M/M/1 Queue System: Calculating A, B, C

[lina29]

Homework Statement

For an M/M/1 queuing system with the average arrival rate of 0.4 min^−1 and the average service time of 2 minutes, compute
A- the expected response time in minutes;
B- the fraction of time when there are more than 5 jobs in the system;
C- the fraction of customers who don't have to wait until their service begins

Homework Equations

The Attempt at a Solution

For A I got that λ_A= 2.5 and that λ_S= 1/2 and then r would be 5. Obviously that's wrong since r has to be less than 1 I just don't understand where I messed up.

 

[Ray Vickson]

Homework Statement

For an M/M/1 queuing system with the average arrival rate of 0.4 min^−1 and the average service time of 2 minutes, compute
A- the expected response time in minutes;
B- the fraction of time when there are more than 5 jobs in the system;
C- the fraction of customers who don't have to wait until their service begins

Homework Equations

The Attempt at a Solution

For A I got that λ_A= 2.5 and that λ_S= 1/2 and then r would be 5. Obviously that's wrong since r has to be less than 1 I just don't understand where I messed up.

λ_A = 0.4, not 2.5.

RGV

 

[lina29]

Thank you!
so for A-10 and for C-.2 which were correct

I'm still stuck on part B though. I know I'm supposed to find out P(X<5) but I don't know how I'm supposed to find it

 

[I like Serena]

Hi lina29!

Did you know that P(X<5)=P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)?

Actually, if you know what the sum of an geometric series is, it turns out that it's even easier to calculate P(X≥5)=P(X=5)+P(X=6)+...

 

[lina29]

Thank you!

 

[B.O.B.]

A slight extension to this question:

Given we know the arrival rate (λ)= 0.4 /min and the average service time (ε) = 0.5/min

How would one go about finding the proportion of customers who are in the shop more than 10min?

I am really unsure about how to approach this.
The probability of a customer being in the shop (ie queueing) for more than 10mins can be solved P(W>10)=exp(-(ε-λ)x10))

But what of the proportion of customers who spend more than 10 mins in the shop?

Thank you for any help!

 

[Ray Vickson]

A slight extension to this question:

Given we know the arrival rate (λ)= 0.4 /min and the average service time (ε) = 0.5/min

How would one go about finding the proportion of customers who are in the shop more than 10min?

I am really unsure about how to approach this.
The probability of a customer being in the shop (ie queueing) for more than 10mins can be solved P(W>10)=exp(-(ε-λ)x10))

But what of the proportion of customers who spend more than 10 mins in the shop?

Thank you for any help!

For a customer named John Smith, what is the probability he spends > 10 min in the system? For a customer named Jennifer Jones, what is the probability she spends > 10 mins in the system? For any customer you can name, what is the probability that he/she spends > 10 mins in the system? So, what proportion of customers spend > 10 mins in the system?