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E_stokpros

# E_stokpros - J Virtamo 38.3143 Queueing Theory Stochastic...

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J. Virtamo 38.3143 Queueing Theory / Stochastic processes 1 STOCHASTIC PROCESSES Basic notions Often the systems we consider evolve in time and we are interested in their dynamic behaviour, usually involving some randomness. the length of a queue the temperature outside the number of students passing the course S-38.143 each year the number of data packets in a network A stochastic process X t (or X ( t )) is a family of random variables indexed by a parameter t (usually the time). Formally, a stochastic process is a mapping from the sample space S to functions of t . With each element e of S is associated a function X t ( e ). For a given value of e , X t ( e ) is a function of time (“a lottery ticket e with a plot of a func- tion is drawn from a urn”) For a given value of t , X t ( e ) is a random variable For a given value of e and t , X t ( e ) is a (fixed) number The function X t ( e ) associated with a given value e is called the realization of the stochastic process (also trajectory or sample path ).

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J. Virtamo 38.3143 Queueing Theory / Stochastic processes 2 State space: the set of possible values of X t Parameter space: the set of values of t Stochastic processes can be classified according to whether these spaces are discrete or con- tinuous: State space Parameter space Discrete Continuous Discrete * ** Continuous * * * * * ** According to the type of the parameter space one speaks about discrete time or continuous time stochastic processes. Discrete time stochastic processes are also called random sequences.
J. Virtamo 38.3143 Queueing Theory / Stochastic processes 3 In considering stochastic processes we are often interested in quantities like: Time-dependent distribution : defines the probability that X t takes a value in a particular subset of S at a given instant t Stationary distribution : defines the probability that X t takes a value in a particular subset of S as t → ∞ (assuming the limit exists) The relationships between X s and X t for different times s and t (e.g. covariance or correlation of X s and X t ) Hitting probability : the probability that a given state is S will ever be entered First passage time : the instant at which the stochastic process first time enters a given state or set of states starting from a given initial state

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J. Virtamo 38.3143 Queueing Theory / Stochastic processes 4 The n th order statistics of a stochastic process X t is defined by the joint distribution F X t 1 ,...,X t n ( x 1 , . . ., x n ) = P { X t 1 x 1 , . . ., X t n x n } for all possible sets { t 1 , . . ., t n } . A complete characterization of a stochastic process X t requires knowing the stochastics of the process of all orders n . 1 st order statistics: Stationary distribution F ( x ) = lim t →∞ P { X t x } Expectation (at time t ) ¯ X t = E[ X t ] 2 nd order statistics: Covariance (autocovariance) R t,s = E[( X t - ¯ X t )( X s - ¯ X s )]
J. Virtamo 38.3143 Queueing Theory / Stochastic processes 5 Stationary process The statistics of all the orders are unchanged by a shift in the time axis F X t 1 + τ ,...,X t n + τ ( x 1 , . . ., x n ) = F X t 1 ,...,X t n ( x 1 , . . ., x n ) n, t 1 , . . ., t n

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