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UNIT IIINFORMATION THEORY, CODING & CRYPTOGRAPHY (MCSE 202)
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STOCHASTIC PROCESS :
In probability theory, a stochastic process or sometimes random process (widely used) is a
collection ofrandom variables;this is often used to represent the evolution of some random value, or
system, over time. This is the probabilistic counterpart to a deterministic process (or deterministic
system). Instead of describing a process which can only evolve in one way (as in the case, for example,
of solutions of an ordinary differential equation), in a stochastic or random process there is some
indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely
many) directions in which the process may evolve.
In the simple case ofdiscrete time,as opposed to continuous time,a stochastic process involves
a sequence of random variables and the time series associated with these random variables (for
example, see Markov chain, also known as discrete-time Markov chain). Another basic type of a
stochastic process is arandom field,whose domain is a region ofspace,in other words, a random
function whose arguments are drawn from a range of continuously changing values. One approach to
stochastic processes treats them asfunctions of one or several deterministic arguments (inputs, in
most cases regarded as time) whose values (outputs) arerandom variables:non-deterministic (single)
quantities which have certain probability distributions. Random variables corresponding to various
times (or points, in the case of random fields) may be completely different. The main requirement is
that these different random quantities all have the same type. Type refers to the codomain of the
function. Although the random values of a stochastic process at different times may beindependent
random variables, in most commonly considered situations they exhibit complicated statistical
correlations.
Stock market fluctuations have been modeled by stochastic processes.
UNIT : II
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Given aprobability space and ameasurable space , an S-valued stochastic
process is a collection of S-valued random variables on , indexed by a totally
ordered set T("time"). That is, a stochastic processXis a collection
where each is an S-valued random variable on . The space S is then called the state
spaceof the process.
STATISTICAL INDEPENDENCE :
In probability theory, to say that two events are independent (alternatively called statistically
independentor stochastically independent)[1]means that the occurrence of one does not affect
the probability of the other. Similarly, tworandom variables are independent if the realization of one
does not affect the probability distribution of the other.
In some instances, the term "independent" is replaced by "statistically independent",
"marginally independent", or "absolutely independent"
For events :
Two events-
Two eventsAand Bare independent if and only if their joint probability equals the product of
their probabilities:
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.
Why this defines independence is made clear by rewriting withconditional probabilities:
and similarly
.
Thus, the occurrence of Bdoes not affect the probability ofA, and vice versa. Although the derived
expressions may seem more intuitive, they are not the preferred definition, as the conditional
probabilities may be undefined if P(A) or P(B) are 0. Furthermore, the preferred definition makes
clear by symmetry that whenAis independent of B, Bis also independent ofA.
More than two events
A finite set of events {Ai} is pairwise independentiff every pair of events is independent.[2]That
is, if and only if for all distinct pairs of indices m, n
.
A finite set of events is mutually independent if and only if every event is independent of any
intersection of the other events.[2]That is, iff for every subset {An}
This is called the multiplication rulefor independent events.
For more than two events, a mutually independent set of events is (by definition) pairwise
independent, but the converse is not necessarily true.
For random variables
Two random variables
Two random variablesXand Yare independentiff the elements of the-system generated by
them are independent; that is to say, for every a and b, the events {X a} and {Y b} are
independent events (as defined above). That is, X and Y with cumulative distribution
functions and , andprobability densities and , are independentif
and only if (iff) the combined random variable (X, Y) has ajoint cumulative distribution function
or equivalently, a joint density
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More than two random variables
A set of random variables is pairwise independent iff every pair of random variables is
independent.
A set of random variables is mutually independentiff for any finite subset and
any finite sequence of numbers , the events are
mutually independent events (as defined above).
The measure-theoretically inclined may prefer to substitute events {XA} for events {Xa} in
the above definition, where A is anyBorel set.That definition is exactly equivalent to the one
above when the values of the random variables arereal numbers.It has the advantage of working
also for complex-valued random variables or for random variables taking values in
anymeasurable space (which includestopological spaces endowed by appropriate -algebras).
Conditional independence
Intuitively, two random variablesXand Yare conditionally independent given Zif, once Zis known,
the value of Y does not add any additional information about X. For instance, two
measurements X and Y of the same underlying quantity Z are not independent, but they
are conditionally independent given Z(unless the errors in the two measurements are somehow
connected).
The formal definition of conditional independence is based on the idea ofconditional distributions.
IfX, Y, and Zarediscrete random variables,then we defineXand Yto beconditionally independent
givenZif
for allx, yand zsuch that P(Z= z) > 0. On the other hand, if the random variables arecontinuous and
have a jointprobability density functionp, thenXand Yareconditionally independent given Zif
for all real numbersx, yand zsuch thatpZ(z) > 0.
IfXand Yare conditionally independent given Z, then
for anyx, yand zwith P(Z= z) > 0. That is, the conditional distribution forXgiven Yand Zis the same
as that given Zalone. A similar equation holds for the conditional probability density functions in the
continuous case.
Independence can be seen as a special kind of conditional independence, since probability can be
seen as a kind of conditional probability given no events.
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Independent -algebras[edit]
The definitions above are both generalized by the following definition of independence for-algebras.
Let (,,Pr) be a probability space and let Aand Bbe two sub--algebras of .Aand Bare said to
be independentif, wheneverAAand BB,
Likewise, a finite family of -algebras is said to be independent if and only if for all
and an infinite family of -algebras is said to be independent if all its finite subfamilies are independent.
The new definition relates to the previous ones very directly:
Two events are independent (in the old sense)if and only if the -algebras that they generate
are independent (in the new sense). The -algebra generated by an eventE is, by
definition,
Two random variablesXand Ydefined over are independent (in the old sense) if and only
if the -algebras that they generate are independent (in the new sense). The -algebra
generated by a random variableX taking values in somemeasurable space Sconsists, by
definition, of all subsets of of the formX1(U), where Uis any measurable subset of S.
Using this definition, it is easy to show that if X and Y are random variables and Y is constant,
thenXand Yare independent, since the -algebra generated by a constant random variable is the
trivial -algebra {, }. Probability zero events cannot affect independence so independence also
holds if Yis only Pr-almost surely constant.
Properties :
Self-dependence
Note that an event is independent of itselfiff
.
Thus if an event or itscomplementalmost surely occurs, it is independent of itself. For example,
ifAis choosing any number but 0.5 from auniform distribution on theunit interval,Ais independent
of itself, even though,tautologically,Afully determinesA.
Expectation and covariance
IfXand Yare independent, then theexpectation operator Ehas the property
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and for thecovariance since we have
so thecovariance cov(X, Y) is zero. (The converse of these, i.e. the proposition that if two random
variables have a covariance of 0 they must be independent, is not true. Seeuncorrelated.)
Characteristic function
Two random variablesXand Yare independent if and only if the characteristic function of the
random vector (X, Y) satisfies
In particular the characteristic function of their sum is the product of their marginal characteristic
functions:
though the reverse implication is not true. Random variables that satisfy the latter condition are
calledsub-independent.
Examples :
Rolling a die
The event of getting a 6 the first time a die is rolled and the event of getting a 6 the second time
are independent. By contrast, the event of getting a 6 the first time a die is rolled and the event
that the sum of the numbers seen on the first and second trials is 8 are not independent.
Drawing cards
If two cards are drawn with replacement from a deck of cards, the event of drawing a red card on
the first trial and that of drawing a red card on the second trial are independent. By contrast, if two
cards are drawn without replacement from a deck of cards, the event of drawing a red card on
the first trial and that of drawing a red card on the second trial are again not independent.
Pairwise and mutual independence
Consider the two probability spaces shown. In both cases, P(A) = P(B) = 1/2 and P(C) = 1/4 The first
space is pairwise independent but not mutually independent. The second space is mutually
independent. To illustrate the difference, consider conditioning on two events. In the pairwise
independent case, although, for example, A is independent of both Band C, it is not independent
of BC:
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In the mutually independent case however:
See also for a three-event example in which
and yet no two of the three events are pairwise independent.
Pairwise independent, but not mutually independent, events.
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Mutually independent events.
BERNOULLI PROCESS :
In probability andstatistics,a Bernoulli process is a finite or infinite sequence of binary random
variables,so it is adiscrete-time stochastic process that takes only two values, canonically 0 and 1.
The component Bernoulli variablesXiare identical andindependent.Prosaically, a Bernoulli process
is a repeated coin flipping, possibly with an unfair coin (but with consistent unfairness). Every
variable Xi in the sequence is associated with a Bernoulli trial or experiment. They all have the
same Bernoulli distribution. Much of what can be said about the Bernoulli process can also be
generalized to more than two outcomes (such as the process for a six-sided die); this generalization
is known as theBernoulli scheme.
A Bernoulli processis a finite or infinite sequence ofindependentrandom variablesX1,X2,X3, ...,
such that
For each i, the value ofXiis either 0 or 1;
For all values of i, the probability thatXi= 1 is the same numberp.
In other words, a Bernoulli process is a sequence ofindependent identically distributedBernoulli
trials.
Independence of the trials implies that the process is memoryless. Given that the probability pis
known, past outcomes provide no information about future outcomes. (Ifpis unknown, however,
the past informs about the future indirectly, through inferences aboutp.)
If the process is infinite, then from any point the future trials constitute a Bernoulli process identical
to the whole process, the fresh-start property.
Interpretation
The two possible values of eachXiare often called "success" and "failure". Thus, when expressed
as a number 0 or 1, the outcome may be called the number of successes on the ith "trial".
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Two other common interpretations of the values are true or false and yes or no. Under any
interpretation of the two values, the individual variables Xi may be called Bernoulli trialswith
parameter p.
In many applications time passes between trials, as the index i increases. In effect, the
trials X1, X2, ... Xi, ... happen at "points in time" 1, 2, ..., i, .... That passage of time and the
associated notions of "past" and "future" are not necessary, however. Most generally,
anyXiandXjin the process are simply two from a set of random variables indexed by {1, 2, ..., n}
or by {1, 2, 3, ...}, the finite and infinite cases.
Several random variables and probability distributions beside the Bernoullis may be derived from
the Bernoulli process:
The number of successes in the first ntrials, which has abinomial distribution B(n,p)
The number of trials needed to get r successes, which has a negative binomial
distribution NB(r,p)
The number of trials needed to get one success, which has ageometric distribution NB(1,p),
a special case of the negative binomial distribution
The negative binomial variables may be interpreted as randomwaiting times.
Formal definition
The Bernoulli process can be formalized in the language of probability spaces as a random
sequence of independent realisations of a random variable that can take values of heads or tails.
The state space for an individual value is denoted by
Specifically, one considers thecountably infinitedirect product of copies of . It is
common to examine either the one-sided set or the two-sided
set . There is a naturaltopology on this space, called theproduct topology.The sets in
this topology are finite sequences of coin flips, that is, finite-lengthstrings ofHand T, with the rest
of (infinitely long) sequence taken as "don't care". These sets of finite sequences are referred to
as cylinder sets in the product topology. The set of all such strings form a sigma algebra,
specifically, a Borel algebra. This algebra is then commonly written as where the
elements of are the finite-length sequences of coin flips (the cylinder sets).
If the chances of flipping heads or tails are given by the probabilities , then one can
define a natural measure on the product space, given by (or
by for the two-sided process). Given a cylinder set, that is, a specific
sequence of coin flip results at times , the probability of observing
this particular sequence is given by
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where kis the number of times that Happears in the sequence, and n-kis the number of times
that Tappears in the sequence. There are several different kinds of notations for the above; a
common one is to write
where each is a binary-valued random variable. It is common to write for . This
probability Pis commonly called theBernoulli measure.[1]
Note that the probability of any specific, infinitely long sequence of coin flips is exactly zero; this
is because , for any . One says that any given infinite sequence
hasmeasure zero.Nevertheless, one can still say that some classes of infinite sequences of coin
flips are far more likely than others, this is given by theasymptotic equipartition property.
To conclude the formal definition, a Bernoulli process is then given by the probability
triple , as defined above.
BINOMIAL DISTRIBUTION :
Thelaw of large numbers states that, on average, theexpectation value of flipping headsfor any
one coin flip isp. That is, one writes
for any one given random variable out of the infinite sequence ofBernoulli trials that compose
the Bernoulli process.
One is often interested in knowing how often one will observe Hin a sequence of ncoin flips. This
is given by simply counting: Given n successive coin flips, that is, given the set of all
possiblestrings of length n, the number N(k,n) of such strings that contain koccurrences of His
given by thebinomial coefficient
If the probability of flipping heads is given by p, then the total probability of seeing a string of
length nwith kheads is
This probability is known as theBinomial distribution.
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Of particular interest is the question of the value of P(k,n) for very, very long sequences of coin
flips, that is, for the limit . In this case, one may make use ofStirling's approximation to
the factorial, and write
Inserting this into the expression for P(k,n), one obtains the Gaussian distribution; this is the
content of thecentral limit theorem,and this is the simplest example thereof.
The combination of the law of large numbers, together with the central limit theorem, leads to an
interesting and perhaps surprising result: the asymptotic equipartition property. Put informally,
one notes that, yes, over many coin flips, one will observe Hexactlypfraction of the time, and
that this corresponds exactly with the peak of the Gaussian. The asymptotic equipartition property
essentially states that this peak is infinitely sharp, with infinite fall-off on either side. That is, given
the set of all possible infinitely long strings of Hand Toccurring in the Bernoulli process, this setis partitioned into two: those strings that occur with probability 1, and those that occur with
probability 0. This partitioning is known as theKolmogorov 0-1 law.
The size of this set is interesting, also, and can be explicitly determined: the logarithm of it is
exactly theentropy of the Bernoulli process. Once again, consider the set of all strings of length n.
The size of this set is . Of these, only a certain subset are likely; the size of this set
is for . By using Stirling's approximation, putting it into the expression for P(k,n),
solving for the location and width of the peak, and finally taking one finds that
This value is the Bernoulli entropy of a Bernoulli process. Here, H stands for entropy; do not
confuse it with the same symbol Hstanding for heads.
von Neumann posed a curious question about the Bernoulli process: is it ever possible that a
given process isisomorphic to another, in the sense of theisomorphism of dynamical systems?
The question long defied analysis, but was finally and completely answered with the Ornstein
isomorphism theorem.This breakthrough resulted in the understanding that the Bernoulli process
is unique anduniversal;in a certain sense, it is the single most random process possible; nothing
is 'more' random than the Bernoulli process (although one must be careful with this informal
statement; certainly, systems that aremixing are, in a certain sense, 'stronger' than the Bernoulliprocess, which is merely ergodic but not mixing. However, such processes do not consist of
independent random variables: indeed, many purely deterministic, non-random systems can be
mixing).
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POISSON PROCESS :
Inprobability theory,a Poisson processis astochastic process that counts the number of event and
the time that these events occur in a given time interval. The time between each pair of consecutive
events has an exponential distribution with parameter and each of these inter-arrival times is
assumed to be independent of other inter-arrival times. The process is named after the French
mathematician Simon Denis Poisson and is a good model of radioactive decay,[1] telephone
calls[2]and requests for a particular document on a web server,[3]among many other phenomena.
The Poisson process is acontinuous-time process;the sum of aBernoulli process can be thought of
as its discrete-time counterpart. A Poisson process is a pure-birth process, the simplest example of
abirth-death process.It is also apoint process on the real half-line.
The basic form of Poisson process, often referred to simply as "the Poisson process", is a
continuous-timecounting process {N(t), t0} that possesses the following properties:
N(0) = 0
Independent increments (the numbers of occurrences counted in disjoint intervals are
independent of each other)
Stationary increments (the probability distribution of the number of occurrences counted in any
time interval only depends on the length of the interval)
Theprobability distribution of N(t) is aPoisson distribution.
No counted occurrences are simultaneous.
Consequences of this definition include:
The probability distribution of the waiting time until the next occurrence is an exponential
distribution.
The occurrences aredistributed uniformly on any interval of time. (Note that N(t), the total
number of occurrences, has a Poisson distribution over (0, t], whereas the location of an
individual occurrence on t(a, b]is uniform.)
Other types of Poisson process are described below.
1. Homogeneous
2. Non- Homogeneous
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Sample Path of a counting Poisson process
Homogeneous :
The homogeneousPoisson process counts events that occur at a constant rate; it is one of the
most well-knownLvy processes.This process is characterized by a rate parameter , also knownas intensity, such that the number of events in time interval (t, t + ] follows a Poisson
distribution with associated parameter . This relation is given as
where N(t+ ) N(t) = kis the number of events in time interval ( t, t+ ].
Just as a Poisson random variable is characterized by its scalar parameter , a homogeneous
Poisson process is characterized by its rate parameter , which is the expected number of
"events" or "arrivals" that occur per unit time.N(t) is a sample homogeneous Poisson process, not to be confused with a density or distribution
function.
Non-homogeneous :
A non-homogeneous Poisson process counts events that occur at a variable rate. In general, the
rate parameter may change over time; such a process is called a non-homogeneous Poisson
processor inhomogeneous Poisson process. In this case, the generalized rate function is
given as (t). Now the expected number of events between time aand time bis
Thus, the number of arrivals in the time interval (a, b], given as N(b) N(a), follows aPoisson
distribution with associated parameter a,b
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A rate function (t) in a non-homogeneous Poisson process can be either a deterministic function
of time or an independent stochastic process, giving rise to a Cox process.A homogeneous
Poisson process may be viewed as a special case when (t) = , a constant rate.
RENEWAL PROCESS :
Renewal theory is the branch of probability theory that generalizes Poisson processes for
arbitrary holding times. Applications include calculating the expected time for a monkey who is
randomly tapping at a keyboard to type the word Macbethand comparing the long-term benefits of
different insurance policies.
A renewal processis a generalization of the Poisson process. In essence, the Poisson process is
a continuous-time Markov process on the positive integers (usually starting at zero) which
has independent identically distributed holding times at each integer (exponentially distributed)
before advancing (with probability 1) to the next integer: . In the same informal spirit, we may
define a renewal process to be the same thing, except that the holding times take on a more general
distribution. (Note however that the independence and identical distribution (IID) property of theholding times is retained).
Let be a sequence of positive independent identically
distributedrandom variables such that
We refer to the random variable as the " th" holding time.
Define for each n> 0 :
each referred to as the " th"jump timeand the intervals
being called renewal intervals.
Then the random variable given by
(where is theindicator function)represents the number of jumps that have occurred by time t,
and is called a renewal process.
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Sample evolution of a renewal process with holding timesSiand jump times Jn.
The renewal equation
The renewal function satisfies :
where is the cumulative distribution function of and is the corresponding probability
density function.
Proof of the renewal equation :
We may iterate the expectation about the first holding time:
But by theMarkov property
So
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as required.
RANDOM INCIDENCE :
The Poisson process is one of many stochastic processes that one encounters in urban servicesystems. The Poisson process is one example of a "point process" in which discrete events(arrivals) occur at particular points in time. For a general point process having its zero tharrival attime T0 and the remaining arrivals at times T1, T2, T3, . . ., the interarrival times are
Such a stochastic process is fully characterized by the family of joint
pdf's for all integer values of p and all possiblecombinations of different n1, n2, . . ., where each n i is a positive integer denoting a particularinterarrival time. Maintaining the depiction of a stochastic process at such a general level,although fine in theory, yields an intractable model and one for which the data (to estimate all the
joint pdf 's) are virtually impossible to obtain. So, in the study of stochastic processes, one ismotivated to make assumptions about this family of pdf's that
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(1) are realistic for an important class of problems and(2) yield a tractable model.
We wish to consider here the class of point stochastic processes for which the marginal pdf's forall of the interarrival times (Yk) are identical. That is, we assume that
Thus, for Yk, if we selected any one of the family of joint pdf's fYn1,Yn2, . . ., Ynp(yn1, yn2, . . . , yk, . . .,
ynP) and "integrated out" all variables except yk, we would obtain fY(.). Note that we have said
nothing about independence of the Yk's
They need not be mutually independent, pairwise independent, or conditionally independent in
any way. For the special case in which the Yk's aremutually independent, the point process is
called a renewal process. The Poisson process is a special case of a renewal process, being
the only continuous-time renewal process having "no memory." However, the kind of process
we are considering can exhibit both memory and dependence among the inter-event times. In
fact, the dependence could be so strong that once we know the value of one of the Y k's we
might know a great deal (perhaps even the exact values) of any number of the remaining Yk's.
Example :
Consider a potential bus passenger arriving at a bus stop. The kth bus arrives Y ktime units after
the (k - 1)st bus. Here the Yk's are called bus headways. The probabilistic behavior of the Yk's will
determine the probability law for the waiting time of the potential passenger (until the next bus
arrives). Here it is reasonable to assume that the Yk's are identically distributed but not
independent (due to interactions between successive buses). One could estimate the pdf fY(.)
simply by gathering data describing bus interarrival times and displaying the data in the form of a
histogram. (This same model applies to subways and even elevators in a multielevator building.)
Suppose that buses maintain perfect headway; that is, they are always T0minutes apart. Then
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That is, the time until the next bus arrives, given random incidence, is uniformly distributed
between 0 and T0, with a mean E[V] = T0/2, as we might expect intuitively.
MARKOV MODULATED BERNOULLI PROCESS :
The Markov-Modulated Bernoulli Process (MMBP) model is used to analyze the delay
experienced by messages in clocked, packed-switched Banyan networks with kx koutput-
buffered switches. This approach allows us to analyze both single packet messages and
multipacket messages with general traffic pattern including uniform traffic, hot-spot traffic,
locality of reference, etc. The ability to analyze multipacket messages is very important for
multimedia applications. Previous work, which is only applicable to restricted message andtraffic patterns, resorts to either heuristic correction factors to artificially tune the model or
tedious computational efforts. In contrast, the proposed model, which is applicable to much
more general message and traffic patterns, not only is an application of a theoretically
complete model but also requires a minimal amount of computational effort. In all cases, the
analytical results are compared with results obtained by simulation and are shown to be very
accurate.
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DTMC - Discrete Time Markov Chains
IRREDUCIBLE FINITE CHAINS WITH APERIODIC STATES :
A Markov chain (discrete-time Markov chain or DTMC) named after Andrey Markov, is a
mathematical system that undergoes transitions from one state to another on a state space. It is
arandom process usually characterized asmemoryless:the next state depends only on the current
state and not on the sequence of events that preceded it. This specific kind of "memorylessness" is
called theMarkov property.Markov chains have many applications asstatistical models of real-world
processes.
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DISCRETE TIME BIRTH DEATH PROCESS :
The birth
death process is a special case of continuous-time Markov process where the statetransitions are of only two types: "births", which increase the state variable by one and "deaths", which
decrease the state by one. The model's name comes from a common application, the use of such
models to represent the current size of a population where the transitions are literal births and deaths.
Birthdeath processes have many applications in demography, queueing theory, performance
engineering, epidemiology or in biology. They may be used, for example to study the evolution
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ofbacteria,the number of people with a disease within a population, or the number of customers in
line at the supermarket.
When a birth occurs, the process goes from state nto n+ 1. When a death occurs, the process goes
from state n to state n 1. The process is specified by birth rates and death
rates .
Example :
A pure birth processis a birthdeath process where for all .
A pure death processis a birthdeath process where for all .
A (homogeneous)Poisson process is a pure birth process where for all
M/M/1 modelandM/M/c model, both used in queueing theory,are birthdeath processes used to
describe customers in an infinite queue.
Use in queueing theory :
In queueing theory the birthdeath process is the most fundamental example of aqueueing model,
the M/M/C/K/ /FIFO(in completeKendall's notation)queue. This is a queue with Poisson arrivals,
drawn from an infinite population, and Cservers withexponentially distributed service time
with Kplaces in the queue. Despite the assumption of an infinite population this model is a good
model for various telecommunication systems.
M/M/1 queue
The M/M/1is a single server queue with an infinite buffer size. In a non-random environment the
birthdeath process in queueing models tend to be long-term averages, so the average rate of
arrival is given as and the average service time as . The birth and death process is aM/M/1 queue when,
Thedifference equations for theprobability that the system is in state kat time tare,
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M/M/c queue
The M/M/c is a multi-server queue with C servers and an infinite buffer. This differs from the
M/M/1 queue only in the service time, which now becomes
and
with
M/M/1/K queue
The M/M/1/K queue is a single server queue with a buffer of size K. This queue has applications
in telecommunications, as well as in biology when a population has a capacity limit. In
telecommunication we again use the parameters from the M/M/1 queue with,
In biology, particularly the growth of bacteria, when the population is zero there is no ability to
grow so,
Additionally if the capacity represents a limit where the population dies from over population,
The differential equations for the probability that the system is in state kat time tare,
MARKOV PROPERTY :
Inprobability theory andstatistics,the term Markov propertyrefers to the memoryless property of
astochastic process.It is named after theRussianmathematicianAndrey Markov.[1]
A stochastic process has the Markov property if theconditional probability distribution of future states
of the process (conditional on both past and present values) depends only upon the present state, not
on the sequence of events that preceded it. A process with this property is called aMarkov process.
The term strong Markov property is similar to the Markov property, except that the meaning of
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"present" is defined in terms of a random variable known as a stopping time.Both the terms "Markov
property" and "strong Markov property" have been used in connection with a particular "memoryless"
property of theexponential distribution.[2]
The term Markov assumptionis used to describe a model where the Markov property is assumed to
hold, such as ahidden Markov model.
A Markov random field[3] extends this property to two or more dimensions or to random variables
defined for an interconnected network of items. An example of a model for such a field is the Ising
model.
A discrete-time stochastic process satisfying the Markov property is known as aMarkov chain.
FINITE MARKOV CHAIN :
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CONTINUOUS-TIME MARKOV CHAIN :
In probability theory, a continuous-time Markov chain (CTMC[1] or continuous-time Markov
process[2])is a mathematical model which takes values in some finite or countable set and for whichthe time spent in each state takes non-negativereal values and has anexponential distribution.It is
acontinuous-time stochastic process with theMarkov property which means that future behaviour of
the model (both remaining time in current state and next state) depends only on the current state of
the model and not on historical behaviour. The model is a continuous-time version of the Markov
chain model, named because the output from such a process is a sequence (or chain) of states.
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A continuous-time Markov chain (Xt)t 0 is defined by a finite or countable state space S,atransition rate matrix Qwith dimensions equal to that of the state space and initial probability
distribution defined on the state space. For ij, the elements qijare non-negative and describe
the rate the process transitions from state ito statej. The elements qiiare chosen such that each
row of the transition rate matrix sums to zero.
There are three equivalent definitions of the process.[3]
Infinitesimal definition
LetXtbe the random variable describing the state of the process at time t, and assume that the
process is in a state iat time t. ThenXt+ his independent of previous values (Xs: st) and as h
0 uniformly in tfor allj
using little-o notation. The qij can be seen as measuring how quickly the transition
from itojhappens
Jump chain/holding time definition
Define a discrete-time Markov chain Yn to describe the nth jump of the process and
variables S1, S2, S3, ... to describe holding times in each of the states where the distribution ofSiis
given by qYiYi.Transition probability definition
For any value n= 0, 1, 2, 3, ... and times indexed up to this value of n: t0, t1, t2, ... and all states
recorded at these times i0, i1, i2, i3, ... it holds that
wherepijis the solution of theforward equation (afirst-order differential equation)
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with initial condition P(0) is theidentity matrix.
HIDDEN MARKOV MODEL :
A hidden Markov model(HMM) is astatisticalMarkov model in which the system being modeled is
assumed to be aMarkov process with unobserved (hidden) states. A HMM can be considered the
simplest dynamic Bayesian network. The mathematics behind the HMM was developed by L. E.
Baum and coworkers.[1][2][3][4][5] It is closely related to an earlier work on optimal nonlinearfiltering
problem (stochastic processes) byRuslan L. Stratonovich,[6]who was the first to describe theforward-
backward procedure.
In simpler Markov models (like a Markov chain), the state is directly visible to the observer, and
therefore the state transition probabilities are the only parameters. In a hiddenMarkov model, the state
is not directly visible, but output, dependent on the state, is visible. Each state has a probability
distribution over the possible output tokens. Therefore the sequence of tokens generated by an HMM
gives some information about the sequence of states. Note that the adjective 'hidden' refers to the
state sequence through which the model passes, not to the parameters of the model; the model is still
referred to as a 'hidden' Markov model even if these parameters are known exactly.
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Hidden Markov models are especially known for their application intemporal pattern recognition such
as speech, handwriting, gesture recognition,[7] part-of-speech tagging, musical score
following,[8]partial discharges[9]andbioinformatics.
A hidden Markov model can be considered a generalization of a mixture model where the hidden
variables (or latent variables), which control the mixture component to be selected for each
observation, are related through a Markov process rather than independent of each other. Recently,
hidden Markov models have been generalized to pairwise Markov models and triplet Markov models
which allow to consider more complex data structures[10][11]and to model nonstationary data.
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