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Jackson's Theorem

Module by: Bart Sinclair. E-mail the author

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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
  • nt= 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=Prnt= 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 =limt→∞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=1M1− 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 Δt 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+Δt=∑j=1MP k 1 k 2 … k j - 1 k j −1 k j + 1 … k M t λ 0 , j Δt+∑i=1MP k 1 k 2 … k i - 1 k i +1 k i + 1 … k M t u i q i , 0 Δt+∑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 Δt+P k 1 k 2 … k M t1−Δt∑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 Δt Δ t , and taking the limit as Δt→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=1M1− 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=1M1− ρ j ρ j k j = λ i ∏j=1j≠iM1− 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

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