Zero-automatic networks
One continues the study of zero-automatic queues introduced in [Adv. Appl. Probab. 39, No. 2, 429--461 (2007; Zbl 1121.60094)] which may be viewed as a synthesis of a queue and a random walk on some infinite group or monoid, or as a synthesis and wide generalization of the simple queue and the G-queue with positive and negative customers. One takes \(N\) zero-automatic queues corresponding to possibly different groups or monoids. For a non-empty subset of queues there is an external arrival process of customers. After completion of its service, a customer is either routed to another queue or leaves the network, two routing policies are assumed: Jackson-like and Kelly-like networks. Customers of different types from the set \(\Sigma \) may enter, the buffer content is given by the word \(\Sigma ^*\), the arrival and service rates are \(\lambda \) and \(\mu \), respectively. The set of possible buffer contents is of the form \(L=\{u_k\ldots u_1\in \Sigma ^*~|~u_i\in Next(u_{i+1})\}\), where \(Next(a)\subset \Sigma \) is the subset of types which do not interact with type \(a\). Let \((X_n)_n\) be the \(L\)-valued random variables corresponding to the successive buffer contents. Let \(|X_n|\) be the length of \(X_n\), set \(\hat{\gamma}=\lim_{n\to \infty}|X_n|/n\). The stability condition for the queue is \(\lambda \hat{\gamma}<\mu \). Under this condition a set of so-called twisted traffic equations is associated with the queue, they have a unique solution \((\rho ,r)\). Set \(q(a)=r(a)/r(Next(a))\) for all \(a\in \Sigma \). The stationary distribution of buffer content satisfies \[ \pi (u)=(1-\rho )\rho ^kq(u_k)\dots q(u_2)q(u_1). \] The term \((1-\rho )\rho ^k\) accounts for the buffer length, and the term \(q(u_k)\dots q(u_2)q(u_1)\) accounts for the precise buffer content. At network level there are two layers: the twisted traffic equations of the individual queues, and the network traffic equations. The stationary distribution has a ``double product form'': with respect to the buffer content for each individual queue according to the above formula and with respect to the different queues. The departure process out of the network is Poisson.
scientific article; zbMATH DE number 5696498
scientific article