Abstract
We consider K nodes in series in condi- tions of identical service : the time of service of one customer on one device of nodes is the same for all different nodes (a.s.) . In every node N devices are located,N ≥ 1. Each node contains the infinite set of places for wait . The tonal number of customers,the time of wait in nodes 2,...,K remains limited for all moments of times in heavy traffic conditions on the first node , if ρ > 1, ρ = 1, ρ → 1 . The fact takes place , if the time of service of one customer by one device of node less than constant. Similar results are got in the general situation and for disciplines of service without interruption.
Highlights
We consider K nodes in series in conditions of identical service : the time of service of one customer on one unit of nodes is the same for all different nodes (a.s.)
In every node N devices are located,N ≥ 1.Each node contains infinite set of places for wait
The identical time of service of any customers in the different nodes by every device is main restriction of the article: ξj1 = ξj2 = . . . = ξjK, j = 1, 2, . . . , where the random value ξji is the time of service of the jth customer on the ith node by one device ; all the different devices on all nodes are identical and {ξj1, j = 1, 2, . . .} are mutually independent random values with the distribution function F (x) = Pr(ξj1 ≤ x)
Summary
We consider K nodes in series in conditions of identical service : the time of service of one customer on one unit ( device) of nodes is the same for all different nodes (a.s.). The main result can be formulated as follows : in the condition of identical service the total quantity of customers on nodes with numbers 2,...,K ( not one ) remains limited for all moments of times t ∈ [0, ∞), if even the classical loading ρ for the first node is more than one ( ρ > 1 ) as so , if ρ = 1, ρ → 1 ( the third and fourth sections and the (8) expression ). If a customers arrives on a node in group (for non-ordinary process ),the customers are disposed in the group in the random order - all results d‘t rely on order in the group.
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