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1 G= V E V={ 2 } E V V 5- servce center CPU I/O E 83

2 5 routng probablty sngle class queueng networks CPU I/O CPU I/O mult-class mult-chan 5.. (open networks) (closed networks) (mxed networks)

3 FCFS PS - LCFS-PR IS n (n) (0) =0 n 0 (n) 0 a sngle server fxed rate SSFR (n)= 85

4 5 b IS (n)= n c queue length dependent QLD (n) a b c FCFS 5.2 = 2 feedforward networks Jackson Jackson Jackson networks 5. Jackson = 2 Jackson QLD 86

5 (n) Jackson S = {(n n ) n 0} n (N N ) n (n) = (n n ) = P (N = n N =n ) Q=(q = ) q q 0 = q 5-2- = q 0 0 Q 0 = 2 q 0 0 q 0 γ = = γ 87

6 5 n (n ) (n) q 5-2 = 2 λ = γ + λ q = = γ = λq ={ } Q I Q I Q { } 88

7 q γ q λ = q 5-2- // Jackson Jackson 89

8 5 5. Jackson Jackson a b (n) n=(n,n 2,,n ) n Jackson 2 η = ( n,n,n ) = ( n ) η L n λ ( ) ( ) ( ) η n = η n µ = Jackson // //N (n ) // (0) (n )(=,2,,) Jackson a b Jackson Jackson Jackson Jackson 90

9 // //N Jackson Jackson Jackson Jackson Lttle SSFR = =P{Q }= // L =E(N ) 9

10 5 q = = ρ L ρ q = E = E = ( N + N 2 + LN ) ( N ) + E( N ) + L+ E( N ) q = = = 2 ρ ρ Lttle T q =q/ γ = γ T q q = = λ µ ( ρ ) T w w ρ = = λ µ ( ρ ) w =q - q T,0 T, 0 = Tq + qt, = I Q T,0 92

11 5.2.4 v v v = v λ v = 5-2- γ 0 v 0 = v vst rato v = q 0 + = v q q 0 v 0 = v = = 0 v q =0,,,

12 5 q 00 =0 D / v D =v / = / = D {D =,2, } q = E [ N ] = L ρ γd = = ρ γd q T q v T q T q = v T q = µ D µ = D ( ρ ) γd Lttle T = γ D q q 94

13 T q T,0 IS IS L = = D T q = D T q =/ D =v / random observer property ROP Jackson Jackson Sevck tran Jackson (n) n n (n) 95

14 5 overtakng Jackson t (n) Gordon Newell 5.2 Gordon-Newell = 2 Gordon-Newell QLD n (n) K K 0 96

15 S L n = K 5-3- = = ( n,n2,,n ) n 0, S S K S K + = n = r n! ( n r)!r! S q = q =, =, 2, L, λ = λ q, =, 2, L, = (I Q)=0 I (I Q) I Q = 0 Q SSFR x =e / = 97

16 5 e e = e C -3-2 = 5.3. Gordon Newell 967 Jackson 5.3 Gordon-Newell K Gordon-Newell QLD () η ( n,n2, L,n ) = x ( n ) G = 98

17 x ( n ) n e =, n n ( ) = µ = G G = n S = ( ) x n = K e = ce ( =, 2, L, ) ( n) G η G ( n) = Gη( n) c n G = Gc η k η ( n) = η( n) { } SSFR convoluton algorthm QLD mean value analyss VA SSFR IS VA 99

18 5 K G=G K S=S K S m = 2 L n 0, n = n = ( m,n) ( n,n,,n ) G ( m,n) n e x, x = ( m,n ) µ = n S = m 0 n 0 n m =0 n m >0 ( ) G m,n = = n S n = 0 m n S m x + ( m,n ) = n S ( m,n ) m n n = n > 0 x + ( m,n ) = n S ( m,n ) k = n ( m) m = m = nm k k 0; k = n = S( m, n ) k m m x n x k G(m,n)=G(m-,n)+x m G(m,n-) m,n> G(m,0)=, m> G(0,n)=0, n (k) // 200

19 n =0 P ( N 0) = G (,K ) n S (,K ) n = 0 = = x n G = (,K ) G(,K ) ( \,K ) (,K ) G P( N = 0 ) =, =, 2, L, G G(\,K) K G ( \,K ) = n S (,K ) = y n y x = x, =, L, =, L, +, ( \,K ) (,K ) G ρ = G 20

20 k= 2 K P ( N k) = G x = G (,K ) n S (,K ) k = n k (,K ) m n ( ) = = m = n k n S (,K ) n k m S(,K-k) P ( N k) (,K k) G(,K ) k G = x, =, 2, L, 5-3- k= (,K ) G(,K ) G ρ = x G K (,K ) = G( \,K ) + x G(,K ),, K > 0 G (,K ) G(,K ) G λ = µ ρ = e Q (k) x n x m Q ( k) = P( N k) P( N k + ) k G(,K k) xg(,k k ) = x G(,K )

21 k=0 K = G =0 Q (k) Q ( k) n x = x = x G = x k (,K ) n S (,K ) = G(,K ) n S (,K k ) G n = k ( \,K k) G(,K ), k = n = 0 =, 2, L, ;k = 0, L,K n q (K) K q ( K ) = P( N = ) = P( N = ) I( k ) = = K K K k = = k P = k = K ( N = ) = P( N k) K k = K q ( K ) = x G (,K ) K k= k G (,K k), =, L, VA Lttle 203

22 T 2 v =Tv v v v 204

23 v v e v 0 v 0 =v a q ab 0 v 0 = v a =/q ab Lttle q = T q =Tv q = Tv T q, =,2,, K q = K = T = = v T T T q T = = K v T q IS T q =/ SSFR Y µ T q ( Y + ) = Y q 205

24 5 K K 5.4 (k n) k= n n K n A (n)= (K-,n) Y (K)=q (K-), K>0 = 2 K 0 T T q µ ( K ) = [ q ( K ) + ] ( K ) = = v T K q ( K ) q (K)=T(K)v T q (K) (0)=0 (0)=0 T q () T() () K=2 2+ K 206

25 O K = 2 Q (K) K D D / Q (K)= v T q (K) D =v / Q (K)=D [q (K-)+] T ( K ) = = K ( ) Q K q (K)=T(K)Q (K) Q(K) K Q ( K ) = v T ( K ) = Q ( K ) = q = Lttle ( K ) λ T v ρ = = = T µ µ = ( K ) D K T ( K ) T(K) mn (/ D ) 207

26 5 5.5 flow equvalent server FES FES FES FES sub-network complement sub-network FES T(k)(k= 2 K) (k)=t(k) QLD FES FES FES

27 n=0 x (n)= 0 {e } { (n )} x (n ) 5.5 Norton FES FES FES FES FES Chandy 975 Norton 5.6 BCP Jackson 209

28 5 I/O CPU BCP Baskett Chandy untz Palacos 975 BCP BCP 5.6. X X varaton coeffcent c 2 Var c = E [ X ] 2 [ X ] svc[x] 20

29 svc[x]= k X k k X k (Erlang-k dstrbuton) k k (convoluton) X (Laplace transform) f ( x) = k ( µ x) ( k ) µ µ x! e, f * ( x) = L[ f ( x) ] µ s = µ + k 5-6- s=0 f(s) k c svc=/k c k= c= k (Gamma dstrbuton) c 2 k X Hypoexponental dstrbuton 2

30 5 f * () s = k µ µ = + s c 3 X k 5-7 α = X Hyperexponental dstrbuton 5-7 X f k k µ α = = µ + µ x * ( x) = α µ e, f ( s) = s c k= c= Cox 22

31 4 Cox Cox Cox 955 method of stages X 5-8 b =0 k + P{X=0}= b 0 b 0 = A A =b 0 b b - b k =0 A k+ =0 5-8 Cox Cox X A ( b ) X X f * k 0 = = µ + () s = b + A ( b ) µ s X E [ X ] = A ( b ) = A ( b ) = = = = k k µ k ( A A + ) = = µ k = k = µ = A µ k k Cox Cox b = =0 k k Cox k=2 23

32 5 Cox Cox Cox Cox Cox Cox Cox Cox Cox BCP FCFS frst come frst served FCFS n (n) FCFS Cox 2 PS FCFS processor sharng PS n 0 (n) /n round robn 0 24

33 Cox 3 IS PS nfnte server IS IS Cox IS 4 LCFS-PR last come frst served pre-emptve resume LCFS-PR IS PS LCFS-PR Cox /G/ r(r=,2,,r) 2 {q r,s } r s r s s r r s nterchangeable 25

34 5 { 2 R} { 2 R} P P routng chan ( r) r {(,r) r r P} (K) K c (K c ) K c (K)= c (K c )= c e r e sq s,r + q0,r = er -6-6 (,s ) c 26

35 q 0,r q 0,r =0 e r e r q 0,r >0 e r γ c γ c q 0,r ( ) γ c e r BCP BCP Jackson BCP Cox Cox J r =Cox r A rs = r s rs = r s r = r 27

36 5 rs = r s r = r = n=(n n 2 n ) n = 2 (n,n 2, n )= η ( n ) ( ρ ) = ρ e ρ, n ρ, n! n FCFS PS LCFS PR IS R γer ρ = µ r = 2 n=(n n 2 n ) n =(n n 2 n r ) r = 2 r=,2,,r G ( K ) η ( n ) η ( n ) Lη ( n ) d 2 2 η( n,n2, L,n ) = G 28

37 d ( k) K k = 0 γ ( k) m c = c= K k = 0 γ c ( k),,, m η R e = r nr! = r µ n r ( n ) n!, FCFS PS LCFS PR r 5-6- R n r er ( n ), IS r= nr! r η = µ n = n r r FCFS r r FCFS / R r n r= µ µ n = 29

38 5 n ( ) = µ () 2 BCP 5.6 BCP n=(n n 2 n ) n = 2 FCFS n =(r,r 2, r n) r FCFS 2 PS 3 IS n =(n,n 2, n r ) n r =(n r,n r2, n rjr ) n rs r s LCFS-PR n ( r,s ),( r,s ), L, ( r, s ) = 2 2 n n r s n (n,n 2, n )=G - d(k) G d(k) e η =, FCFS η ( n ) n r µ = ( ) r ( n ) = n! R J n rs e = = r Ars, 2 PS s nrs! µ re r

39 R J n r rs e ( ) = = r Ars η = n, 3 IS r s nrs! µ re n Ar ( ) = s η n = er, 4 LCFS-PR µ rs Jackson BCP 22

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& &((. ) ( & ) 6 0 &6,: & ) ; ; < 7 ; = = ;# > <# > 7 # 0 7#? Α <7 7 < = ; < ! # %& ( )! & +, &. / 0 # # 1 1 2 # 3 4!. &5 (& ) 6 0 0 2! +! +( &) 6 0 7 & 6 8. 9 6 &((. ) 6 4. 6 + ( & ) 6 0 &6,: & )6 0 3 7 ; ; < 7 ; = = ;# > 7 # 0 7#? Α