## Closed queuing systems with exponential servers |

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ak(nk Arbitrary Asymptotic expressions Central Limit Theorem class of doubly CLOSED QUEUING SYSTEMS closed system completes service customer proceeds customer that completes customers in service Cyclic Queues distribution functions distribution of customers doubly stochastic matrices effective service rate equal equilibrium distribution EQUILIBRIUM EQUATIONS exponentially distributed service feedback Finch finite function G(N independent random variables infinite number input irreducible ith stage J. R. Jackson Koenigsberg large number Let the number M-l stages marginal distribution marginal probability distributions Markov process mean rate mean service rate n<+l normalization number of customers number of stages obtain open system P(ni parallel exponential servers parameter particular customer involved Pi(ni probability matrix P=(pik Q(ni R. R. P. Jackson random variables 17 ratio x,/r separation of variables servers in operation single exponential server Statistical Mechanics SYSTEMS WITH EXPONENTIAL tion of customers TM(N total number transfer probability matrix variance various stages vector