probability in telecom switching
Transcript of probability in telecom switching
Telecom SwitchingGroup Members : Usman Imran(16237) Adnan Safdar(17539)
Statical Parameters
• Parameters are numbers that summarize data for an entire population
• Statistics are numbers that summarize data from a sampleExample• A researcher wants to estimate the average height of
women aged 20 years or older. From a simple random sample of 45 women, the researcher obtains a sample mean height of 63.9 inches
Random /Stochastic process
• random process, is a collection of random variables, representing the evolution of some system of randomvalues over time.
Types of Random Process(a)continuous time continuous state (b) continuous time discrete state (c) discrete time continuous state (d) discrete time discrete state
Random /Stochastic process
Discrete Random process• In telecommunication we deal with discrete-time
random process • A discrete-time random process x(n) is a collection,
or ensemble, of discrete-time signals• A discrete state stochastic process is often called a
chain
Discrete Random process• statistical properties of a random process may be obtained
in two waysTime Average• The average determined by measurements on a single
sample function at successive times will yield a time average
Ensemble average• The statistical average made at some fixed Time on all the
sample functions of the ensample is the ensemble average
label of components x1, x2….x10 and the times t1, t2…t10 the numbers for the ten components at any one time (i.e. a vertical column) add them and divide by ten is Ensemble Average. and average the voltages at the ten different times is Time Average
Ergodicity & Stationary processes
Ergodicity• When the time average of a process is equal to the
ensemble average, it is said to be ergodicStationary• the statistics do not change with time. The behavior is
time-invariant, even though the process is random.
• Telephone traffic is nonstationary. But the traffic obtained during busy hour may be considered as stationary
Pure Chance traffic• pure-chance traffic means that call arrivals and terminations are
independent• user may make a call at any time of the day• number of sources from which calls originate is infinite• All the sources originated calls simultaneously there would be no
liklihood of any further calls occurring.• Propotion sources to the number of calls in progress at any one
time is usually very small• it is reasonable to assume that subscribers' traffic is originated
on a basis of pure chance.
Markov process• Markov process is developed by A.A. Markov on 1907• It can be used to model a random system that changes
states according to a transition rule that only depends on the current state
• (a.) The number of possible outcomes or states is finite.• (b.) The outcome at any stage depends only on the
outcome of the previous stage.• (c.) The probabilities are constant over time.• It is also called markov chains
Continuous-time Markov chain & Discrete-time Markov chain • A continuous-time Markov chain is one in which changes to the system can
happen at any time along a continuous intervalExample: number of cars that have visited a drive-through at a local fast-food
restaurant during the day• A discrete-time Markov chain is one in which the system evolves through
discrete time steps. So changes to the system can only happen at one of those discrete time values
Example This is discrete because changes to the system state can only happen on
someone's turn.
Birth and death process• Every incoming call request is consider as birth • Every user that after being served leave the system
is considered as death. • transition from n to n-1 if any ended condition
(death)• n to n+1 if any just occur condition (birth).
•P(k) is the probability of state k• λk is called the birth rate in state k.•probability of transition from state k to state k– 1 in the time interval ∆t is µk ∆t where µk is called the death rate in state k•The probability in ∆t, from state kto a state other than k+ 1 or k– 1 is zero