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Unformatted text preview: J. Virtamo 38.3143 Queueing Theory / Birthdeath processes 1 Birthdeath processes General A birthdeath (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 / Birthdeath 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 / Birthdeath processes 3 The timedependent 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 ....
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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.
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