MECS 15. Queuing Networks - : Modeling and Evaluation of...

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Unformatted text preview: : Modeling and Evaluation of Computer Systems (MECS) :15 : (Mohammad Abdollahi Azgomi) azgomi@iust.ac.ir : (feed forward QN) (Burke's theorem) (feedback) (Jackson theorem) : CTMC (Gordon and Newell theorem) MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 2 (QN: queueing networks) . (open) . (closed) (open class) . (open QN) . : 1 CPU 2 MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 3 . (closed classes) . (closed QN) . (hybrid QN) (client/server) - : LAN 1 CPU 2 MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 4 . : . (load) (multi-programming) . . . - . . MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 5 . (circulate) . 2 B 1 A D C . 1 2 A C B D : B A . MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 6 (feed forward QN) . (two tandem queues) : 2 1 A B B A . CTMC MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 7 . CTMC : (NA(t), NB(t)) ... - MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 8 : - 0,0 1,0 2 2,0 1 2 2 2 ... ... i,0 1 2 ... 2 1 2 0,1 2 1,1 1 i-1,1 ... i,j-1 i+1,j-1 2 1 0,2 ... ... 2 1 1 0,i 2 ... i-1,j i,j 1 2 i+1,j i-1,j+1 i,j+1 MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 9 (N1(t), N2(t)) : (i, j) = P(N1=i, N2 =j), where Ni = Lim t Ni(t) : (0, 0) = 2(0, 1) <= (+ 2)(0, i) = 2 (0, i+1)+1 (1, i-1) <= (+ 1)(i, 0) = (i-1, 0)+2 (i, 1) <= (+ 1 + 2)(i, j) = 1 (i+1, j-1) + 2 (i, j+1)+ (i-1, j) <= i , j >= 0 (i, j ) = 1 10 MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE : (i, j ) = (1 - 1 ) 1i (1 - 2 ) 2 j i = /i i, j 0 : . MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 11 = : CTMC : (Burke's theorem) M/M/1 : . . . MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 12 = : k : (n1 , n2 ,..., nk ) = P( N1 = n1 , N 2 = n2 ,..., N k = nk ) = (1 - i ) i n i =1 k i ni 0, 1 i k : (Ni) (N) N = k i =1 Ni (Li) (L) : M/M/1 Li . L = L i =1 k i = k i =1 i - i i : w L = w => w = L/ MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 13 : () : L = * w : :L ( ) :w MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 14 . (feedback) (1) . (2) . (2) 1=8 r1= 4 1 0.5 0.25 2=10 r2= 5 2 MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 15 . k . i . ri . (feedback) (routing probability) (branching probability) : . j i :pi,j : i pi , j 1 P(customer leaves system after i) = 1 - pi, j j =1 j =1 k k MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 16 j j : (traffic equation) j = rj + i pi , j i =1 k 1 j k . (j) : (n1 , n2 ,..., nk ) = P( N1 = n1 , N 2 = n2 ,..., N k = nk ) ni 0 . M/M/1 MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 17 . CTMC (Jackson) M/M/1 . : . . . . . MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 18 :(Jackson's theorem) : (n1 , n2 ,..., nk ) = (1 - i ) i n i =1 k i ni 0, 1 i k . . MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 19 . 1=8 r1= 4 1 : 0.5 0.25 2=10 r2= 5 2 : 1 = r1 + 2 p2,1 2 = r2 + 1 p1, 2 1 = 4 + 0.252 2 = 5 + 0.51 1 = 6 2 = 8 MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 20 2 /2 = 8/10 < 1 1 /1 = 6/8 < 1 : . : (n1 , n2 ) = L = Li = i =1 i =1 2 2 13 1 14 44 55 n n2 i - i i = 3+ 4 = 7 w = L / = 7 / 9 = 0.78 MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 21 (1) : (2) w(1) = w1 + 0.5w( 2 ) w( 2 ) = w2 + 0.25w(1) w(1) = 1 /(8 - 6) + 0.5w( 2 ) w( 2 ) = 1 /(10 - 8) + 0.25w(1) w(1) = 1 /( 1 - 1 ) + 0.5w( 2 ) w( 2 ) = 1 /( 2 - 2 ) + 0.25w(1) w(1) = 0.5 + 0.5w( 2 ) w( 2 ) = 0.5 + 0.25w(1) w(1) = 0.86 w( 2 ) = 0.71 MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 22 i : i /i < 1 . MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 23 : . M/M/m . M/M/m/k (PS) . BCMP (extensions) . MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 24 . K : IO1 CPU IO2 K . MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 25 M N : . i . (i, j M) pi,j vi (visit ratio) i ( ) . : vi = v j p j ,i j =1 M i = 2,..., M vi . vi (v1=1 :) MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 26 : i vi = j vj 1 i, j M . MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 27 CTMC CTMC : (N1(t), N2(t), ..., NM(t)) : (N1, N2, ..., NM) = Limt(N1(t), N2(t), ..., NM(t)) : (n1 , n2 ,..., nM ) = P( N1 = n1 , N 2 = n2 ,..., N M = nM ) ni 0, . MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 28 : 2 2 1 1 p1 P2=1- p1 3 3 N=2 CTMC (N1, N2, N3) : MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 29 : CTMC 2 1,1,0 p11 2,0,0 3 p21 1,0,1 p21 3 2 p11 p21 3 0,0,2 0,1,1 0,2,0 p11 2 MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 30 : (p1+p2)1(2, 0, 0) = 2(1, 1, 0)+3(1, 0, 1) (p1+p2)1(1, 1, 0) = 2(0, 2, 0)+3(0, 1, 1) (p1+p2)1(1, 0, 1) = 2(0, 1, 1)+3(0, 0, 2) p11(1, 1, 0) = 2(0, 2, 0) (2+ 3) (0, 1, 1) = p21(1, 1, 0)+ p21(1, 0, 1) p21(0, 0, 2) = 3(0, 0, 2) (2, 0, 0)+(1, 1, 0)+(1, 0, 1)+(0, 2, 0)+(0, 1, 1)+(0, 0, 2)=1 ( ) . MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 31 . (Gordon and Newell) : 1 M vi (n) = G ( N ) i =1 i ni n 0 , ni = N i =1 M : n = ( n1 , n 2 ,..., n M ) : G(N) (n) = 1 MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 32 . . 3=2 2=3 1=2 . p2=1/4 p1=3/4 : v2 = v1 p1 v3 = v1 p2 V2=3/4 V3=1/4 : V1= 1 : (n) = 1 (1 / 2) n1 (1 / 4) n2 (1 / 8) n3 G( N ) MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 33 : G(N) ( n ) = 1 1 (1 / 2) 2 (1 / 4) 0 (1 / 8) 0 G( N ) 1 (1,1,0) = (1 / 2)1 (1 / 4)1 (1 / 8) 0 G( N ) 1 (1,0,1) = (1 / 2)1 (1 / 4) 0 (1 / 8)1 G( N ) 1 (0,2,0) = (1 / 2) 0 (1 / 4) 2 (1 / 8) 0 G( N ) 1 (0,1,1) = (1 / 2) 0 (1 / 4)1 (1 / 8)1 G( N ) 1 (0,0,2) = (1 / 2) 0 (1 / 4) 0 (1 / 8) 2 G( N ) (2,0,0) + (1,1,0) + (1,0,1) + (0,2,0) + (0,1,1) + (0,0,2) = 1 (2,0,0) = G(N) = ? MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 34 Di=vi/i . (service demand) . MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 35 . WinPEPSY . . MECS#15 -Queueing Networks - By: M. Abdollahi Azgomi - IUST-CE 36 ...
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