M/M/1 Systems

  • Thread starter lina29
  • Start date
  • #1
85
0

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.
 

Answers and Replies

  • #2
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,722

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
 
  • #3
85
0
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
 
  • #4
I like Serena
Homework Helper
6,579
176
Hi lina29! :smile:

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)+...
 
  • #5
85
0
Thank you!
 
  • #6
1
0
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!
 
  • #7
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,722
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?
 

Related Threads on M/M/1 Systems

  • Last Post
Replies
6
Views
1K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
2
Views
1K
Replies
10
Views
1K
Replies
1
Views
998
  • Last Post
Replies
14
Views
1K
Replies
1
Views
2K
  • Last Post
Replies
12
Views
2K
  • Last Post
Replies
2
Views
730
Top