{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Ch3_queue_theory_v2009 - Queueing Theory(Delay Models...

Info iconThis preview shows pages 1–20. Sign up to view the full content.

View Full Document Right Arrow Icon
Queueing Theory (Delay Models) Sunghyun Choi Adopted from Prof. Saewoong Bahk’s material
Background image of page 1

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

View Full Document Right Arrow Icon
Introduction
Background image of page 2
Background image of page 3

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

View Full Document Right Arrow Icon
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
Background image of page 4
T : the total delay in the system λ: the customer arrival rate [#/sec]
Background image of page 5

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

View Full Document Right Arrow Icon
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
Background image of page 6
Background image of page 7

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

View Full Document Right Arrow Icon
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 ( ) ( )
Background image of page 8
– 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 ( ) ( )
Background image of page 9

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

View Full Document Right Arrow Icon
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
Background image of page 10
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
Background image of page 11

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

View Full Document Right Arrow Icon
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
Background image of page 12
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 < + <
Background image of page 13

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

View Full Document Right Arrow Icon
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
Background image of page 14
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:
Background image of page 15

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

View Full Document Right Arrow Icon
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
Background image of page 16
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
Background image of page 17

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

View Full Document Right Arrow Icon
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 +
Background image of page 18
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
Background image of page 19

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

View Full Document Right Arrow Icon
Image of page 20
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}