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SUMMARY:Differential equations arrising in network problems - N Vvedenskay
 a (Institute for Information Transmission Problems\, Moscow)
DTSTART:20071121T153000Z
DTEND:20071121T163000Z
UID:TALK9225@talks.cam.ac.uk
CONTACT:Speaker to be confirmed
DESCRIPTION:The talk focuses on large queueing systems and networks\, with
  dependent queues. In particular\, we are interested in the large-time per
 formance and stationary distribution of such networks. Such problems are o
 ften rather complicated\, and one of the ways to study them is via an asym
 ptopical approach. In case where the network is guided by a Markov process
  and the number of its nodes $N$ is very large\, the limiting situation\, 
 as $N\\to \\infty$\, may be described by a system of differential equation
 s. The solution to such asystem gives an accurate assessment of the networ
 k performance. \nI will present several examples where this aproach turnes
  to be successful. First\, we consider fast Jackson networks with dynamic 
 routing\, then a random multiple access system with ALOHA-like protocol. T
 he simplest system with dynamic routing was a system considered by Dobrush
 in\, Karpelevich and Vvedenskaya (and independently\, Mitzenmacher): it co
 ntains $N$ identical servers with IID exponential service times and a Pois
 son flow of tasks. Upon its arrival\, each task randomly selects $m$ serve
 rs and is directed to the one with a shorter queue. The limiting situation
  is described by a rather simple (infinite) system of ODEs\, with a unique
  globally attracting fixed point. The probability of long queus in the lim
 iting system decreases superexponentially. Numerical simulations show that
  in the system with finite $N$ the queue length distribution is close to t
 he limiting one. Later\, Martin--Suhov and Suhov--Vvedenskaya considered o
 pen Jackson-type networks where each node contains $N$ identical servers a
 nd a task selects several servers (from one several nodes) and is directed
  to the one with a shorter queue. \n\nFinally\, in a random multiple acces
 s system with $N$ users and ALOHA-like protocol\, each user tries to trans
 mit its maessage with a friquency determined by its back-off stage (the st
 age is changed after each transmission atempt.) As $N\\to \\infty$\, the s
 ystem performance is described by a solution to a system of ODE with one o
 r seevral fixed points. In case of several solutions the original system i
 s treated as 'metastable'. \n\n
LOCATION:MR4\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
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