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Queueing Models with Spatial Interactions. Challenges and approaches David Gamarnik MIT Markov Lecture Discussion INFORMS 2011 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A AA A
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Renyi parking model Wireless communications Computer systems Reservation systems (hotels) Physics and chemistry Harrison Network Queueing models with interactions
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Baryshnikov, Coffman & Jelenkovic [2004]. Space filling and depletion. Jobs with length arrive at every integer point with Poisson rate Exp service times At most k overlapping jobs can be accepted for service Loss model: jobs not accepted are dropped Queueing model: jobs form queue
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Baryshnikov, Coffman & Jelenkovic [2004]. Space filling and depletion. Questions: loss rate? Stability? Queue length? Wait times? BC&J: solved loss model when k=1 and general k, and length l=1,2. Used generating function method. But alternatively one can view it as a Markov chain (see later). Stability question is wide open even in the uniform case.
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Baccelli & Foss [2011]. Poisson Hail on a Hot Ground Jobs have general (Borel) shape. General interarrival and service times. Ovelapping jobs form queues. Question: stability.
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Baccelli & Foss [2011]. Poisson Hail on a Hot Ground Theorem [BF]. The system is stable if job sizes have exponential tails and the arrival rate is sufficiently small. Tight conditions for stability is open. Performance analysis (queue length, space utilization) open.
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Where can we search for general purpose tools? A small detour into statistical physics.
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Where can we search for general purpose tools? Independent set (hard-core) model When ½ is small, correlations are short-range. When ½ is large correlations are long-range Conjecture: critical ½ *=3.796 Randall [2011]. ½ *<6.18. Restrepo, Shin, Tetali, Vigoda, Yang [2011]. ½ *>2.38. Shah et al. Applications to wireless communications.
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Correlation decay (long-range independence) method The correlation decay method allows computing approximately Weitz [2006]. General graphs, independent sets. G & Katz [2009]. Lattices. Improved earlier estimates for monomer-dimer model with two orders of magnitude. One-dimensional case is solved easily by reduction to a Markov chain. 0.78595 · h(3) · 0.78599 0.7845 · h(3) · 0.7862
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Challenges Non Poisson-Exponential models. Are even 1-dim models solvable? Short range vs long-range for non Poisson/Exp models? Interaction between stability and phase transition (long-range dependence). Which one “acts” first?
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Questions?
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