1. Queuing Theory III

2. 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 ( )

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

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

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

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

7. M/M/2 Queue Rate Transitions
l l l l l l l 1 2 3
n-1 n
m m m m m m m L p jp q j = + å 1 2 3 4 . j p = å 2 ( ) l m

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

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

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

11. M/M/s Queue Rate Transitions
l l l l l l l 1 2 3
n-1 n
m m m m sm sm sm p 1 = ( ) l m p n s = - 1 ! ( ) , l m

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

13. 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.

14. 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

15. 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

16. 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

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

18. Example

19. Example
If Po = 1, P1=(l/m)Po
= 1.5

20. Example
If P1 = 1.5, P2=(l/2m)P1
= 1.125

21. Example

22. Example

23. Example

24. Example