Queuing Theory II.

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Presentation transcript:

Queuing Theory II

Model p = - ( ) l m L = - l m L = - l m ( ) W = - m l 1 W = - l m ( ) n = - ( ) l m L = - l m L q = - l m 2 ( ) W = - m l 1 W q = - l m ( )

M/M/1 Queue Finite Capacity 1 2 3 N-1 N State Balance Eq. 1 N l m p 1 = l m p 2 1 + = l p N 1 - m p N =

M/M/1 Queue Finite Capacity 1 2 3 N-1 N Now, p n N = å 1 p n N = å 1 ( ) l m

M/M/1 Queue Finite Capacity 1 2 3 5 6 Workstation receives parts form a conveyor. Station has buffer capacity for 5 parts in addition to the 1 part to work on (N=6). Parts arrive in accordance with a Poisson process with rate of 1 / min. Service time is exp. with mean = 45 sec. (m = 4/3).

M/M/1 Queue Finite Capacity

M/M/1 Queue Finite Capacity

M/M/1 Queue Finite Capacity

M/M/1 Queue Finite Capacity L Lq

M/M/1 Queue Finite Capacity 1 2 3 N-1 N p n N = å 1 ( ) l m Recall, x n N = + å - 1

M/M/1 Queue Finite Capacity 1 2 3 N-1 N p n N = å 1 ( ) l m 1 = - + p N ( ) l m

M/M/1 Queue Finite Capacity 1 2 3 N-1 N 1 = - + p N ( ) l m m p N 1 = - + ( ) l

M/M/1 Queue Finite Capacity 1 2 3 N-1 N m p N 1 = - + ( ) l m p n N 1 = - + ( ) l

M/M/1 Queue Finite Capacity 1 2 3 N-1 N m L np n N = - + å 1 ( ) l

M/M/1 Queue Finite Capacity 1 2 3 N-1 N Miracle 37 b L N = + - l m [ ( ) )( ] )[ 1

M/M/1 Queue Finite Capacity 1 2 3 5 6 Workstation receives parts form a conveyor. Station has buffer capacity for 5 parts in addition to the 1 part to work on (N=6). Parts arrive in accordance with a Poisson process with rate of 1 / min. Service time is exp. with mean = 45 sec. (m = 4/3). ( ) . l m = 75

M/M/1 Queue Finite Capacity 1 2 3 5 6 ( ) . l m = 75 l = 1 N = 6 m = 4/3 = 1.33 L = + - 1 6 75 7 33 92 [ ( . ) )( ]

Little’s Revisted L W = * . ??? l 1 96 W L = l 1 92 . L W = l 1 21 . m 3 1 2 5 6 W L = l 1 92 . L W q = l 1 21 . m W q = + 1 96 . L W = * . ??? l 1 96

Little’s Revisited l l l l l l 3 1 2 5 6 l = ¥ å n p M / 1 l p n = ¥ å

å Little’s Revisited l = p M / 1 6 l p ( ) 1 = + - l l l l l l 3 1 2 5 3 1 2 5 6 l = ¥ å n p M / 1 6 l p ( ) 1 2 3 4 5 6 = + -

Little’s Revisited l l l l l l 3 1 2 5 6 l = - 1 051 949 6 ( ) . p

Little’s Revisited l l l l l l 3 1 2 5 6 l = 949 . l L = 2 025 . W

Little’s Revisited l = 949 . W = 2 025 . W = - 1 2 025 75 275 . m l l l l l l 3 1 2 5 6 l = 949 . W = 2 025 . W q = - 1 2 025 75 275 . m

Little’s Revisited l = 949 . W = 2 025 . W = 1 275 . l L W = 949 1 275 l l l l l l 3 1 2 5 6 l = 949 . W = 2 025 . W q = 1 275 . l L W q = 949 1 275 210 . ( )