1 / 9

Chap. 20, page 1051 Queuing Theory Arrival process Service process Queue Discipline - PowerPoint PPT Presentation

Chap. 20, page 1051 Queuing Theory Arrival process Service process Queue Discipline Method to join queue. IE 417, Chap 20, Jan 99. Each Distribution for Random Variable Has: Definition Parameters Density or Mass function Cumulative function Range of valid values Mean and Variance.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

PowerPoint Slideshow about ' Chap. 20, page 1051 Queuing Theory Arrival process Service process Queue Discipline' - aisha

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

Queuing Theory

Arrival process

Service process

Queue Discipline

Method to join queue

IE 417, Chap 20, Jan 99

Definition

Parameters

Density or Mass function

Cumulative function

Range of valid values

Mean and Variance

IE 417, Chap 20, Jan 99

Exponential Dist. Poisson Dist.

IE 417, Chap 20, Jan 99

Relation betweenExponential distribution ↔ Poisson distribution

Xi : Continuous random variable, time between arrivals,

has Exponential distribution with mean = 1/4

X1=1/4 X2=1/2 X3=1/4 X4=1/8 X5=1/8 X6=1/2 X7=1/4 X8=1/4 X9=1/8 X10=1/8 X11=3/8 X12=1/8

0 1:00 2:00 3:00

Y1=3 Y2=4 Y3=5

Yi : Discrete random variable, number of arrivals per unit of time, has Poisson distribution with mean = 4. (rate=4) Y ~ Poisson (4)

IME 301

1 / 2 / 3 / 4 / 5 / 6

Arrival / Service / Parallel / Queue / Max / Population

Process Process Servers Discip- Cus- Size

line tomer

M, D, Er, G, GI

IE 417, Chap 20, Jan 99

j = State of the system,

number of people in the system

Pij(t) = Probability that j people are in

the system at time t given that

i people are in the system at time 0

of j people in the system

IE 417, Chap 20, Jan 99

1- : birth rate (arrival) in state j

2- : death rate (service ends) in state j

3- death and births are independent of each

other, no more than 1 event in

M/M/1 is considered a birth-death process

Will not cover mathematical details of Section 20.3

IE 417, Chap 20, Jan 99

Notations used for QUEUING SYSTEM in steady

state (AVERAGES)

= Arrival rate approaching the system

e = Arrival rate (effective) entering the system

= Maximum (possible) service rate

e = Practical (effective) service rate

L = Number of customers present in the system

Lq = Number of customers waiting in the line

Ls = Number of customers in service

W = Time a customer spends in the system

Wq = Time a customer spends in the line

Ws = Time a customer spends in service

IE 417, Chap 20, May 99

Notations used for QUEUING SYSTEM in steady state

= Traffic intensity = /

= P(j) = Probability that j units are in the system

= P(0) = Probability that there are no units (idle)

in the system

Pw = P(j>S) = Probability that an arriving unit has

to wait for service

C = System capacity (limit)

= Probability that a system is full (lost customer)

= Probability that a

particular server is idle

IE 417, Chap 20, Mayl 99