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Lecture 5

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- This lecture is about:
- Introduction to Queuing Theory
- Queuing Theory Notation
- Bertsekas/Gallager: Section 3.3
- Kleinrock (Book I)

- Basics of Markov Chains
- Bertsekas/Gallager: Appendix A
- Kleinrock (Book I)
- Markov Chains by J. R. Norris

- Queuing Theory deals with systems of the following type:
- Typically we are interested in how much queuing occurs or in the delays at the servers.

Server Process(es)

Input

Process

Output

- A standard notation is used in queuing theory to denote the type of system we are dealing with.
- Typical examples are:
- M/M/1Poisson Input/Poisson Server/1 Server
- M/G/1Poisson Input/General Server/1 Server
- D/G/nDeterministic Input/General Server/n Servers
- E/G/Erlangian Input/General Server/Inf. Servers

- The first letter indicates the input process, the second letter is the server process and the number is the number of servers.
- (M = Memoryless = Poisson)

- The simplest queue is the M/M/1 queue.
- Recall that a Poisson process has the following characteristics:
- Where A(t) is the number of events (arrivals) up to time t.
- Let us assume that the arrival process is a Poisson with mean and the service process is a Poisson with a mean

- Interarrival times are i.i.d. and exponentially distributed with parameter .
- tn is the time of packet n and n= tn+1 - tn then:
- For every t 0 and 0:

- If two or more Poisson processes (A1,A2...Ak) with different means(1, 2... k) are merged then the resultant process has a mean given by:
- If a Poisson process is split into two (or more) by independently assigning arrivals to streams then the resultant processes are both Poisson.
- Because of the memoryless property of the Poisson process, an ideal tool for investigating this type of system is the Markov chain.

- You are waiting for a bus. The timetable says that buses are every 30 minutes. (But who believes bus timetables?)
- As a mathematician, you have observed that, in fact, the buses are a Poisson process with a mean arrival rate such that the expectation time between buses is 30 minutes.
- You arrived at a random time at the bus stop. What is your expected wait for a bus? What is the expected time since the last bus?
- 15 minutes. After all, they are, on average, 30 minutes apart.
- 30 minutes. As we have said, a Poisson Process is memoryless so logically, the expected waiting time must be the same whether we arrive just after a previous bus or a full hour since the previous bus.

- Some process (or time series) {Xn| n= 0,1,2,...} takes values in nonnegative integers.
- The process is a Markov chain if, whenever it is in state i, the probability of being in state j next is pij
- This is, of course, another way of saying that a Markov Chain is memoryless.
- pij are the transition probabilities.

A

1/3

1/4

1/2

3/4

2/3

B

C

1/2

A hitchhiking hippy begins at A

town. For some reason he has

poor short-term memory and

travels at random according

to the probabilities shown. What

is the chance he is back at A after 2

days? What about after 3 days? Where is he likely to end up?

A

1/3

1/4

1/2

3/4

2/3

B

C

1/2

- After 1 day he will be in B town with probability 3/4 or C town with probability 1/4
- The probability of returning to A via B after 1 day is 3/12 and via C is 2/12 total 5/12
- We can perform similar
calculations for 3 or 4 days

but it will quickly

become tricky and

finding which city he

is most likely to end up

in is impossible.

A

1/3

1/4

1/2

3/4

2/3

B

C

1/2

- Instead we can represent the transitions as a matrix

Prob of going to B from A

Prob of going to A from C

- pijare the transition probabilities of a chain. They have the following properties:
- The corresponding probability matrix is:

- Define n as a distribution vector representing the probabilities of each state at time step n.
- We can now define 1 step in our chain as:
- And clearly, by iterating this, after m steps we have:

- What does this imply for our hippy?
- We know the initial state vector:
- So we can calculate n with a little drudge work.
- (If you get bored raising P to the power n then you can use a computer)
- But which city is the hippy likely to end up in?
- We want to know

- Assuming the limit exists, the distribution vector is known as the invariant or equilibrium probabilities.
- We might think of them as being the proportion of the time that the system spends in each state or alternatively, as the probability of finding the system in a given state at a particular time.
- They can be found by finding a distribution which solves the equation:
- We will formalise these ideas in a subsequent lecture.

- Formally, a process Xn is Markov chain with initial distribution and transition matrix P if:
- P{X0=i} = i (where i is the ith element of )
- P{Xn+1=j| Xn=i, Xn-1=xn-1,...X0=x0}= P{Xn+1=j| Xn=i }=pij

- For short we say Xn is Markov (,P)
- We now introduce the notation for an n step transition:
- And note in passing that:

This is the Chapman-Kolmogorov equation