The ergodicity of service systems with an infinite number of servomechanisms |
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Authors: | A Yu Veretennikov |
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Institution: | 1. M. V. Lomonosov Moscow State University, Moscow
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Abstract: | Existence, uniqueness, and ergodicity are proved for a stationary distribution for a service system having countably many servomechanisms with input flow rate μk depending on the number k of servomechanisms occupied, and with arbitrary (identical) distribution of the service time with finite mean μ, under the condition \(\mu \mathop {\overline {\lim } }\limits_{k \to \infty } \frac{{\lambda _k }}{{k + 1}}< 1\) . For this system we have, in particular, Erlang's formula $$p_k (t)\mathop \to \limits_{k + \infty } p_k = \frac{{\lambda _0 ...\lambda _{k - 1} \mu ^k }}{{k!}}p_0 ,k = 0,1,...,p_0^{ - 1} = \sum\nolimits_{k = 0}^\infty {\frac{{\lambda _0 ...\lambda _{k - 1} \mu ^k }}{{k!}}} ,\lambda _{ - 1} = 1.$$ |
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