Transient Solution
We can write a probability mass function dependent on t to describe the probability that the M/M/1 queue is in a particular state at a given time. We assume that the queue is initially in state i and write pk(t) for the probability of being in state k at time t. Then
where, and Ik is the modified Bessel function of the first kind. Moments for the transient solution can be expressed as the sum of two monotone functions.
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