final_2006 - MS&E 221 Ramesh Johari Final Examination...

MS&E 221 Final Examination
Ramesh Johari March 22, 2006
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 no notes, books, or calculators.
4. Explain your answers! It helps earn you partial credit.
5. 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 ≥ 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 – Short answer questions. Solve SIX of the following SEVEN questions. (90 points; 15 points
per question)
1. Either justify the following statement if true, or provide a counterexample if false: If a state i is tran-
sient for a continuous time Markov chain