Question

# 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.

- (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?
- To answer the remaining questions, assume that λ1 and λ2 are in this range.
- (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.
- (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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