Just homework 10

IE 425 Homework 10

Submit on Tuesday, 12/10

1.(20 pts) Consider the M/M/1/∞ queuing system descried in Problem 5 in Homework 9. Show that:

(a) (11 pts) The average number of customers in the system is:

L = λ

µ−λ

Hint:

L = ∞∑ n=0

nπn,

∞∑ n=0

nρn−1 = d

dρ

∞∑ n=0

ρn = d

dρ

( 1

1 −ρ

) =

1

(1 −ρ)2

(b) (3 pts) The average waiting time in the system (from entrance to exist) is:

W = 1

µ−λ

(c) (3 pts) The average waiting time in the queue, not including service, is:

W0 = λ

µ(µ−λ)

(d) (3 pts) The average number of customers in the queue, not including service, is:

L0 = λ2

µ(µ−λ)

2.(15 pts) Consider the M/M/c/∞ queuing system descried in Problem 6 in Homework 9. Show that:

L0 = π0 c!

( λ

µ

)c ( λ

cµ

)( 1 −

λ

cµ

)−2 Then, using Little’s law, we can compute:

W0 = L0 λ , W = W0 +

1

µ , L = λW = λ

( W0 +

1

µ

) = L0 +

λ

µ

Hint:

L0 = ∞∑ n=c

(n− c)πn = ∞∑ m=0

mπc+m, ∞∑ m=0

mρm = ρ

(1 −ρ)2

3.(15 pts) Consider the M/M/∞/∞ queuing system descried in Problem 7 in Homework 9. Show that:

L = λ

µ , W =

1

µ , W0 = 0, L0 = 0

4.(15 pts) Consider the M/M/c/c queuing system descried in Problem 8 in Homework 9.

(a) (3 pts) Explain why L0 = 0 and W0 = 0.

(b) (3 pts) Explain why W = 1 µ .

1

(c) (5 pts) Explain why the mean arrival rate to the system is λ(1 −πc) (d) (4 pts) Show that:

L = λ

µ (1 −πc)

5. (20 pts) Consider the M/M/c/k queuing system descried in Problem 9 in Homework 9. Show

that:

(a) (14 pts)

L0 = π0 c!

( λ

µ

)c ( λ

cµ

)( 1 −

λ

cµ

)−2 [ 1 −

( λ

cµ

)k−c − (k − c)

( λ

cµ

)k−c ( 1 −

λ

cµ

)] Hint:

L0 = ∞∑ n=c

(n− c)πn, M∑ m=0

mρm−1 = d

dρ

M∑ m=0

ρm

(b) (2 pts) Explain why:

L = L0 + c−1∑ n=0

nπn + c

( 1 −

c−1∑ n=0

πn

) (c) (2 pts) Explain why the mean arrival rate to the system is λ(1 −πk). (d) (2 pts) Show that:

W0 = L0

λ(1 −πk)

W = L0

λ(1 −πk) +

1

µ

6. (15 pts) For the M/M/c/∞ queuing system with a finite calling population N descried in Problem 10 in Homework 9, it is more convenient to use the generic formulas to compute the queue

length and the number of customers in the system :

L0 = N∑

n=c+1

(n− c)πn

L = L0 + c−1∑ n=0

nπn + c

( 1 −

c−1∑ n=0

πn

) (a) (10 pts) Show that the mean arrival rate to the system is:

N∑ n=0

(N −n)λπn = · · · = λ(N −L)

(b) (5 pts) Show that:

W0 = L0

λ(N −L)

W = L0

λ(N −L) +

1

µ

2