Scarica Catene di Markov a Tempo Continuo: Introduzione e Applicazioni e più Appunti in PDF di Probabilità e Statistica solo su Docsity!
29 11 2023 CONTINUOUS (^) TIME (^) MARKOV (^) CHAIN Dinerete time^ MC Continuous^ True MC
I
HEI Poisson process
INCH toh is the no
of
annals bytimet
It is a^ continuous timeMC P with states^91 We can^ only move faunton i
it is called
apurebirthpo n nn.lu general
an exponentialmodel model withutentmes
cap
which can go
in one transition^ from state
n to^ nos^ a^ n^ i^ is^ called
birtygdeattroces
m 1 Def
Let H to be^ a^ continuous^ time^ stochastic
process (^) taking
values in^ the^ set
of integers twitter.FI E http t^ IfEfocuas IPXHs (^) jIXH i
Def (^) tf IP^ Hs^ s (^) I is (^) independent of s
thenthe^ continuous time MC in said to have
statonorythomogeneonotransitanprobabilita
We will^
consider (^) MC (^) with stationary transition probabilities Suppose that (^) at time o^ the MC^ is^ in^ statei What is the (^) amount
of
time
spent
in state^ i^ before making a (^) transition i LÉA
Ti
the (^) amount of time
spentvi^
state (^) i before making a transition IP (^) ti stt^ Ti s^ IP u^ i^ o^ vastt^ le^ I^ ockes P
u i o cucstt s e^
homogeneous Mako PG u^ i (^) o euct (^) C IP Test Memoryless (^) property Ti is (^) exponentiallydistributed Theory
the intertmes betweenthe^ events^ in a^ continuous
time MC^ are^ exponentiallydistributed
Servicetimes are
indep 2 v^ exponentially distub withnotes μ ardua Customers (^) arriva in accordance (^) to a Poisson^ process
with rated
A customer^ will^ enterthe^ system
only if^ bothservices ore
empty
Salute (^) States O
emptysystem
1 customer^ external^ La cleaning internal
We use def
(^0 1 ) Vo 1
In
BittIIIIIeta
state at
anytime
is represented by the (^) n of people^ inthe
system at^
that time
Supposethatthere^ are^ n^ people in^ the^ system I (^) new (^) arrivals enterthe system at Exp rate (^) In Ii
people leavethe^ system
of
eoproteun n people time until new^ arrival^ Exp in time until^ next^ departure Exp^
a
indep
and
they are indep The (^) parameters in^ o and^4 μmIn are^ calledthe arrival Ibirth and deportureldeathrate Deff A birthand^ deathprocess is^ a^ continuoustime
MC with states 491,
for which transition han state (^) in may goonly tojII Moreover (^) Oo No rieditui viso min arrival departure Poi 1 I 5 3 fin IE rm.p o^ o i Pii e ftp io iii
Example Painompo^ vi^
a
portarla
cose afa birthand death (^) process
pure
birth (^) process di d^ io Mi 0 Viso
rated an^ before
there is an^ exponential rated^ due^ to^ an external
source and^ it^ is^ ridep fan births
Deaths occur at
exp
Time with^
roten for each member of the population t (^) population side^ at^ the^ n t tah
is a^ birth^ and^ death^ process
Yn
nu
min tra itemp
eop
dellemorti
deglim
individui
in mio^ monti E.ca
