Ch3_queue_theory-2008-1-rev

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

Info iconThis preview shows pages 1–15. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Queueing Theory (Delay Models) Sunghyun Choi Adopted from Prof. Saewoong Bahk’s material Introduction • 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 – 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 N(t) of customers in the system during the interval (0, t ], where N(t) = 0 ∫ = < 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] – This relationship holds even in non-FIFO case < = = ∑ T A t T t i i A t 1 1 ( ) ( ) – 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 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 P P i ij ij j ≥ = = ⋅ ⋅ ⋅ = ∞ ∑ 0 1 0 1 , , , , = = = + P X j X i n n { } 1 – 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 = = = + { } – Chapman-Kolmogorov equations – Stationary distribution – For irreducible and aperiodic MCs, there exists and we have (w/ probability 1) ∑ ∞ = + ≥ = ) , , , ( k m kj n ik m n ij j i m n P P P ( 0) 1 j i ij j i j p p P j p ( = = = = 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 ( = – 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...
View Full Document

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.

Page1 / 42

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

This preview shows document pages 1 - 15. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online