12. Networks of Queues

Question 1

N * M/M/1 System

Answer 1

• N different M/M/1 systems with individual queues
• q, rho, and lambda are divided by N per server
• q_total = N * q_individual

Question 2

M/M/N System

Answer 2

• N servers with a shared queue
• rho and lambda are divided by N per server
• Slightly faster than N * M/M/1 at intermediate arrival rates

Question 3

Pipelining

Answer 3

• System at steady state
• Rate in = rate out for every queue
• If the arrival process to a queue is Poisson, so is the departure process

Question 4

Splitting

Answer 4

• Path A is taken with probability p
• Path B is taken with probability 1- p
• If the arrivals are Poisson, a fixed portion p of arrivals on path A are Poisson

Question 5

Merging

Answer 5

• If two flows are joined, the resulting flow is the sum of the rates
• If the arrival rates are Poisson(a) and Poisson(b), the combined arrival rate is Poisson(a + b)

Question 6

Jackson’s Theorem

Answer 6

• System in steady state
• Outside arrivals are Poisson
• Service times are exponential
• All queues are infinite
• Theorem: in a network of queues, each queue can be analyzed in isolation