ch6 - AB HELSINKI UNIVERSITY OF TECHNOLOGY Networking...

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A B HELSINKI UNIVERSITY OF TECHNOLOGY Networking Laboratory 1 ch6.ppt S-38.215 – Applied Stochastic Processes – Spring 2004 Lawler (1995) Chapter 6 Renewal Processes 6.1 Introduction 6.2 Renewal Equation 6.3 Discrete Renewal Processes 6.4 M/G/1 and G/M/1 queues 2 Chapter 6: Renewal Processes 6.1 Introduction Definition : – Let T 1 , T 2 , … be independent and identically distributed (i.i.d.) nonnegative random variables with mean μ = E [ T i ] > 0 . The renewal process associated with renewal sequence T i is the process N t with – Thus, N t denotes the number of events occurred up to time t . Definition : – Let T 1 , T 2 , … be independent and identically distributed (i.i.d.) nonnegative random variables with mean μ = E [ T i ] > 0 . Furthermore, let Y another nonnegative random variable indepedndent of the sequence T i . The renewal process associated with random variable Y and sequence T i is the process N t with + + = < = 1 1 1 }, | , 2 , 1 max{ , 0 T t t T T n T t N n t K K > + + + = < = Y t t T T Y n Y t N n t }, | , 2 , 1 min{ , 0 1 K K (6.1)
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3 Chapter 6: Renewal Processes 6.1 Introduction Definitions : – Let N t be a renewal process associated with renewal sequence T i . The age associated with renewal process N t is the process A t with –T h e residual life associated with renewal process N t is the process B t with h e total lifetime associated with renewal process N t is the process C t with Note : – Renewal process N t alone is not a Markov process (unless the interarrival times T i be exponential) but the pair ( N t , A t ) constitutes a Markov process + + - < = 1 1 1 ), ( , T t T T t T t t A t N t K } : inf{ 1 t s t t N t N N s A T B t > = - = + + t t N t B A T C t + = = + 1 4 Chapter 6: Renewal Processes 6.1 Introduction Strong Law of Large Numbers (SLLN) for renewal processes : – Let N t be a renewal process associated with renewal sequence T i with mean μ .
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ch6 - AB HELSINKI UNIVERSITY OF TECHNOLOGY Networking...

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