ISEN 609 Lecture 9

ISEN 609 Lecture 9 - Renewal Processes Suppose cfw_N (t), t...

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Renewal Processes Suppose is a counting process where the times between arrivals of events are independent, identically distributed random variables. In this case, we do not require that the times between events be exponentially distributed. Let F be the distribution function of the times between events, and suppose the mean of this d.f. is and the variance is (historically, “events” are referred to as “renewals” because at times of events, the process starts over, or “renews” itself) { N ( t ) , t 0 } μ σ 2
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Notation (similar to Poisson process) N(t) Number of events in [0, t ] Time between ( n -1)-st and n- th event (non-negative r.v.) Time of the n- th event Assume and X n F (0) = P ( X n = 0) < 1 F ( ) = P ( X n < ) = 1 E [ X n ] = μ V ar ( X n ) = σ 2 S n
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Distribution of N ( t ) Note that The distribution of is an n-fold convolution of F with itself; P ( N ( t ) = n ) = F n ( t ) - F n +1 ( t ) N ( t ) n ⇐⇒ S n t P ( N ( t )= n P ( N ( t ) n ) - P ( N ( t ) n + 1) = P ( S n t ) - P ( S n +1 t ) S n
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Computing Convolutions Suppose the interrenewal times are exponentially distributed You should recognize this as the gamma cdf. P ( S 2 t )= P ( X 1 + X 2 t ± t 0 P ( X 1 + X 2 t | X 2 = u ) dF ( u ) = ± t 0 P ( X 1 t - u ) dF ( u ) = ± t 0 (1 - e - λ ( t - u ) ) λ e - λ u du = ± t 0 λ e - λ u dt + ± t 0 λ e - λ t dt =1 - e - λ t + e - λ t λ t
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Computing Convolutions Suppose the interrenewal times are uniformly distributed on [0, b ] Note we can also do this using the mgf! P ( S 2 t )= P ( X 1 + X 2 t ± t 0 P ( X 1 + X 2 t | X 2 = u ) dF ( u ) = ± t 0 P ( X 1 t - u ) dF ( u ) = ± t 0 ( t - u ) b 1 b du = t 2 2 b 2
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Mean of N ( t ) (the renewal function)
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This note was uploaded on 04/28/2011 for the course ISEN 609 taught by Professor Klutke during the Spring '08 term at Texas A&M.

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ISEN 609 Lecture 9 - Renewal Processes Suppose cfw_N (t), t...

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