Queuing Theory III

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/2 Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m 2m 2m 2m 2m 2m 2m In steady state, rate at which process enters a state = rate at which a process leaves a state state 0 l m p 1 =

M/M/2 Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m 2m 2m 2m 2m 2m 2m In general, P 1 = ( ) l m P n 2 = ( ) l m r

M/M/2 Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m 2m 2m 2m 2m 2m 2m In general, P 1 = ( ) l m P o = - + 1 r P n 2 = ( ) l m r

M/M/2 Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m 2m 2m 2m 2m 2m 2m P 2 1 = + - l m r ( ) [ ]

M/M/2 Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m 2m 2m 2m 2m 2m 2m L P jP q j = + ¥ å 1 2 3 4 . j P = ¥ å 2 ( ) l m

M/M/2 Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m 2m 2m 2m 2m 2m 2m L q j P = ¥ å 2 ( ) l m L j P q = ¥ å 2 ( ) r Aside

M/M/2 Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m 2m 2m 2m 2m 2m 2m L P j q = ¥ å 2 ( ) r P = - 2 1 ( ) r

M/M/2 Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m 2m 2m 2m 2m 2m 2m L P q = - 2 1 ( ) r W L q = / l W q = + m 1 L W q = + / l m 1

M/M/s Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m 2m 3m 4m sm sm sm P 1 = ( ) l m P n s = £ ³ - 1 ! ( ) , l m

M/M/s Queue Rate Transitions l l l l l l l 1 2 3 n-1 n m 2m 3m 4m sm sm sm L P s q = - 2 1 ( ) ! r W L q = + / l m 1

Example A 2 server queue has Poisson arrivals at rate l=1/minute. Service times are exponential with mean 45 seconds. Determine the average number in the queue, and the probability that one or more servers will be idle.

Example P 75 5 375 4545 = + - [ ( ) ] . (. l m r l m r = 1 4 3 2 8 , / A 2 server queue has Poisson arrivals at rate l=1/minute. Service times are exponential with mean 45 seconds. Determine the average number in the queue, and the probability that one or more servers will be idle. l m r = 1 4 3 2 8 , / P 2 1 75 5 375 4545 = + - [ ( ) ] . (. l m r

Example P 4545 = . P 75 4545 341 = ( ) (. )(. . l m l m r = 1 4 3 2 8 A 2 server queue has Poisson arrivals at rate l=1/minute. Service times are exponential with mean 45 seconds. Determine the average number in the queue, and the probability that one or more servers will be idle. l m r = 1 4 3 2 8 , / P 4545 = . P 1 75 4545 341 = ( ) (. )(. . l m

Example P 4545 = . L P = - 1 4545 375 1227 ( ) . )(. r l m r = 1 4 3 2 A 2 server queue has Poisson arrivals at rate l=1/minute. Service times are exponential with mean 45 seconds. Determine the average number in the queue, and the probability that one or more servers will be idle. l m r = 1 4 3 2 8 , / P 4545 = . L P q = - 2 1 4545 375 1227 ( ) . )(. r

Example Suppose we have a feeder providing parts at rate l=2. Each station operates at m = 4/3 parts per min. Buffer = 5 N = 5+2 = 7.

Example

Example

Example

Example

Example