MECS 14. Single Queuing Models

MECS 14. Single Queuing Models - Modeling and Evaluation of...

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Unformatted text preview: : Modeling and Evaluation of Computer Systems (MECS) :14 : (Mohammad Abdollahi Azgomi) [email protected] M/M/1 M/M/m M/M/ M/M/1/B M/M/m/B M/M/m/m MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE 2 . - . . MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE 3 : - M/M/1 M/M/m M/M/ MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE 4 M/M/1 M/M/1 : M/M/1 . . . . FCFS . CTMC . (n) : CTMC - 0 1 2 n 5 MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE M/M/1 . =/<1 M/M/1 .- (traffic intensity) 1 . . <1 . n n n pn . MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE 6 M/M/1 : <1 0 1 2 n 0 = 1 0 + 2 = ( + ) 1 ... n -1 + n+1 = ( + ) n ... . : i =1 i=0 MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE 7 M/M/1 : 0 = 1 1 = / 0 = 0 0 + 2 = ( + ) 1 2 = 2 0 ... n -1 + n +1 = ( + ) n n = n 0 , n = 1,2,... 0 + 1 + 2 + ... + n + ... = 1 0 + 0 + 2 0 + ... + n 0 + ... = 1 0 = 1 1 + + + ... + n + ... 2 : : <1 0 = 1 = 1- 1 /(1 - ) 8 MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE M/M/1 : n ... 2 1 1 = 0 = (1 - ) 2 = 2 0 = 2 (1 - ) ... n = n 0 = n (1 - ), n = 1,2,... L n ... 2 1 0 : L = i i = i i (1 - ) L = i =1 i =1 1- L . MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE 9 M/M/1 : w L = w w = L / w= (1 - ) / w= 1 1 = (1 - ) - : LQ LQ = (i - 1) i = (i - 1) i (1 - ) LQ = i =1 i =1 2 1- : wQ LQ = wQ wQ = LQ / wQ = 2 / wQ = = ( - ) (1 - ) (1 - ) 10 MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE M/M/1 : M/M/1 = / , L= n = n (1 - ) = 2 2 = - 1- ( - ) 1 - 1 1 w= = , wQ = = - (1 - ) ( - ) (1 - ) , LQ = MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE 11 M/M/m M/M/m : M/M/m . . m . . FCFS : 1 2 m 12 MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE M/M/m . CTMC . (n) : CTMC - 0 1 2 2 3 m m m m 1 m - i i . (state-dependent) . m .(i=1...m ) MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE 13 M/M/m . =/m<1 M/M/m : <1 0 = 1 0 + 2 2 = ( + ) 1 ... n -1 + ( n + 1) n +1 = ( + n ) n , n < m n -1 + m n +1 = ( + m ) n , n m i=0 i =1 : 0 -1 m -1 ( / ) n m 1 m 0 = + n =0 n! m! m - MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE 14 M/M/m M/M/m 0 : = / m m -1 ( / ) n m 1 m 0 = + m! m - n =0 n! L = m + w= L (m ) m +1 0 m(m!)(1 - ) 2 -1 1 m wQ = w - LQ = wQ MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE 15 M/M/2 :1 ( ) . ( )( ) /2 1 /2 2 . (w) . . 16 MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE M/M/2 :2 M/M/2 . . (heterogeneous) : 1 2 CTMC . MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE 17 M/M/ M/M/ : M/M/ . . . . (self-service ) ( ) . FCFS : CTMC 0 1 2 2 3 n n (n+1) MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE 18 M/M/ : < 0 = 1 0 + 2 2 = ( + ) 1 ... n -1 + ( n + 1) n +1 = ( + n ) n , n = 1, 2 ,... (LBE: local : ( ) balance equations) n -1 = n n , n = 1, 2 ,... = 1 ( ) n n -1 n n = 0( n 1 ) n! 19 MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE M/M/ : 1 0 + = 1 0 + 0 ( )n = 1 n! n =1 n =1 n i= 0 i = 1 1 0 = 1 + ( ) n = e n =0 n! -1 - . e 1 L = n n = n 0 ( ) n L = n! n =1 n=0 : L L = w w = 1 / LQ = wQ = 0 : w : . MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE 20 : - M/M/1/B M/M/m/B M/M/m/m . (reject) (rejection . probablility) MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE 21 M/M/1/B M/M/1/B : M/M/1/B . . . . FCFS (B) : CTMC 0 1 2 B MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE 22 M/M/1/B : 0 = 1 0 + 2 = ( + ) 1 ... n -1 + n +1 = ( + ) n , n = 1,2,..., B - 1 B -1 = B i=0 B i =1 : ( ) n -1 = n , n = 1, 2 ,..., B n = 0( n ) 23 MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE M/M/1/B : 0 + = 1 0 + 0 ( )n = 1 n =1 n =1 B n B i= 0 i =1 B 0 = 1 + ( ) n n =1 -1 . B = 1 + ( ) n ( ) B n =1 B -1 : : (e) e = (1 - B ) . 24 MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE M/M/m/B M/M/m/B : (m<B) M/M/m/B . . . m . FCFS (B) : CTMC 0 1 2 2 3 m m m m B MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE 25 M/M/m/B : 0 = 1 0 + 2 2 = ( + ) 1 ... n -1 + (n + 1) n +1 = ( + n ) n , n = 1,2,..., m - 1 n -1 + m n +1 = ( + m ) n , n = m,.., B - 1 B -1 = m B i=0 B i =1 MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE 26 M/M/m/B : =/m (m ) n 0 , n = 1,2,..., m - 1 n = n n!m m 0 , n = m, m + 1,..., B m! : (1 - B - m +1 )( m ) m m -1 ( m ) n 0 = 1 + + m!(1 - ) n! n =1 -1 1 . MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE 27 M/M/m/m M/M/m/m : M/M/m/m . . . m . (m) . . : CTMC 0 1 2 2 3 m m MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE 28 M/M/m/m : 0 = 1 0 + 2 2 = ( + ) 1 ... n -1 + (n + 1) n +1 = ( + n ) n , n = 1,2,..., m - 1 m -1 = m m i=0 m i =1 : (LBE) n -1 = n n , n = 1, 2 ,..., m n = 0( n 1 ) n! 29 MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE M/M/m/m : m 1 0 = ( ) n n! n =0 ( )m -1 m = : 1 n! m! ( ) n n =0 m . (Erlang B Formula) B . MECS#14 - Solution of Single Queueing Models - By: M. Abdollahi Azgomi - IUST-CE 30 ...
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