13.19 Suppose orders are coming to a market exchange according to a Poisson process with rate λ from high-frequency traders. However, the exchange can only process K orders at the same time, so an order sent when the system is processing K orders is lost. Furthermore, assume that the order processing time is an exponential random variable with rate μ (mean 1/μ). Show that the number of orders being processed at time t is a birth and death process with transition rates qi,i−1 = iμ, for i ∈ {1, 2,...,K}

qi,i+1 = λ, for i ∈ {0, 1,...,K − 1}

Calculate the stationary distribution and the probability that an order is going to be lost (thus the long-term proportion of orders lost). This is called the Erlang formula.