 /( ρ) for ρ < so that we have p = ( ρ )ρ for ( ρ = λ / µ < ) (C.) C Measures of effectveess The steady-state probablty dstrbuto for the system sze allows us to calculate the system's measures of effectveess. Two of mmedate terest are the expected umber the system ad the expected umber the queue at steady state. To derve these, let N represet the radom varable umber of customers the system steady state ad L represet ts expected value. We ca the wrte L E N = [ ]= p =0 =0 Cosder the summato = ( ρ ) p (C.) p = ρ + ρ + 3ρ = ρ(+ ρ + 3ρ +L) = ρ ρ. =0 = Sce ρ s the dervatve of ρ wth respect to ρ ad = =0 ρ =/( ρ) =0 we have ρ = d [/( ρ)] =/( ρ) = d ρ ad ρ ( ρ ) ρ λ L = = = (C.3) ρ µ λ ( ρ ) 4

5 If the radom varable umber queue steady state s deoted by N q ad ts expected value by Lq, the we have L q = ( )p = p p = L ( p 0 ) = ρ ρ = = = ρ L q = L ( p 0 ) holds for all sgle-chael, oe-at-a-tme servce queues, sce o assumptos were made the dervato as to the put ad servce dstrbutos. Thus the mea queue legth s ρ λ L q = = ρ µ µ λ ( ) (C.4) We are also terested the expected queue sze of oempty queues, whch we deote by L'q; that s, we wsh to gore the cases where the queue s empty. We ca wrte ' ' ' L q = E N q N q 0 = ( ) p = ( ) p = = ' where p s the codtoal probablty dstrbuto of the system ' gve the queue s ot empty, or p = Pr{ system }. From the laws of codtoal probablty, ' Pr{ systemad } p p = = ( ) Pr{ } p p p = ( ρ) ( = ρ) ρ ρ ' = The probablty dstrbuto {p } s the dstrbuto {p }ormalzed by omttg the cases = 0 ad. Thus L q ' = ( ) p = L p ( p 0 p ). = ρ ρ Hece ' L q = = µ. (C.5) ρ µ λ 5

6 The expected steady-state system watg tme W ad le delay Wq ca be foud easly from L ad Lq by usg Lttle's formulas, L = λw ad L q = λw q. I the case of the M/M/ queue, we have L ρ W = = = (C.6) λ λ( ρ) µ λ ad W q = L q ρ = λ λ( ρ) = ρ µ λ. (C.7) C3 Watg-tme dstrbutos We ow cosder the radom varable tme spet watg the queue T q ad ts cumulatve probablty dstrbuto W q (t). The frstcome, frst-served (FCFS) queug dscple s assumed. Watg tme s, for the most part, a cotuous radom varable, except that there s a ozero probablty that the delay wll be zero, that s, a customer eterg servce mmedately upo arrval. Hece we have W q ( 0 )= Pr{ T q 0}= Pr{T q = 0}= Pr{system empty at a arrval} = q 0. Let the codtoal probablty of the system gve that a arrval s about to occur be q. These probabltes are ot always the same as the p wth whch we have bee worg, sce the p are ucodtoal probabltes of the system at a arbtrary pot tme. To fd the dstrbuto of vrtual watg tme (.e. the tme a fcttous customer would have to wat were t to arrve at a arbtrary pot tme), we would use p. However, for Posso put, q = p, as we wll show later whe dervg q case where q p. Thus W q (0) = p 0 = ρ. for the trucated M/M/c/K It the remas to fd W q (t)for t > 0. Cosder W q (t), the probablty of a customer watg a tme less tha or equal to t for servce. If there are uts the system upo arrval, the order for the customer to go to servce at a tme betwee 0 ad t, all uts must have bee served by tme t. Sce the servce dstrbuto s memoryless, the dstrbuto of the tme requred for completos s depedet of 6

7 the tme of the curret arrval ad s the covoluto of expoetal radom varables, whch s a Erlag type dstrbuto. I addto, sce the put s Posso, the arrval pots are uformly spaced ad hece the probablty that a arrval fds the system s detcal to the statoary dstrbuto of system sze. Therefore we may wrte W ()= t Pr{T t} = q W 0 + q Pr completos t arrvalfoud system p q ( ) { } = t = ρ + ( ρ) ρ µ(µx) e µx dx = 0 ( )! (µ ) = ρ + ( ρ) µe dx = ρ + ρ( ρ ) µe t t µx xρ 0 = ( )! 0 = ρe µ ( ρ )t (t>0). (C.8). µx( ρ ) dx Smlarly, we ca get the probablty dstrbuto of the total tme (cludg servce) that a customer has to sped a M/M/ system. Deote ths radom varable by T, ts CDF by W(t), ts desty by w(t), ad ts expected value by W. We have already derved W, ad t ca further be show that W ( t ) = e (µ λ )t (t > 0) w( t ) = ( µ λ)e (µ λ )t (t > 0) (C.9) The dervato of Equato (C.9) very much follows that of the le delay dstrbuto except that + servce completos are requred tme t. C4 Queues Wth Parallel Chaels (M/M/c) Next we cosder the case whch there are c servers. I ths multserver M/M/c model each server has a depedetly ad detcally dstrbuted expoetal servce-tme dstrbuto, wth the arrval process aga assumed to be Posso. Sce the put s Posso 7

8 ad servce expoetal, we have a brth-death process. Hece λ = λ for all, ad t remas to determe µ pror to beg able to use the prevous results. If there are more tha c customers the system, all c servers must be busy wth each puttg out at a mea rate µ, ad the mea system output rate s thus equal to cµ. Whe there are fewer tha c customers the system, say < c, oly of the c servers are busy ad the system s puttg out at a mea rate of µ. Hece µ may be wrtte as µ µ = cµ ( ) ( < c), (C.0) c. Utlzg Equato (C.0) Equato (C.7) ad the fact that λ = λ for all, we obta λ p c!µ 0 ( < ) p = (C.) λ p ( c ) c c c!µ 0 I order to fd p0, we aga use the codto that probabltes must sum to, whch gves c p 0 = λ! µ + λ c c c!µ =0 =c If we ow use otato r = λ / µ, ad ρ = r / c = λ / cµ. we have c p = + 0 r r! c c c! =0 =c Now cosder the fte seres the above equato 8

9 c c c m r r r r r c = = = c c c! c! = c c c! m=0 c r c = c! r/ c ( r/ c = ρ < ). whch results c r r c p 0 = + =0! c!( ρ) (r / c = ρ < ) (C.) To cosder measure of effectveess we frst cosder the expected queue sze L q, as t s computatoally easer to determe tha L, sce we have oly to deal wth p for c. Thus L q = ( c) p = r ( c ) c = + = c+ c c! c c c p 0 = r p 0 mρ m = r ρ p 0 mρ m ( C.3) c! m= c! m= c c c r ρ p 0 d m r ρ p 0 d r ρ p = ρ = 0 =. c! d ρ m= c! dρ ρ c! ρ ( ) To fd L ow, we employ Lttle's formula to get W q, the use W q to fd W = W q +/ µ, ad fally employ Lttle's formula aga to calculate L = λw. Thus we get W q = L q r c = p C.4 λ c ( c )( ρ )! µ 0 ( ) r c W = + p 0 ( C.5) µ c ( c )( ρ )! µ ad r c ρ L= r+ p 0 (C.6) c! ρ ( ) 9

10 Lettg T q represet the radom varable tme spet watg queue ad W q (t) ts CDF, we have 0 } W q ( )= Pr{T q = 0} = Pr{ c c c system = p = p 0 r =0 =0! Now to evaluate r /!, recall that t appears the expresso for p 0 as gve Equato (C.), so that thus gvg W q ( 0 ) = p 0 p 0 c r r c = =0! p0 c! ( ρ ) c c r r p ( ) = 0 c! ρ c! ( ρ ) ForT q > 0 ad assumg FCFS, W () (C.7) q t = Pr{T q t} = W q ( 0 )+ Pr{ c + =c completos t arrval foud system} p. Now whe c, the system output s Posso wth mea rate cµ, so that the tme betwee successve completos s expoetal wth mea /( cµ ), ad the dstrbuto of the tme for the -c+ completos s Erlag type - c +. Thus we ca wrte ( c x) c e c x dx t c Wq ()= t W q ( 0) + p 0 r µ µ µ c c c c! ( c)! = 0 µ µ ( rx) c W q ( )+ r c p t µ 0 = 0 e c x dx (c )! = ( c)! r p = W q ( ) 0 ( ) 0 c c t c µx c r 0 ( ) r p 0 ( ) ( (cµ λ 0 + µe dx = W q ( 0 )+ e c! c! ρ )t ) 0

11 Puttg ths result together wth (C.7), we have c r p W q () t = 0 c! ( ρ) e (cµ λ )t. From Equato (C.8), c r p 0 )t Pr{ T q > t} = Wq ()= t e (cµ λ c! ( ρ ) so that the codtoal probablty Pr{T q > t T q > 0}= e (cµ λ )t. (C.8) To fd the formula for the CDF of the system watg tme, we frst splt the stuato to two separate possbltes, amely, those customers havg o le wat [probablty W q (0)] ad those whose system wat s a le delay plus a servce tme [probablty - W q (0)]. The frst of these two classes of customers has a CDF whch s detcal to the expoetal servce-tme dstrbuto, wth mea/ µ ; the secod has a CDF foud as the covoluto of the servce-tme dstrbuto wth a secod expoetal dstrbuto havg mea /(cµ λ), the latter represetg the CDF of the le watg tme, gve that T q > 0 (see earler ths secto). Ths covoluto ca be also wrtte as the dfferece of the two expoetal fuctos ( ) ( e µt ) c( c ρ Pr {T t} = ( c( ρ ) ρ ) e (cµ λ)t ) Thus the overall CDF of M/M/c system wats may be wrtte as ()= W (0) e µt + q 0 W t q W ( ) c ρ ( ) ( e µt ) ( e ( (cµ λ)t ) ( C.9) c ρ c ρ = ( ) ) c( ρ ) W 0 W q ( ) c( ρ ) C5 M/M/c/K queue ( ) ( e ) c( ρ ) ( e ) q 0 µt (cµ λ)t

12 We ow cosder a M/M/c/K queue, whch there s a lmt K placed o the umber allowed the system at ay tme. The approach here s detcal to that of the fte-capacty M/M/c except that the arrval rate λ, must ow be 0 wheever K. Equato (C.) ow becomes λ p c!µ 0 ( < ) p = (C.30) λ (c K ) c p 0 c c!µ The usual boudary codto that the probabltes must sum to wll yeld p 0. Aga, the computato s early detcal to that for the M/M/c, except that ow both seres the computato are fte ad thus there wll to be o requremet that the traffc testy ρ be less tha. So c λ K λ p 0 = + c =0!µ = c c c!µ To smplfy, cosder the secod summato above, wth r = λ / µ ad ρ = r / c : K r = r c K ρ c = c! ρ c c = c c! c! c resultg + r c ρ K c (ρ ) = r c ( K c + ) (ρ =) c! p 0 = c + r r c ρ K c + (ρ ) =0! c! ρ (C.3) c r r c ( + ) ρ = + K c ( ) =0! c! To fd the expected queue legth (ρ ) we start wth

13 pr K c K 0 ( c)r c q c c c c! = c+ L = ( c) p = = + pr c ρ K pr c ρ K c ( 0 = c c! = c+ c! = ) ρ c = 0 ρ K c pr c ρ d ρ = 0 c! d ρ ρ or c pr 0 ρ q c! ( ρ ) K c L = ρ + + ) ρ K c ( ρ )(K c (C.3) For ρ =, t s ecessary to employ L'Hoptal's rule (dfferetato) twce. To obta the expected system sze, we use, le the urestrcted M/M/c model, L = L q + r. However, for the fte-watg-space case, we eed to adjust ths result (ad Lttle's formula, as well), sce a fracto p of the arrvals do ot jo the system, because they have come whe there s o watg space left. Thus the actual rate of arrvals to jo the system must be adjusted accordgly. Sce Posso arrvals see tme averages, t follows that the effectve arrval rate see by the servers s λ( p K ). We heceforth deote ay such adjusted put rate as λ eff. The relatoshp betwee L ad L q must therefore be reframed for ths model to be L = L + q λ eff / µ = L q + λ( p K )/ µ = L q + r( p K ).We ow that the quatty r( p K ) must be less tha c, sce the average umber of customers servce must be less tha the total umber of avalable servers. Ths suggests the defto of a parameter ρ eff = λ eff / cµ, whch would thus have to be less tha for ay M/M/c model eve though o such restrcto exsts o the value of ρ = λ / cµ. Expected values for watg tmes ca ow be obtaed by Lttle's formula as 3

14 L L W = = λ eff λ ( p K ) (C.33) L q W q = W = µ λ eff For M/ M//K, all of the above measures of effectveess reduce to cosderably smpler expressos, wth ey results of ρ (ρ K + ) ρ p 0 = (C.34) (ρ =) K + ad ) ρ (ρ ) K + ( ρ ρ p = K + (ρ =) ( C.35) ρ ρ (K ρ K +) + (ρ ) ρ ρ L q = K( K ) (K +) ( ρ = ) ( C.36) wth L = L + ( p 0 ). The dervato of the watg-tme CDF s ow q complcated, sce the seres are fte, although they ca be expressed terms of cumulatve Posso sums, as we shall show. Also, t s ow ecessary to derve the arrval-pot probabltes { q }, sce the put s o loger Posso because of the sze trucato at K, ad q p. We use Bayes' theorem to determe the q, so that q = Pr{ system arrval about to occur } 4

15 Pr{arrvalabout tooccur system } p = =0 Pr {arrvalabout tooccur system } p λ t ο ( ) p λ+ ο( )/ t p = lm + t K = lm t K 0 t =0 λ t ο ( ) p t 0 + t =0 λ+ ο( t )/ t p λ p = K λ =0 = ( K ) p p p K To get the CDF W q (t) we ote that W q ()= t Pr{ T q t } = W q ( 0 ) + K Pr{-c+ completos t arrval foud system } q =c sce there caot be arrvals jog the system wheever they ecouter K customers. It follows that K W ()= t W ( )+ q t c µ ( c µ c x) e cµ x q q 0 dx 0 ( c)! by usg = W q ( ) t ( ) = c 0 + ( x) c c x K cµ c µ µ q = e dx t c ( c )! x m λ λ λx t e e dx m! =0! ad m = - c ad λ = cµ gves t m = ( λ ) λt cµ(cµx) c c (cµx) e cµt ( c)! ad hece e cµx dx =! =0 5

16 K K c W q ()= t W q ( 0 )+ q q (cµt) e cµt =c =c =0! cµt e K c cµt = q ( ) =c =0! ( C.37) C6 Erlag's Formula (M/M/c/c) The specal case of the trucated queue M/M/c/K for whch K = c, that s, where o le s allowed to form, results a statoary dstrbuto whch s ow as Erlag's frst formula ad ca be readly obtaed from Equatos (C.30) ad (C.3) wth K =c (λ/ µ) p = c (! λ / µ ) (0 c) (C.38) =0! The resultat formula for p c s tself called Erlag's loss formula ad correspods to the probablty of a full system at ay tme the steady state, amely, c r / c! p c = c (r = λ / µ ) (C.39) =0 r /! Sce the put to the M/M/c/c s Posso, the probablty that a arrval s lost s equal to the probablty that all chaels are busy. C7 Queues wth ulmted servce (M/M/ ) If a fte umber of servers s avalable we have a model for selfservce stuato. Usg the geeral brth-death results wth λ = λ ad µ = µ, for all, yelds p = r p 0, p 0 = r! =0! The fte seres the expresso for p 0 s detcal to the represetato of e r resultg 6

17 r p = re ( 0) (C.40)! so that the steady-state probablty dstrbuto of the system s Posso wth parameter r = λ / µ. C8 Fte source queues I ths secto we assume that the callg populato s fte, say of sze M, ad future evet occurrece probabltes are fuctos of system state. A typcal applcato would be aalyss of a etwor wth relatvely small umber of termals. We assume c servers (e.g. shared wreless chael capacty s c) are avalable, that the servce tmes are detcal expoetal radom varables wth mea / µ, ad that the arrval process s descrbed as follows. If a callg ut s ot the system at tme t, the probablty t wll have etered by tme t + t s λ t+ o( t ) ; that s, the tme a callg ut speds outsde the system s expoetal wth mea / λ. Because of these assumptos, we ca use the brth-death theory developed prevously, wth the modfed brth ad death rates gve by (M )λ (0 λ = 0 ( M ad µ (0 < c ) < M ) ) µ = c µ ( c ) Usg Equato (C.5) wth r= λ / µ yelds!( )! M M r p 0 ( < c )! p = M!( M )! r p (c M c 0 c c! or equvaletly, ) 7

18 M ( )r p 0 ( <c) p = (C.4) M! ( ) r p (c M ) c 0 c c! The algebrac form of the { p } does ot allow the closed-form calculato of p 0, L q, L, W ad W q. Istead, we must calculate each of the coeffcets multplyg p 0 Equato (C.4) (call them { a, =.,3,..., M}) ad the complete the computato as p = ( + a + a + a a ) 0 M To fd the average umber of customers the system ( our example the umber of termals wth baclogged pacet watg for trasmsso), we get M M L = p = p 0 a = = However, to obta L q ad W ad Wq, we must frst fd the effectve mea rate of arrvals to the system. The mea arrval rate whe the system s state s (M - ) λ ad we get M λ eff = (M )λ p = λ (M L) =0 From our earler wor we have L q = λ eff L = L r (M L). (C.4) µ ad from Lttle's formula L W = (C.43) λ (M L) ad W q = λ (M L q L) For the sgle-server verso of ths problem, Equato (C.4) reduces to M p = ( )r! p 0 (0 M ) ad the rest of the aalyss s detcal. 8

19 C9 Networ wth dyamc spectra sharg As metoed Chapter, 4G etwors wll allow for spectra (resource) sharg betwee dfferet systems (operators). Let us assume that up to Y addtoal actve termals from aother system ca be temporally permtted to coted for c chaels the system. The sharg model s defed by modfyg λ as Mλ (0 < Υ) λ = (M + Υ)λ (Υ < Υ + M ) 0 ( Υ + M ) For c avalable chaels, we have µ (0 < c) µ = cµ ( c) For c Y Equato (C.5) wth r = λ / µ gves M r p 0! (0 < c) M p = c r p 0 (c < Υ) (C.44) c c! (Υ Υ + M ) M Υ M! r p c 0 (M + Υ)!c c! If Y s very large, we essetally have a fte callg populato wth mea arrval rate M λ. Lettg Y go to fty Equato (C.44) yelds the M / M / c / results of Equato (C.) wth M λ for λ. Whe c > Y, we have 9

20 M r p 0 (0 Υ)! Υ M M! p = (M + Υ)!! r p 0 (Υ + < c) (C.45) M Υ M! r p c 0 (c Υ + M ) (M + Υ)!c c! The empty-system probablty, p 0, ca be foud as prevously by oce more usg the fact that the probabltes must sum to, so that the computato of p 0 s made up of fte sums. The same s true for L ad L q. To obta results for W ad W q we must aga obta the effectve mea arrval rate λ eff. To obta λ eff, oe ca use Equato (C.4) drectly or obta t usg logc smlar to that used prevously amely, Υ Υ+M Υ+M λ eff = M λ p + (M + Υ)λ p = λ M ( Υ) p =0 =Υ =Υ (C.46) C0 Watg tme dstrbuto for fte source queues Before we ca obta the CDF of the le wat, we have to relate the geeral-tme probablty p to the probablty q that a arrval fds the system. For the geeral fte-source queue, the two probabltes are related as q = (M ) p /, where s a approprate ormalzg costat determed from summg the {q } to. To prove ths, we aga use Bayes theorem as the M/M/c/K stuato. Pr{ system arrval s about to occur} 0

21 Pr{ system}pr{arrval s about to occur system} = (Pr{ system}pr{arrvals about to occur system}) p [(M )λ t + o( t)] = lm t 0 p [(M )λ t + o( t)] (M ) p (M ) p = = (M ) p M L For the spectra sharg system, q (M) ca be show to be Mp + M y M ( Y) p = y (0 Y ) q = ( M + Y) p ( Y Y + M ) y+ M M ( Y) p = y Now we ca wrte W () t = Pr {T t} = W (0) + q q q Y+ M [Pr{ c+ completos t arrval foud system} q ] = c Y+ M c µ ( µ ) c x e µ dx 0 ( c)! = W q (0) + q t c c x = c Y+ M c c ( c x ) c x e µ dx = c ( c)! = W q (0) + q t µ µ = Y+ M c µ c t q (c t) e µ = c =0! (C.47) C Recofgurable systems ad state-depedet servce As a respose to a creased cogesto the system a advaced wreless commucatos system may adapt the requred servce tme ether by mag the pacets shorter at the MAC layer or by chagg the source codg rate o applcato layer.

22 The frst model we cosder s oe whch a sgle server has two mea rates, say slow ad fast. Wor s performed at the slow rate utl there are the system, at whch pot there s a swtch to the fast rate. We stll assume the servce tmes are Marova, but the mea rate µ ow explctly depeds o the system state. Furthermore, o lmt o the umber the system s mposed. Thus µ s gve as < µ = µ ( ) µ ( ) (C.48) Assumg the arrval process s Posso wth parameter λ ad utlzg Equato (C.5) we have < p = ρ p (0 ) 0 + ρ ρ p 0 ( ) (C.49) where ρ = λ / µ ad ρ = λ / µ <. Because the probabltes must sum to, we have ad 0 + p = ρ + ρ ρ =0 = ρ ρρ + (ρ, ρ <) ρ ρ p 0 = ρ + (ρ =, ρ <) + ρ To fd the expected system sze, we assume ρ (C.50)

23 = 0 + L = p p ρ + ρ ρ =0 =0 = ρ = p ρ ρ + ρ ρ 0 =0 ρ = = p d 0 ρ ρ + ρ ρ d ρ dρ =0 ρ dρ = d ρ ρ d ρ = p 0 ρ + ρ dρ ρ ρ dρ ρ ρ Ths results ( ) ρ] ρ + ρ ρ ( L p 0 ρ ρ [ ) = + (C.5) ( ρ ) ( ρ) We ca fd L q formulas as W = L / λ as L q = L ( p 0 ), ad W ad W q from Lttle s ad W q = L q / λ. Note that the relato W = W q +/ µ caot be used here, sce µ s ot costat but depeds o the system-state swtch pot. However, by combg the above equatos, we see that W = Wq + ( p 0 ) / λ whch mples that the expected servce tme s ( p 0 )/ λ. C QoS provsog ad queues wth mpatece Customers are sad to be mpatet f they ted to jo the queue oly whe a short wat s expected ad ted to rema le f the wat has bee suffcetly small. These stuatos are typcal for the systems wth costraed delays where QoS parameter s the maxmum allowable delay. Impatece geerally taes three forms. The frst s balg, the reluctace of a customer to jo a queue upo arrval; the secod reegg, the reluctace to rema le after 3

24 jog ad watg; ad the thrd joceyg betwee les whe each of a umber of parallel les has ts ow queue. C. M/M/ Balg I real practce, t ofte happes that arrvals become dscouraged whe the queue s log ad do ot wsh to wat. Oe such model s the M/M/c/K; that s, f the messages see K ahead of them the system, they do ot jo. Rarely do all messages have exactly the same dscouragemet lmt all the tme. Ths mght deped o the remag umber of hops to ther fal destato. A alteratve approach to balg s to employ a seres of mootocally decreasg fuctos of the system sze multplyg the average rate λ. Let b be ths fucto, so that λ = b λ ad 0 b + b ( > 0, b 0 ). Now Equato (C.5) whe c = gves λ λ p = p 0 = p 0 b (C.5) = µ µ = Possble examples that may be useful for the dscouragemet fucto b, are /( + ), /( + ), ad e α. Messages are ot dscouraged oly because of queue sze, but may rather attempt to estmate how log they would have to wat. If the queue s movg qucly, the the message may jo a log oe. O the other had, f the queue s slowmovg, a message may become dscouraged eve f the le s short. Now f messages are the system, a estmate for the average watg tme mght be /µ, f the customer had a dea of µ. We usually do, so a plausble balg fucto mght thus be b = e α / µ. Also ote that the M/M//K model s a specal case of balg where b = for 0 K ad 0 otherwse. C. M/M/ Reegg Messages that ted to be mpatet may ot always be dscouraged by excessve queue sze, but may stead jo the queue to see how log ther wat may become, all the tme retag the 4

25 prerogatve to reege f ther estmate of ther total wat s tolerable. We ow cosder a sgle-chael brth-death model where both reegg ad the smple balg of the prevous secto exst, whch gves rse to a reegg fucto r() defed by Pr{ut reeges durg t customers preset} r( )= lm t 0 t r (0) = r () 0 Ths ew process s stll brth-death, but the death rate must ow be adjusted to µ = µ + r(). Thus t follows from Equato (C.45) that where p = p 0 λ b = p 0 λ ( ) = µ = µ + r () p 0 b = + λ = = µ + r () α µ (C.53) A good possblty for the reegg fucto r() s e /,. A watg customer would probably estmate the average system watg tme as /µ f customers were frot of hm, assumg a estmate for µ were avalable. Aga, the probablty of a reege would be estmated by a fucto of the form e α / µ. C3 Traset behavor of M / M // {p (t)} are probabltes that at arbtrary tme t there are customers a sgle-chael system wth Posso put, expoetal servce, ad o watg room. I ths system, p (t) = 0 for all >. We start wth Equato (C.3), wth λ 0 = λ, λ = 0, > 0, ad µ = µ : d p ( t ) = µ p () d t t + λ p (t 0 (C.54) 5

26 d p 0 () t = λ p 0 () t + µ p () t d t Sce p 0 () t + p () d p () t d t resultg t = Equato (C.54) s equvalet to p ( t ) = µ p () t + λ [ p () t ] p ( t ) + ( λ + µ) p () t = λ Ths s a ordary frst-order lear dfferetal equato wth costat coeffcets, whose soluto s λ p () t = C e (λ +µ )t + λ + µ To determe C, we use the boudary value of p (t) at t = 0, whch s p 0 () t resultg C p 0 λ = ( ) λ + µ ad p ()= t () λ (λ + µ)t λ µ e + p ( 0 ) e ( + )t λ µ + + )t (C.55) λ µ (λ + µ)t p t = µ 0 ( ) (λ µ 0 e + p 0 e + sce p 0 () t = p () t for all t. The statoary soluto ca be foud drectly from Equato (C.54) the usual way by lettg the dervatves equal zero ad the, wth the use of the fact that p 0 + p =, solvg for p 0 ad p. Also, the lmtg (steady-state, equlbrum) soluto ca be foud as the lmt of the traset soluto of Equato (C.55) as t goes to, resultg 6

27 ρ p = ad p 0 = ρ + ρ + C4 Traset behavor of M/M/l/ Let us assumed that the tal system sze N(0) = where N( t ) deotes the umber the system at tme t. The dfferetaldfferece equatos goverg the system sze are gve Equato (C.3) as p ()= t ( λ + µ ) p() t + λ p ( t ) + µ p+ ( t ) ( > 0) (C.56) () t = λ p 0 () t + µ p () t p 0 We start wth probablty geeratg fucto defed as P( z,t ) = p ( ) =0 t z (z complex) such that the summato s coverget for z, wth ts Laplace trasform defed as ( )= e s t P( z,t ) d t ( Re s > 0) P z,t 0 Startg from Equato (C.56), we get (, ) = (λ + + P z t ) (s) z + µ ( z p0 (C.57) µ s ) z µ λ z where p 0 (s) s the Laplace trasform of p 0 (t). Sce the Laplace trasform P(z,t) coverges the rego z, Re s > 0, wherever the deomator of the rght-had sde of Equato (C.57) has zeros that rego, so must the umerator. Ths 7

28 fact s used to evaluate p 0 (s). The deomator has two zeros, sce t s a quadratc z ad they are (as fuctos of s) s) λ + µ + s (λ + µ + 4λµ z = λ s) 4 λ + µ + s + (λ + µ + λµ z = λ Oe ca chec that z < z, z + z = (λ + µ + s)/ λ, ad z z = µ / λ. At ths pot Rouche's theorem s used that ca be expressed as: If f(z) ad g(z) are fuctos aalytc sde ad o a closed cotour C ad f gz ( ) < f ( z ) o C, the f(z) ad f(z) + g(z) have the same umber of zeros sde C. For z = ad Re s > 0 we see that f ( ) s) z z ( λ + µ + = λ + µ + s > λ + µ µ + λ z g ( z) Hece, from Rouche's theorem, (λ + µ + s)z µ λz has oly oe zero the ut crcle. Ths zero s obvously z, sce z < z. Thus equatg the umerator of the rght-had sde of Equato (C.57) to zero for z = z gves p0 () s = + z µ( ) z Whe ths trasform for p 0 (t) s serted to Equato (C.57) ad the result wrtte fte seres form, we fd + j j z z z P( z, s ) = z z + ( z / z < ) λ z j= 0 =0 z λ z ( z ) =0 z 8

29 where = N( 0). Now, the trasform of p ( t ), p (s) s the coeffcet of z the Laplace trasform of the geeratg fucto P( z,t), P(z, s). So the ext step the process s to fd p (s), ad the ths s, tur, verted to get p (t), utlzg ey propertes of the trasforms of Bessel fuctos ths last step. The fal result s, terms of modfed Bessel fuctos of the frst d, I (y), ad s p () t = e (λ +µ )t λ ( )/ I ( λµ t)+ λ ( l )/ I ++ ( λµ t)+ µ µ j / + λ λ λ I ( λµ t) µ µ j =++ µ (C.58) for all 0, where I ( ) y = =0 ( y /!( ) + + )! ( > ) (C.59) C5 Traset behavor of M/M/ Let us assume that the tal system sze at tme 0 s 0, so that N (0) = 0. The dfferetal-dfferece equatos goverg the system sze are derved from Equato (C.3) wth λ = λ ad µ = µ as p ()= (λ t + µ ) p t ( t ) = λp 0 () t + µp () t p o ( ) + λ p (t) + ( +) µ p (t) ( > 0) + (C.60) It ca be show that the geeratg fucto of the probablte {p () t } s P z t ( ) e µt ) λ (, ) = p t z = exp (z )( o = µ (C.6) 9

30 So f we expad Equato (C.6) a power seres, a 0 + a z + a z +... the coeffcets a the are the traset probabltes p () t that we desre. To do ths we ca use a Taylor seres expaso about zero (Maclau-r seres), ad we fd that ()=! ( e µt ) λ exp λ p t ( exp µt µ ) 0 (C.6) µ C6 Composte queueg models for multmeda systems I multmeda wreless commucatos le WCDMA, a customer may be usg a hgh data rate requrg a multcode trasmsso. Ths ca be see as usg a umber of parallel chaels smultaeously. A adaptve system may eve chage the rate o pacet by pacet bass. To model these applcatos, ths secto we assume that the actual umber of customers ay arrvg module s a radom varable X, whch may tae o ay postve tegral value less tha wth probablty c x. Ths queueg problem, referred to as x M [ ] / M /, s stll Marova the sese that future behavor s a fucto oly of the preset ad ot the past. If λ x s the arrval rate of a Posso process of batches of sze X, the c x = λ x / λ where λ s the composte arrval rate of all batches ad s equal to = λ. Ths total process, whch arses from the overlap of the set of Posso processes wth rates {λ x, x =,,...}, s a multple or compoud Posso process. By usg the same logc as dervato of Equatos (C.)-( C.6) equato (C.6) ow becomes 0 = + ) (λ µ) p + µ p + + λ p c ( = (B.63) 0 = λ p 0 + µ p The last term Equato (C.63) states that the process ca get to state from ay state - wth probablty c. I the sequel we wll eed to defe 30

31 ad z = p z ( z ) =0 P( ) ( )= c z ( z C z = ) as the geeratg fuctos of the steady-state probabltes { p } ad the batch-sze dstrbuto {c } respectvely. By multplyg Equato (C.63) by the approprate z,ad summg up we have 0 = λ p z µ p z + µ p z + λ p =0 = z = = = Because C z P z p c z = c z p z = ( ) ( ) = = = = c z (C.64) Equato (C.64) may be rewrtte as 0 P z µ P z P z ] C z P z z ad thus µp ( )= 0 ( z) P z ( z ) µ ( z) λz[ C( z )] = λ ( ) [ ( ) p 0 ]+ µ [ ( ) p o + λ ( ) ( ) (C.65) To fd p 0 from P() ad the average umber the system, L, from P ( ), let us rewrte the geeratg fucto Equato (C.65), wth r = λ µ, as p0 P( z ) = C z. rz z, C ( (z) )= z C ( ) C ( z )= [ C( z )/ ] [ z] s the geeratg fucto of the complemetary batch-sze probabltes Pr{X f x}= C x = C x, sce /( z) s the geeratg fucto of ad C(z)/( z)s the geeratg fucto of the cumulatve probabltes C, whch ca be x see from 3

32 cz = z = cz x x = = = z x x x x Cz x = c z = = = = x It follows after oe applcato of l Hoptal s rule that C ( ) = E[ X ]ad after two applcatos that C () = E[X (X )]/, resultg p0 = P( )= ad P ()= r () Therefore = re C p 0 [ X ] = ρ ad { [ ] } ρ + re X ( ) () () r C + C rc r E X + E X L = = ( ρ ) ( ρ ) (ρ = λ E[ X ]/ µ ) (C.66) The remag usual measures of effectveess may be foud by usg L q = L ( p 0 ) = L ρ ad the Lttle's formulas. The dvdual state probabltes { p } ca ofte also be obtaed by the drect verso of the geeratg fucto of Equato (C.65). Two terestg useful examples of these results would be to let the batch szes be ether costat (each of sze K) or geometrcally dstrbuted. For the costat case, L = ρ + Kρ = K + ρ ( ρ) ρ (ρ = λ K µ) (C.67) ad L q = L ρ = ρ + (K ) ρ ( ρ ) The verso of P( z)to get the dvdual {p } (C.68) s a reasoable tas whe K s small. Whe the batch szes are stead dstrbuted geometrcally, c = α α x ; 0 < α < t follows that x ( ) ( ) ρ = λe[ X ]/ µ = r /( α ), C( z ) = ( α ) α z = z ( α ) /( α z) ad from Equato (C.65) = 3

33 ( ρ )( z) P( z)= z rz C( z ) ( ρ )( z) ( ρ )( α z) z rz z ( α )/( α z) z α + ( α) ρ = = = ( ρ ) α z z α + ( α) ρ z α ( ) + α ρ Utlzg the formula for the sum of a geometrc seres, we have + P( z ) = ( ρ ) α + { ( α) ρ z } { α + ( α ) p z } =0 =0 from whch we get p = ρ α + α ρ α α + α ρ ( ){ ( ) ( ) } (C.69) = ( ρ) α + ( α) ρ ( α ρ ) ( > 0) C6. Bul servce M / M [X ] / Let us assume that arrvals occur at a sgle-chael faclty as a ordary Posso process, that they are served FCFS, that there s o watg-capacty costrat, ad that these customers are served K at a tme, except whe less tha K are the system ad ready for servce, at whch tme all uts are served. Ths would correspod to the case whe a umber of sgals are multplexed pror to trasmsso. Further, f less tha K are servce, ew arrvals mmedately eter servce up to the lmt K, ad fsh wth the others, regardless of the tme to servce after servce begs. The amout of tme requred for the servce of ay batch s a expoetally dstrbuted radom varable, whether or ot the batch s of full sze K. For ths model we use otato M / M [K ] /. A specal case s a model whch the batch sze for servce must be exactly K ad f the umber preset whe the server becomes dle s less tha K, t wats utl K accumulates. The basc model s, of 33

34 course, a o-brth-death Marova problem. The stochastc balace equatos are ow gve as 0 = ( λ + µ ) p + µ p +K + λ p ( ) (C.70) 0 = λp 0 + µ p + µ p +L+ µ p K + µ p K By usg the operator D defed as p = Dp (B.70) becomes µd K + (λ + µ ) D + λ p = 0 ( 0 ); (C.7) If (r,kr K + )are the roots of the operator equato, the we have K + p = Cr ( 0) =0 Sce p =, each r must be less tha or C = 0 for all r ot =0 less tha. So let us ow determe the umber of roots less tha oe. By usg Rouche's theorem t s foud that there s, fact, oe ad oly oe root (say r 0 ) (0, ). So we have p = Cr 0 ( 0, 0 < r 0 < ). Usg the boudary codto that p must total, we fd that C = p 0 = r 0 ; hece p = ( r 0 )r 0 ( 0), (0 < r 0 < ) (C.7) Measures of effectveess for ths model ca be obtaed the usual maer. Sce the statoary soluto has the same geometrc form as that of the M / M / (wth r 0 replacg ρ ), we ca mmedately wrte = 0 /( r 0 ), L q = L λ / µ, W = r 0 / λ ( r 0 ), ad W q = W / L r C6. Modfed bul servce Let us ow assume that the batch sze must be exactly K, ad f ot, the server wats utl such tme to start. The Equato (C.70) becomes µ 34

35 0 = (λ + µ) p + µ p + K + λ p ( K ) 0 = λp + µ p K + λ p ( K ) (C.73) + 0 = λp 0 + µ p K The frst le of Equato (C.73) s detcal to Equato (C.70); hece p = Cr 0, ( K, 0 < r < ). The obtag of C (ad p 0 ) s a more complcated procedure here, sce the foregog formula for p s vald oly for K.From the steady-state K equatos, p K = ( λ / µ ) p 0 = Cr 0, ad therefore C = (λ p 0 )/( µr K 0 ). To get p 0 ow, we must use the K - statoary equatos gve Equato (C.73) as 0 = µ p K λ p + λ p. + Substtutg the above geometrc formula for p, K, to these K equatos, t ca be see that p r = p p, 0 K. 0 0 ( ) These equatos ca be solved by terato startg wth = or we ca ote that these are ohomogeeous lear dfferece equatos whose solutos are p = C + C r 0, where C = p 0 C ad C = p 0 r 0 /( r 0 ), ad + p ( r ) 0 0 ( K) p = r 0 (C.74) p r K 0λ 0 ( K) µ To get p we use p 0 =. Hece from Equato (C.74), 0 = 35

36 p 0 K r + λ 0 K = + + r 0 = r 0 µ =K K K r 0 ( r 0 ) λ = + + r µ ( r ) ( r 0 ) 0 = µr K + ( + ) r 0 λ µ + λ + µ K ( r 0 0 ) µ ( r 0 ) K From Equato (C.7) we ow that µr + 0 (λ + µ ) + λ = 0. So, the prevous equato becomes ) ) = µ ( r p 0 = 0 r 0 (C.75) µ K ( r 0 K The formulas for the { p }could have also bee obtaed va the probablty geeratg fucto as the prevous secto. C6.3 Erlaga Models (M / E /, E / M /, E j / E /) I ths secto, we allow for a more geeral probablty dstrbuto for descrbg the put process or the servce mechasm. I may practcal stuatos, the expoetal assumptos used so far may be rather lmtg, especally the assumpto cocerg servce tmes beg dstrbuted expoetally. The Erlag dstrbuto To start wth, cosder a radom varable T whch has the gamma probablty desty f ( t ) = Γ( ) α tα e t / β (α, β > 0 0 > t > ), α β where Γ( α ) = t α e t dt s the usual gamma fucto, ad α ad β 0 are the parameters of the dstrbuto. The mea E[T ] ad varace 36

37 Var [T ] are gve as ET [ ] = αβ ad Var [T ] = αβ. If we further cosder a specal class of these dstrbutos where α ad β are related by α = ad β =/ µ, where s ay arbtrary postve teger ad µ ay arbtrary postve costat, we obta the Erlag famly of probablty dstrbutos, amely, () ( µ ) µ t ) e t (0 < t < ) f t = (! The parameters of the Erlag are thus ad µ, ad the mea ad varace are gve by E T [ ] =/ µ ad Var [T ] = / µ. Fgure C. Erlag type ( E ) dstrbuto. For a gve, the resultg Erlag s referred to as a Erlag type or E dstrbuto. Fgure C. llustrates the effect of o the Erlag famly of dstrbutos. Parameter s ofte called the shape parameter. The Erlag famly of probablty dstrbutos provdes much more modelg flexblty tha does the expoetal. I fact, the expoetal s a specal Erlag, amely type. As creases, the Erlag becomes more symmetrcal, ad as t approaches, the Erlag becomes determstc wth value / µ. Thus practcal stuatos where observed data mght ot bear out the expoetal 37

38 dstrbuto assumpto, the Erlag ca provde greater flexblty by beg better able to represet the real world. Aother reaso why the Erlag s useful queueg aalyses s ts relato to the expoetal dstrbuto ad the Marova property. The Erlag dstrbuto tself s of course o-marova. However, as show later, the sum of d expoetal radom varables wth mea / µ yelds a Erlag type- dstrbuto. It s ths relato, as we shall see a lttle later, that allows the aalyss of queueg models wth Erlaga put or servce to be performed. Erlag servce model (M/E /) We ow cosder a model whch the servce tme has a Erlag type- dstrbuto. Eve though the servce may ot actually cosst of phases, t s coveet aalyzg ths model to cosder the Erlag as beg made up of expoetal phases, each wth mea / µ. Let p, (t) represet the probablty of the system ad the customer servce beg phase ( =,,..., ), where we ow umber the phases bacward; that s, s the frst phase of servce ad s the last (a customer leavg phase actually leaves the system). We ca wrte the followg steady-state balace equatos 0 = -( λ + µ )p, + µ p,+ + λ p, (, ), 0 = -( λ + µ )p, + µ p +, + λ p, ( ) 0 =-( λ + µ )p, + µp,+ ( < ) (C.76) 0 = -( λ + µ )p, + µp, + λ p 0 0=- λp 0 + µp,. It s easer to obta the expected measures of effectveess (L, L q, W, W q ) ad the state probabltes from Equato (C.76) by worg drectly o the pot process whch stead couts phases of servce the system [wth the bvarate state par (, ) trasformed to (-) + ] ad the coverts bac to customers. Ths ca be terpreted as the umber of phases the system requrg servce, sce - customers are watg ( ), each requrg phases of servce, ad the customer servce requres phases more. Ths s essetally equvalet to modelg the Erlag servce problem as a costat bul 38

39 put model (the M [ K ] /M/), where each put ut s cosdered to brg K = phases ad the (phase) servce rate µ s to be replaced by µ. (The phase approach to the Erlag servce problem ad the aalogous bul-put model are ot detcal, sce completed termedate phases of servce caot leave the system Erlag queue, ule what happes to a dvdual completed customer the bul model.) The frst ey result of ths coecto s that the average le delay for a M/E / customer ca be foud from the average system sze of the bul-put model gve Equato (C.67) multpled by / µ [sce the tme to serve L phases s L(/ µ )] wth the servce rate µ of the bul model tself replaced by the phase servce rate µ. Ths yelds + ρ W q =, (ρ = λ / µ) µ( ρ) (C.77) It follows that + ρ L q = λw q = (C.78) ρ ad L = L q + ρ, W =W q +/ µ. Regardg the steady-state probabltes themselves, we ca mmedately get the empty probablty, sce we ow for all sglechael, oe-at-a-tme-servce queues that p 0 = ρ. Next, we shall covert Equato (C.76) to a (batch-put) system based o a sgle varable coutg phase the system usg the aforemetoed trasformato (, )=(-)+. Mag the above trasformato Equato (C.76) yelds 0 = -( λ + µ) p ( ) + + µp ( ) ++ + λp ( ) + (, ) (C.79) 0 =- λp 0 + µp, where ay p turg out to have a egatve subscrpt s assumed to be zero. By wrtg out the top equato Equato (C.79) sequetally startg at =, =, Equato (C.79) ca be smplfed to 39

40 0=-( λ + µ) p + µp + + λp ( ), (C.80) 0 =- λp 0 + µp Equato (C.63) would gve the same result for a costat batch sze ( P) ad servce rate µ. Now f we let p represet the probablty of the phase or bul-put system defed by Equat (C.80), the t follows that the probablty of the Erlag servce system p s gve by p = p ( P) j ( ). (C.8) j =( ) + The watg-tme CDF wll be dscussed later wth the geeral theory of G/G/ queues. Whle we have here utlzed the relato betwee the M [ ] /M/ ad the M/E / t s mportat to ote that a smlar partershp holds betwee the M/M [ ] / queue ad the E /M/, so that the prevous bul results ca be useful dervg results about the Erlag arrval model to be treated the followg secto. Erlag arrval model (E /M/) As metoed the prevous secto, we ca use the results of the bul-servce model to develop results for the Erlag put model. We assume that the terarrval tmes are Erlag type dstrbuted, wth a mea of / λ. We ca loo, therefore, at a arrval havg passed through phases, each wth a mea tme of / λ, pror to actually eterg the system. Here we umber the phases frotwards from 0 to -. Aga, oe should eep md that ths s a coveet cocept for aalyss that does ot have to correspod to the actual arrval mechasm; the oly assumpto o terarrval tmes s that they follow a Erlag type- dstrbuto wth mea / λ. We defe the state varable as the umber of arrval phases the system, so that we wat to fd the probablty of arrval phases the system the steady state, whch we deote by p ( P). Oce we have 40

41 ths we use a relato smlar to Equato (C.8), to obta the probablty of customers the system, that s, + p = p j P j= ( ) (C.8) We ca get p ( P) from p ( B), the steady-state probablty of the bul-servce model gve by Equato (C.74) but wth λ replaced by λ. We ow from Equatos (C.74) ad (C.75) that So, for > 0, + p = p ( j= P) j p (P) = λp ( P) 0 r j (j ) j 0 µ = ρ( r 0 )r j 0 ( ρ = λ / µ ) ρ( r 0 )(r 0 + r 0 + +K+ r 0 ) = ρ( r 0 )r 0 (+ r 0 +K+ r 0 ) = ρ( r 0 )(r 0 ) (C.83) Ths s a geometrc dstrbuto (as M/M/), but wth r 0 multpler stead of ρ. It follows from Equato (C.83) that m L = ρ( r 0 ) (r 0 ) = ρ( r 0 ) = ρ = ( r 0 ) r 0 from whch we ca get L=L q +ρ,w=l/ λ ad W q = W-/ µ. as the To derve the watg-tme dstrbuto for ths model deote the probablty that a arrval to the system fds customers already there by q, q = Pr{ system arrval about to occur} = Pr { system arrval phase -} ( P ) p + Pr{ system ad arrval phase -} q = = Pr{arrval phase -} / sce t s equally lely that a arrval s ay oe of the phases wth probablty /. Thus we have 4

42 ( P) ρ( r 0 ) q = p = r = ( r ) r (C.84) r 0 where the fal step follows from the characterstc Equato (C.7) wth λ replaced by λ. Now f there are customers the system upo arrval, the codtoal watg tme s the tme t taes to serve these customers, whch s the covoluto of expoetals, each wth mea / µ, Ths yelds a Erlag type dstrbuto, ad the ucodtoal le-delay dstrbuto fucto ca thus be wrtte as W q (t) = q 0 + µ ( x ( µ ) t )! e µx µ ( µ ) dx = q 0 + ( r 0 )r 0 0 = = ( (µxr 0 ) t µx = q 0 + r 0 ( r 0 )µe dx 0 t ( r0 )µe = ( )! µ ( r 0 ) x µ ( r t dx = q 0 + r 0 [ e 0 ) ] x )! e µx dx = q 0 + r 0 0 The probablty of o wat for servce upo arrval s gve by, Equato (C.84) as q 0 = r 0, ad thus W q (t) = r 0 e µ ( r 0 )t, ( t 0 ) (C.85) Queueg wth dfferetated QoS We wll start wth a dscusso of a sgle expoetal chael wth prortes. 4G systems wll be also characterzed by dfferetated QoS (DQoS), so that queues wth prortes wll be a realty. Let us beg by assumg that customers arrve as a Posso process to a sgle expoetal chael ad that upo arrval to the system each ut wll be desgated to be a member of oe of two prorty classes. A typcal example s data ad voce trasmsso where voce traffc s gve hgher prorty due to the costrats o delay. The usual coveto s to umber the prorty classes so that the smaller the umber, the hgher the prorty. Let t further be assumed that the (Posso) arrvals of the frst or hgher prorty (voce) have mea rate λ ad that those of the secod (data) or lower prorty have mea rate λ, such that λ = λ + λ. We also suppose that the frst-prorty 4

43 tems have the rght to be served ahead of the others, but that there s o preempto. I vew of the foregog assumptos, a system of steady-state balace equatos may be establshed for p mr = Pr{ steady state, m uts of prorty ad uts of prorty are the system, ad a ut of prorty r = or s servce}. These the lead to the followg dfferece equatos the evet that ρ = λ µ <: 0 = (λ + µ) p m + λ p m,, + λ p m,, (m>0, >) 0 = (λ + µ) p m + λ p m,, + λ p m,, + µ( p m+,, + p m,+, ) 0 = (λ + µ) p m + λ p m,, (m>,>0) 0 = (λ + µ) p + λ p,, + µ( p + p,+,) 0 = (λ + µ) p 0 + λ p 0,, + µ( p + p 0, +, ) (C.86) 0 = (λ + µ) p m0 + λ p m,0, + µ( p m+,0, + p m ) 0 = (λ + µ) p 0 + λ p 0 + µ( p + p 0 ) 0 = (λ + µ) p 0 + λ p 0 + µ( p 0 + p ) 0 = λ p 0 + µ( p 0 + p 0 ) p 0 s stll ρ, sce the orderg of servce o way affects the probablty of dleess, also p = ( p m,m, + p m, m, ) = ( ρ)ρ (>0) m=0 Sce the percetage of tme the system s busy s ρ, the percetage of tme t s busy wth a type-r customer wll be ρλ r λ, so that p m = λ ad p m = λ m= =0 µ m=0 = µ The most we ca do ths case comfortably s obta expected values va two-dmesoal geeratg fuctos defed as P (z) = z p P z z m ( ) = m, m pm =0 = 43

44 m H (y, z) = y P m (z) [wth H (,) = λ µ ] m= m H ( y.z) = y P m (z) [wth H (,) = λ µ ] m=0 ad H (y, z) = H ( y, z) + H ( y, z) + p 0 m m = y z p m + y z p m + p 0 m= =0 m=0 = m m = y z ( p m + p m ) + y p m0 m= = m= = + z p 0 + p 0 where H ( y, z) s the jot geeratg fucto for the two prortes, regardless of whch type s servce. Note that H ( y, y) = p 0 ( ρy) [wth H (,) =], sce H ( y, z) collapses to the geeratg fucto of M M whe z s set equal to y ad thus o prorty dstcto s made. Hece f L ad L are used to deote the mea umber of customers preset system for each of the two prorty classes, the H ( y, z) = L = L + λ q = λw y µ y =z= ad H ( y, z) λ = L = L q + = λ W z y=z= µ where L q ad L q are the respectve mea queue legths. If we multply Equato (C.86) by the approprate powers of y ad z ad sum accordgly, we get (+ ρ λ y λ z )H ( y, z) = H ( y, z) + λ yp 0 P (z) P 0 (z) µ µ y z µ z ad 44

45 ( ρ ) (, ) = P ( ) λ y λ z H y z z P ( z) (ρ λ z ) p 0 (C.87) µ µ z µ To fd H ad H, we eed to ow the values of P (z), P 0 (z), ad p 0. Oe equato relatg P ( z), P 0 ( z ) ad p 0 may be foud by summg z ( =, 3,...) tmes Equato (C.86), whch volves p 0, ad the usg the fal three equatos of Equato (B.86), resultg ( ) ( ρ λ z )P ( ) (ρ λ z P z = + 0 z + ) p0 µ z µ Usg ths equato Equato (B.87) gves H ad H as fuctos of p 0 ad P, ad thus also H ( y, z) as 0 ( y) p H ( y, z) = H ( y, z) + H ( y, z) + p 0 = 0 y ρy( z λ y λ + λ z λ) + (+ ρ ρy + λ z µ)(z y)p 0 (z) z[+ ρ λ y µ λ z µ][ y ρy( z λ y λ + λ z λ)] By employg the codto that H (,) =, we have λ p 0 µ P 0 () = (ρ = λ µ) (+ λ µ )( ρ) We ext tae the partal dervatves of H wth respect to both y ad z ad the evaluate at (,) to fd the meas L ad L. I so dog, the exact fuctoal relatoshp for P 0 (z) turs out ot to be eeded, ad P 0 () aloe suffces. Wthout presetg the detals of the dfferetato, the fal results are 45

46 L = (λ µ)( + ρ λ µ) λ µ ρλ µ L q = λ µ ρ W = q µ λ (C.88) L = (λ µ)(+ ρλ µ λ µ) ( ρ)( λ µ) ρλ µ L q = ( ρ)( λ µ) ρ W q = ( ρ )( µ λ ) Usg the more sophstcated theory of multdmesoal brth-death processes, oe ca fd the state probabltes for prorty- customers as [] = ( ρ) λ + λ λ (λ µ) µ λ µ (+ λ µ) + p ( 0 ) (C.89) Some systems wth dfferetated QoS or dyamc spectra sharg my be better modeled by the slght geeralzato of the prevous problem by servg the prorty- customers at the rate µ ad the prorty- customers at µ.these results are [] 46

47 λρλ λ + ( λ λ )( µ µ L = q µ λ µ (λ µ ) ρ λ λ + ( λ λ )( µ µ L = q λ µ λ µ λ µ L q = L q + L q ) λρ µ + λλ µ ( λ µ ) [ λ λ + ( λ λ )( µ µ = λ µ λ µ λ µ (ρ = λ µ ) ) )] (C.90) Usg the theory of multdmesoal brth-death processes, the state probabltes for prorty- customers are [] λ p = ( ρ * ) µ + λ λ + µ µ λ µ µ λ (λ + µ ) + ( 0 ) Fally, for a model havg o prortes but uequal servce rates for customers of two major types we have λρ ( µ µ )( λ µ L q = µ λ µ λ µ ) λ ρ µ µ + ( µ µ )( λ µ L = q µ λ µ λ µ (λ µ ) ρ + ( λ µ ) ρ(µ µ ) ( ρ * ) λ λ + ( λ λ )( µ = µ ) L q = λ µ λ µ ρ * λ λ + ( λ λ µ µ ) ] [ )( * ρ λ µ ρ λ µ λ µ ) ( =, = + (C.9) ) 47

48 Queueg wth multple dfferetated QoS For the modelg of 4G etwors multple dfferetated QoS (MDQoS) models wll be eeded. Ufortuately whe the umber of prortes exceeds two, the determato of statoary probabltes a opreemptve Marova system s a exceedgly dffcult problem. I ths case, a alteratve approach to obtag the meavalue measures L ad W ca be used, amely, a drect expectedvalue procedure. Suppose that tems of the th prorty (the smaller the umber, the hgher the prorty) arrve before a sgle chael accordg to a Posso dstrbuto wth parameter λ ( =,,...,r) ad that these customers wat o a frst-come, frst-served bass wth ther respectve prortes. Let the servce dstrbuto for the th prorty be expoetal wth mea µ.whatever the prorty of a ut servce, t completes ts servce before aother tem s admtted. We beg by defg ρ = / λ µ ( r), σ = ρ (σ 0 0,σ r ρ). = The system s statoary for σ r = ρ <. The suppose that a customer of prorty arrves at the system at tme t 0 ad eters servce at tme t. Its le wat s thus T q = t t 0. At t 0 assume that there are customers of prorty the le ahead of ths ew arrval, of prorty, 3 of prorty 3, ad so o. Let S 0 be the tme requred to fsh the tem already servce, ad S be the total tme requred to, serve. Durg the ew customer's watg tme T q, tems of prorty < wll arrve ad go to servce ahead of ths curret,, arrval. If S s the total servce tme of all the the, T q = S + S S 0. + = = By tag the expected values of both sdes of the above equato + E S 0 [ ] [ ] (), W E T = E q q S + E S = = Sce σ < σ for all, ρ < mples that σ < for all. 48