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Unformatted text preview: MS&E 221 Final Examination Ramesh Johari March 14, 2005 Instructions 1. Take alternate seating if possible. 2. Answer all questions in the spaces provided on these sheets. If needed, additional paper will be avail able at the front of the room. Answers given on any other paper will not be counted. 3. You may use a calculator, but no notes or books. 4. The examination begins at 8:30 am, and ends at 11:30 am. Honor Code In taking this examination, I acknowledge and accept the Stanford University Honor Code. NAME (signed) NAME (printed) 1 Useful Formulas 1. If T is an exponentially distributed random variable with mean 1 /λ , then the density of T is given by f T , where: f T ( t ) = λe λt , t ≥ . 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 ≥ . 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 – Short answer questions. (60 points; 12 points per question) 1. Either justify the following statement if true, or provide a counterexample if false: If a state i is recur rent for a continuous time Markov chain X t , then it is also recurrent for the jump chain corresponding to X t . 2. Suppose that X n is a discrete time Markov chain on the finite state space X = { 1 , . . . , K } . Assume that all states are aperiodic. Suppose also that this Markov chain has exactly two closed communicat ing classes, A 1 and A 2 (although there may be other communicating classes that are not closed). For any two states i and j does the limit: lim n →∞ P ( X n = j  X = i ) always exist? Justify your answer. 3 3. Consider an infinite server queue, where customers arrive according to a Poisson process of rate λ ....
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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.
 Winter '11
 Ramesh

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