1. 1. STOCHASTIC MODELLING AND ITS APPLICATIONS
2. 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. 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. 4. Real life example of stochastic process
5. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.