Lecture 7 Poisson Processes a reminder Some simple

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

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

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.

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 < tn+1 would be just the same as if we start counting from the previous arrival.

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

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.

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.

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 )

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

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

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

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:

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

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)

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

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:

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:

Average Queue Length in M/M/1 E[N] Utilisation