2. Stochastic process A stochastic process or sometimes random
process (widely used) is a collection of random variables,
representing the evolution of some system of random values over
time. This is the probabilistic counterpart to a deterministic
process . Instead of describing a process which can only evolve in
one way, in a stochastic or random process there is some
indeterminacy: even if the initial condition is known, there are
several directions in which the process may evolve.
3. Mathematical Representation Given a probability space and a
measurable 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 process X is a collection where
each is an S-valued random variable on . The space S is then called
the state space of the process.
4. Real life example of stochastic process
5. A method of financial modeling in which one or more
variables within the model are random. Stochastic modeling is for
the purpose of estimating the probability of outcomes within a
forecast to predict what conditions might be like under different
situations. The random variables are usually constrained by
historical data, such as past market returns. Stochastic
Modelling
6. Real life application The Monte Carlo Simulation is an
example of a stochastic model used in finance. When used in
portfolio evaluation, multiple simulations of the performance of
the portfolio are done based on the probability distributions of
the individual stock returns. A statistical analysis of the results
can then help determine the probability that the portfolio will
provide the desired performance. stochastic modelling as applied to
the insurance industry, telecommunication , traffic control
etc
7. telecommunication When messages flow from a source to a
destination (end-to-end) through a network, parts of a message or
the whole message may be dropped due to unavailable resources (buer
capacity) to store the messages. The probability of delivering a
message with some data loss is termed as loss probability. The time
between the source sending a message and the destination receiving
it is called latency or delay. The message flow (will be called
trac henceforth) and the network conditions are ex- tremely
stochastic in nature. Other applications of stochastic processes in
communications include coding theory, signal
8. Token rings Consider N independent and identical users that
are arranged logically in the form of a ring In this model at most
one user is allowed to generating a message over the cable or ring.
Wiring center A B C D E
9. When a user with a message to transmit receives the free
token, the user holds on to the token and transmits the message
onto the ring or cable. Frame circles the ring and is removed by
the transmitting station. Each station interrogates passing frame,
if destined for station, it copies the frame into local buffer.
Once the user completes transmission, the busy token is converted
into a free token and passed along the ring.
10. Re-inserting token on the ring Choices: 1. After station
has completed transmission of the frame. 2. After leading edge of
transmitted frame has returned to the sending station
11. Networks: Token Ring and FDDI 11 A A A A A A A t=0, A
begins frame t=90, return of first bit t=400, transmit last bit A
t=490, reinsert token t=0, A begins frame t=400, last bit of frame
enters ring t=840, return of first bit t=1240, reinsert token
12. In probability theory, a continuous-time Markov chain
(CTMC) is a mathematical model which takes values in some finite or
countable set and for which the time spent in each state takes
non-negative real values and has an exponential distribution. It is
a continuous-time stochastic process with the Markov property which
means that future behaviour of the model depends only on the
current state of the model and not on historical behaviour.
13. To model the system as a CTMC, one could assume that the
packets are generated according to a Poisson process, the length of
the packets are exponentially distributed. The propagation time is
also exponentially distributed. Since all the users are identical,
a CTMC of the form {(X(t), Y (t), t 0} model where X(t) is the
number of messages in the network and Y (t) is the status of the
token (free or busy) at time t. Using the steady state distribution
of the CTMC, performance measures such as
14. Traffic Models Trac flowing through the networks can be
classified into several types. Depending on the network segment,
all messages are broken down into either packets or cells. Packets:
The length or size of a packet ranges anywhere from 60 bytes to
1500 bytes and generally follows a bimodal distribution. ATM Cells:
The length of ATM cells is fixed at 53 bytes.
15. Hierarchical Networks Telecommunication networks are
typically hierarchical in nature. Some frequently used stochastic
models for trac flow are explained in this section. Trac can be
classified into four level 1. Application Level The trac generated
by an application, say, http or telnet or ftp which can vary
significantly based on the protocols they follow
16. 2. Source Level : Each workstation or computer can be
thought of as a source that generates trac. This trac comprises of
the trac generated by dierent applications that are running on the
source. Therefore the trac that flows on a link that exits the
computer is a mixture of the dierent applications. The process of
mixing is known as multiplexing.
17. 3. Aggregate Level Several computer, printers, etc are
connected together to form a local area network (LAN). The trac on
a LAN pipe is the aggregated trac that is multiplexed from all the
sources. 4. Backbone Level The LANs are connected together by means
of a backbone (say, the Internet backbone), and this forms the
Metropolitan Arean Networks (MANs) or the Wide Area Networks
(WANs). The trac on a MAN/WAN pipe is the combination of the trac
from several LANs.
18. In the fluid-flow models it is assumed that trac is in the
form of fluid which flows through a pipe at dierent rates at
dierent times. For example, fluid flows at rate r(1) bytes per
second for a random amount of time t1, then flows at rate r(2)
bytes per second for a random amount of time t2, and so on. This
behaviour can be captured as a discrete stochastic process that
jumps from one state to another whenever the trac flow rate
changes. This can be formalized as a stochastic process {Z(t), t 0}
that is in state Z(t) at time t. Fluid flows in the pipe at rate
r(Z(t)) at time t. Fluid-flow Traffic Models
19. Aggregate Dynamic Stochastic Model For ATS Air traffic
control can be simplified using stochastic modelling. Here we
assume the aircrafts arriving at an airport as a Poisson
distribution and compute the average delay incurred due to
constraints of landing aircraft we assume that each aircraft in
Centre i independently travels to Centre j (or leaves the airspace
for j = 0) between time-steps k and k + 1 with probability pij[k].
We denote the total number of aircraft that flow from Centre i to
Centre j between times k and k + 1 by Uij[k]. For small enough T,
it can be shown that the conditional distribution for the flow
Uij[k] given the Centre count si[k] is well- approximated by a
Poisson random variable, with mean pij[k] si[k] .
20. Now that we have characterized the flows of aircraft in our
model, the state variable update can be specified by accounting for
the number of aircraft entering and leaving each Centre i between
times k and k + 1: 1) This update rule defines the temporal
evolution of our aggregate stochastic model. In our application of
the aggregate model, it is not Equation 1 that we propagate
forwards in time. Instead, we propagate expectations and variances
of the si[k], using equations that are derived from Equation 1, and
that have a very simple structure (Uji[k](Uij[k)-si[k]1]si[k
21. the conditional expectation for the number of aircraft in
Center i is which is a linear function of the time-k Centre counts.
Finally, by taking the expectation with respect to the time-k
Centre counts s[k], given the initial Centre counts s[0], we nd
that E( si[k + 1] | s[0] ) = E( si[k]|s[0]) - Thus, we see that the
expected number of aircraft in Centre i at time k+1givens[0] can be
written as a linear function of the expected Centre counts at time
k given s[0]. ),)i[k][k]pji[k]sj((]si[k]pij[k)s[k]|1]s[kE(
)i[k]]s[0])pji[k|(E(sj[k]])s[0])pij[k|(E(si[k]
22. The U.S. ATS is subject to disturbances that change rates
of aircraft flow in parts of the network. Many of these ow-altering
disturbances, which are often inclement weather events in parts of
the airspace, cannot accurately be predicted in advance.
Furthermore, although the disturbance event may directly aect only
a small part of the airspace, the resulting changes in ows and
Sector/ Centre counts may propagate throughout the network. Since
our model for the U.S. ATS is stochastic, we can naturally in-
corporate stochastic disturbances that alter ows in the model. By
computing the expected behaviour and variability of Centre counts
and ows in the model, regions of the airspace that may be prone to
capacity excesses due to the weather events can be identied. In
turn, the model may suggest improved methods for managing trac ow
in response to weather disturbances. Disturbances:
23. Given that a particular set of disturbances has occurred,
we can calculate statistics of Centre counts with our basic model,
using the appropriate set of model parameters (which are modied
from their nominal values based on the particular disturbances that
have occurred). In turn, we can calculate statistics of Centre
counts without prior knowledge of the disturbances, by scaling the
predicted statistics for each set of disturbances with the
probability that these disturbances occur, and then summing these
scaled statistics. In this way, the dynamics of an ATS that is
subject to stochastic disturbances can be modelled and analyzed.
One possible shortcoming of this approach for modelling stochastic
disturbances is the computational complexity resulting from the
large number of disturbances that may need to be considered. (For
example, if there are 10 dierent weather events that may or may not
be present on a given day, we must consider 2^10 = 1024 possible
combinations of disturbances.) Given certain special conditions on
the location of disturbances, the computational complexity can
sometimes be reduced by considering the change in the systems
dynamics due to each disturbance separately, and then combining
these individual responses.
24. Wireless Network Models One of hottest research topics in
telecommunications is wireless communications technology and a
survey paper would certainly be incomplete without describing some
of the on-going research work in mobile communications. However,
the field is relatively new and most of the techniques are not
well-established. Therefore only a brief summary of some of the
current papers in the area of stochastic models in wireless
networks are presented here. Almost all the forementioned trac
models, performance analysis, flow control, congestion control, etc
do not make any assumptions about whether the networks are at least
partially wireless or not. It is to be noted that mobile
communications where the users (sources and destinations) are
mobile are called wireless communication here. Since the sources
and destinations are not static an important problem is to locate
the users to send and receive messages.
25. Awduche et al describe location management issues that
involve tracking compo-nents that maintain dynamic data on the
locations of mobile stations through a distributed database. The
main focus is on a search component that prescribes the manner in
which the wireless network is to be paged so as to determine the
location of mobile stations whose whereabouts are unknown. The
methods used are based on search theory where a stochastic
sequential framework that systematically determines the location of
mobile stations situated within a group of cells. Search algorithms
are hence developed.
26. A Poisson-arrival location model (PALM) was introduced in
which customers arrive according to a non-homogeneous Poisson
process and move independently through a general location state
space according to a location stochastic process. That was extended
to a version of PALM to study communicating mobiles on a highway.
Leung et al stress the need for combining tele-traffic theory and
vehicular traffic theory. Their numerical results indicate that
both the time-dependent behaviour and the mobility of vehicles play
important roles in determining the system performance.
27. Other Topics One of the most critical factor that will
enable QoS provisioning in high- speed networks is pricing. F.P.
Kelly and colleagues have developed some optimal pricing models .
ATM switch design and router design involve significant amount of
stochastic modeling, particularly queueing. All the multiclass
scheduling policies (polling, static priority, waited fair
queueing, etc) can be implemented on the currently available
switches and routers. All the models considered here were unicast
where trac flows from a single source to a single destination.
There are interesting stochastic models for multicasting (single
source and a few destinations like an Internet classroom with
students globally located) and for broadcasting (single source and
all nodes as destinations) applications. Several scenarios in
telecommunication networks (such as client-server systems) can be
modeled as Queueing Networks. Walrand [76] provides several
applications of Queueing Networks in Telecommunications.