A queue with a total capacity of $5$ customers has $2$ servers who serve at rates $\mu_1 = 1$ customer/hour and $\mu_2 = 2$ customers/hour respectively. Service times are exponentially distributed. Assume that customers are served on a $FCFS$ (first-come, first-served) basis with customers arriving according to a Poisson process with $\lambda = 3$ customers/hour. In addition, assume that when a customer first arrives in the (empty) queue, they are assigned to either one of the servers with $\frac{1}{2}$ probability.

How do I find the limiting probability distribution of the number of customers in the queue? I think I have to set up the state space as $S = \{0, 1_A, 1_B, 2, 3, 4, 5\}$ to account for the fact that the $1$ state differs depending on which server is operating. But I do not know how to account for the fact that it is a $FCFS$ basis & the probability of the first customer being assigned either server is $1/2$.


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