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# Packets arrive to the queue from sources 1 and 2 according to a Poisson process with rate λ1 and λ2,

respectively. The service times of packets from sources 1 and 2 are all i.i.d. and exponentially distributed with mean 1/μ. A packet from source 1 is always accepted to the queue. A packet from source 2 is accepted (into the queue) only if the number of packets in the system (queue+service) is less than a given number K > 0; otherwise, it is assumed to be lost (denied). Assume that the size of "waiting room" in the queue is infinite.

1. (a) (5 points) What is the range of values of λ1 and λ2 for which there exists a steady-state probability distribution of the system?
2. To answer the remaining questions, assume that λ1 and λ2 are in this range.
3. (b) (10 points) Draw the state transition diagram for this system. Find the steady-state proba- bility of having n packets (Pn) in the system (in terms of system parameters and P0). Also, express P0 in terms of system parameters λ1, λ2, μ, K.
4. (c) (10 points) Assume now that λ1 = λ2 = λ and this λ is within the stability range (as obtained in (a)). Define ρ = λ/μ. Find the probability that a packet from source 2 is lost (not accepted to the queue) in terms of ρ and P0 (where P0 is from (b)).

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