{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Ch3_queue_theory_v2009

# Ch3_queue_theory_v2009 - Queueing Theory(Delay Models...

This preview shows pages 1–20. 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
Applications of Little’s Theorem Example 3.1 : The average number of packets waiting in queue : The average time spent by a packet waiting in queue : The average transmission time : The arrival rate in a transmission line : The line’s utilization factor Q N W λ = X ρ λ = Q N W X λ ρ

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

View Full Document
Example 3.2 : The average delay : The average number of packets inside the network : The arrival packet rate of node i : The average number in the network of packets arriving at node i : The average delay of packets arriving at node i 1 n i i N T λ = = T N i λ i i i N T λ = i N i T
Example 3.3: regular packet arrivals and departures : The packet arrival interval : The packet tx time ( ) : The processing and propagation delay per packet 1 K λ = K α T K P α = + P N T K λ α = = + K P 1 α < 2 K K P K α < + <

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

View Full Document
Example 3.5: “Closed system” with K servers and N customers : The number of servers : The average customer time in the system : The average customer service time N T λ = K X λ = N X T K = K T N K ( X
If customers are blocked (and lost) if the system is full ( 29 1 K X β λ = - 1 K X β λ = - 1 K X β λ - Since , K K ( The average number of busy servers: The blocking probability:

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

View Full Document
Example 3.6: a round-robin system for m users : The time interval used for overhead of user i : The average cycle length 1 2 m A A A A = + + + L 1 1 m i i i A X L λ = = + 1 1 m i i i A L X λ = = - i A L
Example 3.7 - Combined with / N T λ = T R D = + R P T R NP + + N N R NP R P λ + + 1 P λ

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

View Full Document
The # of terminals becomes the bottleneck The processing power becomes the bottleneck 1 min , N N R NP P R P λ ° + + { } max , NP R P T R NP + + 1 / N R P < + 1 / N R P +
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

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 ]}