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- W155726497 abstract "A formula of Erlang type, denoted by Cd (r, rρ) and called 3rd (non-immediate-service) Erlang formula, is defined for the M|M|r|r+d system. This formula represents the equilibrium probability of non-immediate-service and is a function of three parameters: intensity of traffic “ρ”, number of servers “r”, and number of waiting positions “d”. When d = 0, Co (r, rρ) is the classical 1st Erlang formula and the limit of Cd (r, rρ) as d goes to +∞ is the classical 2nd Erlang formula. The extra parameter d allows a unified study of the three most important Markovian queueing system — M|M|r|r, M|M|r|r+d and M|M|r — and introduces a greater flexibility in its design. It is shown that, when in equilibrium, the number of customers waiting, the number of busy servers, the waiting time in the queue and the total sojourn time in the system are distributed as a sum of two well-known random variables, weighted by the 3rd Erlang formula. Some of these results also show the influence on one hand of the 1-server-system and on the other of the infinite-server-system in the study of the r-server-system, where the 3rd Erlang formula, can indicate a “heavy” / “low” traffic situation. The equilibrium loss probability, denoted by Bd (r, rρ), and the equilibrium delay probability, denoted by Dd (r, rρ), are also rewritten in terms of Cd (r, rρ). It is shown that Bd (r, rρ) is convex in d. It is reported that, Bd (r, rρ) is convex in ρ if ρ is less than some ρ* and concave if ρ is greater than ρ*, for a broad range of integer values of r and d, and that the M|M|1|1 system is the only M|M|r|r+d system for which the loss probability is concave in pin all its range." @default.
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- W155726497 date "1994-01-01" @default.
- W155726497 modified "2023-10-05" @default.
- W155726497 title "Generalizations of Erlang Formulae and Some of Their 2nd Order Properties" @default.
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- W155726497 doi "https://doi.org/10.1007/978-3-642-46955-8_101" @default.
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