minimum delay routing

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• minimum delay routing
• congestion control
• modeling
• resource allocation
• simulation
• network measurement management

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Minimum delay routing
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Problem formulation

• network represented by graph G (V,E)
• traffic matrix given by
• rs(d) traffic entering s destined for d
• r ?s,d?V rs(d)
• - expected traffic (bps) on link (i,k) for
  source/dest. pair s,d
• fik expected traffic (bps) on link (i,k)

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• Tsd - delay of msg from s to d
• T - delay of random message
• DT(fik) ? ET r-1 ?s,d?V rs(d) ETsd
• minimize DT(fik)
• s.t. flow constraints

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Digression - network performance analysis
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Littles law

• N number of customers in queue at steady state
• T customer delay at steady state
• l throughput

EN l ET
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• N number of pkts in network
• Nik number of pkts in (i,k) ? E
• T pkt network delay DT ET
• Tik pkt delay on (i,k) ? E Dik ETik
• EN ?(i,k)?E ENik ?(i,k)?E fik ETik
• r ET
• or
• ET (?(i,k)?E fik ETik)/r

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ETi - M/M/1 queue

• Poisson arrivals with rate l
• A(t,ts) no. arrivals in t,ts)
• P(A(t,ts) k) (ls)ke-ls/k!
• exponential interarrival times, mean 1/l
• one server
• exponential service times with mean 1/m
• S - service time
• FS(x) P(Sltx) 1 - e-ms

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• model as continuous time Markov process
• state N(t) - number in system at time t
• assume steady state behavior (lltm)
• pn - steady state probabilityof N n N
  limt?8N(t)

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• balance equations

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• balance equations
• have solution

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• balance equations
• which have solution
• where r l/m.

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• mean number of customers in system
• EN r/(1-r)
• mean sojourn time
• ET 1/(m-l)

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