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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.

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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

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


Kendell-Lee Notation for Queuing System

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


Queuing System

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

Steady state probability

of j people in the system

IE 417, Chap 20, Jan 99


Laws of Birth-Death Process

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


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