Lecture16_2up

Lecture16_2up - ECE-547 Introduction to Computer...

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1 Purdue University ECE ECE -547 547 Introduction to Computer Introduction to Computer Communication Networks Communication Networks Instructor: Instructor: Xiaojun Xiaojun Lin Lin Lecture 16 Lecture 16 Purdue University Derivation of P Derivation of P -K formula K formula ¾ Can we use continuous time Markov chain analysis? ¾ No, because service time distribution is not exponential ¾ Focus on the departure instances in queue Service time of jth packet 1 j N j N 1 + j N 1 + j t j t 1 j t ν j arrive N j : # of packets that remain in queue just after the departure of packet j v j : # of packets that arrive in the service time of packet j t j : departure time of packet j
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2 Purdue University Derivation of P Derivation of P -K formulas K formulas 1j - 1 11 1, if N 1 ,otherwise () 1, if x 1 0, otherwise jj j j j j N N NN U N Ux ν −− +− = =− + = ¾ First, the statistics of ν j can be expressed as functions of the statistics of the service time. (1) Purdue University Derivation of P Derivation of P -K formula K formula ¾ For example, to calculate σ v 2 , we first calculate: 22 0 2 0 0 { } () , where is the r.v. denoting the service time k k Ek P k kP vk t f t d t τ νν = = == = τ dt t f t t dt t f k t e k E k k t ) ( ) ( ) ( ! ) ( ) ( 2 2 0 0 0 2 2 λ = + = = Poisson distributed, because it is the number of arrivals in time t ¾ Now, interchanging summations and integration: 2 nd moment of a Poisson r.v.
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3 Purdue University Derivation of P Derivation of P -K formula K formula ¾ Thus, 22 2 00 2 ( ) () E tf t dt t f t dt EE
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Lecture16_2up - ECE-547 Introduction to Computer...

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