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Unformatted text preview: Queueing Theory (Delay Models) Sunghyun Choi Adopted from Prof. Saewoong Bahk’s material Introduction • Total delay of the ith customer in the system T i = W i + τ i – N(t) : the number of customers in the system – N q (t) : the number of customers in the queue – N s (t) : the number of customers in the service – W : the delay in the queue – τ : the service time – T : the total delay in the system – λ: the customer arrival rate [#/sec] Little’s Theorem E[N] = λE[T] • Number of customer in the system at t N(t)=A(t)D(t) where – D(t) : the number of customer departures up to time t – A(t) : the number of customer arrivals up to time t – Time average of the number of customers in the system N(t) during the interval (0, t ] ∫ = < t t dt t N t N ' ) ' ( 1 = = ∑ 1 1 t T i i A t ( ) < = λ t A t t ( ) < =< = ∑ N A t T t t i i A t λ 1 1 ( ) ( ) – Let <T> t be the average of the times spent in the system by the first A(t) customers Then <N> t =<λ> t <T> t – Assume an ergordic process and t →∞, then – E[N] = λE[T] – E[N q ] = λE[W] < = = ∑ T A t T t i i A t 1 1 ( ) ( ) • Server utilization E[N s ] = λE[τ] = ρ for one server where ρ=λ/μ<1 to be stable =λE[τ]/c for c server M/M/1 queueing system • Arrival statistics: stochastic process taking nonnegative integer values is called a Poisson process with rate λ if – A(t) is a counting process representing the total number of arrivals from 0 to t – arrivals are independent – probability distribution function { ( ) } A t t ≥ • Characteristics of the Poisson process – Interarrival times are independent and exponentially distributed – If t n denotes the nth arrival time and the interval τ n = t n+1 t n , the probability distribution is P s e s s [ ] , τ λ ≤ =  ≥ 1 0 ,... 2 , 1 , , ! ) ( ] ) ( ) ( [ = = = + n n e n t A t A P n λτ τ λτ – The interarrival probability density function – mean: 1/λ, variance: 1/λ 2 – for every t, δ≥0 where n e p n λ τ λ τ = ) ( ) ( ] 1 ) ( ) ( [ ) ( ] 1 ) ( ) ( [ ) ( 1 ] ) ( ) ( [ δ δ δ λδ δ δ λδ δ o t A t A P o t A t A P o t A t A P = + + = = + + = = + lim ( ) δ δ δ → = o – If A 1 , A 2 , …, A k are merged into a process A, A is Poisson with a rate equal to • Service statistics – The service times are exponentially distributed with parameter μ . The service time of the nth customer s n : where μ is the service rate k λ λ λ + + + ... 2 1 P s s e s n [ ] , ≤ =  ≥ 1 μ s • Poisson Process (mδ=T) – P[“1 arrival in mth interval”] ~λδ – P[“no arrival in mth interval”] ~1λδ – P[“ k arrivals in (0, T )”] ~ k m k k m C ) 1 ( ) ( λδ λδ P N k C T m T m T m m k k m k [ ] lim ( ) ( ) = =  → ∞ λ λ 1 = ⋅ ⋅ ⋅ + → ∞ lim ( ) !...
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This note was uploaded on 06/16/2008 for the course COMPUTER S 853 taught by Professor Choi during the Spring '08 term at Seoul National.
 Spring '08
 Choi

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