1. Queuing Theory

2. M/M/1 Queue Rate Transitions1 2 3
n-1 n E t
times
process
enters
state n L
leaves ( ) # = - 1

3. M/M/1 Queue Rate Transitions1 2 3
n-1 n
lim ( ) t n E L - = 1
lim ( ) t n E L =
rate in = rate out

4. M/M/1 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 =

5. M/M/1 Queue Rate Transitions
l l l l l l l 1 2 3
n-1 n
m m m m m m m
State Balance Eq. l m p 1 =

6. M/M/1 Queue Rate Transitions
l l l l l l l 1 2 3
n-1 n
m m m m m m m
State Balance Eq. 1 l m p 1 = l m p 2 1 + =

7. M/M/1 Queue Rate Transitions
l l l l l l l 1 2 3
n-1 n
m m m m m m m
State Balance Eq. 1 n l m p 1 = l m p 2 1 + = l m p n 1 + = -

8. M/M/1 Queue Rate Transitions
l l l l l l l 1 2 3
n-1 n
m m m m m m m
Balance Eq. p 1 = l m l m p 1 = 2 + n -

9. M/M/1 Queue Rate Transitions
l l l l l l l 1 2 3
n-1 n
m m m m m m m
Balance Eq. p 1 = l m l m p 1 = 2 + n - l m p 2 + = ( )

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

11. M/M/1 Queue Rate Transitions
l l l l l l l 1 2 3
n-1 n
m m m m m m m l m p 2 + = - ( )
/ m p 1 = l m

12. M/M/1 Queue Rate Transitions
l l l l l l l 1 2 3
n-1 n
m m m m m m m l m p 2 + = - ( )
/ m p 1 = l m l m p 2 = ( )

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

14. M/M/1 Queue Rate Transitions
l l l l l l l 1 2 3
n-1 n
m m m m m m m p n = ( ) l m
In general,
We also know that in steady state, we must be
somewhere. That is,

15. M/M/1 Queue Rate Transitions
l l l l l l l 1 2 3
n-1 n
m m m m m m m p n = ( ) l m
In general,
We also know that in steady state, we must be
somewhere. That is, p n = å 1

16. M/M/1 Queue Rate Transitions
l l l l l l l 1 2 3
n-1 n
m m m m m m m p n = å 1 p n = å 1 ( ) l m p = - 1 ( ) l m

17. M/M/1 Queue p = - 1 ( ) l m p 1 = - ( ) l m 1 p n 1 = - (
)n ( ) l m

18. Performance Criteria m L
average in
system np n = - å # ( ) 1 l

19. Performance Criteria å å m L average in system np n = - # ( ) 1 l Re ,å # ( ) 1 l Re , ( )
call nx x n = å - 2 1

20. Performance Criteria å L = - ( ) 1 l m Re , ( ) call nx x - 1 2 ¥ n 22 1

21. Performance Criteria l W
average
wait in
system L = - m 1

22. Performance Criteria W avg wait time in system service E S = - [ ] ( )q = - [ ] ( ) 1 m l

23. Performance Criteria L
avg in
queue w q = - # ( ) l m 2

24. Performance Criteria L W q = - l m 1 2 ( )