Hw7_ORIE3510_slns

Hw7_ORIE3510_slns - ORIE3510 Introduction to Engineering...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
ORIE3510 Introduction to Engineering Stochastic Processes Spring 2010 Homework 7: CTMCs and the Poisson Process Not to be handed in. 1. (a) Let S n be the time that the n th rider departs, then S n is the sum of n independent exponentials with rate λ and thus it has a gamma distribution with parameters n and λ . This is given by f S n ( t ) = λe - λt ( λt ) n - 1 ( n - 1)! for t 0. (b) Recall that if { N ( t ) ,t 0 } is a Poisson process with rate λ and points { S i ,i 0 } , then { ( S 1 ,S 2 ,...,S n ) | N ( t ) = n ) } is equal in distribution to { U (1) ,U (1) ,...,U ( n ) } . Therefore, each of these riders will still be walking at time t with probability p = R t 0 e - μ ( t - s ) ds t =. Hence the probability that none of the riders are walking at time t is (1 - p ) n - 1 . 2. X n = Number of customers in the system immediately before the n-th arrival Y n = Number of customers that remain the system when the n -th customer departs If X n = i then X n +1 ∈ { i + 1 - j ; j 0 } (j is the number of departures between the arrivals)
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 10/07/2010 for the course OR&IE 3510 taught by Professor Lewis during the Spring '10 term at Cornell.

Page1 / 2

Hw7_ORIE3510_slns - ORIE3510 Introduction to Engineering...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online