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Lecture 7

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

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Lecture 7

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  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. 0 1 k-1 2 k ... ... k 2 0 1 1 2 k k+1 2 The General Birth-Death Process as a Markov Chain 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 E[N] Utilisation

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