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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.
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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 )]
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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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