Poisson flows in systems with retrial queues

Queuing systems with retrial queues represent large interest in manifold applications of the queuing theory. Especially last years it is connected with infinite server queuing systems and retrial queues. These models may be considered in terms of queuing networks which have a structure of directed tree. An analysis of such networks may be based on the Burke theorem and the graph theory concepts in a definition of independence of random flows. In this paper an attempt to solve this problem in the most general terms is realized. Consider infinite server queuing system with Poisson input flow and exponentially distributed service times. Assume that each customer after a service may be served iteratively. The different protokols of functioning for such system may be realized. For example, a distribution of repeated service time may differ from a distribution of initial service time, a number of repeated service may be different and so on. The most general model of such system is a model of queuing network in which each node has infinite number of servers with exponential distribution of service times. Such network is described by a directed tree D in which any edge is directed from a tree root r (a node which have not incoming edges) to its leaves (nodes which have not outgoing edges). On service finishing a customer may be directed to some nodes where its outcoming edges show. Natural generalization of this model is based on an asumption that service times have hyperexponential distributions (probability mixtures of exponential distributions). This asumption does not change a view of this network in which transitions between nodes are discribed by oriented tree also. From the Burke theorem we have that if the network is in a stationary regime then its outcomig and incoming flows of each node are Poisson. But a problem is on dependent or independent for these flows. This problem is solved as follows. Consider now Poisson point flow Л = {0 < t₁ < t₂ < ...} with the intensity X . Assume that each point of the flow Л independently with a probability л ,0< p <1, becomes a point of a flow Л^ and with a probability p₂ =1 - p₁ - a point of a flow Л2 It is well known that the flows Л₁, Л₂ are Poisson flows also with intensities Xpj, Xp₂, accordingly. Theorem 1. Flows Л₁, Л₂ are independent. On a set of the directed tree D (in which each node has a single incoming edge) define a relation of partial order U У U2, if in the tree D there is a way from the node to the node U2 (here U У U ). Contrast each node U of the tree D the nodes sets P(u) = {v: u у v}, Q(u) = {v: vу u}иP(u). Theorem 2. Assume that nodes Щ,..., Ukofthe tree D satisfy the condition that the sets P(u₁),. ,,P(u_k) do not intersect.Then for any nodes v₁ e P(u₁),..., v_k e P(u_k ) flows Л(^),..., Л(vk ), incoming to the nodes v_1;..., v_k, appropriately are independent. Obtained results may be spread onto networks with final number of servers in their nodes and to transit from these networks to networks with infinite nimber of servers by analyzing synergetic effects. If input network flow is recurrent then it is possible to use results on asymptotic analysis of numbers of occupied servers in network nodes.

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