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Part BCSIR NET December 2024the-service-rate-scales-with-busy-servers

The service rate scales with busy servers

Consider an M/M/3 queuing system with arrival rate λ=2\lambda = 2 and service rate μ=3/2\mu = 3/2. Define, for i=1,2,…,Ii=1i = 1, 2, \dots, I_{i} = 1 if the first transition is from i to i + 1, and Ii=0I_{i} = 0 if the first transition from i is i − 1. Then, Var(I4)Var(I_{4}) equals

  1. A.33/169
  2. B.34/169
  3. C.35/169
  4. D.36/169✓ Correct

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests hypothesis dropped.

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50 are analysed free — try those first.

The trap it tests

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this

Related counterexample: An irreducible chain with a stationary distribution converges to it

More on this topic

The chapter behind this: Markov chains: classification and stationary behaviour — free to read

From Limit Theorems and Markov Chains › Markov chains: classification of states, stationary distributions

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