MSO Week10
Revision
Exponential RV
Poisson RV
Poisson Process
combining poisson processes
splitting poisson processes
CTMC Modelling
define X(t), state of system at time t
write down S, the state space
generator matrix, Q, and draw rate diagram
features of Q
row sum must be equal to 0
differential of the transition probability matrix where you set t = 0
the values of lambda and miu in the diagram dont have to be between 0 and 1
diagonals (downward right) must be less than or equal to 0
common features of both distributions
memoryless
X ~ Exp(lambda) where X refers to the waiting time and lambda refers to the arrival rate
Mean = 1/lambda
Variance = 1/lambda^2
CDF and PDF of Exponential RV
min of two independent exponentials = Exp (lambda + miu)
smaller average waiting time = 1/(lambda + miu) < 1/lambda
prob of two service times being equal = 0 for continuous distribution : P(X= A) when X is a continuous variable = 0
PMF and CDF of Possion
E(X) = Var(X) = lambda where lambda refers to the arrival rate
sum of poissons --> new distribution = Poisson(lambda1 + lambda2)
note lambda 1 and lambda 2 must follow the same time scale
E(X(t)) = Var(X(t)) = lambda * t
difference between Poisson(lambda) and PP(lambda)
one is about the unit time with mean lambda
while the other is about t with mean lambda * t
PP(lambda) = Poisson(lambda*t)
events arrives one by one, and inter arrival times, T, follow a exponential distribution with parameter lambda
arrival rate = lambda, mean of inter arrival time = 1/lambda
PP(lambda1 + lambda2)
split based on probability, so new lambda = lambda*p
definition of CTMCs
time invariant
memoryless
lambda could also refer to failure rate, and 1/lambda refers to average lifetime of the product
notes
redo the inclass activites again (both 1 and 2)
Queues
MM1 Queue
arrival rate lambda, service rate miu
retrial queues
X(t) refers to total no of customers in system
arrival rate lambda, service rate miu
retrial rate lambda: means finish waiting with rate lambda and go back to to server
two X(t)s, one X(t) for those in orbit, second X(t) for those at server
CTMCs steady state
no one state transition prob in CTMC
note rate != probability
infinite number of ways to go from one state to another state within a period t
methods
steady state analysis
pi * Q = 0
sum of pis = 1
cut analysis
pi lambda = pi miu
N many states: N-1 many cuts
long run prob that you have idle server = 1 - lambda/miu
long run prob that you have busy server = 1 - pi (nought) = lambda/miu
need to try the hw questions again
remember they sum to 0 AND NOT 1
remember, one equation will be useless
flow in mean rate = flow out mean rate
take incremental steps: always express pi 1 in terms of pi 0 ... express pi 2 in terms of pi 1 --> then sub in
limiting distribution --> find pi i and pi nought (rmbr to express pi i fully , not with the pi nought term)