Article

# A note on insensitivity in stochastic networks

Journal of Applied Probability (Impact Factor: 0.69). 12/2006; DOI: 10.1239/jap/1175267175

Source: arXiv

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**ABSTRACT:**We consider a family of discrete time multihop switched queueing networks where each packet moves along a fixed route. In this setting, BackPressure is the canonical choice of scheduling policy; this policy has the virtues of possessing a maximal stability region and not requiring explicit knowledge of traffic arrival rates. BackPressure has certain structural weaknesses because implementation requires information about each route, and queueing delays can grow super-linearly with route length. For large networks, where packets over many routes are processed by a queue, or where packets over a route are processed by many queues, these limitations can be prohibitive. In this article, we introduce a scheduling policy for FIFO networks, the Proportional Scheduler, which is based on the proportional fairness criterion. We show that, like BackPressure, the Proportional Scheduler has a maximal stability region and does not require explicit knowledge of traffic arrival rates. The Proportional Scheduler has the advantage that information about the network's route structure is not required for scheduling, which substantially improves the policy's performance for large networks. For instance, packets can be routed with only next-hop information and new nodes can be added to the network with only knowledge of the scheduling constraints.12/2014; - [Show abstract] [Hide abstract]

**ABSTRACT:**We deal with additive functionals of stationary processes. It is shown that under some assumptions a stationary model of the time-changed process exists. Further, bounds for the expectation of functions of additive functionals are derived. As an application we analyze virtual sojourn times in an infinite-server system where the service speed is governed by a stationary process. It turns out that the sojourn time of some kind of virtual requests equals in distribution an additive functional of a stationary time-changed process, which provides bounds for the expectation of functions of virtual sojourn times, in particular bounds for fractional moments and the distribution function. Interpreting the GI(n)/GI(n)/∞ system or equivalently the GI(n)/GI system under state-dependent processor sharing as an infinite-server system where the service speed is governed by the number n of requests in the system provides results for sojourn times of virtual requests. In the case of M(n)/GI(n)/∞, the sojourn times of arriving and added requests equal in distribution sojourn times of virtual requests in modified systems, which yields several results for the sojourn times of arriving and added requests. In case of positive integer moments, the bounds generalize earlier results for M/GI(n)/∞. In particular, the mean sojourn times of arriving and added requests in M(n)/GI(n)/∞ are proportional to the required service time, generalizing Cohen’s famous result for M/GI(n)/∞.Queueing Systems 04/2012; 70(4). · 0.60 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Polynomial convergence rates in total variation are established in Erlang--Sevastyanov's type problem with an infinite number of servers and a general distribution of service under assumptions on the intensity of serving.Queueing Systems 10/2013; 76(2). · 0.60 Impact Factor

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