1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)


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1 6.3 /  Communcaton Networks II (Görg) SS Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes (SP & MP) 4. Fnte State Markovan Processes 5. Analyss of Markovan servce systems 6. Queues for modelng of communcaton networks 7. M/G/ model 8. The model M/G//FCFS/NONPRE 9. The model M/G//FCFS/PRE
2 6.3 / 2  Communcaton Networks II (Görg) SS BCMP networks Baskett, Chandy, Muntz and Palacos extended n 975 the class of networks, for whch product form soluton s vald, to nclude also the networks wth: any fnte number of statons N any fnte number of user classes R arbtrary servce tme dstrbutons p, customers changng ther class wth probablty, r; j s p, r ; j, s : probablty, that a class r job after recevng ts servce at staton changes to class s and requres servce at staton j n open networks p 0; j, s denotes the arrval probablty (from outsde) to staton j, class s and p rn,; denotes the departure probablty from staton, class r.
3 6.3 / 3  Communcaton Networks II (Görg) SS The matrx P p r,;, js of the transton probabltes defnes a Markov chan wth states, that s determned through the par (,r). It s possble to map ths chan nto m ergodc (=rreducble) subchans E, E 2, K, E m, where 2 states t of a job bbelong to the same chan, only f the job can take both states wth a probablty >0. k r s the number of class r jobs n staton at network state S (see below). The number of jobs n each of the ergodc sub chans n the state S s gven by (6.3.34) 34) j r,2, K, K S E k j m r, E j ths quantty K S E j s always a constant tn closed networks. The total t number of jobs n the network at the state S can be calculated usng m K S E j K S j (6.3.35) 35) In open networks two dfferent generally state dependent arrval processes are assumed:
4 6.3 / 4  Communcaton Networks II (Görg) SS a) Jobs enter the network accordng to a nonhomogeneous Posson process, whose arrval rate KS depends on all the combnatons of the system states correspondng to the total number of jobs n the system. A newly arrvng job enters the staton n class r wth fxed probablty, p 0;, r where t holds N R (6.3.36) p0 ;, r r b) Correspondng to the m transton subchans, the jobs enter the network over j ndependent nonhomogeneous Posson processes wth the mean arrval rate K S E j, that depends on the network status that agan depends on the jobs already n the chan. An ncomng job of the jth stream enters staton wth constant probablty p 0; r, where t holds: p (6.3.37) 0;, r j 2,, K, m r, E j,, K, S The network status S s defned through the vector S S S S, where 2 S N summarzes the condton, that s present at staton. These condtons are defned through the type of the staton.
5 6.3 / 5  Communcaton Networks II (Görg) SS Staton types Type : M/M/n  FCFS: Such statons contan n servers workng under the strategy FCFS. All the classes of jobs have the same servce tme dstrbuton. In addton, the mean servce tme of all the classes are equal. Type 2: M/G/  PS: Such statons contan a sngle server, workng under the strategy processor sharng,.e. all the new ncomng jobs are served at the rate /k sec. Jobs belongng to dfferent classes are allowed to have dfferent and arbtrary servce tme dstrbutons. Type 3: M/G/  no watng: Such statons have a free server (nfnte server) avalable for each of the new comng jobs, so that they never have to wat. The servce tme dstrbuton can be arbtrary and specfc to the class. Type 4: M/G/  LCFSPRE: Here the only avalable server works under the preemptve LCFSstrategy. The servce tme dstrbuton can be arbtrary and specfc to the class.
6 6.3 / 6  Communcaton Networks II (Görg) SS Gservce tme dstrbuton b For staton types 2, 3, 4, t s desrable for the servce tme to have a general (=G) servce tme dstrbuton b t wth a ratonal Laplace Transform. Snce Cox (955) t s known, that any dstrbuton wth a ratonal Laplace Transform can be reconstructed through a network of exponental phases. (compare secton 552).5.5.2). Fgure 6.6: 6: Phase model for Gdstrbuton The Laplace transform of the probablty densty of a random varable that conssts of phases s gven by,
7 6.3 / 7  Communcaton Networks II (Görg) SS n j 0 2 j j j l s aaa K a b l s l the ndces,r are omtted here for the sake of smplcty. It means: rl servce rate of M dstrbuton at the lth step, l 2,, K, n for class r r n staton n r total number of exponental steps for class r n staton a rl probablty for job also gong through the (l+)th step b probablty blt for completng the job after step l b rl Soluton for closed network problem In order to get the soluton for the statonary state probablty P(S), t s necessary to solve the followng system of equlbrum equatons, see equaton (4.3): S ' S P S' transton rate from S' to S P S transton rate out of S (6.3.4)
8 6.3 / 8  Communcaton Networks II (Görg) SS In addton, the condton of total probablty holds P S. For each transt (=routng) sub chan, the followng system of equatons hold for the unknowns : e r er p0; r ejs pjs; r (, r) E, e r js E k k S (6.3.42) where s nterpreted to be the relatve arrval rate of jobs n class r at staton. These equatons correspond to flow equatons n Jackson networks. When the sub chan E k s closed, then p 0 ;, r 0 for all, r E k and the number of jobs s constant. E k Theorem of BCMP For open, closed or partally closed networks, n whch each staton s of types to 4, the dstrbuton of the statonary state probabltes take the product form gven below, when they satsfy the system of equatons descrbed above:, 2, K, PS S S S GK ds F S N N (6.3.43)
9 6.3 / 9  Communcaton Networks II (Görg) SS where G(K) s the normalzaton constant, so that all the probabltes sum up to. N G( K) d( S) F ( S ), wth K K, K2, L, K R, K j s the total number N S K of jobs n class j n the network, j 2,, L, R. d(s) s a functon of the number of jobs n the network, and t s gven by: d S K S at arrval process a) 0 m K S E j at arrval process b) j =0 at closed network * : holds for open networks * * (6.3.44) F ( S)s a functon, that depends on type and state of the staton. The explct presentaton of the functons F ( S) not gven here, as ths product form soluton contans more detaled nformaton than often requred, e.g. the sequence of job classes n the queue of each staton. It also contans
10 6.3 / 0  Communcaton Networks II (Görg) SS more nformaton than meanngful, e.g. the probablty for crossng of each phase of the Coxmodel. The two smplfed versons shown below, descrbe only the boundary dstrbutons, that result once the states that are not nterestng es are added together. e These smplfcatons s make t easy to calculate especally the normalzaton constant G(K) and allows to make nterestng conclusons. BCMP  erson We defne a summarzed state for a closed BCMP network, namely the number of jobs n each class at each staton: S ( S, S2,... S N ) wth S k, k, K, k 2 R (6.3.46) (6.3.47) where the number of jobs n class r n staton s denoted by k r. Under these condtons the statonary state probablty dstrbuton s gven by:
11 6.3 /  Communcaton Networks II (Görg) SS P S d S g S G K where t holds: N wth GK g S N ( ) ( ) N S K g S k R k r k! e Type r r b ( k ) k r! k r R e r k! Type 2, 4 r k r! r k r R e r Type 3 r k! r r R (6.3.49) wth b( k) obtaned from the equaton (6.2.28). k kr gves here the r total number of jobs belongng to all the classes n staton. The vstng frequences e r s determned from equaton (6.3.42) wth p 0, r 0. We can see, that P(S) depends only on the expected values of the servce tme dstrbuton. b t
12 6.3 / 2  Communcaton Networks II (Görg) SS BCMP erson 2 We consder an open BCMP network wth the lmtaton, that the parameters of the arrval process and the servce rate of the statons reman constant, that s ndependent of the queue length. S k, k 2, K, k N s the network state, where s the total number of jobs belongng to any class at staton. The statonary dstrbuton of state probabltes s now k pk PS N (6.3.50) wth p ( k ) k ( ) Typ, 2, und 4 ( n ) k e Typ 3 k! (6.3.5)
13 6.3 / 3  Communcaton Networks II (Görg) SS and s the load at the staton wth R r r (6.3.52) r L r er r er r r r Typ ( n ) Typ 2, 3, 4 ;! (6.3.52a) Statons of types, 2 and 4 behave as N ndependent M/M/ models. Type 3 as an M/G/ model. For type statons wth more than one server n, equatons (6..6) and (6..9) can be used to calculate the probabltes p( k).
14 6.3 / 4  Communcaton Networks II (Görg) SS Results From the statonary state probabltes we can determne the followng performance measures as expected values: Load at ndvdual statons Throughput at each staton, for each of the job classes mean queue length L for each class and staton t L Q mean watng and system tme for each class and staton The models can be very complcated n these cases. In order to obtan numercal values for the statonary state probabltes, frst G(K) must be calculated. Ths needs complcated calculatons, because the total state space should be taken nto consderaton. In the lterature there are correspondng proposals, that dffer n algorthm and thus also n the volume of mathematcal operatons. However, such algorthms are not handled here.
15 6.3 / 5  Communcaton Networks II (Görg) SS For the calculaton of the throughput, system load and the mean system tme we also need the normalzaton constants for dfferent numbers of jobs n the network, n addton to,, K, G K G K K2 K R a) Throughput In a closed network wth K r class r jobs r, K,, r e r GK, K, K R GK K K K. R s the throughput of class r jobs at staton. The soluton of the system of equatons for the chans s here denoted by e r. If the chan s open, the results from the Jackson network contnues to hold. (6.3.53) For state ndependent,.e. constant servce rates of all 4 types, t holds: r r (6.3.55) r wth r beng the throughput as shown above. For statons consstng of n > dentcal servers wth constant servce rates, the followng holds
16 6.3 / 6  Communcaton Networks II (Görg) SS r r n r (6.3.56) r where s determned d from the above equaton for statons t of type 2. For statons of type or type 4 t s necessary to go through the relatonshp for. b) Mean system tme, watng tme and queue length r The mean system tme for class r jobs n staton s the mean tme span from the arrval of job to staton to ts completon at the same staton. r s the throughput of class r jobs at staton and L r s the mean number of class r jobs at staton. From Lttle s law, we get: r Lr r (6.3.57) If staton s of type 3 and has a server for each of the class r jobs, then the jobs don t have to wat there. Then /. From equaton ( ) follows: L r r r r r
17 6.4 / 7  Communcaton Networks II (Görg) SS r L r G K, K, K G K, K, K, K, K e R r R r (6.3.58) Thus the mean watng tme works out to: W r r r (6.3.59) From ths, usng Lttle s law, we get the mean number n system to be: L W Q r r r 6.4 Mean value analyss Applcable for the models, that have product form solutons, especally ll for the closed networks wth many chans h and djb jobs Calculaton of normalzaton constants become redundant Calculaton of product terms s not requred
18 6.4 / 8  Communcaton Networks II (Görg) SS As the product terms are not calculated, followng changes n comparson to the earler methods become evdent: State probabltes bl and the mean values can be obtaned. Dsadvantage: large memory requrements compared to product form soluton usng sutable algorthms. All ntermedate t results need to be smultaneously present n determnaton of the fnal results. Applcable also for nonproduct form networks. The results are accurate only asymptotcally, are usually good n closed networks. Smplfcaton: recursve soluton s replaced by an teratve soluton Smplfcaton: recursve soluton s replaced by an teratve soluton. Ths leads to a smple calculaton and to good results after a relatvely small number of teratons.
19 6.4 / 9  Communcaton Networks II (Görg) SS Ths recursve technque s llustrated here for the smple case of a sngle server (no parallel servers) per staton. Three equatons buld the base for ths: k L k k (6.4.60) k k k N (6.4.6) L k k k (6.4.62) The second equaton s obtaned by applyng Lttle s law n general ( L ) and the thrd equaton s also from Lttle, as appled to the th staton.
20 6.4 / Communcaton Networks II (Görg) SS Further: k k k number of avalable jobs n network, k=,2,...,k mean sojourn tme (response tme) n staton for k jobs n network. Servce rate of staton as a functon of the number of jobs L L k mean number of jobs n staton, =,2,...,N mean number of jobs n staton for the case, that there s one job less n the network k throughput accordng to Lttle for k crculatng jobs N k total system tme for statons, that are n a sngle chan, n case of k jobs n network From these three equatons, we can buld the recursve algorthm for calculaton of. We always start wth. k, k und L k L 0 0 We frst begn for k= and proceed one by one up to k=k.
21 6.4 / 2  Communcaton Networks II (Görg) SS Example 64: 6.4.: The followng cyclc network s gven wth 2s, 2 3s and K=3. Fgure 6.7: Cyclc queung network. Iteraton. k= L L L () () L
22 6.4 / Communcaton Networks II (Görg) SS Iteraton 2. k=2 2 L L ( 2) ( 2) L L Iteraton 3. k= L L ( 3 ) ( 3 ) L L
23 6.4 / Communcaton Networks II (Görg) SS Applcaton for more general networks The recursve method presented above can be appled to more general networks through some modfcatons to the three basc equatons. Then the followng equatons become vald: k L k type, 2, 4 type 3 ( n ) (6.4.63) e N e p j j j (6.4.64) k Gk ( ) k k N G ( k ) ( k) L k k k (6.4.65) (6.4.65a) (6.4.66) In addton to the already ntroduced quanttes followng quanttes are also needed: e total servce tme at staton
24 6.4 / Communcaton Networks II (Görg) SS e mean number of vsts made by a job to staton between two consecutve vsts to any other staton * These three equatons, along wth the same set of ntal assumptons L ( 0) 0, G( 0) allow recursve calculatons l for any network. After completon of the teraton, we get the normalzng constant G(K), that, makes the calculaton of state probabltes possble usng equaton (6.2.29) Pk ( ) GK ( ) N k. Now let us take R dfferent closed streams (Routng chans) n place of one. However, there s the lmtaton, that the jobs cannot change ther class wthn the chan. That means, jobs n a chan always have at staton the same servce tme dstrbuton (wthout loosng the generalty!). After completon of chan ndces r, we get the recursve equatons n the followng form (r=,2,...,r): r r ( k ) r L k br for statons of type, 2, 4 ( n= ) r for statons of type 3 k r r Q( r) ( k) N k r r ( k) (6.4.67) (6.4.68)
25 6.4 / Communcaton Networks II (Görg) SS L k ( k) ( k) r r r L ( k ) = L ( k) L ( k) rr() r R r r (6.4.69) (6.4.70) It also holds r k mean system tme of a job n chan r at staton between two consecutve vsts of an arbtrarly selected staton * L ( k b ) r mean number of jobs n staton wth one job less n chan r L L r ( k ) ( k ) mean number of jobs n chan r n staton wth populaton k n network mean number of jobs n staton wth populaton k n network r e r r total servce tme of a job n chan r at staton e N e p j, compare e, relatve arrval rate of jobs n chan r at r jr jr r
26 6.4 / Communcaton Networks II (Görg) SS k b k, k,, k,, k r 2 r R e r staton, can be obtaned from ths equaton apart from a proportonalty constant; no change of classes! K K populatons vector from network wth one job less n chan r r k Q(r) R() r =,2,...,R throughput n chan r wth populaton k n network set of statons vsted by chan r set of chans that vst staton countng ndex for chans
27 6.4 / Communcaton Networks II (Görg) SS Example A closed network wth N = 4 statons and R = 2 classes. Jobs are not allowed to change ther classes. Fgure 6.8: Closed network of queues.
28 6.4 / Communcaton Networks II (Görg) SS staton type 2 M/G/PS 4 M/G/LCFSPRE 4 M/G/LCFSPRE 3 M/G/no watng r r r= r= N, From the system of equatons er e p (mean number of vsts of a j jr jr r class r job to the th staton) we determne:
29 6.4 / Communcaton Networks II (Görg) SS Class : Class 2: e e p e p e p 2 2; 3 3; 4 4; e2 e2 p2;2 e22 p22;2 e32 p32;2 e42 p42;2 e e p 2 ; e e p ; e e p 3 ; e e p ; e4 e p ; e e p. Iteraton k, k2 0 Class : Mean system tme e r,0 wth r r,0 k0,0,0 2 k 20,0,0 3 k 30,0, ; wth k ,0 0
30 6.4 / Communcaton Networks II (Görg) SS Throughput: Mean number of jobs: 0, r L L L L 0, , , 0, 0, 022., 0, 0, Iteraton 2 k 0, k Class 2: Mean system tme , 2 0, , , 4. 8
31 6.4 / 3  Communcaton Networks II (Görg) SS Throughput: 2 Mean number of jobs: L L L L 0, , , r 0, 0, 0, , 0, 0, Iteraton 3 k ; k 2 ;, (wg.typ 3 Staton) 24. ;, L 0, 695. ;, L 0, , L 0, 87. ;, L 0, , L 0, 4. 03;, L 0, , (wg.typ 3 Staton) ,, ; 2,,
32 6.4 / Communcaton Networks II (Görg) SS L L,,, ; L,,, L,,, ; L,,, L,,, ; L,,, L Iteraton 4 k 2 ; k 0,, ;,,, , 0 L, ; 2, 0 L, , L 0, 448. ; 20, , 4 20, L 20, 20, 20, ; L 20, 20, 20, L 20, 20, 20, 09. ; L 20, 20, 20, Iteraton 5 k 0, k , L 0, ; 02, L 0, , 2 L 0, ; 0,
33 6.4 / Communcaton Networks II (Görg) SS , , 05.,,, ;,,, L 02, 02, 02, ; L 02, 02, 02, L 02, 02, 02, ; L 02, 02, 02, Iteraton 6 k, k 2; 2 2 (, ) 2 2 (, ) = L 2 ( 02, ) =,353; 2 2 = 2 L (,) L 2 (,) = 2,577 (, ) 3 2 = L ( 02, ) = 2,65; 22 2 = 22 L 2(,) L 22 (,) =2, (, ) (, ) = L ( 02, ) = 5,07; 32 2 = L (,) L (,) = 5, (, ) (, 2) = = 2,4; 42 (, 2) = 42 = 4,8 4 (, 2 ) 2 = 4 = 0,094; 094; = = 03 0,3 2 (, ) 2 (, 2) 4 2(,2)
34 6.4 / Communcaton Networks II (Görg) SS L (,) 2 L 2 (, 2 ) L 3 (,) 2 3 =,2) (,2) = 0,237; L 2 (,) 2 = = 2 2 0,34 ( (,2) ( 2 (,2) (,) 2 (,) 2 2(,2) 22 (,2) = = 09 0,979; 9 L 22 (,) 2 = =0, = ( 2, ) 3 ( 2, ) = 0,4588; L 32 ( 2, ) = 2 ( 2, ) 32 ( 2, ) = 0,68 L 4 (,) 2 (,) 2 (,) 2 = (,) 2 4 (,) 2 = 0,29; L 42 (,) 2 = 2 42 = 0,62 0,9994 And so on, tll the requred number of jobs for each chan s reached. Ths procedure s applcable only for cases wth a lmted number of chans and jobs per chan. Otherwse, the memory requrement and the calculaton R effort become too heavy. Both these are of the order k r r. The calculaton effort s as same as that s requred n soluton of the product form expresson wth applcaton of effcent algorthms. Here, we have no accuracy problems, n contrast t to the other case. 2
35 6.4 / Communcaton Networks II (Görg) SS20 
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