Article

# A note on insensitivity in stochastic networks

Journal of Applied Probability (Impact Factor: 0.55). 12/2006; DOI: 10.1239/jap/1175267175

Source: arXiv

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**ABSTRACT:**We address a conjecture introduced by Massouli\'e (2007), concerning the large deviations of the stationary measure of bandwidth-sharing networks functioning under the Proportional fair allocation. For Markovian networks, we prove that Proportional fair and an associated reversible allocation are geometrically ergodic and have the same large deviations characteristics using Lyapunov functions and martingale arguments. For monotone networks, we give a more direct proof of the same result relying on stochastic comparisons that hold for general service requirement distribution. These results comfort the intuition that Proportional fairness is 'close' to allocations of service being insensitive to the service time requirement.Mathematics of Operations Research 07/2012; · 0.90 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We consider a system with Poisson arrivals and independent and identically distributed service times, where requests in the system are served according to the state-dependent (Cohen's generalized) processor-sharing discipline, where each request receives a service capacity that depends on the actual number of requests in the system. For this system, we derive expressions as well as tight insensitive upper bounds for the moments of the conditional sojourn time of a request with given required service time. The bounds generalize and extend corresponding results, recently given for the single-server processor-sharing system in Cheung et al. (2006) and for the state-dependent processor-sharing system with exponential service times by the authors (2008). Analogous results hold for the waiting times. Numerical examples for the M/M/m-PS and M/D/m-PS systems illustrate the given bounds.Advances in Applied Probability 01/2010; 42(2010). · 0.90 Impact Factor -
##### Chapter: A Queueing Analysis of Data Networks

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**ABSTRACT:**In packet-switched networks, resources are typically shared by a dynamic set of data flows. This dynamic resource sharing can be represented by a queueing network with state-dependent service rates. For a specific resource allocation we refer to as balanced fairness the corresponding queueing network is a Whittle network and has an explicit stationary distribution. We give some key properties satisfied by balanced fairness and compare the resulting throughput performance to those obtained under the max-min fair and proportional fair allocations.11/2010: pages 729-765;

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