BCMP network

The paper is well known, and the theorem was described in 1990 as "one of the seminal achievements in queueing theory in the last 20 years" by J. Michael Harrison and Ruth J. Williams.

Definition of a BCMP network

A network of m interconnected queues is known as a BCMP network if each of the queues is of one of the following four types:

1. FCFS discipline where all customers have the same negative exponential service time distribution. The service rate can be state dependent, so write ${\displaystyle {\mu }_j}$ for the service rate when the queue length is j.
2. Processor sharing queues
3. Infinite-server queues
4. LCFS with pre-emptive resume (work is not lost)

In the final three cases, service time distributions must have rational Laplace transforms. This means the Laplace transform must be of the form

${\displaystyle L(s)=\frac{N(s)}{D(s)}.}$

Also, the following conditions must be met.

1. external arrivals to node i (if any) form a Poisson process,
2. a customer completing service at queue i will either move to some new queue j with (fixed) probability ${\displaystyle P_{ij}}$ or leave the system with probability ${\displaystyle 1-\sum _{j=1}^m{P}_{ij}}$, which is non-zero for some subset of the queues.