### CS558 Computer Networks, Homework 2

• Question 1

• Give the state transition diagram for the M/M/2 (2 servers with infinite queue size) queueing system (10 pts)

• Give the set of equilibrium equations for the M/M/2 queueing system (10 pts)

• Give an expression the steady state probabilities

 p1 pk, for k ≥ 2

in terms of p0 (10 pts)

• Give an expression for p0 in terms of the arrival rate λ and service rate μ (without a sum of infinite number of terms - work out the summation !!!) (10 pts)

• Give an expression for the average queue length in terms of the arrival rate λ and service rate μ (without a sum of infinite number of terms - work out the summation !!!) (10 pts)

• Question 2: Repair shop

• An office operates 5 high speed printers.

Each printer has a exponentially distributed operational lifetime with an average life time of 1/λ. This will result in a Poission process with rate λ. Printer failures are independent; so the printer "failure rate" when k printers are operational is equal to k × λ

The office has a printer repait shop with one printer repair technician. The technician can repair a printer in an average repair time of 1/μ. This will result in a Poission process with rate μ.

• Questions:

 Give the state transition diagram for this Birth-death process (10 pts) Give the set of equilibrium equations (10 pts) Solve the equilibrium equations ( numerically ) and give the steady state probabilities ( numerical values ) for the case when λ = 1 and μ = 2 (10 pts) Compute (numerically) the average queue length for the case when λ = 1 and μ = 2 (10 pts) Compute (numerically) the average arrival rate of the printers to the repair shop for the case when λ = 1 and μ = 2 (10 pts)