Pollaczek–Khinchine formula

In queueing theory, a discipline within the mathematical theory of probability, the Pollaczek–Khinchine formula states a relationship between the queue length and service time distribution Laplace transforms for an M/G/1 queue (where jobs arrive according to a Poisson process and have general service time distribution). The term is also used to refer to the relationships between the mean queue length and mean waiting/service time in such a model. In queueing theory, a discipline within the mathematical theory of probability, the Pollaczek–Khinchine formula states a relationship between the queue length and service time distribution Laplace transforms for an M/G/1 queue (where jobs arrive according to a Poisson process and have general service time distribution). The term is also used to refer to the relationships between the mean queue length and mean waiting/service time in such a model. The formula was first published by Felix Pollaczek in 1930 and recast in probabilistic terms by Aleksandr Khinchin two years later. In ruin theory the formula can be used to compute the probability of ultimate ruin (probability of an insurance company going bankrupt). The formula states that the mean queue length L is given by