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Overview of the development of the theory

Basic Formulas This section is devoted to the most well-known formulas of queueing theory. The selection is subjective, but I think these are the ones from which many others have been derived. 4.1. Erlang’s Formulas As we mentioned in the earlier section the whole theory started with a practical problem. Erlang’s task can be formulated as follows: What fraction of the incoming calls is lost because of the busy line at the telephone exchange office. Of course, the answer is not so simple, since we first should know the inter-arrival and service time distributions. After collecting data Erlang verified that the Poisson-process arrival 18 J. Sztrik and exponentially distributed service were appropriate mathematical assumptions. He considered the M/M/n/n and M/M/n cases, that is the system where the arriving calls are lost because all the servers are busy, and where the calls have to wait for service, respectively. Assuming that the arrival intensity is λ, service rate is µ he derived the famous formulas for loss and delay systems, called Erlang B and C ones, respectively. Denoting ρ = λ/µ, the steady state probability that an arriving call is lost can be obtained in the following way Pn = ρ n n! Xn k=0 ρ k k! = B(n, ρ), where B(n, ρ) is the well-known Erlang B- formula, or loss formula. It can easily be seen that the following recurrence relation is valid B(n, ρ) = ρB(n − 1, ρ) n + ρB(m − 1, ρ) n = 2, 3, … B(1, ρ) = ρ 1 + ρ . Similarly, by using the B-formula the steady state probability that an arriving customer has to wait can be written as C(n, ρ) = nB(n, ρ) n − ρ(1 − B(n, ρ) , which is called Erlang C-formula, or Erlang’s delay formula. It should be mentioned that the B-formula is insensitive to the service time distribution, in other words it remains valid for any service time distribution with mean 1/µ.