Bernoulli Process Statistics

[lina29]

Homework Statement

Consider Bernoulli single-server queuing process with an arrival rate of 2 jobs per minute, a service rate of 4 jobs per minute, frames of 0.2 minutes, and a capacity limited by 2 jobs. Compute the steady-state distribution of the number of jobs in the system.

Homework Equations

The Attempt at a Solution

For P I got
.6 .4 0
.48 .44 .08
0 .48 .44

and my system of equations became

.6a + .48b = a
.4a+ .44b + .48c =b
.08b + .44c = c

and then by simplifying I got

.48b = .4a
.08b = .56c

But I'm stuck right there and don't know where to go. Any help would be appreciated. thanks!

 

[mighty2000]

Hi there,

To solve this problem, we can use the steady-state distribution formula for a single-server queuing process:

P(n) = (1-r/r)^n * r/r

where P(n) is the probability of having n jobs in the system, r is the arrival rate, and r is the service rate.

In this case, r=2 and r=4, so we have:

P(0) = (1-2/4)^0 * 2/4 = 1/2
P(1) = (1-2/4)^1 * 2/4 = 1/4
P(2) = (1-2/4)^2 * 2/4 = 1/8

Therefore, the steady-state distribution of the number of jobs in the system is:

P(0) = 1/2
P(1) = 1/4
P(2) = 1/8

I hope this helps! Let me know if you have any further questions.