POISSON PROCESSES:
exponential distribution, pdf: f(x) = lambda*exp(lambda*x). Exponential distribution has memoryless property: P{X>(s+t)  X>t} = P{X>s}. If
X~exp(lambda) and Y~exp(mu), then P{X<Y} = lambda/(lambda + mu); the min{X,Y}~exp(lambda +mu). Definition of a Poisson Process: {N(t),t>=0} is said to be a Poisson
process having rate lambda if (1) N(0) = 0, (2) the process has independent increments, (3) the number of events in any interval of length t is Poisson distributed with mean
(lambda*t): P{N(t+s) – N(s) = n} = exp(lambda*t)*(lambda*t)^n/(n!); E[N(t)] = (lambda*t). Monotone increasing Markov Chain: at each time step, increase by 1. Poission
Process: continuous (events can occur at any time), jumps occur one at a time, interevent time is exponentially distributed, and there are independent and stationary increments.
LITTLE o(h):
f() is o(h) if the lim as h
0 of f(h)/h = 0.
CTMC:
I = set of state space; i
stay at state i for an exponential amount of time with parameter v(i) and then P(ij)/q(ij)
is the probability of going from state i to state j. A Poisson Process is an example of a CTMC: it is a pure birth process because it always goes from state n to n+1. Interevent times
are exponential random variables with parameter lambda.
Sojourn time:
stay in state i for exponential length of time, and with probability q(ij), the process jumps from state i to
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 Spring '07
 LEWIS,M.
 Probability theory, Cycle Time, Poisson process, rate lambda, pure birth process

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