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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?

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1 Answer 1

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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.

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