We revisit a single-server retrial queue with two independent Poisson streams (corresponding to two types of customers) and two orbits. The size of each orbit is infinite. The exponential server (with a rate independent of the type of customers) can hold at most one customer at a time and there is no waiting room. Upon arrival, if a type $i$ customer $(i=1,2)$ finds a busy server, it will join the type $i$ orbit. After an exponential time with a constant (retrial) rate $μ_i$, an type $i$ customer attempts to get service. This model has been recently studied by Avrachenkov, Nain and Yechiali~\cite{ANY2014} by solving a Riemann-Hilbert boundary value problem. One may notice that, this model is not a random walk in the quarter plane. Instead, it can be viewed as a random walk in the quarter plane modulated by a two-state Markov chain, or a two-dimensional quasi-birth-and-death (QBD) process. The special structure of this chain allows us to deal with the fundamental form corresponding to one state of the chain at a time, and therefore it can be studied through a boundary value problem. Inspired by this fact, in this paper, we focus on the tail asymptotic behaviour of the stationary joint probability distribution of the two orbits with either an idle or busy server by using the kernel method, a different one that does not require a full determination of the unknown generating function. To take advantage of existing literature results on the kernel method, we identify a censored random walk, which is an usual walk in the quarter plane. This technique can also be used for other random walks modulated by a finite-state Markov chain with a similar structure property.
The generalized join the shortest orbit queue system: Stability, exact tail asymptotics and stationary approximations
我们介绍了\textit{generalized join the shortest queue model with retrials}和两个无限容量的轨道队列。三个独立的Poisson作业流（即一个\textit{smart}和两个\textit{dedicated}流）流入最多可容纳一个作业的单台服务系统。发现服务器被占用的到达作业被路由到轨道，如下所示：来自\textit{smart}流的阻塞作业被路由到最短轨道队列，并且在平局的情况下，它们随机选择一个轨道。来自\textit{dedicated}流的阻塞作业被直接路由到其轨道。轨道作业以不同的重试率重试与服务器连接，即，异构轨道队列这种系统的应用中发现的无线合作网络的建模。我们感兴趣的是这个模型的平稳分布的渐近行为，只要系统是稳定的。更确切地说，我们调查的条件下，最小轨道队列长度的尾部渐近是完全几何。此外，我们应用启发式渐近方法来获得近似的稳态联合轨道长度分布。给出了数值算例，结果表明渐近分析和启发式方法所得结果是一致的。
Stability of multidimensional skip-free Markov modulated reflecting random walks: Revisit to Malyshev and Menshikov's results and application to queueing networks
On the Age of Information in Single-Server Queues with Aged Updates
Public University of Navarre（纳瓦拉公立大学）；University of the Basque Country, EHU（巴斯克大学）；IkerBasque - Basque Foundation for Science（伊克尔巴斯科——巴斯克科学基金会）；CNRS, IRIT（法国国家科学研究中心，IRIT）；Institut National Polytechnique, Toulouse（图卢兹国立理工学院）；Department of Mathematics, University of the Basque Country, EHU（巴斯克大学数学系）
