BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//talks.cam.ac.uk//v3//EN
BEGIN:VTIMEZONE
TZID:Europe/London
BEGIN:DAYLIGHT
TZOFFSETFROM:+0000
TZOFFSETTO:+0100
TZNAME:BST
DTSTART:19700329T010000
RRULE:FREQ=YEARLY;BYMONTH=3;BYDAY=-1SU
END:DAYLIGHT
BEGIN:STANDARD
TZOFFSETFROM:+0100
TZOFFSETTO:+0000
TZNAME:GMT
DTSTART:19701025T020000
RRULE:FREQ=YEARLY;BYMONTH=10;BYDAY=-1SU
END:STANDARD
END:VTIMEZONE
BEGIN:VEVENT
CATEGORIES:Optimization and Incentives Seminar
SUMMARY:Differential equations arrising in network problem
 s - N Vvedenskaya (Institute for Information Trans
 mission Problems\, Moscow)
DTSTART;TZID=Europe/London:20071121T153000
DTEND;TZID=Europe/London:20071121T163000
UID:TALK9225AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/9225
DESCRIPTION:The talk focuses on large queueing systems and net
 works\, with dependent queues. In particular\, we 
 are interested in the large-time performance and s
 tationary distribution of such networks. Such prob
 lems are often rather complicated\, and one of the
  ways to study them is via an asymptopical approac
 h. In case where the network is guided by a Markov
  process and the number of its nodes $N$ is very l
 arge\, the limiting situation\, as $N\\to \\infty$
 \, may be described by a system of differential eq
 uations. The solution to such asystem gives an acc
 urate assessment of the network performance. \nI w
 ill present several examples where this aproach tu
 rnes to be successful. First\, we consider fast Ja
 ckson networks with dynamic routing\, then a rando
 m multiple access system with ALOHA-like protocol.
  The simplest system with dynamic routing was a sy
 stem considered by Dobrushin\, Karpelevich and Vve
 denskaya (and independently\, Mitzenmacher): it co
 ntains $N$ identical servers with IID exponential 
 service times and a Poisson flow of tasks. Upon it
 s arrival\, each task randomly selects $m$ servers
  and is directed to the one with a shorter queue. 
 The limiting situation is described by a rather si
 mple (infinite) system of ODEs\, with a unique glo
 bally attracting fixed point. The probability of l
 ong queus in the limiting system decreases superex
 ponentially. Numerical simulations show that in th
 e system with finite $N$ the queue length distribu
 tion is close to the limiting one. Later\, Martin-
 -Suhov and Suhov--Vvedenskaya considered open Jack
 son-type networks where each node contains $N$ ide
 ntical servers and a task selects several servers 
 (from one several nodes) and is directed to the on
 e with a shorter queue. \n\nFinally\, in a random 
 multiple access system with $N$ users and ALOHA-li
 ke protocol\, each user tries to transmit its maes
 sage with a friquency determined by its back-off s
 tage (the stage is changed after each transmission
  atempt.) As $N\\to \\infty$\, the system performa
 nce is described by a solution to a system of ODE 
 with one or seevral fixed points. In case of sever
 al solutions the original system is treated as 'me
 tastable'. \n\n
LOCATION:MR4\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
CONTACT:Speaker to be confirmed
END:VEVENT
END:VCALENDAR