1. Queuing Theory Non-Markov Systems

2. Motivation
What happens if the system is not markovian; that is, we do not have exponential inter-arrival times and/or exponential service times.
Three possible approaches
Simulate the system with appropriate distributions
Use other analytical approaches that approximate solutions through bounding techniques
Ignore the underlying assumptions and approximate as an M/M/s/K model anyway

3. Motivation
What happens if the system is not markovian; that is, we do not have exponential inter-arrival times and/or exponential service times.
Three possible approaches
Simulate the system with appropriate distributions
Use other analytical approaches that approximate solutions through bounding techniques
Ignore the underlying assumptions and approximate as an M/M/s/K model anyway

4. Pollaczek-Khintchine Formulation
M/G/1 system
Exponential inter-arrival times
General service distribution with mean and standard deviation

5. P-K Formula

6. Suppose Service is Exponential
If service is actually exponential, then
Which is the formula for the M/M/1 model

7. Suppose Service is Exponential
Further, if service is actually exponential, then
Which again is the formula for the M/M/1 model

8. Non-Poisson System Approximation
Suppose we have a general inter-arrival time and a general service distribution

9. Non-Poisson System Approximation
define
Then,

10. Non-Poisson System Approximation
define
Then,
Note these are
formulas for
M/M/1 queue

11. Non-Poisson Approximation
Suppose we do have exponential inter-arrival and exponential service times.
Then,

12. Non-Poisson Approximation

13. Non-Poisson Approximation
Again, if we have exponential inter-arrival and exponential service times, then
and

14. M/D/1 Queue
With the M/D/1 queue, we have exponential inter-arrival but deterministic service times
Then,

15. M/D/1 Queue

16. M/D/1 Queue
This is half the queue length and half the queue wait time as that of an M/M/1 queue. Why does this make sense?

17. M/D/1 Queue Because there is no variability in the service time.
This is half the queue length and half the queue wait time as that of an M/M/1 queue. Why does this make sense?
Because there is no variability in the service time.

18. Non-Poisson System Approximation Multiple Servers
For a general inter-arrival time, general service distribution, and multiple servers