Week 4 - Steady State Analysis, Classification Of Queueing Systems And Approximations

Question 1

Conditions for steady state behaviour (7)

Answer 1

• Probability distribution for Lq does not depend on time
• The arrival process is stationary
• Service rate is not dependant on time
• Number of servers (m) - constant
• Every server works independently
• One customer is served at one time by each server
• Has a state probability

Question 2

What is state probability (Pn)?

Answer 2

Is the probability that there are n customers in the system, meaning the system is in a particular state

Question 3

Example of state probability (2)

Answer 3

• If you have 2 states the probability would be 1/2
• If you have 3 states the probability would be 1/3
Etc etc

Question 4

Formula for state probability

Answer 4

1/Ns (Ns is the number of states)

Question 5

Formula for Ls using Pn + example

Answer 5

Question 6

Formula for Lq using Pn

Answer 6

Question 7

Formula for Wq using Little’s formula

Answer 7

Wq =Lq / λ

Question 8

What is service rate (μ)

Answer 8

Is the amount of customer served by time period

Question 9

Formula for average service time

Answer 9

1/μ
1 / service rate

Question 10

What do you get when you minus average service time (1/μ) from average time in system (Ws)

Answer 10

Average time in the queue

Question 11

Formula for absolute utilisation of a system (ρ)

Answer 11

λ/μ

Question 12

Formula linking Ls, Lq, ρ and λ/μ

Answer 12

ρ = Ls - Lq = λ/μ

Question 13

Formula for proportional utilisation

Answer 13

Absolute utilisation / number of servers
ρ/m

Question 14

Four characteristics of queueing systems

Answer 14

• Arrival process A(t) - the pattern of interatrial times
• Service process - S(t) - the pattern of service times
• Number of servers - m
• Queue size/capacity - b

Question 15

Markovian distribution (M)

Answer 15

Is exponential, typical for poison arrivals

Question 16

Gamma distribution (En)

Answer 16

Has a shape parameter n and is normally used to describe service process (normal distribution in a bell shape)

Question 17

General distribution (G)

Answer 17

Employs approximations and is vastly applicable

Question 18

Potential X1 or X2 distributions

Answer 18

• Markovian distribution (M)
• Gamma distribution (En)
• General distribution (G)
• Deterministic distribution (D)

Question 19

Deterministic distribution

Answer 19

Values are constant