# final - MS&E 221 Ramesh Johari Final Examination...

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Final Examination Ramesh Johari March 22, 2007 Instructions 1. Take alternate seating. 2. Answer all questions in the spaces provided on these sheets. If needed, additional paper will be available at the front of the room. Answers given on any other paper will not be counted. 3. The examination begins at 8:30 am, and ends at 11:30 am. 4. No notes, books, calculators, or other aids are allowed. 5. Show your work! Partial credit will be given for correct reasoning. Honor Code In taking this examination, I acknowledge and accept the Stanford University Honor Code. NAME (signed) NAME (printed) 1

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Useful Formulas 1. If T is an exponentially distributed random variable with mean 1 , then the density of T is given by f T , where: f T ( t ) = λe - λt , t 0 . 2. If S is a random variable with a gamma distribution of parameters n and λ , where n is a positive integer, then the density of S is given by f S , where: f S ( s ) = λe - λs ( λs ) n - 1 ( n - 1)! , s 0 . 3. For an M/M/ 1 queue with arrival rate λ and service rate μ with λ < μ , the equilibrium distribution is: P ( Q = j ) = (1 - ρ ) ρ j , j = 0 , 1 , 2 , . . . , where ρ = λ/μ . 4. For an M/M/ queue with arrival rate λ and service rate μ , the equilibrium distribution is: P ( Q = j ) = e - ρ ρ j j ! , j = 0 , 1 , 2 , . . . , where ρ = λ/μ . 5. For an M/M/K/K queue with arrival rate λ and service rate μ , the equilibrium distribution is: P ( Q = j ) = ρ j /j ! 1 + ··· + ρ K /K ! , j = 0 , 1 , 2 , . . . , K, where ρ = λ/μ . 2
Part A (5 questions; 25 points) Answer each of the following short answer questions. Explain your answers! (5 points per question) 1. True or false: a finite communicating class of a continuous time Markov chain is transient if and only if it is transient for the associated jump chain. 2. A stock is currently at price

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## This note was uploaded on 02/03/2011 for the course MS&E 221 taught by Professor Ramesh during the Winter '11 term at Stanford.

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final - MS&E 221 Ramesh Johari Final Examination...

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