1. Lecture 7 Poisson Processes (a reminder)
Some simple facts about Poisson processes
The Birth/Death Processes in General
Differential-Difference Equations
The Pure Birth Process (Generating our own Poisson process from scratch)
How are queues related to Markov chains
The M/M/1 queue
A Poisson arrival and Poisson departure queue modelled using Markov chains

2. The Poisson Distribution (reminder)
A random variable X is said to be Poisson if it has the following distribution:
We can calculate the mean as follows:
beginner exercise –
prove it IS a distribution

3. A Poisson Process
A Poisson process is a process where the number of arrivals in a time interval (size ) has a Poisson distribution:
Where A(t) is the number of events (arrivals) up to time t.
Note that this is a Poisson distribution with mean t
The parameter  is known as the rate of the process – because in t time units, t arrivals will occur.

4. Poisson Interarrival Times
tn is the time of packet n and n = tn+1 – tn
How is n distributed? The probability n > s is the probability that there are 0 arrivals in the period tn to tn+s
Note that a similar derivation proves the “memoryless” property of the Poisson process. The distribution of the time to next arrival starting from any time t where tn < t < tn+1 would be just the same as if we start counting from the previous arrival.

5. Approximating a Poisson Process
For every t  0 and   0:
The third property follows from the first two.

6. The Birth Death Process
A birth-death process is a Markov process in which transitions from state k can only be made from the adjacent states k-1 and k+1
Think of a transition from k to k+1 as a birth and in the reverse direction as a death.
More importantly, we could consider it as arrivals and departures from a queue where the arrivals and departures are Poisson processes.
Firstly, we must consider why a Markov chain is appropriate to the modelling.

7. Continuous Time Markov Chains
Note that the Markov chains we have talked about before were “discrete time” – there were discrete steps which occurred at given times.
Here we need to think about continuous time Markov chains – those where transitions between states could occur at any time.
The technicalities of continuous time Markov chains are beyond the scope of this course.
Therefore, we will ignore this technicality and pretend we are dealing with discrete time Markov chains with very small times between states.

8. The General Birth-Death Process
When the pop. = k, births and deaths happen as Poisson processes: birth rate k and death rate k (0 =0)
B(t,t ) is the number of births in the period (t,t+t )
D(t,t ) is the number of deaths in the period (t,t+t )
P{B(t,t )= 0 | Pop. = k} = 1 - k t +o (t )
P{B(t,t )= 1 | Pop. = k} = k t +o (t )
P{B(t,t )> 1 | Pop. = k} = o (t )
P{D(t,t )= 0 | Pop. = k} = 1 - k t +o (t )
P{D(t,t )= 1 | Pop. = k} = k t +o (t )
P{D(t,t )> 1 | Pop. = k} = o (t )

9. Differential Difference Equations
Define the probability that the pop. is k at time t as Pk(t). Now, for k > 0 we have:
Now, taking the limit as t0
These are known as
differential difference equations

10. A Quick Aside – The Pure Birth Process
Consider process k= and k=0
Which is the original Poisson process we started with (no surprise)!

11. The General Birth-Death Process as a Markov Chain1 2 k 0 1 2 1 2
k-1 k k
k+1
...
Note that we number the states from 0 so that the state
number is the same as the population.

12. Equilibrium Probabilities
We are often interested in questions of the form: “What is the average size of the population?” or “What is the probability that the population is of size k at time t?”
We are therefore interested in the equilibrium probabilities.
Recall our balance equations:

13. Equilibrium Probabilities(2)
In the case of our Birth-Death process these are:
rearrange to:
compare with:

14. Solving the problem (1) (2) Rearrange (2):
Substitute into (1) with k=1
Rearrange:
We suspect (correctly) the following relation:
(proving this is part of your coursework)

15. Completing the Birth-Death Process
from other balance equation:
Rearranging

16. Finally, the M/M/1 process
The M/M/1 queue is simply a birth death process with k=  and k=. Substituting into our previous equations we get:
where =/ is known
as the utilisation factor for
a stable system this is < 1
From the geometric series:
Therefore:

17. The M/M/1 Process Solved
We now want to get the expected queue length:
we use a familiar trick to get:
From Little’s Theorem the average delay:

18. Average Queue Length in M/M/1
Utilisation