pk(t) = P{X(t)=k}

 

                                                          = Probability that system is in state k at time t

State Transition Diagram for a Birth-Death Process

Note that when the system is in state k, we assume that the arrival rate to the system is λ_k and the departure rate from the system is μ_k. Therefore, in a time interval Δt, the system will go from state k to state k +1 with probability λ_kΔt and will go from state k to state k -1 with probability μ_kΔt. This allows us to write the state transition equations given in the next slide.

The state transitions from time t to t + Δt will then be governed by the following equations -

(2.1)