This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 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 nonFIFO 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 – nstep 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 = = = + { } – ChapmanKolmogorov 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.
 Spring '08
 Choi

Click to edit the document details