Homework 4

Question 1

Consider an 𝑀/𝑀/1 model with 𝜆 , 𝜇 , and FIFO queue discipline.
Compute p,𝑝0,pn, 𝐿, 𝐿𝑞, 𝑊, 𝑊𝑞 assuming that the queue can be infinitely long and the queue
system is in steady state.

Answer 1

p = 𝜆/𝜇
p0 = 1 - p
pn = pp0
L = sum from n = 0 to infinity npn = sum from n = 0 to infinity p0np^n = 𝜆/(𝜇−𝜆) = p/p0
𝑊 = 1/(𝜇−𝜆) = L/𝜆
Lq = 𝐿 −p
Wq = 𝐿𝑞/𝜆

Question 2

Consider an 𝑀/𝑀/s model with 𝜆 , 𝜇 , and FIFO queue discipline.
Compute p,𝑝0,pn, 𝐿, 𝐿𝑞, 𝑊, 𝑊𝑞 assuming that the queue can be infinitely long and the queue
system is in steady state.

Answer 2

p = cnp0
p0 = (1+sum(from n=1 to infinity) cn)^-1
L = p0sum(from n=0 to infinity) ncn
Lq = p0sum(from n=s to infinity) (n-s)cn
W = L/ 𝜆
Wq = Lq/ 𝜆

Question 3

cn for M/M/s/K model

Answer 3

if n<=s: cn= (1/n!)(𝜆/𝜇)^n
if s<=n<=K: cn = (1/s!)(𝜆/𝜇)^2(𝜆/s𝜇)^(n-s)
n>K: xn= 0

Question 4

sum from n=0 to infinity for n(x)^n

Answer 4

x/(1-x)^2

Question 5

sum from n=0 to infinity for (x)^n

Answer 5

1/(1-x)