Ch3_queue_theory-2008-1-rev - Queueing Theory(Delay Models...

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Queueing Theory (Delay Models) Sunghyun Choi Adopted from Prof. Saewoong Bahk’s material
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Introduction
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Total delay of the i-th 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
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T : the total delay in the system λ: the customer arrival rate [#/sec]
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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
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Time average of the number N(t) of customers in the system during the interval (0, t ], where N(t) = 0 = < t t dt t N t N 0 ' ) ' ( 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 ( ) ( )
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– 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] This relationship holds even in non-FIFO case < = = T A t T t i i A t 1 1 ( ) ( )
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E[N q ] = λE[W] Server utilization E[N s ] = λE[τ] where utilization factor ρ=λ/(μc) < 1 to be stable for c server case
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Review of Markov chain theory Discrete time Markov chains – discrete time stochastic process { X n | n =0,1,2,..} taking values from the set of nonnegative integers Markov chain if where P P X j X i X i X i ij n n n n = = = = ⋅⋅⋅ = + - - { , , , } 1 1 1 0 0 P P i ij ij j = = ⋅⋅⋅ = 0 1 0 1 0 , , , , = = = + P X j X i n n { } 1
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The transition probability matrix n-step transition probabilities = ... ... ... ... 12 11 10 02 01 00 P P P P P P P P P X j X i ij n n m m = = = + { }
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Chapman-Kolmogorov equations Stationary distribution For irreducible and aperiodic MCs, there exists and we have (w/ probability 1) = + = 0 ) 0 , , , ( k m kj n ik m n ij j i m n P P P 0 0 ( 0) 1 j i ij j i j p p P j p ( = = = = 0 lim { | } ( 0) j n n p P X j X i i ( = = = # of visits to state up to time lim j k j k p k ( =
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Theorem. In an irreducible, aperiodic MC, there are two possibilities for no stationary distribution unique stationary distribution of the MC Example for case 1: a queueing system with arrival rate exceeding the service rate Case 2: global balance equation At equilibrium, frequencies out of and into state j are the same 0 lim { | } j n n p P X j X i ( = = = , 0 j j p 2200 = , 0 j j p 2200 0 0 0 ( 0) since 1 j ji i ij ji i i i p P p P j P ( = = = = =
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