Probs and Stats problem with Queuing systems

[caliboy]

1. Homework Statement [/b]
A barber shop has two chairs to cut hair and 10 people per hour enter the barbershop to get a haircut. . The average time it takes to get a haircut is 6 minutes. On this particular day, only one barber is cutting hair. Customers that enter the barber shop and use the other chair to wait in. Customers who see both chairs occupied, leave.
A) What is the system state probabilities?
B) What is the average number of customers that get a haircut in an hour
C) What is the average number of customers that get a haircut in an hour if both barbers are now working? There are no waiting chairs


2. Homework Equations



3. The Attempt at a Solution [/b]
A) I am really stumped by this one and would like some help. I believe this is a M/M/1/GD/c/∞ system; the formula I would use would be:

∏₂=(1-ρ)/(1-ρ^c+1)

c=2
ρ=1

B) λ= 10
µ=10 people/hr ρ=10/10; =1
(10)*1=10 customers/hr

C) λ= 10
µ=20 people/hr ρ=10/20; =1/2
(20)*1/2=10 customers/hr

 

[Ray Vickson]

1. Homework Statement [/b]
A barber shop has two chairs to cut hair and 10 people per hour enter the barbershop to get a haircut. . The average time it takes to get a haircut is 6 minutes. On this particular day, only one barber is cutting hair. Customers that enter the barber shop and use the other chair to wait in. Customers who see both chairs occupied, leave.
A) What is the system state probabilities?
B) What is the average number of customers that get a haircut in an hour
C) What is the average number of customers that get a haircut in an hour if both barbers are now working? There are no waiting chairs


2. Homework Equations



3. The Attempt at a Solution [/b]
A) I am really stumped by this one and would like some help. I believe this is a M/M/1/GD/c/∞ system; the formula I would use would be:

∏₂=(1-ρ)/(1-ρ^c+1)

c=2
ρ=1

B) λ= 10
µ=10 people/hr ρ=10/10; =1
(10)*1=10 customers/hr

C) λ= 10
µ=20 people/hr ρ=10/20; =1/2
(20)*1/2=10 customers/hr

You can model it as a finite-state continuous-time Markov chain, and get the equilibrium distribution using the standard methods. Of course, it is just a special case of a birth-death process, so you can specialize the general formulas for that case. Surely your textbook or course notes must have that material. If not, it is widely available on-line.

I really do not understand question (C): over the long-run, sometimes both barbers are idle, sometimes only one is working and sometimes both are busy (so customers are turned away). You just need the long-run rate at which customers exit the system (after being served, not turned away); this is also the long-run rate at which customers enter the shop. Are you sure you have written question (C) correctly?

RGV