In this task, you need to simulate and analyze an M/M/m/n queue with arbitrary m and n. You can reuse the codes in Week 7 Lab to simulate an M/M/m/n queue. You Python code must be submitted as supplementary material. You also need to theoretically compute the stationary distribution to verify your simulation. Finally, you will understand how queueing theorem will help design and install real-world telephone and cellular networks.

Throughout this question, let the arrival rate be λ. Let the service rate of each server be μ.

(1) Let λ = 1 and μ = 2. What is the stationary distribution of an M/M/2/6 queue. Please figure out this result by both

simulation and analysis.

(2) Let λ = 1 and μ = 1. What is the stationary distribution of an M/M/5/10 queue. Please figure out this result by both simulation and analysis.

(3) Use theoretical analysis only. What is the stationary distribution of an M/M/m/m queue (now n = m). What is the probability that a new arrival is blocked (dropped) by the system? This probability is defined as pb.

(4) Follow (3). Let μ = 1 and m = 10. If the blocking probability pb must be limited to 0.001, what is the maximum value ofλ?

(5) Follow (3). Let λ = 10 and μ = 1. If the blocking probability pb must be limited to 0.001, what is the minimum value ofm, i.e., the number of servers?

(6) Telecom/cellular companies usually use methods in (4) and (5) to design their telephone/cellular networks. In (4), given the required blocking probability pb and the number of telephone lines m, they want to know the maximum possible traffic intensity λ they can accommodate. In (5), given the required blocking probability pb and the traffic intensity λ, they want to know how many telephone lines they shall install. The method to compute pb is also well known as *Erlang Formula*. Google*Erlang Formula *and answer the following questions.

(a) What are Erlang-B and Erlang-C Formulas? When are they used? What is the difference between them? (b) In (4) and (5), do you use Erlang-B or Erlang-C Formula? Why?

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