Handout 8 introduction to queueing theory
Download
1 / 9

Handout # 8: Introduction to Queueing Theory - PowerPoint PPT Presentation


  • 65 Views
  • Uploaded on

Handout # 8: Introduction to Queueing Theory. CSC 2203 – Packet Switch and Network Architectures. Professor Yashar Ganjali Department of Computer Science University of Toronto [email protected] http://www.cs.toronto.edu/~yganjali Thanks to Monia Ghobadi.

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

PowerPoint Slideshow about ' Handout # 8: Introduction to Queueing Theory' - mandek


An Image/Link below is provided (as is) to download presentation

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
Handout 8 introduction to queueing theory

Handout # 8:Introduction to Queueing Theory

CSC 2203 – Packet Switch and Network Architectures

Professor Yashar Ganjali

Department of Computer Science

University of Toronto

[email protected]

http://www.cs.toronto.edu/~yganjali

Thanks to Monia Ghobadi

TexPoint fonts used in EMF.

Read the TexPoint manual before you delete this box.:


A simple queue model
A Simple Queue Model

One server, infinite number of waiting positions

Arrival rate: l,

∆t0, P{one arrival in time interval ∆t} = l ∆t

P{more than one arrival in ∆t} ~ 0 (negligible)

Average service rate: m

∆t0, P{one departure from the system in ∆t} = m ∆t

P{more than one departures from the system ∆t} ~ 0 (negligible)

Arrivals

Departures

Waiting

positions

Server(s)

University of Toronto – Fall 2012


Markov process
Markov Process

  • Markov property: Memoryless

  • For a stochastic process X(t) and any choice of time instants ti, i=1,…,n, we have

  • P{X(tn+1)=xn+1|X(tn)=xn……. X(t1)=x1}=

    P{X(tn+1)=xn+1|X(tn)=xn}

    • The state of the process/system at time instant tn+1 depends only on the state of the process/system at the previous instant tn and not on any of the earlier time instants.

  • Markov process: Give the present of state of the process, its future evolution is independent of the past of the process (one-step dependency feature).

University of Toronto – Fall 2012


State transition diagram
State Transition Diagram

Arrivals

Departures

Waiting

positions

Server(s)

No arrival

no departure

arrival

arrival

No arrival

no departure

No arrival

no departure

k-1

k

k+1

departure

departure

University of Toronto – Fall 2012


System state

The system state at any time instant may be taken as the number in the system at that instant.

pN(t) = P{system in state N at time t}

p0(t+ ∆t) = p0(t) [1-l∆t] + p1(t) m∆tN=0

pN(t+ ∆t) = pN(t) [1-l∆t-m ∆t] + pN-1(t) l∆t + pN+1(t)m ∆tN>0

Subject to the normalisation condition that:

∑ipi(t) = 1 for all t ≥0

Take limit as ∆t0

System State

University of Toronto – Fall 2012


Equilibrium solution
Equilibrium Solution number in the

  • These differential equations along with the normalization condition may be used to get the equilibrium solutions.

  • The conditions invoked are:

  • Define ρ=λ/μ, with ρ < 1 for stability we get:

  • p1= ρ p0 (eq. 1)

  • pN+1= (1+ρ) pN - ρ pN-1= ρpN=ρN+1 p0 N≥1

University of Toronto – Fall 2012


System state probabilities
System State Probabilities number in the

  • Solving eq.1 we get the system state probabilities:

  • pi= ρi(1- ρ) i=0,1,……

  • Note: The summation in normalization condition would only have a finite value when ρ<1.

    • This condition is therefore required for the queue to be stable.

  • Once we know the equilibrium state probabilities, we can use them to compute various mean performance parameters for this simple queue.

University of Toronto – Fall 2012


Performance parameters
Performance Parameters number in the

  • Mean number in system, N

  • Mean number waiting in queue, Nq

  • Mean time spent in system, W

  • Mean time spent waiting in queue, Wq

University of Toronto – Fall 2012


Lessons learned
Lessons Learned number in the

  • The basic approach to the analysis of simple queuing models would begin by defining an appropriate system state for the queue.

  • The analysis of queue would then essentially be the study of the way this system state would evolve.

  • Interested in the performance analysis of the queue once equilibrium conditions have been reached.

  • Review some of the basics of the theory of Markov Chains.

University of Toronto – Fall 2012


ad