E_bdpros - J. Virtamo 38.3143 Queueing Theory / Birth-death...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: J. Virtamo 38.3143 Queueing Theory / Birth-death processes 1 Birth-death processes General A birth-death (BD process) process refers to a Markov process with- a discrete state space- the states of which can be enumerated with index i=0,1,2,. . . such that- state transitions can occur only between neighbouring states, i → i + 1 or i → i- 1 l m 1 1 l 1 m 2 2 l 2 m 3 i+1 l i+1 m i+2 i l i m i+1 . . . Transition rates q i,j =              λ i when j = i + 1 μ i if j = i- 1 otherwise vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle probability of birth in interval Δ t is λ i Δ t probability of death in interval Δ t is μ i Δ t when the system is in state i J. Virtamo 38.3143 Queueing Theory / Birth-death processes 2 The equilibrium probabilities of a BD process We use the method of a cut = global balance condition applied on the set of states 0 , 1 , . . ., k . In equilibrium the probability flows across the cut are balanced (net flow =0) λ k π k = μ k +1 π k +1 k = 0 , 1 , 2 , . . . We obtain the recursion π k +1 = λ k μ k +1 π k By means of the recursion, all the state probabilities can be expressed in terms of that of the state 0, π , π k = λ k- 1 λ k- 2 ··· λ μ k μ k- 1 ··· μ 1 π = k- 1 productdisplay i =0 λ i μ i +1 π The probability π is determined by the normalization condition π π = 1 1 + λ μ 1 + λ λ 1 μ 1 μ 2 + ··· = 1 1 + ∞ summationdisplay k =1 k- 1 productdisplay i =0 λ i μ i +1 J. Virtamo 38.3143 Queueing Theory / Birth-death processes 3 The time-dependent solution of a BD process Above we considered the equilibrium distribution π of a BD process. Sometimes the state probabilities at time 0, π (0), are known- usually one knows that the system at time 0 is precisely in a given state k ; then π k (0) = 1 and π j (0) = 0 when j negationslash = k and one wishes to determine how the state probabilities evolve as a function of time π ( t )- in the limit we have lim t →∞ π ( t ) = π ....
View Full Document

This note was uploaded on 01/05/2010 for the course ECE 01 taught by Professor All during the Spring '09 term at Aarhus Universitet.

Page1 / 12

E_bdpros - J. Virtamo 38.3143 Queueing Theory / Birth-death...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online