Skip to content Skip to navigation

Connexions

You are here: Home » Content » BCMP Theorem

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

    • External bookmarks
  • E-mail the author

Recently Viewed

This feature requires Javascript to be enabled.

BCMP Theorem

Module by: Bart Sinclair

Summary: (Blank Abstract)

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.

Comments, questions, feedback, criticisms?

Send feedback