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 ${\displaystyle \scriptstyle {\mu _{j}}}$ 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[3]

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

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.

For a BCMP network of m queues which is open, closed or mixed in which each queue is of type 1, 2, 3 or 4, the equilibrium state probabilities are given by

${\displaystyle \pi (x_{1},x_{2},\ldots ,x_{m})=C\pi _{1}(x_{1})\pi _{2}(x_{2})\cdots \pi _{m}(x_{m}),}$

where C is a normalizing constant chosen to make the equilibrium state probabilities sum to 1 and ${\displaystyle \scriptstyle {\pi _{i}(\cdot )}}$ represents the equilibrium distribution for queue i.