Skip to content Skip to navigation

Connexions

You are here: Home » Content » Jackson's Theorem

Navigation

Recently Viewed

This feature requires Javascript to be enabled.
 

Jackson's Theorem

Module by: Bart Sinclair. E-mail the author

Summary: (Blank Abstract)

Jackson's Theorem is the first significant development in the theory of networks of queues. It assumes an open queueing network of single-server queues with the following characteristics:

  • M=# of queues in the system, not counting queue 0 which represents the outside world M # of queues in the system, not counting queue 0 which represents the outside world
  • μ i =service rate at queue i μ i service rate at queue i
  • λ j = total rate at which jobs arrive at queue j λ j total rate at which jobs arrive at queue j
  • ∀i,1≤i≤M: ρ i =utilization of at queue i= λ i μ i <1 i 1 i M ρ i utilization of at queue i λ i μ i 1
  • n i ⁢t=# of jobs in queue i at time t n i t # of jobs in queue i at time t
  • n ⁢t= n 1 ⁢t n 2 ⁢t… n M ⁢tT=the system state at time t n t n 1 t n 2 t … n M t the system state at time t
  • P⁢ k 1 k 2 … k M t=Pr⁢ n ⁢t= k 1 k 2 … k M T P k 1 k 2 … k M t Pr n t k 1 k 2 … k M
  • P⁢ k 1 k 2 … k M =limit  t→∞P⁢ k 1 k 2 … k M t P k 1 k 2 … k M t P k 1 k 2 … k M t
Arrivals from the outside world are Poisson. All queues have exponential service time distributions.

Theorem 1: Jackson's Theorem

P⁢ k 1 k 2 … k M =∏i=1M(1− ρ i )⁢ ρ i k i P k 1 k 2 … k M i 1 M 1 ρ i ρ i k i

Proof

Let

  • λ i , j = rate at which jobs leaving queue i go to queue j λ i , j rate at which jobs leaving queue i go to queue j
  • q i , j = probability that a job departing queue i goes directly to queue j q i , j probability that a job departing queue i goes directly to queue j
During an interval of length Δ⁢t Δ t , only four possible events may occur:
  1. a job arrives from the outside world
  2. a job departs to the outside world
  3. a job leaves on queue and enters another
  4. none of the above
These four possibilities are incorporated into the following equation:
P⁢ k 1 k 2 … k M t+Δ⁢t=∑j=1MP⁢ k 1 k 2 … k j - 1 k j −1 k j + 1 … k M t⁢ λ 0 , j ⁢Δ⁢t+∑i=1MP⁢ k 1 k 2 … k i - 1 k i +1 k i + 1 … k M t⁢ u i ⁢ q i , 0 ⁢Δ⁢t+∑i=1M∑j=1MP⁢ k 1 k 2 … k i - 1 k i +1 k i + 1 … k j - 1 k j −1 k j + 1 … k M t⁢ u i ⁢ q i , j ⁢Δ⁢t+P⁢ k 1 k 2 … k M t⁢(1−Δ⁢t⁢∑j=1M λ 0 , j + μ j ) P k 1 k 2 … k M t Δ t j 1 M P k 1 k 2 … k j - 1 k j 1 k j + 1 … k M t λ 0 , j Δ t i 1 M P k 1 k 2 … k i - 1 k i 1 k i + 1 … k M t u i q i , 0 Δ t i 1 M j 1 M P k 1 k 2 … k i - 1 k i 1 k i + 1 … k j - 1 k j 1 k j + 1 … k M t u i q i , j Δ t P k 1 k 2 … k M t 1 Δ t j 1 M λ 0 , j μ j (1)
Moving the P⁢ k 1 k 2 … k M t P k 1 k 2 … k M t term to the left hand side of the equation, dividing both sides by Δ⁢t Δ t , and taking the limit as Δ⁢t→0 Δ t 0 gives us
ddtP⁢ k 1 k 2 … k M t=∑j=1MP⁢ k 1 k 2 … k j - 1 k j −1 k j + 1 … k M t⁢ λ 0 , j +∑i=1MP⁢ k 1 k 2 … k i - 1 k i +1 k i + 1 … k M t⁢ u i ⁢ q i , 0 +∑i=1M∑j=1MP⁢ k 1 k 2 … k i - 1 k i +1 k i + 1 … k j - 1 k j −1 k j + 1 … k M t⁢ u i ⁢ q i , j −∑j=1MP⁢ k 1 k 2 … k M t⁢( λ 0 , j + μ j )=0 t P k 1 k 2 … k M t j 1 M P k 1 k 2 … k j - 1 k j 1 k j + 1 … k M t λ 0 , j i 1 M P k 1 k 2 … k i - 1 k i 1 k i + 1 … k M t u i q i , 0 i 1 M j 1 M P k 1 k 2 … k i - 1 k i 1 k i + 1 … k j - 1 k j 1 k j + 1 … k M t u i q i , j j 1 M P k 1 k 2 … k M t λ 0 , j μ j 0 (2)
in steady state, which will exist as long as λ j < μ j λ j μ j , 1≤j≤M 1 j M . Hence,
∑j=1MP⁢ k 1 k 2 … k M t⁢( λ 0 , j + μ j )=∑j=1MP⁢ k 1 k 2 … k j - 1 k j −1 k j + 1 … k M t⁢ λ 0 , j +∑i=1MP⁢ k 1 k 2 … k i - 1 k i +1 k i + 1 … k M t⁢ u i ⁢ q i , 0 +∑i=1M∑j=1MP⁢ k 1 k 2 … k i - 1 k i +1 k i + 1 … k j - 1 k j −1 k j + 1 … k M t⁢ u i ⁢ q i , j j 1 M P k 1 k 2 … k M t λ 0 , j μ j j 1 M P k 1 k 2 … k j - 1 k j 1 k j + 1 … k M t λ 0 , j i 1 M P k 1 k 2 … k i - 1 k i 1 k i + 1 … k M t u i q i , 0 i 1 M j 1 M P k 1 k 2 … k i - 1 k i 1 k i + 1 … k j - 1 k j 1 k j + 1 … k M t u i q i , j (3)
Assume that the network of queues has a product-form solution; i.e., assume P⁢ k 1 k 2 … k M =∏i=1M(1− ρ i )⁢ ρ i k i P k 1 k 2 … k M i 1 M 1 ρ i ρ i k i Substitute this solution into the steady state equation and cancel common terms:
∑j=1M λ 0 , j + μ j =∑j=1M λ 0 , j ρ j +∑i=1M ρ i ⁢ μ i ⁢ q i , 0 +∑i=1M∑j=1M ρ i ρ j ⁢ μ i ⁢ q i , j j 1 M λ 0 , j μ j j 1 M λ 0 , j ρ j i 1 M ρ i μ i q i , 0 i 1 M j 1 M ρ i ρ j μ i q i , j (4)
Looking at the individual terms, ∑j=1M λ 0 , j ρ j =∑j=1M λ 0 , j ⁢ μ j λ j j 1 M λ 0 , j ρ j j 1 M λ 0 , j μ j λ j ∑i=1M ρ i ⁢ μ i ⁢ q i , 0 =∑i=1M λ i ⁢(1−∑i=1M q i , j )=∑i=1M λ i −∑i=1M∑j=1M λ i ⁢ q i , j =∑j=1M λ 0 , j i 1 M ρ i μ i q i , 0 i 1 M λ i 1 i 1 M q i , j i 1 M λ i i 1 M j 1 M λ i q i , j j 1 M λ 0 , j ∑i=1M∑j=1M ρ i ρ j ⁢ μ i ⁢ q i , j =∑j=1M μ j λ j ⁢∑i=1M λ i ⁢ q i , j =∑j=1M μ j λ j ⁢( λ j − λ 0 , j )=∑j=1M μ j −∑j=1M μ j ⁢ λ 0 , j λ j i 1 M j 1 M ρ i ρ j μ i q i , j j 1 M μ j λ j i 1 M λ i q i , j j 1 M μ j λ j λ j λ 0 , j j 1 M μ j j 1 M μ j λ 0 , j λ j Substituting these terms back into the previous equation gives us
∑j=1M λ 0 , j +∑j=1M μ j =∑j=1M λ 0 , j ⁢ μ j λ j +∑j=1M λ 0 , j +∑j=1M μ j −∑j=1M λ 0 , j ⁢ μ j λ j j 1 M λ 0 , j j 1 M μ j j 1 M λ 0 , j μ j λ j j 1 M λ 0 , j j 1 M μ j j 1 M λ 0 , j μ j λ j (5)
proving the theorem.

One consequence of this theorem is that a Jackson network satisfies local balance; i.e., μ i ⁢P⁢ k 1 k 2 … k i … k n = λ i ⁢P⁢ k 1 k 2 … k i −1… k n μ i P k 1 k 2 … k i … k n λ i P k 1 k 2 … k i 1 … k n This can easily be verified:

μ i ⁢∏j=1M(1− ρ j )⁢ ρ j k j = λ i ⁢∏j=1,j≠iM(1− ρ j )⁢ ρ j k j ⁢((1− ρ i )⁢ ρ i k i −1) μ i j 1 M 1 ρ j ρ j k j λ i j 1 M j i 1 ρ j ρ j k j 1 ρ i ρ i k i 1 (6)
μ i ⁢(1− ρ i )⁢ ρ i k j = λ i ⁢(1− ρ i )⁢ ρ i k i −1 μ i 1 ρ i ρ i k j λ i 1 ρ i ρ i k i 1 μ i ⁢ ρ i = λ i μ i ρ i λ i

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

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

What is in a lens?

Lens makers point to 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 member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks