Suppose you have a system of 2 servers with service times exponentially distributed with service rate $\mu_1 =2$ jobs/s and $\mu_2= 4$ jobs/s. Whenever a job arrives in the system, it is served from one of the available servers. If both servers are available, the faster server is chosen with q probability. If both servers are busy, the job is lost.

  1. Create a model of the system (graph and analytical) when the job arrival process is Poissonian with arrival rate $\lambda=1$ job/s. (Write the system of equations that describe the system at steady state).

  2. Find an analytical expression of the average throughput at steady state.

I created the CTMC model as shown here:

hand-drawn state diagram

The generator matrix $Q$ is: \begin{bmatrix}-\lambda&\lambda q&\lambda(1-q)&0 \\\mu_2&-(\lambda+\mu_2)&0&\lambda\\\mu_1&0&-(\lambda+\mu_1)&\lambda\\0&\mu_1&\mu_2&-(\mu_1+\mu_2)\end{bmatrix}

In this kind of exercises, my professor usually deletes one of the equations and replaces it with the normalization condition that the probability sum is 1. I thought that was due to the linear dependence of rows, but in this case there's no linear dependence among the rows. (The determinant is not zero.) So how am I supposed to continue? Even if I compute $pQ=0$, which of the four equations should be replaced with normalization condition?


1 Answer 1


This matrix is built such a way that always a sum of all columns is zero, so determinant is zero, so you can proceed with it as usual.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.