## Cyclic queuing systems with restricted length queues |

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actual service rate average number average speed complete service conditional probability configuration is self-dual configuration mi,m consider counter-sequential motion customer at stage customers and holes cyclic system decreases differential-difference equations distribution of customers dual configuration dual problem dual system SD duality effect of blocking equal equi equilibrium distribution equilibrium equations equilibrium probability distribution exceed mini exponential servers finite capacity queue flow increases monotonically flow of customers flow rate given configuration im—N implies initial value problem ith server ith stage joint distribution joint equilibrium probability large number librium probability distribution marginal distributions directly mean rate mean service rates mi—N mi,nM mM,nM-i mM,tiM motion of customers N customers nM;t number of cus number of customers number of holes number of stages open system Pi(ni primal configuration primal system queue lengths rate at stage rate of flow sequential service at stage solution stage i+1 steady-state system of tandem tandem queuing time-dependent total number