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BCMP Theorem

Module by: Bart Sinclair. E-mail the author

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Jackson's Theorem is concerned only with networks of single-server queues having exponentially distributed service times. The theorem states that the steady state queue occupancy distribution is the product of the individual queue distributions when each queue is treated as an independent, M/M/1 queue with the appropriate arrival rate. For this reason, networks of single server queues with exponential service times and Poisson arrival rates from the "outside world" are called product-form or separable queueing networks.

The BCMP Theorem takes this idea much farther. It proves a similar result for a much larger class of queueing networks. In particular, queues are not constrained to have exponentially distributed service rates, although if the service time distribution for a queue in not exponential, the queue can have only one of three queueing disciplines not including FCFS.

Assume a computer network consists of MM queues. Each queue is one of the following:

  1. FCFS with class-independent exponentially distributed service time
  2. PS (processor sharing)
  3. IS (infinite server)
  4. LCFSPR (last come first served preemptive resume)
For the last three, the service time distribution is Coxian; i.e., the service time distribution has a Laplace transform of the form Ls=NsDs L s N s D s where Ns N s and Ds D s are polynomials in ss, degNs<degDs deg N s deg D s , all roots of Ds D s are real, and L0=1 L 0 1 (this last condition is necessary for the transform of any probability distribution function).

Also assume that routing of jobs among queues is state-independent. That is, jobs are routed among the queues according to fixed probabilities (which may be different for different classes of jobs) and not based on the number of jobs in the queues.

Then the steady state probability distribution π n 1 n 2 … n M π n 1 n 2 … n M , the probability of being in the state with n m n m jobs in queue mm, 1≤m≤M 1 m M , is of the form

π n 1 n 2 … n M = π 1 n 1 π 2 n 2 ⋯ π M n M K π n 1 n 2 … n M π 1 n 1 π 2 n 2 ⋯ π M n M K (1)
where π m n m π m n m is the probability of n m n m jobs in queue mm if the queue were in isolation with a Poisson input process having the same rate as the throughput for queue mm, and KK is a normalization constant. In the case of an open network KK, is always 1. For a closed network, KK is determined by the constraint that the state probabilities sum to 1; i.e.,
K=∑all states∈ n 1 ′ n 2 ′ … n M ′ n M ′ ≥0∧1≤m≤M∧∑m=1M n m ′ =N π 1 n 1 ′ π 2 n 2 ′ ⋯ π M n M ′ K all states n 1 ′ n 2 ′ … n M ′ n M ′ 0 1 m M m 1 M n m ′ N π 1 n 1 ′ π 2 n 2 ′ ⋯ π M n M ′ (2)
with NN being the total number of jobs in the system.

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