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Problem

Consider a single-server system that operates as follows:

• There are two types of customers: each type arriving at the system in a Poisson manner at rates 𝜆1 and 𝜆2, respectively; each type has exponential service time with rate 𝜇1 and 𝜇2, respectively.

• The system has capacity of 3 customers, including the one receiving service. In other words, whenever three customers are in the system (one receiving service and two waiting) the system is full and additional arriving customers are rejected and go elsewhere.

I am trying to find the proportion of customers that are rejected. I need to find the limiting probabilities of the states to do this.

Attempt

So far, I have thought to represent the queue states as the possible sequence of arrivals to the queue. The first position in this sequence indicates the customer type being currently processed. State $0$ corresponds to an empty system. State $1$ means I have a customer of type 1 being processed. State $12$ means I have a type 1 customer being processed and a type $2$ customer waiting.

Altogether I have state space, $S = \{0, 1, 11, 111, 12, 122, 121, 112, 2, 22, 222, 21, 211, 212, 221\}$

I believe the way solve this is to form my generator matrix, $A$, and use the expression $\pi A = 0$, where $\pi$ is the vector of limiting probabilities. However, using this approach is an algebraic nightmare. Is there a simpler/quicker way to solve this?

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