Birth Death Process

Birth Death Process is a Random Process that follows Memoryless property of Continuous Parameter Markov Chain. The random process X(t) with State Space S = {0,1,2,3,.....} is called Birth-Death Process.

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Let queuing System is in State Sn if there are 'n' customers in the system including those being served.
Let N(t) be Markov Process that takes on the value 'n' when Queuing System is in State 'Sn'.


Assumptions: 

• If System is in State Sn, it can make transition only to S_n-1 or  S_n+1 , n ≥1 i.e either a customer completes service and leaves the system or while the present customer is still being served, another customer arrives in the system. So next state can only be S₁ .
• If the system is in State Sn at time t, the probability of transition to S_n+1 in the time interval  (t, t+Δt)  =  λ.Δt   , where λ is the arrival parameter (birth parameter).
• If the system is in State Sn at time t, the probability of transition to S_n-1 in the time interval  (t, t+Δt)  =  μ.Δt   , where μ is the departure parameter (death parameter).

Process N(t) is the Birth-Death Process.

Let P_n(t)  be the probability that the Queuing System is in state Sn at time t,
P_n(t) = P{N(t) = n}

At steady state, (no transition) i.e 't' tends to infinity
P_n(t) = Pn

States: 0, 1, 2, 3,......, n-1, n, n+1
Arrival rate: λ₀ = λ1 = λn .......= λ  
Departure rate: μ₁ =  μ₂ = μ_n+1 = μ

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