Ch3_queue_theory-2008-1-rev

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

This preview shows pages 1–16. Sign up to view the full content.

Queueing Theory (Delay Models) Sunghyun Choi Adopted from Prof. Saewoong Bahk’s material

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

View Full Document
Introduction

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

View Full Document
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]

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

View Full Document
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

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

View Full Document
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 ( ) ( )
– 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 ( ) ( )

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

View Full Document
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 0 0 P P i ij ij j = = ⋅⋅⋅ = 0 1 0 1 0 , , , , = = = + P X j X i n n { } 1

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

View Full Document
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) = + = 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 ( =

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

View Full Document
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 ( = = = = =

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern